Cross-posted from https://wsmoses.com/blog/2018/12/18/boaz/

Lecturer: Aram Harrow

Scribes: Sinho Chewi, William S. Moses, Tasha Schoenstein, Ary Swaminathan

November 9, 2018

January 20, 2019

(The following blog post serves as an introduction to the following notes:)

Black Holes, Hawking Radiation, and the Firewall

There are many different types of “theoretical physicists.” There are theoretical astrophysicists, theoretical condensed matter physicists, and even theoretical biophysicists. However, the general public seems to be most interested in the exploits of what you might call “theoretical high energy theorists.” (Think Stephen Hawking.)

The holy grail for theoretical high energy physicists (who represent only a small fraction of all physicists) would be to find a theory of quantum gravity. As it stands now, physicists have two theories of nature: quantum field theory (or, more specifically, the “Standard Model”) and Einstein’s theory of general relativity. Quantum field theory describes elementary particles, like electrons, photons, quarks, gluons, etc. General relativity describes the force of gravity, which is really just a consequence of the curvature of spacetimes.

Sometimes people like to say that quantum field theory describes “small stuff” like particles, while general relativity describes “big stuff” like planets and stars. This is maybe not the best way to think about it, though, because planets and stars are ultimately just made out of a bunch of quantum particles.

Theoretical physicists are unhappy with having two theories of nature. In order to describe phenomena that depend on both quantum field theory and general relativity, the two theories must be combined in an “ad hoc” way. A so-called “theory of everything,” another name for the currently unknown theory of “quantum gravity,” would hypothetically be able to describe all the phenomena we know about. Just so we’re all on the same page, they don’t even have a fuly worked out hypothesis. (“String theory,” a popular candidate, is still not even a complete “theory” in the normal sense of the word, although it could become one eventually.)

So what should these high energy theoretical physicists do if they want to discover what this theory of quantum gravity is? For the time being, nobody can think up an experiment that would be sensitive to quantum gravitation effects which is feasible with current technology. We are limited to so-called “thought experiments.”

This brings us to Hawking radiation. In the 1970’s, Stephen Hawking considered what would happen to a black hole once quantum field theory was properly taken into account. (Of course, this involved a bit of “ad hoc” reasoning, as mentioned previously.) Hawking found that, much to everybody’s surprise, the black hole evaporated, realeasing energy in the form of “Hawking radiation” (mostly low energy photons). More strangely, this radiation comes out exactly in the spectrum you would expect from something “hot.” For example, imagine heating a piece of metal. At low temperatures, it emits low energy photons invisible to the human eye. Once it gets hotter, it glows red, then yellow, then perhaps eventually blue. The spectrum of light emitted follows a very specific pattern. Amazingly, Hawking found that the radiation which black holes emit follow the exact same pattern. By analogy, they have a temperature too!

This is more profound than you might realize. This is because things which have an temperature should also have an “entropy.” You see, there are two notions of “states” in physics: “microstates” and “macrostates.” A microstate gives you the complete physical information of what comprises a physical system. For example, imagine you have a box of gas, which contains many particles moving in a seemingly random manner. A “microstate” of this box would be a list of all the positions and momenta of every last particle in that box. This would be impossible to measure in pratice. A “macrostate,” on the other hand, is a set of microstates. You may not know what the exact microstate your box of gas is in, but you can measure macroscopic quantities (like the total internal energy, volume, particle number) and consider the set of all possible microstates with those measured quantities.

The “entropy” of a macrostate is the logarithm of the number of possible microstates. If black holes truly are thermodynamic systems with some entropy, that means there should be some “hidden variables” or microstates to the black hole that we currently don’t understand. Perhaps if we understood the microstates of the black hole, we would be much closer to understanding quantum gravity!

However, Hawking also discovered something else. Because the black hole is radiating out energy, its mass will actually decrease as time goes on. Eventually, it should disappear entirely. This means that the information of what went into the black hole will be lost forever.

Physicists did not like this, however, because it seemed to them that the information of what went into the black hole should never be lost. Many physicists believe that the information of what went into the black hole should somehow be contained in the outgoing Hawking radiation, although they do not currently understand how. According to Hawking’s original calculation, the Hawking radiation only depends on a parameters of the black hole (like its mass) and have nothing to do with the many finer details on what went in, the exact “microstate” of what went in.

However, physicists eventually realized a problem with the idea that the black hole releases its information in the form of outgoing Hawking radiation. The problem has to do with quantum mechanics. In quantum mechanics, it is impossible to clone a qubit. That means that if you threw a qubit in a black hole and then waited for it to eventually come out in the form of Hawking radiation, then the qubit could no longer be “inside” the black hole. However, if Einstein is to be believed, you should also be able to jump into the black hole and see the qubit on the inside. This seems to imply that the qubit is cloned, as it is present on both the inside and outside of the black hole.

Physicists eventually came up with a strange fix called “Black Hole Complementarity” (BHC). According to BHC, according to people outside the black hole, the interior does not exist. Also, according to people who have entered the black hole, the outside ceases to exist. Both descriptions of the world are “correct” because once someone has entered the black hole, they will be unable to escape and compare notes with the person on the outside.

Of course, it must be emphasized that BHC remains highly hypothetical. People have been trying to poke holes in it for a long time. The largest hole is the so called “Firewall Paradox,” first proposed in 2012. Essentially, the Firewall paradox tries to show that the paradigm of BHC is self-contradictory. In fact, it was able to use basic quantum mechanics to show that, under some reasonable assumptions, the interior of the black hole *truly* doesn’t exist, and that anyone who tries to enter would be fried at an extremely hot “firewall!” Now, I don’t think most physicists actually believe that black holes really have a firewall (although this might depend on what day of the week you ask them). The interesting thing about the Firewall paradox is that it derives a seemingly crazy result from seemingly harmless starting suppositions. So these suppositions would have to be tweaked in the theory of quantum gravity order to get rid of the firewall.

This is all to say that all this thinking about black holes really might help physicists figure out something about quantum gravity. (Then again, who can really say for sure.)

If you would like to know more about the Firewall paradox, I suggest you read my notes, pasted at the top of this post!

The goal of the notes was to write an introduction to the Black Hole Information Paradox and Firewall Paradox that could be read by computer scientists with no physics background.

The structure of the notes goes as follows:

- Special relativity
- General relativity
- Quantum Field Theory (in which I ambitiously tell you what QFT actually is!)
- Statistical Mechanics (this is the best part)
- Hawking radiation
- The information paradox

Because the information paradox touches on all areas of physics, I thought it was necessary to take “zero background, infinite intelligence” approach, introducing the all the necessary branches of physics (GR, QFT, Stat Mech) in order to understand what the deal with Hawking radiation really is, and why physicists think it is so important. I think it is safe to say that if you read these notes, you’ll learn a non-trivial amount of physics.

January 3, 2019

Nilin Abrahamsen nilin@mit.edu

Daniel Alabi alabid@g.harvard.edu

Mitali Bafna mitalibafna@g.harvard.edu

Emil Khabiboulline ekhabiboulline@g.harvard.edu

Juspreet Sandhu jus065@g.harvard.edu

Two-prover one-round (2P-1R) games have been the subject of intensive study in classical complexity theory and quantum information theory. In a 2P-1R game, a *verifier* sends questions privately to each of two collaborating *provers* , who then aim to respond with a compatible pair of answers without communicating with each other. Sharing quantum entanglement allows the provers to improve their strategy without any communication, illustrating an apparent paradox of the quantum postulates. These notes aim to give an introduction to the role of entanglement in nonlocal games, as they are called in the quantum literature. We see how nonlocal games have rich connections within computer science and quantum physics, giving rise to theorems ranging from hardness of approximation to the resource theory of entanglement.

- Questions are asked according to some distribution (e.g. uniform).
- Answers are provided by players (call them Alice and Bob).
- The verifier computes a predicate used to decide acceptance/rejection.

□

- If , Alice measures in basis . If , Alice measures in . Alice answers bit if outcome is and answers otherwise.
- If , Bob measures in basis . If , Bob measures in .
- Each player responds with their respective measurement outcome.

- () The three cases are all analogous: in each case Alice an Bob must output the same answer, and in each case Bob’s measurement basis is almost the same as Alice’s except rotated by a small angle . Of the three above cases we consider the one where and check that indeed the two measurement outcomes agree with probability : When Alice measures her qubit and obtains some bit , the shared pair collapses to . Indeed, since the question was , Alice measures her qubit in the basis . This means that Alice applies the measurement on the global state. The post-measurement state is the normalization of because can be viewed as a Kronecker delta of and . In particular, Bob is now in the pure state . Because Bob received question he measures in the basis Therefore his probability of correctly outputting is
- () Now consider the case where Alice and Bob are supposed to give different answers. Alice measures in basis consisting of and . If Alice gets outcome then the post-measurement global state is . Therefore when Bob applies the measurement in basis he mistakenly outputs only with probability .

□

Lemma 2 implies a lower bound on the value of the CHSH game.
It turns out that this lower bound is sharp, that is, the strategy just described is optimal.

□

- There exist such that for any . Further this would imply that ;
- There exist real unit vectors for such that ;

Theorem 4

The algorithm of theorem 4 proceeds by relaxing the set of quantum strategies to a larger convex set of

Definition 5 (Block-matrix form)

Definition 5 is simply a convenient change of notation and we identify with , using either notation depending on the context.

Definition 6 (Pseudo-strategies)

Proof.

□

Lemma 7 means is an (efficiently computable) upper bound for :
To finish the proof of theorem 4 we need to show that any pseudo-strategy can be

, (*)

where the bar represents entrywise complex conjugation, is the entrywise dot product of matrices, and the entrywise complex inner product (Hilbert-Schmidt inner product). We now choose the measurements. Given question , Alice measures in the PVM with Similarly, Bob on question applies the PVM with The condition 3 in definition 6 ensures that for any question , the vectors are orthogonal so that this is a valid PVM. The measurement outcome “” is interpreted as “fail”, and upon getting this outcome the player attempts the measurement again on their share of a fresh copy of . This means that the strategy requires many copies of the entangled state to be shared before the game starts. It also leads to the complication of ensuring that with high probability the players measure the same number of times before outputting their measurement, so that the outputs come from measuring the same entangled state. By (*), at a given round of measurements the conditional distribution of answers is given by We wish to relate the LHS to , so to handle the factor each prover performs repeated measurements, each time on a fresh copy of , until getting an outcome . Moreover, to handle the factor , each prover consults public randomness and accepts the answer with probability and respectively, or rejects and start over depending on the public randomness. Under a few simplifying conditions (more precisely, assuming that the game is
At this stage it is important that we are dealing with a *unique game* . Indeed, by (4) we have for every and ,
where the last inequality follows from concavity. Taking the expectation over and implies the bound (3), thus concluding the proof of theorem 4.

□

- The referee (verifier) randomly sends a clause to Alice (first prover) and a variable to Bob (second prover).
- Alice and Bob reply with assignments.
- The referee accepts if Alice’s assignment satisfies the clause and Bob’s answer is consistent with Alice’s.

Theorem 8 ([6])

Theorem 9 ([2])

Proposition 11 (adapted from [13])

□

Person | Strategy | Bound(entangled bits) |
---|---|---|

Slofstra | (Possibly) Non-Clifford | |

Tsirelson | Clifford |

Definition 13 (Solution Algebra)

Corollary 15

Lemma 16

□

Having proved this theorem, we now obtain Corollary 15, which is the main desired result. To see how it subsumes Tsirelson’s result as a special case, we use a simple fact from Representation Theory:
[1] David Avis, Sonoko Moriyama, and Masaki Owari. From bell inequalities to tsirelson’s theorem. IEICE Transactions, 92-A(5):1254–1267, 2009.

[2] Lance Fortnow, John Rompel, and Michael Sipser. On the power of multi-prover interactive protocols. Theoretical Computer Science, 134(2):545 – 557, 1994.

[3] T. Ito, H. Kobayashi, and K. Matsumoto. Oracularization and Two-Prover One-Round Interactive Proofs against Nonlocal Strategies. ArXiv e-prints, October 2008.

[4] J. Kempe, H. Kobayashi, K. Matsumoto, B. Toner, and T. Vidick. Entangled games are hard to approximate. ArXiv e-prints, April 2007.

[5] Julia Kempe, Oded Regev, and Ben Toner. Unique games with entangled provers are easy. SIAM Journal on Computing, 39(7):3207– 3229, 2010.

[6] S. Khanna, M. Sudan, L. Trevisan, and D. Williamson. The approximability of constraint satisfaction problems. SIAM Journal on Computing, 30(6):1863–1920, 2001.

[7] Anand Natarajan and Thomas Vidick. Two-player entangled games are NP-hard. arXiv e-prints, page arXiv:1710.03062, October 2017.

[8] Anand Natarajan and Thomas Vidick. Low-degree testing for quantum states, and a quantum entangled games PCP for QMA. arXiv e-prints, page arXiv:1801.03821, January 2018.

[9] William Slofstra. Lower bounds on the entanglement needed to play xor non-local games. CoRR, abs/1007.2248, 2010.

[10] B.S. Tsirelson. Quantum analogues of the bell inequalities. the case of two spatially separated domains. Journal of Soviet Mathematics, 36(4):557–570, 1987.

[11] Thomas Vidick. Three-player entangled XOR games are NP-hard to approximate. arXiv e-prints, page arXiv:1302.1242, February 2013.

[12] Thomas Vidick. Cs286.2 lecture 15: Tsirelson’s characterization of xor games. Online, December 2014. Lecture Notes.

[13] Thomas Vidick. Cs286.2 lecture 17: Np-hardness of computing . Online, December 2014. Lecture Notes.

December 23, 2018

author: Beatrice Nash

Abstract

In this blog post, we give a broad overview of quantum walks and some quantum walks-based algorithms, including traversal of the glued trees graph, search, and element distinctness [3; 7; 1]. Quantum walks can be viewed as a model for quantum computation, providing an advantage over classical and other non-quantum walks based algorithms for certain applications.

We begin our discussion of quantum walks by introducing the quantum analog of the continuous random walk. First, we review the behavior of the classical continuous random walk in order to develop the definition of the continuous quantum walk.

Take a graph with vertices and edges . The adjacency matrix of is defined as follows:

And the Laplacian is given by:

The Laplacian determines the behavior of the classical continuous random walk, which is described by a length vector of probabilities, **p**(t). The th entry of **p**(t) represents the probability of being at vertex at time . **p**(t) is given by the following differential equation:

which gives the solution .

Recalling the Schrödinger equation , one can see that by inserting a factor of on the left hand side of the equation for **p**(t) above, the Laplacian can be treated as a Hamiltonian. One can see that the Laplacian preserves the normalization of the state of the system. Then, the solution to the differential equation:

,

which is , determines the behavior of the quantum analog of the continuous random walk defined previously. A general quantum walk does not necessarily have to be defined by the Laplacian; it can be defined by any operator which “respects the structure of the graph,” that is, only allows transitions to between neighboring vertices in the graph or remain stationary [7]. To get a sense of how the behavior of the quantum walk differs from the classical one, we first discuss the example of the continuous time quantum walk on the line, before moving on to the discrete case.

An important example of the continuous time quantum walk is that defined on the infinite line. The eigenstates of the Laplacian operator for the graph representing the infinite line are the momentum states with eigenvalues , for in range . This can be seen by representing the momentum states in terms of the position states and applying the Laplacian operator:

Hence the probability distribution at time , , with initial position is given by:

While the probability distribution for the classical continuous time

random walk on the same graph approaches, for large , , or a Gaussian of width . One can see that the quantum walk has its largest peaks at the extrema, with oscillations in between that decrease in amplitude as one approaches the starting position at . This is due to the destructive interference between states of different phases that does not occur in the classical case. The probability distribution of the classical walk, on the other hand, has no oscillations and instead a single peak centered at , which widens and flattens as increases.

A *glued tree* is a graph obtained by taking two binary trees of equal height and connecting each of the leaves of one of the trees to exactly two leaves of the other tree so that each node that was a leaf in one of the original trees now has degree exactly . An example of such a graph is shown in Figure 2.

The time for the quantum walk on this graph to reach the right root from the left one is exponentially faster than in the classical case. Consider the classical random walk on this graph. While in the left tree, the probability of transitioning to a node in the level one to the right is twice that of transitioning to a node in the level one to the left. However, while in the right tree, the opposite is true. Therefore, one can see that in the middle of the graph, the walk will get lost, as, locally, there is no way to determine which node is part of which tree. It will instead get stuck in the cycles of identical nodes and will have exponentially small probability of reaching the right node.

To construct a continuous time quantum walk on this graph, we consider the graph in terms of *columns*. One can visualize the columns of Figure 2 as consisting of all the nodes equidistant from the entrance and exit nodes. If each tree is height , then we label the columns , where column contains the nodes with shortest path of length from the leftmost root node. We describe the state of each column as a superposition of the states of each node in that column. The number of nodes in column , , will be for and for . Then, we can define the state as:

The factor of latex ensures that the state is normalized. Since the adjacency matrix of the glued tree is Hermitian, then we can treat as the Hamiltonian of the system determining the behavior of the quantum walk. By acting on this state with the adjacency matrix operator , we get the result (for ):

Then for , we get the same result, because of symmetry.

For :

The case of is symmetric. One can see that the walk on this graph is equivalent to the quantum walk on the finite line with nodes corresponding to the columns. All of the edges, excluding that between columns and , have weight . The edge between column and has weight .

The probability distribution of the quantum walk on this line can be roughly approximated using the infinite line. In the case of the infinite line, the probability distribution can be seen as a wave propagating with speed linear in the time . Thus, in time linear in , the probability that the state is measured at distance from the starting state is . In [3] it is shown that the fact that the line is finite and has a single differently weighted edge from the others (that between and ) does not change the fact that in polynomial time, the quantum walk will travel from the left root node to the right one, although in this case there is no limiting distribution as the peaks oscillate. This was the first result that gives an exponential speed up over the classical case using quantum walks.

In this section, we will first give an introduction to the discrete quantum walk, including the discrete quantum walk on the line and the Markov chain quantum walk, as defined in [7]. Next, we discuss how Grover search can be viewed as a quantum walk algorithm, which leads us into Ambainis’s quantum-walks based algorithm from [1] for the element distinctness problem, which gives a speed up over classical and other quantum non-walks based algorithms.

The discrete time quantum walk is defined by two operators: the *coin flip* operator, and the *shift* operator. The coin flip operator determines the direction of the walk, while the shift operator makes the transition to the new state conditioned on the result of the coin flip. The Hilbert space governing the walk is , where corresponds to the space associated with the result of the coin flip operator, and corresponds to the locations in the graph on which the walk is defined.

For example, consider the discrete time walk on the infinite line. Since there are two possible directions (left or right), then the Hilbert space associated with the coin flip operator is two dimensional. In the unbiased case, the coin flip is the Hadamard operator,

and shift operator that produces the transition from state to or ,

conditioned on the result of the coin flip, is .

Each step of the walk is determined by an application of the unitary

operator . If the walk starts at position

, then measuring the state after one application of gives with probability and with probability . This is exactly the same as the case of the classical random walk on the infinite line; the difference between the two walks becomes apparent after a few steps.

For example, the result of the walk starting at state after 4 steps gives:

One can see that the distribution is becoming increasingly skewed

towards the right, while in the classical case the distribution will be

symmetric around the starting position. This is due to the destructive

interference discussed earlier. The distribution after time

steps is shown in Figure 4.

Now, consider the walk starting at state :

This distribution given by this walk is the mirror image of the first.

To generate a symmetric distribution, consider the start state . The resulting distribution after steps will be , where is the probability distribution after steps resulting from the start state and is the probability distribution after steps resulting from the start state . The result will be symmetric, with peaks near the extrema, as we saw in the continuous case.

A reversible, ergodic Markov chain with states can be represented by a transition matrix with equal to the probability of transitioning from state to state and . Then, , where is an initial probability distribution over the states, gives the distribution after one step.

Since for all , is stochastic and thus preserves normalization.

There are multiple ways to define a discrete quantum walk, depending on the properties of the transition matrix and the graph on which it is defined (overview provided in [4]). Here we look at the quantum walk on a Markov chain as given in [2]. For the quantum walk on this graph, we define state as the state that represents currently being at position and facing in the direction of . Then, we define the state as a superposition of the states associated with position :

The unitary operator,

,

acts as a coin flip for the walk on this graph. Since is reversible, we can let the shift operator be the unitary operator:

.

A quantum walk can also be defined for a non-reversible Markov chain using a pair of reflection operators (the coin flip operator is an example of a reflection operator). This corresponds to the construction given in [7].

Given a black box function and a set of inputs with , say we want to find whether an input exists for which equals some output value. We refer to the set of inputs for which this is true as marked. Classically, this requires queries, for nonempty . Using the Grover search algorithm, this problem requires quantum queries. In this section, we give a quantum walks based algorithm that also solves this problem in time. If we define a doubly stochastic matrix with uniform transitions, then we can construct a new transition matrix from as:

Then, when the state of the first register is unmarked, the operator defined in the previous section acts as a diffusion over its neighbors. When the state in the first register is marked, then will act as the operator , and the walk stops, as a marked state has been reached. This requires two queries to the black box function: one to check whether the input is marked, and then another to uncompute. By rearranging the order of the columns in so that the columns corresponding to the non-marked elements come before the columns corresponding to the marked elements, we get:

where gives the transitions between non-marked elements and gives the transitions from non-marked to marked elements.

We now look at the hitting time of the classical random walk. Assume

that there is zero probability of starting at a marked vertex. Then, we

can write the starting distribution , where the last elements of , corresponding to the marked elements, are zero, as

, where are the eigenvalues of , and are the corresponding eigenvectors, with the last entries zero. Let be the principal (largest) eigenvalue. Then, the probability that, after steps, a marked element has not yet been reached will be . Then, the

probability that a marked element has been reached in that time will be

. Setting

gives probability that a marked element will be reached in that time.

The eigenvalues of will be and

. Then, the classical hitting time will be:

It can be showed that for a walk defined by a Markov chain, the

classical hitting time will be , where , the *spectral gap*, and [2].

Magniez *et al* proved in [6] that for a reversible, ergodic

Markov chain, the quantum hitting time for a walk on this chain is

within a factor of the square root of the classical hitting time. Since

the walk on this input acts as a walk on a reversible Markov chain until

a marked element is reached, then this is also true for a walk defined

by our transition matrix . This arises from the fact that the

spectral gap of the matrix describing the quantum walk corresponding to

stochastic matrix is quadratically larger than the spectral gap of

the matrix describing the classical random walk corresponding to , the proof of which is given in [2]. Thus, the quantum hitting time

is , which exactly matches the quantum query complexity of Grover search.

Now, we describe Ambainis’s algorithm given in [1] for solving

the *element distinctness problem* in time, which

produces a speed up over the classical algorithm, which requires queries, and also over other known quantum algorithms that do not make use of quantum walks, which require queries. The element distinctness problem is defined as follows: given a function on a size set of inputs

,…,,

determine whether there exists a pair for which . As in the search problem defined in the previous section, this is a decision problem; we are not concerned with finding the values of these pairs, only whether at least one exists.

The algorithm is similar to the search algorithm described in the previous section, except we define the walk on a *Hamming graph*. A Hamming graph is defined as follows: each vertex corresponds to an -tuple, (,…,), where for all and repetition is allowed (that is, may equal for ), and is a parameter we will choose. Edges will exist between vertices that differ in exactly one coordinate (order matters in this graph). We describe the state of each vertex as:

,…,,…,

Then, moving along each edge that replaces the th coordinate with such that requires two queries to the black box function to erase and compute . In the case, the marked vertices will be those that contain some for . Since the function values are stored in the description of the state, then no additional queries to the black box are required to check if in a marked state.

The transition matrix is given by . is the all one matrix, and the superscript denotes the operator acting on the th coordinate. The factor of normalizes the degree, since the graph is regular. We can compute the spectral gap of this graph to be (for details of this computation, see [2]). Then, noting that that the fraction of marked vertices, , is

, classically, the query complexity is , where is the queries required to construct the initial state. Setting the parameters equal to minimize with respect to gives classical query complexity , as expected.

Then in the quantum case, queries are still required to set up the state. queries are required to perform the walk until a marked state is reached, by [6]. Setting parameters equal gives queries, as desired.

[1] Ambainis, A. Quantum walk algorithm for element distinctness, SIAM Journal on Computing 37(1):210-239 (2007). arXiv:quant-ph/0311001

[2] Childs, A. Lecture Notes on Quantum Algorithms (2017). https://www.cs.umd.edu/ amchilds/qa/qa.pdf

[3] Childs, A., Farhi, E. Gutmann, S. An example of the difference between

quantum and classical random walks. Journal of Quantum Information

Processing, 1:35, 2002. Also quant-ph/0103020.

[4] Godsil, C., Hanmeng, Z. Discrete-Time Quantum Walks and Graph Structures

(2018). arXiv:1701.04474

[5] Kempe, J. Quantum random walks: an introductory overview, Contemporary

Physics, Vol. 44 (4) (2003) 307:327. arXiv:quant-ph/0303081

[6] Magniez, F., Nayak, A., Richter, P.C. et al. On the hitting times of

quantum versus random walks, Algorithmica (2012) 63:91.

https://doi.org/10.1007/s00453-011-9521-6

[7] Szegedy, M. Quantum Speed-up of Markov Chain Based Algorithms, 45th

Annual IEEE Symposium on Foundations of Computer Science (2004).

https://ieeexplore.ieee.org/abstract/document/1366222

[8] Portugal, R. *Quantum Walks and Search Algorithms*. Springer, New York, NY (2013).

December 22, 2018

By Abhijit Mudigonda, Richard Wang, and Lisa Yang

*This is part of a series of blog posts for CS 229r: Physics and Computation. In this post, we will talk about progress made towards resolving the quantum PCP conjecture. We’ll briefly talk about the progression from the quantum PCP conjecture to the NLTS conjecture to the NLETS theorem, and then settle on providing a proof of the NLETS theorem. This new proof, due to Nirkhe, Vazirani, and Yuen, makes it clear that the Hamiltonian family used to resolve the NLETS theorem cannot help us in resolving the NLTS conjecture.*

We are all too familiar with **NP** problems. Consider now an upgrade to **NP** problems, where an omniscient prover (we’ll call this prover Merlin) can send a polynomial-sized proof to a **BPP** (bounded-error probabilistic polynomial-time) verifier (and we’ll call this verifier Arthur). Now, we have more decision problems in another complexity class, **MA** (Merlin-Arthur). Consider again, the analogue in the quantum realm where now the prover sends over qubits instead and the verifier is in **BQP** (bounded-error quantum polynomial-time). And now we have **QMA** (quantum Merlin-Arthur).

We can show that there is a hierarchy to these classes, where **NP** **MA** **QMA**.

Our goal is to talk about progress towards a **quantum PCP theorem** (and since nobody has proved it in the positive or negative, we’ll refer to it as a quantum PCP *conjecture* for now), so it might be a good idea to first talk about the PCP theorem. Suppose we take a Boolean formula, and we want to verify that it is satisfiable. Then someone comes along and presents us with a certificate — in this case, a satisfying assignment — and we can check in polynomial time that either this is indeed a satisfying assignment to the formula (a correct certificate) or it is not (an incorrect certificate).

But this requires that we check the entire certificate that is presented to us. Now, in comes the **PCP Theorem** (for *probabilistically checkable proofs*), which tells us that a certificate can be presented to us such that we can read a constant number of bits from the certificate, and have two things guaranteed: one, if this certificate is correct, then we will never think that it is incorrect even if we are not reading the entire certificate, and two, if we are presented with an incorrect certificate, we will reject it with high probability [1].

In short, one formulation of the PCP theorem tells us that, puzzingly, we might not need to read the entirety of a proof in order to be convinced with high probability that it is a good proof or a bad proof. But a natural question arises, which is to ask: is there a quantum analogue of the PCP theorem?

The answer is, we’re still not sure. But to make progress towards resolving this question, we will present the work of Nirkhe, Vazirani, and Yuen in providing an alternate proof of an earlier result of Eldar and Harrow on the NLETS theorem.

Before we state the quantum PCP conjecture, it would be helpful to review information about local Hamiltonians and the -local Hamiltonian problem. A previous blog post by Ben Edelman covers these topics. Now, let’s state the quantum PCP conjecture:

**( Quantum PCP Conjecture)**: It is QMA-hard to decide whether a given local Hamiltonian (where each ) has ground state energy at most or at least when for some universal constant .

Recall that MAX--SAT being NP-hard corresponds to the -local Hamiltonian problem being QMA-hard when . (We can refer to Theorem 4.1 in these scribed notes of Ryan O’Donnell’s lecture, and more specifically to Kempe-Kitaev-Regev’s original paper for proof of this fact.) The quantum PCP conjecture asks if this is still the case when the gap is .

Going back to the PCP theorem, an implication of the PCP theorem is that it is NP-hard to approximate certain problems to within some factor. Just like its classical analogue, the qPCP conjecture can be seen as stating that it is QMA-hard to approximate the ground state energy to a factor better than .

Let’s make the observation that, taking to be the ground state energy, the qPCP conjecture sort of says that there exists a family of Hamiltonians for which there is no trivial state (a state generated by a low depth circuit) such that the energy is at most above the ground state energy.

Freedman and Hastings came up with an easier goal called the **No Low-Energy Trivial States conjecture**, or **NLTS conjecture**. We expect that ground states of local Hamiltonians are sufficiently hard to describe (if NP QMA). So low-energy states might not be generated by a quantum circuit of constant depth. More formally:

**( NLTS Conjecture)**:

To reiterate, if we did have such a family of NLTS Hamiltonians, then it we wouldn’t be able to give “easy proofs” for the minimal energy of a Hamiltonian, because we couldn’t just give a small circuit which produced a low energy state.

-error states are states that differ from the ground state in at most qubits. Now, consider -error states (which “agree” with the ground state on most qubits). Then for bounded-degree local Hamiltonians (analogously in the classical case, those where each variable participates in a bounded number of clauses), these states are also low energy. So any theorem which applies to low energy states (such as the NLTS conjecture), should also apply to states with -error (as in the NLETS theorem).

To define low-error states more formally:

**Definition 2.1** (-error states): *Let (the space of positive semidefinite operators of trace norm equal to 1 on ). Let be a local Hamiltonian acting on . Then:*

- is an -error state of if of size at most s.t. .
- is an -error state for if s.t. and is an -error state for .

Here, see that is just the partial trace on some subset of integers , like we’re tracing out or “disregarding” some subset of qubits.

In 2017, Eldar and Harrow showed the following result which is the NLETS theorem.

**Theorem 1** (NLETS Theorem): *There exists a family of 16-local Hamiltonians s.t. any family of -error states for requires circuit depth where .*

In the next two sections, we will provide background for an alternate proof of the NLETS theorem due to Nirkhe, Vazirani, and Yuen. After this, we will explain why the proof of NLETS cannot be used to prove NLTS, since the local Hamiltonian family we construct for NLETS can be linearized. Nirkhe, Vazirani, and Yuen’s proof of NLETS makes use of the Feynman-Kitaev clock Hamiltonian corresponding to the circuit generating the cat state (Eldar and Harrow make use of the Tillich-Zemor hypergraph product construction; refer to section 8 of their paper). What is this circuit? It is this one:

First, we apply the Hadamard gate (drawn as ) which maps the first qubit . Then we can think of the CNOT gates (drawn as ) as propagating whatever happens to the first qubit to the rest of the qubits. If we had the first qubit mapping to 0, then the rest of the qubits map to 0, and likewise for 1. This generates the cat state , which is highly entangled.

Why do we want a highly entangled state? Roughly our intuition for using the cat state is this: if the ground state of a Hamiltonian is highly entangled, then any quantum circuit which generates it has non-trivial depth. So if our goal is to show the existence of local Hamiltonians which have low energy or low error states that need deep circuits to generate, it makes sense to use a highly entangled state like the cat state.

(We’ll write that the state of a qudit – a generalization of a qubit to more than two dimensions, and in this case dimensions – is a vector in . In our diagram above, we’ll see 4 qudits, labelled appropriately.)

Let’s briefly cover the definitions for the quantum circuits we’ll be using.

Let be a unitary operator acting on a system of qudits (in other words, acting on ), where . Here, each is a unitary operator (a gate) acting on at most two qudits, and is a product of such operators.

If there exists a partition into products of non-overlapping two-qudit unitaries (we call these layers and denote them as , where each here is in layer ) such that then we say has layers.

In other words, has size and circuit depth .

Consider and an operator.

For define as the gates in layer whose supports overlap that of any gate in , …, or with .

**Definition 3.1** (lightcone): *The lightcone of with respect to is the union of : .*

So we can think of the lightcone as the set of gates spreading out of all the way to the first layer of the circuit. In our diagram, the lightcone of is the dash-dotted region. We have , , and .

We also want a definition for what comes back from the lightcone: the set of gates from the first layer (the widest part of the cone) back to the last layer.

Define . For , let be the set of gates whose supports overlap with any gate in .

**Definition 3.2** (effect zone): *The effect zone of with respect to is the union .*

In our diagram, see that , , and . The effect zone of is the dotted region.

**Definition 3.3** (shadow of the effect zone): *The shadow of the effect zone of with respect to is the set of qudits acted on by the gates in the effect zone.*

In our diagram, the first three qudits are effected by gates in the effect zone. So .

Given all of these definitions, we make the following claim which will be important later, in a proof of a generalization of NLETS.

**Claim 3.1** (Disjoint lightcones): *Let be a circuit and operators. If the qudits acts on are disjoint from , then the lightcones of and in are disjoint.*

Now we’ll give some definitions that will become necessary when we make use of the Feynman-Kitaev Hamiltonian in our later proofs.

Let’s define a unary clock. It will basically help us determine whatever happened at any time little along the total time big . Let . For our purposes today, we won’t worry about higher dimensional clocks. So we’ll write , but we’ll really only consider the case where , which corresponds to . For simplicity’s sake, we will henceforth just write .

Our goal is to construct something a little similar to the tableaux in the Cook-Levin theorem, so we also want to define a history state:

**Definition 4.1** (History state): *Let be a quantum circuit that acts on a witness register and an ancilla register. Let denote the sequence of two-local gates in . Then for all , a state is a -dimensional history state of if:*

where we have the clock state to keep track of time and is some state such that and . With this construction, we should be able to make a measurement to get back the state at time .

We provide a proof of (a simplified case of) the NLETS theorem proved by Nirkhe, Vazirani, and Yuen in [2].

**Theorem 2** (NLETS): *There exists a family of -local Hamiltonians on a line (Each Hamiltonian can be defined on particles arranged on a line such that each local Hamiltonian acts on a particle and its two neighbors) such that for all , the circuit depth of any -error ground state for is at least logarithmic in .*

First, we’ll show the circuit lower bound. Then we’ll explain why these Hamiltonians can act on particles on a line and what this implies about the potential of these techniques for proving NLTS.

*Proof*: We will use the **Feynman-Kitaev clock construction** to construct a -local Hamiltonian for the circuit : .

Fix and let have size . The Hamiltonian acts on qubits and consists of several local terms depending on :

We can think of a qubit state as representing a step computation on qubits (i.e. for each time , we have a bit computation state of ). Intuitively, a qubit state has energy with respect to iff it is the history state of . This is because checks that at time , consists of the input to . Each checks that proceed correctly from (i.e. that the th gate of is applied correctly). Then checks that at time , the output is . Finally, checks that the qubit state is a superposition only over states where the first qubits represent “correct times” (i.e. a unary clock state where time is represented by zeros followed by ones).

Therefore, has a unique ground state, the history state of , with energy :

Later we will show how to transform into a Hamiltonian on qutrits on a line. Intuitively, the structure of allows us to fuse the time qubits and state qubits and represent unused state qubits by . For the Hamiltonian , the ground state becomes

For the rest of this proof, we work with respect to .

Let be an -error state and let be the subset of qutrits such that . We define two projection operators which, when applied to alone, produce nontrivial measurements, but when applied to together, produce trivial measurements.

**Definition 5.1**: *For any , the projection operator*

*projects onto the subspace spanned by on the th qutrit.*

*For any , the projection operator*

*projects onto the subspace spanned by on the th qutrit.*

**Claim 5.1**:
*For , . For , . Note that these values are positive for any .*

*Proof*: If , then measurements on the th qutrit are the same for and .

If , then any qutrit pure state cannot have nonzero weight in both and (every pure state ends in some number of s which tells which (if any) it can be a part of). Therefore,

If , then projecting onto the th qutrit gives with probability . Therefore, .

Similarly, .

**Claim 5.2**: *For such that , .*

*Proof*:
As before, we can calculate

If , then the th qutrit of is so . If , then because the first qutrits of contain the state so under any measurement, the and th qutrits must be the same.

Now we use these claims to prove a circuit lower bound. Let be a circuit generating (a state with density matrix) . Let be the depth of .

Consider some . For any operator acting on the th qutrit, its lightcone consists of at most gates so its effect zone consists of at most gates which act on at most qudits (called the shadow of the effect zone).

Assume towards contradiction that . Then the shadow of any operator acting only on the th qutrit has size at most since . So there is some outside of the shadow which is in the complement of . By Claim 3.1, we have found two indices such that any pair of operators acting on and have disjoint lightcones in . WLOG let . The lightcones of are disjoint which implies

By the two claims above, we get a contradiction.

Therefore, . We can take any constant epsilon: letting , we get

This analysis relies crucially on the fact that any -error state matches the groundstate on most qudits. However, NLTS is concerned with states which may differ from the groundstate on many qudits, as long as they have low energy.

**Remark 2.1**: *The paper of Nirkhe, Vazirani, and Yuen [2] actually proves more:
*

- A more general lower bound: logarithmic lower bound on the circuit depth of any -approximate ( far in L1 norm) -noisy state (probability distribution over -error states).
- Assuming QCMA QMA (QCMA takes a bit witness string instead of a qubit state as witness), they show a superpolynomial lower bound (on the circuit depth of any -approximate -noisy state).
- “Approximate qLWC codes”, using techniques from their superpolynomial lower bound.

So far, we’ve shown a local Hamiltonian family for which all low-error (in “Hamming distance”) states require logarithmic quantum circuit depth to compute, thus resolving the NLETS conjecture. Now, let’s try to tie this back into the NLTS conjecture. Since it’s been a while, let’s recall the statement of the conjecture:

**Conjecture** (NLTS): *There exists a universal constant and a family of local Hamiltonians where acts on particles and consists of local terms, s.t. any family of states satisfying requires circuit depth that grows faster than any constant.*

In order to resolve the NLTS conjecture, it thus suffices to exhibit a local Hamiltonian family for which all low-energy states require logarithmic quantum circuit depth to compute. We might wonder if the local Hamiltonian family we used to resolve NLETS, which has “hard ground states”, might also have hard low-energy states. Unfortunately, as we shall show, this cannot be the case. We will start by showing that Hamiltonian families that lie on constant-dimensional lattices (in a sense that we will make precise momentarily) cannot possibly be used to resolve NLTS, and then show that the Hamiltonian family we used to prove NLTS can be linearized (made to lie on a one-dimensional lattice!).

**Definition 6.1**: *A local Hamiltonian acting on qubits is said to lie on a graph if there is an injection of qubits into vertices of the graph such that the set of qubits in any interaction term correspond to a connected component in the graph*.

**Theorem 2**: *If is a local Hamiltonian family that lies on an -dimensional lattice, then has a family of low-energy states with low circuit complexity. In particular, if is a local Hamiltonian on a -dimensional lattice acting on qubits for large enough , then for any , there exists a state that can be generated by a circuit of constant depth and such that where is the ground-state energy.*

*Proof*: In what follows, we’ll omit some of the more annoying computational details in the interest of communicating the high-level idea.

Start by partitioning the -dimensional lattice (the one that lives on) into hypercubes of side length . We can “restrict” to a given hypercube (let’s call it ) by throwing away all local terms containing a qubit not in . This gives us a well-defined Hamiltonian on the qubits in . Define to be the -qubit ground state of , and define

where is an -qubit state. Each can be generated by a circuit with at most gates, hence at most depth. Then, can be generated by putting all of these individual circuits in parallel – this doesn’t violate any sort of no-cloning condition because the individual circuits act on disjoint sets of qubits. Therefore, can be generated by a circuit of depth at most . and are both constants, so can be generated by a constant-depth circuit.

We claim that, for the right choice of , is also a low-energy state. Intuitively, this is true because can only be “worse” than a true ground state of on local Hamiltonian terms that do not lie entirely within a single hypercube (i.e. the boundary terms), and by choosing appropriately we can make this a vanishingly small fraction of the local terms of . Let’s work this out explicitly.

Each hypercube has surface area , and there are hypercubes in the lattice. Thus, the total number of qubits on boundaries is at most . The number of size -connected components containing a given point in a -dimensional lattice is a function of and . Both of these are constants. Therefore, the number of size -connected components containing a given vertex, and hence the number of local Hamiltonian terms containing a given qubit, is constant. Thus, the total number of violated local Hamiltonian terms is at most . Taking to be , we get the desired bound. Note that to be fully rigorous, we need to justify that the boundary terms don’t blow up the energy, but this is left as an exercise for the reader.

Now that we have shown that Hamiltonians that live on constant-dimensional lattices cannot be used to prove NLTS, we will put the final nail in the coffin by showing that our NLETS Hamiltonian (the Feynman-Kitaev clock Hamiltonian on the circuit ) can be made to lie on a line (a -dimensional lattice). To do so, we will need to understand the details of a bit better.

**Proposition 6.1**: * for the circuit is -local.*

*Proof*: Recall that we defined

Let’s go through the right-hand-side term-by-term. We will use to denote the qubit of the time register and to denote the qubit of the state register.

- needs to serially access the qubit pairs for all and ensure that they are all set to . Thus, is -local.
- Each term needs to access the states , and and ensure that the state transitions are correct. Thus, is -local.
- needs to access the states and ensure that the progression of the time register is correct. Thus, is -local.

Now, we follow an approach of [3] to embed into a line.

**Theorem 3**: *The Feynman-Kitaev clock Hamiltonian can be manipulated into a -local Hamiltonian acting on qutrits on a line.*

*Proof*: Rather than having act on total qubits ( time qubits and state qubits), let’s fuse each and pair into a single qudit of dimension . If we view as acting on the space of particles , we observe that, following Proposition 6.1, each local term needs to check at most the particles corresponding to times , , and . Therefore, is -local and on a line, as desired.

To see that we can have act on particles of dimension (qutrits) rather than particles of dimension , note that the degree of freedom corresponding to is unused, as the qubit of the state is never nonzero until timestamp . Thus, we can take the vectors

as a basis for each qutrit.

Even though we’ve shown that the clock Hamiltonian for our original circuit cannot be used to prove NLTS (which is still weaker than the original Quantum PCP conjecture) this does not necessarily rule out the use of this approach for other “hard” circuits which might then allow us to prove NLTS. Furthermore, NLETS is independently interesting, as the notion of being low “Hamming distance” away from vectors is exactly what is used in error-correcting codes.

- [1] Sanjeev Arora and Boaz Barak.
*Computational complexity: a modern approach.*Cambridge University Press, 2009. - [2] Chinmay Nirkhe, Umesh Vazirani, and Henry Yuen. Approximate low-weight check codes and circuit lower bounds for noisy ground states.
*arXiv preprint arXiv:1802.07419*, 2018. - [3] Dorit Aharonov, Wim van Dam, Julia Kempe, Zeph Landau, Seth Lloyd, and Oded Regev. Adiabatic quantum computation is equivalent to standard quantum computation.
*SIAM J. Comput.*, 2007.

December 22, 2018

Quantum computers have demonstrated great potential for solving certain problems more efficiently than their classical counterpart. Algorithms based on the quantum Fourier transform (QFT) such as Shor’s algorithm offer an exponential speed-up, while amplitude-amplification algorithms such as Grover’s search algorithm provide us with a polynomial speedup. The concept of “quantum supremacy” (quantum computers outperforming classical computers) has been explored for three general groups of problems:

- Structured problems, such as factoring and discrete logarithm. Out quantum computer takes advantage of the structure of these classes of problems to offer an exponential speedup compared to the best known classical alternative. While these speedups are the most promising, they require a large number of resources and are cannot be feasibly implemented in the near future.
- Quantum Simulations, originally proposed by Richard Feynman in the late 80s was thought to be the first motivation behind exploring quantum computation. Due to the fact that the space of all possible states of the system scales exponentially with the addition of a new element (eg. an atom), complex systems are very difficult to simulate classically. It has been shown that we can use a quantum computer to tackle interesting problems in quantum chemistry and chemical engineering. Furthermore, there are results on sampling the output of random quantum circuits which have been used for “quantum supremacy experiments”.
- General constraint satisfaction and optimization problems. Since these problems are NP-hard it is widely believed that we cannot gain an exponential speedup using a quantum computer, however, we can obtain quadratic speedup but utilizing a variation of Grover’s algorithm.

While these quantum algorithms are very exciting, they are beyond the capabilities of our near-term quantum computers; for example, any useful application of Shor’s factoring algorithm requires anywhere between tens of thousands to millions of qubits with error correction compared to quantum devices with hundreds of qubits that we might have available in the next few years.

Recently there has been increasing interest in hybrid classical-quantum algorithms among the community. The general idea behind this approach is to supplement the noisy intermediate-scale quantum (NISQ) devices with classical computers. In this blog post, we discuss the Quantum Approximate Optimization Algorithm (QAOA), which is a hybrid algorithm, alongside some of its applications.

QAOA is used for optimizing combinatorial problems. Let’s assume a problem with bits and clauses. Each clause is a constraint on a subset of the bits which satisfies a certain assignment. We can define a cost function as follows:

where is the bit string. In this article we consider a minimization problem, therefore we want if satisfies clause and 1 otherwise. Note that in the case of a maximization problem we only need to switch the value assigned to a satisfactory clause to 1. Our objective is to find a (qu)bit string that minimizes (or maximizes) our cost function.

At a higher level, we start with a quantum state in a uniform superposition of all possible inputs . This can be accomplished with qubits which span a space of size . Our goal is to come up with a series of operations that would evolve our initial quantum state into a superposition of states in which the valid solutions would have a significantly higher probability than other states. In manner, upon sampling the quantum state we are likely to get the correct solution with high probability. QAOA uses the cost function to construct a set of operations that would be able to efficiently map the unifrom superposition state into the desired quantum state. These operators involve single qubits rotations around the x-axis, and multiqubit rotations around the z-axis of our qubits.

Now let’s discuss the details of QAOA. For this algorithm we assume that our quantum computer works in the computation basis of . We start by setting our initial state to a uniform superposition over computational basis states:

Next, we define a unitary operator using the cost function as follows:

Here we convert every clause to a Hamiltonian consisting of Pauli Z ($\sigma^z$) operators. Just as a review, the two Pauli operators (X and Z) used in this blog post are representated as follows:

For example if we can map the clause to for a minimization problem. If , then will return a value of 1, and if the operator will return -1. The same applies to qubit as well. Therefore it is not hard to see that if and have the same value, then the operator as defined above will result in a 1, and it’ll result in 0 otherwise. Furthermore, since has integer eigenvalues we can restrict the angle to lie in .

Next, we define the admixing Hamiltonian:

and use it to define a unitary operator which consists of a product of commuting one qubit operations:

where . It’s easy to see that couples 2 or more qubits, while performs a single qubit rotation on the qubits in our system. Using these unitaries and our initial state we define a QAOA angle-dependent “ansatz” state as follows:

Here is the “depth” of our QAOA circuit, and , are each a vector of length controlling the angles for each layer. In the worst case scenario this state can be produce by a quantum circuit of depth , however by taking advantage of the structure of the instance we can further reduce the number of layers required. Let be the expectation of in our ansatz:

and let be the minimum of over angles,

Note that minimization at layers can be viewed as a constrained minimization at layers, therefore

Using an adiabatic approach [1] We can show that

Based on these results our QAOA algorithm will look like the following:

- c: pick a
- c: choose a set of angles
- q: prepare
- q: compute
- c: perform gradient descend/ascend on and get a new set of angles
- repeat from step 3 till convergence
- report the measurement result of in computational basis

If does not asymptotically grow with can be efficiently computed in

In this section we apply the QAOA algorithm to the MaxCut problem with bounded degree. MaxCut is an NP-hard problem that asks for a subset of the vertex set such that the number of edges between and the complementary subset is as large as possible. While QAOA does not offer a theoretical guarantee to solve MaxCut in polynomial time, it offers a path to utilizing NISQ devices for tackling such optimization problems and discuss patterns in such problems that can be used for reducing the number of steps required.

For this section, let’s assume , and we have a graph with vertices and an edge set of size . We can construct a cost function to be maximized as follows:

We can the compute the angle dependent cost of our ansatz as follows:

Let’s consider the operation associated with some edge :

Since QAOA consists of local operations, we may take advantage by thinking about the problem in terms of subproblems (or subgraphs) involving certain nodes. This property will allow us to simplify our clauses even further depending on the desired depth of our quantum circuit, therefore decreasing the amount of resources necessary to implement the algorithm.

The operator includes qubits (nodes) and , therefore the sequence of operators above will only involve qubits that are at most distance away from qubits and . Let’s consider the example of :

It’s easy to see that any factor of that does not depend on or will commute through and cancel out. Since the degree is bounded, each subgraph contains a number of qubits that is independent of , which allows for the evaluation of in terms of subsystems of size independent of .

For an subgraph define:

We can define our total cost as a sum over the cost of each subgraph:

where is a subgraph of type and “…” is used to omit the sequence of angle depending unitaries constructed using the elements of and . is then

where is the number of occurrence of the subgraph in the original edge sum. The function does not depend on and , and the only dependence on these variables comes through the weights from the original graph. The maximum number of qubits that can appear in our sequence of operators comes when the subgraph is a tree. For a graph with maximum degree , the number of qubits in this tree is

(or if ), which is independent of and . Therefore we can see that for constant can be efficiently computed.

Next, let’s consider the spread of C measured in the state .

For fixed and we see that the standard deviation of is upper-bounded by . Using this fact and the appropriate probability bounds we can see that the result of measuring the cost function of the state will be very close to the intended value of which bounds the uncertainty present in quantum measurement.

[1] E. Farhi, J. Goldstone, and S. Gutmann, “A Quantum Approximate Optimization Algorithm,” 2014.

[2] J. S. Otterbach, et. al, “Unsupervised Machine Learning on a Hybrid Quantum Computer,” 2017.

**by Fred Zhang**

*This is the second installment of a three-part series of posts on quantum Hamiltonian complexity based on lectures given by the authors in Boaz and Tselil’s seminar. For the basic definitions of local Hamiltonians, see Ben’s first post. Also check out Boriana and Prayaag’s followup note on area laws.*

This post introduces tensor networks and matrix product states (MPS). These are useful linear-algebraic objects for describing quantum states of low entanglement.

We then discuss how to efficiently compute the ground states of the Hamiltonians of D quantum systems (using classical computers). The density matrix renormalization group (DMRG), due to White (1992, 1993), is arguably the most successful heuristic for this problem. We describe it in the language of tensor networks and MPS.

**1. Introduction **

We are interested in computing the ground state—the minimum eigenvector—of a quantum Hamiltonian, a complex matrix that governs the evolution of a quantum system of qubits. We restrict our attention to the local Hamiltonian, where the matrix is a sum of Hamiltonians each acting only on qubits. In the previous article, we discussed some hardness results. Namely, a local Hamiltonian can be used to encode SAT instances, and we further gave a proof that computing the ground state is QMA-Complete.

Despite the hardness results, physicists have come up with a variety of heuristics for solving this problem. If quantum interactions occur locally, we would hope that its ground state has low entanglement and thus admits a succinct classical representation. Further, we hope to find such a representation efficiently, using classical computers.

In this note, we will see *tensor networks* and *matrix product states* that formalize the idea of succinctly representing quantum states of low entanglement. As a side remark for the theoretical computer scientists here, one motivation to study tensor network is that it provides a powerful visual tool for thinking about linear algebra. It turns indices into edges in a graph and summations over indices into contractions of edges. In particular, we will soon see that the most useful inequality in TCS and mathematics can be drawn as a cute tensor network.

In the end, we will discuss the density matrix renormalization group (DMRG), which has established itself as “the most powerful tool for treating 1D quantum systems” over the last decade [FSW07]. For many 1D systems that arise from practice, the heuristic efficiently finds an (approximate) ground state in its matrix product state, specified only by a small number of parameters.

**2. Tensor Networks **

Now let us discuss our first subject, *tensor networks*. If you have not seen *tensors* before, it is a generalization of matrices. In computer scientists’ language, a matrix is a two-dimensional array, and a tensor is a multi-dimensional array. In other words, if we think of a matrix as a square, then a 3 dimensional tensor looks like a cube. Formally, a (complex) n dimensional tensor maps indices to complex values, namely, to its entries:

The simplest tensor network is a graphical notation for a tensor. For an -dimensional tensor , we draw a star graph and label the center as and the edges as the indices. To evaluate this tensor network, we put values on the edges, *i.e.*, indices, and then the tensor network would spit out its entry specified by the indices.

Notice that the degree of the center is the number of indices. Hence, a tensor network of degree is a vector, and that of degree is a matrix, and so forth.

How is this related to quantum information? For the sake of genearlity we will deal with qudits in , instead of qubits in . Now recall that a quantum state of qudits can be encoded as an dimensional tensor. It can be written as

It is easy to see that all the information, namely, the amplitudes, is just the tensor . In the later sections, we will see more powerful examples of using tensor networks to represent a quantum state.

So far our discussion is focused merely on these little pictures. The power of tensor networks come from its composition rules, which allow us to join two simple tensor networks together and impose rich internal structures.

** 2.1. Composition Rules **

We introduce two ways of joining two simple tensor networks. Roughly speaking, they correspond to multiplication and summation, and I will give the definitions by showing examples, instead of stating them in the full formalism

**Rule #1: Tensor Product.** The product rule allows us to put two tensor networks together and view them as a whole. The resulting tensor is the tensor product of the two if we think of them as vectors. More concretely, consider the following picture.

The definition of this joint tensor is

**Rule #2: Edge Contractions**. At this moment, we can only make up disconnected tensor networks. Edge contractions allow us to link two tensor networks. Suppose we have two dimensional tensor networks. Contracting two edges, one from each, gives us a tensor network of *free edges*. This now corresponds a tensor of dimensions.

We name the contracted edge as . The definition of is

** 2.2. Useful Examples **

Before we move on, let’s take some examples. Keep in mind that the degree of the vertex determines the number of indices (dimensions of this tensor).

Here, one needs to remember that an edge between two tensor nodes is a summation over the index corresponding to the edge. For example, in the vector inner product picture, , where edge is labeled as . Now you would realize that this picture

is the famous

For us, the most important building block is matrix multiplication. Let . By definition

This is precisely encoded in the picture below.

We are ready to talk about matrix product states. In the language of tensor network, a matrix product state is the following picture.

As the degrees indicate, the two boundary vertices represent matrices and the internal vertices represent -dimensional tensors. We can view each matrix as a set of (column) vectors and each -dimensional tensor as a stack of matrices. Then each one of the free edges picks out a vector or a matrix, and the contracted edges multiply them together which gives out a scalar. If this confused you, move on to the next section. I will introduce the formal definition of matrix product states, and you will see that it is just the picture above.

**3. Matrix Product States **

Before giving the definition, let’s talk about how matrix product state (MPS) naturally arises from the study of quantum states with low entanglement. Matrix product state can be viewed as a generalization of *product state*—(pure) quantum state with no entanglement. Let’s consider a simple product state of qubits. It can be factorized:

This state is described by complex scalars , and there is nothing quantum about it. However, if the state has entanglement among its qubits, then we know that it is impossible to be factorized and thereby written as (1). MPS generalizes the form of (1) by replacing the scalars with matrices and vectors.

More formally, a matrix product state starts with the following setup. For an -qudit system, we associate

- a qudit in with vectors ; and
- a qudit in with matrices .

Here, range from to , and is called *bond dimension*. One can think of the set of vectors as a by matrix and the set of matrices as a by by three-dimensional tensor. Then let them correspond to the vertices in MPS picture. With this setup, a quantum state is in matrix product state if it can be written as

It is important to keep in mind that are two vectors, and the other inner terms are matrices, and we get a scalar from the product. Thus, this represents the tensor .

Now back to the picture,

notice that each amplitude from the equation above is an output of the tensor in the picture, where the free edges take values . Also, as discussed earlier, the contracted edges in MPS tensor network correspond to matrix and vector multiplications, so the tensor is precisely represented by the picture.

The complexity of the MPS is closely related to the bond dimension . In particular, the number of parameters in this model is . We would expect that with higher , we may describe quantum states of more entanglement. In other words, the representation power of an MPS increases with . In principle, one can represent any quantum state as an MPS; however, can be exponentially large. See, *e.g.*, Section 4.1.3 of~\cite{schollwock2011density} for a proof. On the other extreme, the product state example shows that if , one can represent and *only* represent unentangled states. To summarize, here is the picture you should keep in mind.

**4. Density Matrix Renormalization Group **

We are now ready to describe Density Matrix Renormalization Group, proposed originally in [Whi92, Whi93]. As mentioned earlier, it does not come with provable guarantees. In fact, one can construct artificial hard instances such that the algorithm get stuck at certain local minima [SCV08]. However, it has remained one of the most successful heuristics for D systems. We refer the readers to [Sch11] for a complete survey.

DMRG is a simple alternating minimization scheme for computing the ground state of a D Hamiltonian. We start with an arbitrary MPS. Then each step we optimize over the set of matrices associated with site , while fixing everything else, and iterate until convergence. (You may wonder if one can simultaneously optimize over multiple sites. It turns out that it is an NP-hard problem [Eis06].)

Formally, the Hamiltonian problem can be phrased as a eigenvalue problem given a Hermitian matrix , and thus we want to optimize over all in MPS of a fixed bond dimension

Here, we assume that the input Hamiltonian is in the product form. In particular, it means that it can be written as a tensor network as

so the numerator of the optimization objective looks like

The DMRG works with the Langrangian of the objective. For some , we will consider

DMRG optimizes over the set of matrices associated with one qudit. Both terms in (2) are quadratic in this set of matrices.

Now to optimize over the set of parameters associated with one site, calculus tells you to set the (partial) derivative to , and the derivative of a quadratic thing is linear. Without going through any algebra, we can guess that the derivative of with respect to a particular site, say the second one, is the same picture except removing the second site on one side.

Notice that the unknown is still there, on the bottom side of each term. The trick of DMRG is to view the rest of the network as a linear map applied to the unknown.

Given and , we now have a clean numerical linear algebra problem of solving

This is called a generalized eigenvalue problem, and it is well studied. Importantly, for D systems, is typically very sparse, which enables very fast solvers in practice. Finally, DMRG sweeps over the sites one after another and stops until convergence is achieved.

**5. Concluding Remarks **

Our presentation of tensor networks and MPS roughly follows [GHLS15], a nice introductory survey on quantum Hamiltonian complexity.

The notion of tensor networks extends well beyond 1D systems, and a generalization of MPS is called tensor product state. It leads to algorithms for higher dimensional quantum systems. One may read [CV09] for a comprehensive survey.

Tensor network has been interacting with other concepts. Within physics, it has been used in quantum error correction [FP14, PYHP15], conformal field theory [Orú14], and statistical mechanics [EV15]. In TCS , we have found its connections with Holographic algorithms [Val08, CGW16], arithmetic complexity [BH07, CDM16, AKK19], and spectral algorithms [MW18]. In machine learning, it has been applied to probabilistic graphical models [RS18], tensor decomposition [CLO16], and quantum machine learning [HPM18].

For DMRG, we have only given a rough outline, with many details omitted, such as how to set and and how to obtain the Hamiltonian in the matrix product form, and how to compute the linear maps and for each iteration. An interested reader may read [Sch05, Sch11].

**References**

[AKK19] Per Austrin, Peeri Kaski, and Kaie Kubjas. Tensor network complexity of multilinear maps. In Proceedings of the 2019 Conference on Innovations in Theoretical Computer Science. ACM, 2019.

[BH07] Martin Beaudry and Markus Holzer. The complexity of tensor circuit evaluation. Computational Complexity, 16(1):60, 2007.

[CDM16] Florent Capelli, Arnaud Durand, and Stefan Mengel. e arithmetic complexity of tensor contraction. eory of Computing Systems, 58(4):506{527, 2016.

[CGW16] Jin-Yi Cai, Heng Guo, and Tyson Williams. A complete dichotomy rises from the capture of vanishing signatures. SIAM Journal on Computing, 45(5):1671{1728, 2016.

[CLO16] Andrzej Cichocki, Namgil Lee, Ivan V Oseledets, A-H Phan, Qibin Zhao, and D Mandic. Low-rank tensor networks for dimensionality reduction and large-scale optimization problems: Perspectives and challenges part 1. arXiv preprint arXiv:1609.00893, 2016.

[CV09] J Ignacio Cirac and Frank Verstraete. Renormalization and tensor product states in spin chains and laices. Journal of Physics A: Mathematical and Theoretical, 42(50):504004, 2009.

[Eis06] Jens Eisert. Computational difficulty of global variations in the density matrix renormalization group. Phys. Rev. Le., 97:260501, Dec 2006.

[EV15] G. Evenbly and G. Vidal. Tensor network renormalization. Phys. Rev. Le., 115:180405, Oct 2015.

[FP14] Andrew J Ferris and David Poulin. Tensor networks and quantum error correction. Phys. Rev. Le., 113(3):030501, 2014.

[FSW07] Holger Fehske, Ralf Schneider, and Alexander Weie. Computational Many-Particle Physics. Springer, 2007.

[GHLS15] Sevag Gharibian, Yichen Huang, Zeph Landau, and Seung Woo Shin. Quantum Hamiltonian complexity. Foundations and Trends in Theoretical Computer Science, 10(3):159, 2015.

[HPM18] William James Huggins, Piyush Patil, Bradley Mitchell, K Birgia Whaley, and Miles Stoudenmire. Towards quantum machine learning with tensor networks. Quantum Science and Technology, 2018.

[MW18] Ankur Moitra and Alexander S Wein. Spectral methods from tensor networks. arXiv preprint arXiv:1811.00944, 2018.

[Orú14] Román Orús. Advances on tensor network theory: symmetries, fermions, entanglement, and holography. e European Physical Journal B, 87(11):280, 2014.

[PYHP15] Fernando Pastawski, Beni Yoshida, Daniel Harlow, and John Preskill. Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence. Journal of High Energy Physics, 2015(6):149, 2015.

[RS18] Elina Robeva and Anna Seigal. Duality of graphical models and tensor networks. Information and Inference: A Journal of the IMA, 2018.

[Sch05] Ulrich Schollwöck. The density-matrix renormalization group. Rev. Mod. Phys., 77(1):259, 2005.

[Sch11] Ulrich Schollwöck. The density-matrix renormalization group in the age of matrix product states. Annals of Physics, 326(1):96, 2011.

[SCV08] Norbert Schuch, Ignacio Cirac, and Frank Verstraete. Computational difficulty of finding matrix product ground states. Phys. Rev. Le., 100(25):250501, 2008.

[Val08] Leslie G Valiant. Holographic algorithms. SIAM Journal on Computing, 37(5):1565, 2008.

[Whi92] Steven R. White. Density matrix formulation for quantum renormalization groups. Phys. Rev. Le., 69:2863, Nov 1992.

[Whi93] Steven R. White. Density-matrix algorithms for quantum renormalization groups. Phys. Rev. B, 48:10345, Oct 1993.

Cross-posted from https://wsmoses.com/blog/2018/12/18/boaz/

Lecturer: Aram Harrow

Scribes: Sinho Chewi, William S. Moses, Tasha Schoenstein, Ary Swaminathan

November 9, 2018

Sampling from thermal states was one of the first and (initially) most important uses of computers. In this blog post, we will discuss both classical and quantum Gibbs distributions, also known as thermal equilibrium states. We will then discuss Markov chains that have Gibbs distributions as stationary distributions. This leads into a discussion of the equivalence of mixing in time (i.e. the Markov chain quickly equilibrates over time) and mixing in space (i.e. sites that are far apart have small correlation). For the classical case, this equivalence is known. After discussing what is known classically, we will discuss difficulties that arise in the quantum case, including (approximate) Quantum Markov states and the equivalence of mixing in the quantum case.

We have already learned about phase transitions in a previous blog post, but they are important, so we will review them again. The **Gibbs** or **thermal distribution** is defined as follows: Suppose that we have an **energy function** , which takes -bit strings to real numbers. Usually, , where each term depends only on a few bits. For example, the energy might be the number of unsatisfied clauses in a 3-SAT formula, or it may arise from the Ising model. The Gibbs distribution is

where the normalization factor in the denominator, also called the **partition function**, is . Another, perhaps more operational, way to define the Gibbs distribution is:

In this expression, is the set of probability distributions on , is the Shannon entropy, and is a constant representing the average energy. We are thinking of probability distributions and as vectors of size . It turns out that if we solve this optimization problem, then the Gibbs distribution is the unique solution.

Why is it useful to work with Gibbs distributions?

Gibbs distributions arise naturally in statistical physics systems, such as constraint satisfaction problems (CSPs), the Ising model, and spin glasses. One approach to deal with Gibbs distributions is through belief propagation (BP), which yields exact inference on tree graphical models and sometimes phase transition predictions on loopy graphs. Instead, we will focus on a different approach, namely,

*sampling*from the Gibbs distribution.If we want to minimize (say, to find a 3-SAT solution), we can use

**simulated annealing**. The idea of annealing is that we want to produce a crystal; a crystal is the lowest energy configuration of molecules. If we heat up the substance to a liquid and then cool it quickly, we will not get a nice crystal, because little bits of the material will point in different directions. In order to form a crystal, we need to cool the system slowly.In computer science terms, we take a sample from a high temperature because sampling is generally easier at a higher temperature than at a lower temperature. We then use that sample as the starting point for an equilibration process at a slightly lower temperature, and repeat this procedure. If we reach zero temperature, then we are sampling from the minimizers of . In practice, the system will usually stop mixing before we get to zero temperature, but this is a good heuristic. You can think of this process as gradient descent, with some additional randomness.

Gibbs distributions are used to simulate physical systems.

Gibbs distributions are used in Bayesian inference due to the Hammersley-Clifford theorem, which will be discussed next.

Gibbs distributions are also connected to multiplicative weights for linear programming (not discussed in this blog post).

In order to present the Hammersley-Clifford theorem, we must first discuss Markov networks. For this part, we will generalize our setup to a finite alphabet , so the energy function is now a function .

First, let us recall the idea of a **Markov chain** with variables , , .

The random variables , , form a Markov chain if their joint distribution can be written in a factored way: . For example, imagine that , , represent the weather on Monday, Tuesday, and Wednesday respectively. These random variables form a Markov chain if, conditioned on the weather on Tuesday, we have all of the information we need to forecast the weather on Wednesday. Another way to say this is that conditioned on the weather on Tuesday, then the weather on Monday and the weather on Wednesday are **conditionally independent**. Note that the weather on Monday and the weather on Wednesday are *not* independent; there can be correlations, but these correlations are mediated through the weather on Tuesday. It is important to note that the definition of a Markov chain is symmetric with respect to going forwards or backwards in time, so we can also write the conditional independence condition as .

The conditional independence condition can also be written as Recall that for two random variables and with joint distribution , they are independent, i.e., , if and only if , where here denotes the mutual information. Similarly, conditional independence is equivalent to the **conditional mutual information** equaling zero. This quantity is defined as .

Keep in mind that conditional independence is characterized in two equivalent ways: via an algebraic condition on the distributions, and via mutual information.

A **Markov network** is like a Markov chain, but with more random variables and a more interesting structure. Imagine that we have a graph, where each node is associated with a random variable and the edges encode possible correlations. A Markov network has the property that if we take any disjoint collection of nodes , , and such that and are fully separated by (that is, any path from to must go through , or alternatively, removing leaves and disconnected), then . The notation here means the collection of random variables associated with the nodes in .

For example:

Here, if , , and , then separates and .

A Markov network is also called a **graphical model** or a **Markov random field**; and yet another name for them is *Gibbs distribution*, which is the content of the following theorem:

**Theorem 1** (Hammersley-Clifford Theorem): *Let be a strictly positive distribution on . Then, can be represented as a Markov network with respect to a graph if and only if can be expressed as a Gibbs distribution , where is the set of cliques (fully connected subsets) of . *

This theorem says that Markov networks are the same as Gibbs states, *with the same notion of locality*.

The Hammersley-Clifford theorem implies an area law for mutual information; we will explain what this is and sketch why this is true. Divide a system into two disjoint pieces and . We want to know about the mutual information between and , . The Hammersley-Clifford theorem gives us a bound which depends only on the size of the boundary between these sets. For simplicity, assume . Also, assume that the interactions have bounded range; then, the Hammersley-Clifford theorem tells us that .

Now, we will use the fact . We can see this by writing out the expressions, but the intuition is that the term on the left asks about how much knows about , having already known about . This equals how much knows about and combined, minus how much knows about alone. In this case, since we said , then is the same as . In general, however, we have an upper bound:

In this calculation, we have used (the information between and is the amount by which the entropy of gets reduced once we know ) and (which is true classically).

Since the mutual information only scales with the *surface area* of the boundary and not with the area of the two regions and , this is known as an *area law* [1].

In Bayesian inference, we have a model for a system which can be very complicated. The model represents our assumptions on how parts of the system are causally related to the rest of the system. We have some observations, and we want to sample from a distribution conditionally on the fixed observations. Sampling from a conditional distribution is not the same as sampling from the original distribution, but we can still formally represent the conditional distribution as a Markov network. Therefore, sampling from Markov networks is a broadly useful task.

As an example of a complicated Bayesian model, consider a *hierarchical Bayesian model* [2]. Bayesian statistics requires choosing a prior distribution, and when there is a natural parameterized family of priors that a statistician can use, it may make sense to introduce a distribution over the priors; this is known as *introducing a hyperparameter*, and inference in the resulting hierarchical model (including computation of the posterior distribution) is frequently intractable. However, it is still desirable to work with these models because they are often more accurate than models in which the prior is handpicked by a statistician.

The task of sampling from an arbitrary Gibbs distribution is MA-complete [3], and it is not hard to see that at low enough temperatures this problem is at least NP-hard. So, how do we sample from these distributions?

This section will discuss Monte Carlo Markov chain (MCMC) methods, namely the Metropolis-Hastings algorithm and Glauber dynamics. Readers familiar with these methods may wish to skip to the discussion of mixing in time. For readers who wish to build more intuition about Markov chains before proceeding, see the Appendix, where the simple example of the random walk on a cycle is treated in detail.

The general approach is to use a Markov chain. Let be the possible states of the system. Effectively, a Markov chain is a way of doing a random walk over .

The transition probabilities of the Markov chain are^{1} Here, is the **transition probability matrix**. The column at index of is the probability distribution of the next state of the Markov chain, if the current state is . The row at index is a row of probability values which give the probabilities of jumping into state from every other state. It has the properties that its entries are non-negative and for every , . These properties say that is a (column) **stochastic matrix**.

Suppose we start at a state ; or, more generally, we will start with a distribution over . If we move according to the chain once, the distribution will be . If we move agian, the distribution will be . In general, after movements, the distribution is . So, we can express the dynamics of the chain as matrix-vector multiplication.

It is worth mentioning that if we are simulating the chain on a computer and we are manipulating -bit numbers, then these probability vectors are of size so it becomes impractical to store the entire probability distributions.

The justification for our algorithms is the following theorem.

**Theorem 2** (Perron-Frobenius Theorem): *If is a stochastic aperiodic matrix, then one of the eigenvalues is , and all other eigenvalues have magnitude strictly less than . There is a unique probability distribution such that . *

The theorem implies that will converge to the stationary distribution as . So, if we want to sample from a distribution, this provides a method of doing so: cook up a Markov chain that equilibrates to the desired distribution, and then run the Markov chain until convergence. *A priori*, it is not obvious how we can design the Markov chain. At first, our problem was to sample from a probability distribution (a vector), and now we have changed the problem to designing an entire matrix, which does not appear to make our task easier.

Now, the question becomes: how does one come up with Markov chains that give you the desired stationary distribution?

The first algorithm we will introduce is the **Metropolis-Hastings algorithm**. One more desirable feature of a Markov chain is that it satisfies **detailed balance**, which says for all and . This condition says that if we pick a point with probability according to the stationary distribution and transition, the probability of picking and then moving to should be the same as picking and then moving to .

For a Markov chain in equilibrium, the total amount of probability flowing out of must equal the total amount of probability flowing into . For example, the United States might export products to Europe and import from China. Detailed balance says that the flow along each edge must balance, which is a more demanding condition. In the example with country trade deficits, we are requiring that all bilateral trade deficits must be zero.

Mathematically, detailed balance implies that can be transformed, via similarity transformations, into a symmetric matrix. The Metropolis-Hastings algorithm says that we should choose with the property Suppose that we have an underlying graph on our state space, and suppose that we are at a state . The algorithm chooses a random neighbor, say , and then accepts or rejects this move with some probability. If the move is accepted, then we move to and continue the algorithm from there. Otherwise, if the move is rejected, then we stay at . We are free to choose any underlying graph (as long as it is connected and has a self-loop), and then we will tune the acceptance probability so that detailed balance holds.

Look at the trial move . One way we can accomplish detailed balance is by looking at the ratio . If , then always accept the move. If , then accept the move with probability .

To get an idea for how the algorithm works, suppose that our underlying graph is -regular. Then, for neighbors and ,

**Claim**: which is manifestly symmetric in and ; thus, we have reversibility. This is the basic idea of the Metropolis-Hastings algorithm.

How does it work for a Gibbs distribution , where the energy function might, for example, count the number of violated clauses in a 3-SAT formula? In this case, we might be a little worried. The numerator of is pretty easy to compute (we can count how many violated constraints there are), but the denominator is hard to compute. In general, it is #P-hard to compute the denominator, because as drops to , the partition function in this case approaches the number of 3-SAT solutions. So, how do we calculate the ratios that the algorithm requires? We’re able to do this because the ratio does not depend on :

Suppose that the energy is a sum of local terms, and the underlying graph corresponds to modifying one site at at a time. What this means is that the graph is and the edges in the graph correspond to flipping exactly one bit. In this case, it becomes very easy to evaluate the computations needed for the algorithm; in fact, we can even do them in parallel.

How do we choose the underlying graph? The key idea is that we do not want the majority of our moves to be rejected. A good example to keep in mind is the **Ising model**, where the configurations are and the energy is . If for all , , then we say that the model is **ferromagnetic** (we obtain lower energy by making the sites agree with each other). Of course, an **antiferromagnetic** model is just the opposite of this.

Assume that the bits are laid out in a square and if and are neighbors on the square, and if they are not. As we vary the quantity , we observe a *phase transition*. If is small, then the coupling between the random variables is weak and the different parts of the system are almost independent; we call this the **disordered phase**. If is large, then the spins want to align in the same direction and the Gibbs distribution will look almost like the following: with probability , all spins are , and with probability , all spins are ; we call this the **ordered phase**.

In the disordered phase, when the spins do not need to align so closely, the Metropolis-Hastings algorithm will work well. In the ordered phase, the algorithm is doomed. Indeed, suppose that most of the spins are . As time proceeds, any s will switch to . There may be islands of spins initially, but it will be energetically favorable for these islands to shrink over time. Therefore, there will be an exponentially small chance for the system to switch to a configuration with mostly ’s, and thus the chain takes exponentially long to mix. Here, people are interested in understanding the *autocorrelation time*, because the goal is to run the chain for some time, get one sample, run the chain for some more time, get another sample, etc.

This next method (**Glauber dynamics**) is essentially the same as Metropolis-Hastings, but this is not immediately obvious. We are at a state . (For the Metropolis-Hastings algorithm, we could be walking on a state space without a product structure. However, Glauber dynamics requires a product structure.) Then, we update to with chance . In other words, we hold all other bits fixed, and conditioned on those other bits, we resample the th bit. Like Metropolis-Hastings, is stationary for this chain.

It is not obvious that these conditional distributions can be computed efficiently, but it is possible since normalizing the conditional distribution only requires summing over the possible configurations for a single random variable. On a Markov network, the conditional probability is , where denotes the set of neighbors of . This makes the computation a constant-sized calculation (i.e., does not depend on the size of the system).

For example, in the Ising model, suppose we are at state . In Glauber dynamics, we pick a vertex u.a.r. and update it to with probability

Mixing in time means that the dynamics will equilibrate rapidly. It turns out that this is equivalent to mixing in space, which means that itself has decaying correlations. For example, the Ising model at low temperature has a lot of long-range correlations, but at high temperature it does not. For the high temperature regime, we can prove that mixing in time occurs. We will prove this for the ferromagnetic Ising model. The result is known more generally, but the proofs are much easier for the Ising model.

People have known about the Metropolis-Hastings algorithm since the 1950s, but only recently have researchers been able to prove convergence guarantees for the 2D Ising model. There is a large gap between theory and practice, but in some situations we can prove that the algorithm works.

Sampling from the distribution is roughly equivalent to estimating the partition function (sampling-counting equivalence). There have been many papers addressing tasks such as estimating the non-negative permanent, the number of colorings of a graph, etc.^{2} A dominant way of accomplishing these tasks is proving that the Metropolis-Hastings algorithm converges for these problems. It is easy to find algorithms for these problems that converge to Gibbs distributions, but the convergence may take exponential time.

We will look at the situation when the energy function looks like the Ising model, in the sense that the interactions are local and reflect the structure of some underlying space. Also, assume that the interactions are of size and that the scaling comes from the size of the system. When can we expect that our algorithms work? There are two main cases when we can argue that there should be rapid mixing.

- High temperature regime: The system is very disordered, and in the limit as the temperature approaches infinity, we get the uniform distribution.
- One-dimension: In 1D, we can exactly compute the partition function using dynamic programming. Before, we mentioned that if there are a sea of s and an island of s, then it is energetically favorable for the island to shrink; note that this is no longer true in 1D. In a way, 1D systems are more “boring” because they cannot exhibit arbitrarily long-range correlations.

In this part of the blog post, we will try to be more proof-oriented. We will start by explaining why it is plausible that high temperature means that the chain will mix rapidly in time.

One method of proving rates of convergence for Markov chains is by analzying the spectral gap. Another method is the **coupling method**.

The idea behind the coupling method is to start with two configurations . We want each one to evolve under the Markov chain.

The key part is that there is still some freedom with respect to what the dynamics looks like. In particular, we are allowed to correlate the and processes. Thus, we are defining a joint transition probability . We want to design the process such that and are closer together than and . Imagine that we have two particles bouncing around. Each particle follows the dynamics of , but they are correlated so that they drift together, and once they meet, they stick together. It turns out that the mixing time can be upper bounded by the time it takes for the particles to meet each other.

Assume we have some sort of distance function on the underlying space and we can prove that . Then, it turns out that the mixing time , i.e. the time required to get within of the stationary distribution, is upper bounded as

Initially, the two particles can be apart, but the expected distance is exponentially shrinking as we run the coupling, so the mixing time is logarithmic in the diameter.

The distance between probability distributions is defined as follows. Let and be two probability distributions on . Then, the metric is:^{3}

In this expression, and denote the first and second marginals of respectively. The minimum is taken over all *couplings* of and . This is the correct way to measure the distance between distributions. To give some intuition for this quantity, the quantity on the right represents the best *test* to distinguish the two distributions. If and are the same, we can take a coupling in which and are always identical. If and have disjoint supports, then no matter what coupling we use, and will never be equal.

It suffices to consider when and are neighbors, i.e. at distance apart. This is because if we have and far apart, then we could look at the path between them and reduce to the case when they are neighbors. Formally, this is known as *path coupling*. The formal statement is in Theorem 12.3 of [4]:

**Theorem 3**: *Let be a connected weighted graph on the state space, where no edge has weight less than . Let be the length of the shortest path from to in and let be the diameter of . Suppose there is a coupling such that for some *,

*for all neighboring pairs , , i.e., those pairs connected by an edge in . Then, the mixing time is bounded by *

Recall that in Glauber dynamics, we pick a site randomly and then update the site conditioned on its neighbors. The first way we will couple together and is by picking the *same* site for both of them.

- Pick a random .
- If , then set (if the neighborhoods of the two points agree, then update them the same way). Otherwise, update them using the best possible coupling, i.e., pick a coupling for which minimizes .

So if , then the points will never drift apart. The reason why analyzing this coupling is non-trivial is because there is a chance that the distance between the two points can *increase*.

Assume that the degree of the graph is . Suppose that , that is, there is a single such that . What will happen to and ? We start by picking a random . There are three cases:

- (with probability ): Nothing changes; and agree at , and and will also agree at . The distance remains at .
- (with probability ): We picked the one spot in which the two configurations differ. The neighborhoods of are the same for and , so we update in the same way for both processes, and the distance drops to .
- (with probability ): We could have different updates. Here, we have to use the high temperature assumption, which says that if we change one bit, the probability of a configuration cannot change too much.In the Ising model, . Changing can bias the energy by at most , so the expected distance afterwards is .

Adding these cases up to get the overall expected distance gives

for large enough, so the expected distance will shrink. This argument also tells us how large the temperature must be, which is important for applications. This gives us Notice that this is the same dependence as the coupon collector problem. Therefore, in the high temperature regime, the system behaves qualitatively as if there are no correlations.

The analysis of Glauber dynamics at high temperature is already a version of the equivalence between mixing in time and mixing in space. It says that if the correlations even with the immediate neighbors of a node are weak, then Glauber dynamics rapidly mixes.

Now, we want to consider the situation in which there can be strong correlations between immediate neighbors, but weak correlation with far away sites. We want to show that spatial mixing implies temporal mixing.

We will give a few definitions of correlation decay. (Note: The definitions of correlation decay below are not exactly the ones from Aram’s lecture. These definitions are from [5] and [6].)

For non-empty and , let be the distribution of the spins in conditional on the spins in being fixed to . For , let be the marginal of on the spins in . We will assume that the interactions between the spins have finite range , and denotes the -boundary of , i.e., .

- (
**Weak decay of correlations**) Weak spatial mixing holds for if there exist constants such that for any subset , - (
**Strong decay of correlations**) Strong spatial mixing holds for if there exist constants such that for every and every differing only at site , - (
**Strong decay of correlations**) Strong spatial mixing in the*truncated*sense holds for if there exist such that for all functions which depend only on the sites at and respectively and such that ,

Here, is the **correlation length** (in physics, it is the characteristic length scale of a system). In the disordered phase, the correlation length is a constant independent of system size. For our purposes, the main consequence of these definitions is that the effective interaction range of each spin is . For the Ising model, there is a key simplification due to *monotonicity*. Namely, the ferromagnetic Ising model has the nice property (which is not true for other models) that if we flip a sign from to , this only makes more likely everywhere. This is because the spins want to agree. There are a lot of boundary conditions to consider, but here, due to monotonicity, we only need to consider two: all of the spins are , and all of the spins are . All spins will give the highest probability of a spin, and all spin will give the lowest probability of a spin. This monotonicity property is generally not required for time-space mixing equivalence to hold, but it greatly simplifies proofs.

It is a very non-obvious fact that all of these notions of spatial mixing are equivalent. We will sketch a proof that strong correlation decay implies that .

The idea is to use another coupling argument. Let differ in one coordinate, i.e., and for . We want to argue that the expected distance between the processes will decrease. The proof uses a generalization of Glauber dynamics called **block Glauber dynamics**. In Glauber dynamics, we take a single spin and resample it conditioned on its neighbors. In block Glauber dynamics, we take an box and resample it conditioned on its neighbors. There is an argument, called *canonical paths*, which can be used to show that if block Glauber dynamics mixes, then regular Glauber dynamics also mixes (slightly more slowly; we lose a factor, but anyway will be a large constant) so analyzing block Glauber dynamics is fine.

If lies in the box, then the expected change in distance is . If is far away from the box, then there is no change. If is in the boundary of the box, then it is possible for the distance to increase. However, strong spatial mixing allows us to control the influence of a single site, so the expected change in distance is bounded by . Now, since is a constant, if we choose sufficiently large, then we will have the same situation as in the high temperature case: the expected distance will exponentially shrink over time.

The quantum version of Markov chains has many more difficulties. The first difficulty is that the Hammersley-Clifford theorem (which we have been relying on throughout this blog post) fails.

To properly discuss what we mean, let’s set up some notation. Readers already familiar with density matrices, quantum entropy, and quantum mutual information may wish to skip to the next subsection. Most of the time we discuss quantum objects here, we’ll be using density matricies, often denoted . A density matrix can be thought of as an extension to regular quantum states , where there is some classical source of uncertainty.

A density matrix is a positive semidefinite matrix with trace . This extends the notion of a classical probability distribution; in the quantum setting, a classical probability distribution (thought of as a vector whose entries sum to ) is represented as the density matrix .

For example, we can consider a situation in which there is a probability that we started with the quantum state and a probability that we started with the quantum state . This would be denoted as follows:

Density matricies are generally useful for a lot of tasks, but for our purposes a density matrix will be used to discuss both the classical and quantum “uncertainty” we have about what state we have.

Now let’s also talk about a second important piece of notation: the tensor product. Often when discussing quantum states, it is important to discuss multiple quantum states simultaneously. For example, Alice has one system and Bob has another system . However, these systems might be entangled, meaning that the results of the two systems are correlated.

For instance, let us consider the following state:

This particular state has the property that Alice and Bob will always both measure or they will both measure . The notation for tensors is often ambiguous in the literature as there are many ways of specifying tensors. For instance, above we used subscripts to explicitly denote which particle was in system and which was in system . One may also choose to simply use the index of the system as below. The symbol is used to denote a tensor between states (where it is assumed that the first state is system and the second, system ). Gradually folks may shorten the notation as follows:

These are all notations for the same state. Let’s now talk about this state in the context of a density matrix. The density matrix of this state is as follows:

Writing the density matrix as makes explicit that this is the density matrix over systems and .

A crucial operation that one will often perform using density matricies is the partial trace. The partial trace is a way of allowing us to consider only a smaller part of the larger part of the system, while taking into account the influence of the larger system around it.

Here’s an example: Suppose Bob wants to know what his state is. However, Bob really doesn’t care about Alice’s system and just wants to know what the density matrix for his system is. Bob’s density matrix is simply the following density matrix (a 50% chance of being in and a 50% chance of being in ).

More explicitly, we could write the following:

The partial trace is an operation that will let us take our original density matrix and generates a new density matrix that ignores system . This is specifically called the partial trace over , or .

So how do we do this? We simply sum over the state (effectively taking a trace, but only along one axis):

This is easier to evaluate using certain choices of notation:

This gives us the answer that we had expected.

We now have all of the tools we need to talk about quantum entropy. Intuitively, entropy can be thought of as the amount of uncertainty we have for our system, or equivalently the amount of information it takes to define our system. The entropy for a quantum system is defined as follows:

Note that here we use the shorthand to denote . Here, writing without the subscript indicates that this is the full or normal trace that one might expect (or equivalently performing the partial trace over all systems). We can now define the conditional entropy of a system as follows:

This definition intuitively makes sense since we can think of conditional entropy as the amount of information it takes to describe our joint system , given that we already know what is.

We can now discuss quantum mutual information, the amount of information that measuring system will provide you about system . Like the classical case, this is defined as follows:

We can now finally discuss **quantum mutual information (QCMI)**, defined as follows: . With some algebraic simplifications, one can arrive at the expression:

The QCMI equals if and only if is a **quantum Markov state**. Classically, the entropic characterization of conditional independence corresponds to an algebraic characterization.

Here, the algebraic characterization is more grueling. We have

Equivalently,

Here, is called the **Petz recovery map**,^{4} . One can think of a recovery may as a way that we can reconstruct the entire system using just system . It is not obvious that this is a quantum channel, but it is.

Suppose is a probability distribution, so for some vector . Then, all of the density matrices are diagonal and commuting. Then, the recovery map means that we divide by and multiply by , i.e., multiply by . This is the natural thing to do if we lost our information about and were trying to figure out what was based on our knowledge of . This is why is known as a *recovery* map, and it is used to discuss conditional distributions in the quantum setting. In the classical case, if we start with , look only at , and use this to reconstruct , then we would have the whole state in a Markov chain. That is why this is a plausible quantum version of being a Markov chain.

However, quantum Gibbs states are not, in general, quantum Markov chains. The failure of this statement to hold is related to *topological order*, which is similar to the degrees of freedom that show up in error correcting codes.

Here, we will formally define a quantum Markov network. The reference for this is [7].

Let be a finite graph. We associate with each vertex a Hilbert space and we consider a density matrix acting on . Then, is a **quantum Markov network** if for all , is conditionally independent of given , where the conditional independence statement is w.r.t. and means that the corresponding QCMI satisfies .

A quantum Markov network is called **positive** if has full rank. (Recall that in the statement of the Hammersley-Clifford Theorem, , it is assumed that the distribution is strictly positive.)

Now, consider the following example. First, we introduce the Pauli matrices

We define a Hamiltonian on three qubits , , by

(Juxtaposition in the above expression signifies the tensor product as discussed before.) Finally, for , we define the Gibbs state

The Hamiltonian here has local terms which correspond to interactions , . However, it can be shown that the QCMI between and conditioned on w.r.t. is non-zero, which means that this is not a quantum Markov network w.r.t. the line graph . This demonstrates the failure of the Hammersley-Clifford Theorem in the quantum setting.

We will briefly discuss the results of two papers.

- [8] This paper shows that mixing in space implies mixing in time in the quantum case. However, the result of the paper only applies to commuting Hamiltonians. For commuting Hamiltonians, it turns out that quantum Gibbs states are quantum Markov networks. They use a version of Glauber dynamics, which can be simulated on a quantum computer but are also plausible dynamics for a physical system in nature. This is a difficult paper to read, but it is worth digesting if you want to work in the field.
- [9] This second paper is much easier and more general, covering non-commuting Hamiltonians, but it requires more conditions. They give a method of preparing the Gibbs state which can run on a quantum computer, but the dynamics are not plausible as a physical system because they are too complicated. The more complicated dynamics allows them to make the proof work. The paper also uses QCMI.They have two assumptions. The first assumption looks like mixing in space (weak correlation decay). The second assumption is that the state looks approximately like a quantum Markov network (this is definitely not met in general). A very important paper in this space is a recent breakthrough ([10]) which characterizes quantum Markov chains. They show that if the QCMI is bounded by , then the recovery map is -close to , i.e., low QCMI implies that the recovery map works well. This is trivial to prove classically, but very difficult in the quantum world.The algorithm in [9] is very elegant. Essentially, we take the entire system and punch out constant-sized boxes. If we can reconstruct the region outside of the boxes, then we can use the recovery maps to reconstruct the regions inside of the boxes, and the boxes are far apart enough so they are almost independent. For this argument, we must assume that the QCMI decays exponentially. Whenever we have exponential decay, we get a correlation decay that sets the size of the boxes. It is very difficult to condition on quantum states, but recovery maps provide a sense in which it is meaningful to do so. The paper gives an efficient method of preparing Gibbs states and simulating quantum systems on quantum computers.

The standard treatment of information theory is [11]. This book contains definitions and properties of entropy, conditional entropy, mutual information, and conditional mutual information.

To see a treatment of the subject of Markov chains from the perspective of probability theory, see [12] or the mathematically more sophisticated counterpart [13]. An introduction to coupling can be found in [14], as well as [4] (the latter also contains an exposition to spatial mixing). The connection between Markov chain mixing and the so-called *logarithmic Sobolev inequality* is described in [15].

We have points on the cycle, . At each step, we move left or right with probability . We can write the transition matrix as

where is the shift operator . The matrix is diagonalized by the Fourier transform. Define, for ,

We have the same amount of amplitude at every point, but there is a varying phase which depends on . If , we get the all-ones vector. If is small, then the phase is slowly varying. If is large, then the phase is rapidly varying. Look at what happens after we apply the shift operator:

After the shift, we pick up an additional phase based on how rapidly the phase is varying. From this, we get:

The eigenvalues are

Only will give me an eigenvalue of .

How do we analyze ? We should Fourier transform the distribution.

If is odd, then as , for all , so . Whatever you put into this operator, you get out.

The example of the random walk on the cycle shows that there is generally a unique stationary distribution and suggests that the speed of convergence is determined by how close the other eigenvalues are to . Specifically, suppose for simplicity that the eigenvalues of are (real and positive). Then, the convergence time is on the order of .

Typically, the distance of the eigenvalues from reflects the size of the physical system. Even from the simple example, we can get some physical intuition from this. If is small, then the spectral gap is . Thus, the convergence time is , which is indeed the convergence time for a random walk on a cycle.

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- This is the opposite of the probabilists’ convention, i.e., the transition probability matrix that we define here is the
*transpose*of the one usually found in most probability theory textbooks. ↩ - As a side note, it may be a good research question to investigate to what extent quantum algorithms can be used to compute summations whose terms are possibly negative. In quantum Monte Carlo, the quantum Hamiltonian is converted to a classical energy function; this conversion always works, but sometimes you end up with complex energies, which is terrible for estimating the partition function because terms can cancel each other out. ↩
- You may recognize this as the total variation norm. ↩
- Petz wrote about quantum relative entropy in 1991, way before it was cool. ↩