## Aaronson and Arkhipov’s Result on Hierarchy Collapse

Scott Aaronson gave a thought-provoking lecture in our Theory seminar three weeks ago.  (Actually, this was eleven months ago.) The slides are here . The lecture discussed two results regarding the computational power of quantum computers. One result from this paper gives an oracle-evidence that there are problems in BQP outside the polynomial hierarchy.  The method is based on “magnification” of results on bounded depth circuits. (It is related to the Linial-Nisan conjecture.)

The second result that we are going to discuss in this post (along with some of my provoked thoughts) is a recent result of Scott Aaronson and Alex Arkhipov which asserts that if  the power of quantum computers can be simulated by digital computers  then the polynomial hierarchy collapses.  More precisely, their result asserts that if sampling  probability distribution created by quantum computers (this is abbreviated as QSAMPLE) is in P then the polynomial hieararchy collapses to its third level.

The beautiful and important paper of Aaronson and Arkhipov is now out. Its main point of view is related to what I describe in the sixth section about “photon machines”. Update: Let me mention related idependent results by Michael J. Bremner, Richard Jozsa, Dan J. Shepherd in the paper entitled: “Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy“.

Here is the plan for this post

1) The polynomial hierarchy and results about hierarchy collapse

2) The Aaronson Arkhipov (AA) theorem and its proof

3) Two problems posed by AA

4) Does fault tolerance allow QSAMPLE (on the nose)? (Answer: yes)

5) Is there a quantum analog to Aaronson and Arkhipov’s result and what is the computational power of quantum computers?

6) Three Two competing scenarios

7) Aaronson and Arkhipov photon machines and the complexity class BOSONSAMPLING.

## 1) The polynomial hierarchy and results about hierarchy’s collapse.

If P=NP then the polynomial hierarchy collapses to the 0th level (P). If NP=coNP then the polynomial hierarchy collapses to the first level. Karp and Lipton proved that if there is a polynomial-size circuit for NP complete problems then the hierarchy collapses to the second level. There are various other results whose conclusion is a collapse of the hierarchy. Feige and Lund proved that if we can extract one bit in computing a certain permanent then this already leade to hierarchy collapse.

How should we interpret a result of the form “If X is in P then the polynomial hierarchy collapses”? Certainly it is  strong evidence that X is not in P, or, in other words, that X  is computationally difficult. Perhaps it means something even stronger like: “In some rough scale of computational hardness, X is almost NP-hard. (In such a scale, being NP hard and being coNP hard are comparably hard.) For example, an interpretation of the Feige-Lund theorem would be: Even if you cannot solve an NP complete problem using a subroutine that extracts one bit of the permanent,  extracting one bit is nonetheless almost as difficult as giving the full answer. Under the second interpretation you would not expect hierarchy collapse results from assumptions like ”factoring is in P” or “graph isomorphism is in P” or even from a statement like “(NP intersection coNP)=P”.

## 2) The AA’s result and how its proof roughly goes

A quantum computer with n qubits creates after a polynomial time computation certain states on these qubits. Once we measure these qubits, each such state translates into a probability distribution on 0-1 strings of length n. QSAMPLE is the name for sampling (precisely) from this probability distribution. (We measure each qubit separately in a fixed way.)

AA’s theorem: If  QSAMPLE is in P then the polynomial hierarchy collapses to $BPP^{NP}$.

The proof goes roughly as follows. The point is that a permanent of an arbitrary matrix can be described as the probability p that the outcome of a measurement of a computable quantum state is 0. (This probability is exponentially small so even quantum computers will not be able to compute it.) If M is the classical sampling algorithm that simulates the quantum circuit., then M takes as input a random string r, and outputs a sample M(r) from the quantum computer’s output distribution.  By the definition of p, p is the probability that   M(r) = (0,0,0,…,0), and to estimate p, it suffices to estimate the fraction of r’s such that M(r) = (0,0,0,…,0).  We know that any such approximate counting problem is in $BPP^{NP}$, by the 1983 result of Stockmeyer which can be found in this post entitled “Stockmeyers’ approximate counting method” on Lipton’s blog.

## 3) Two problems AA posed

Problem 1: Is it possible to replace QSAMPLE by  BQP ?

Specifically, is the following conjecture true? Conjecture: If  BQP is in P then the polynomial hierarchy collapses.

Problem 2: Is it the case that if APPROXIMATE-QSAMPLE is in Q then this already leads to hierarchy collapase.

By APPROXIMATE-QSAMPLE we refer to the ability to sample the distribution up to a small constant error in terms of total variation distance. (The AA’s result requires much stronger approximation.) AA were able to show that their result extends to APPROXIMATE-SAMPLE if a certain natural problem regarding approximation of random permanents is #P hard.

A natural question that arises (yet again) from this work is: What is really the power of (noiseless) quantum computers?

In particular,

Problem 3 : Do BQP and QSAMPLE represent (more or less) the same computational power?

## 4) Does Fault Tolerance allows QSAMPLE (on the nose)?

With a couple of savvy fault tolerance quantum people in the audience (Michael Ben-Or and Dorit Aharonov) and a novice one (me) the question of what happens for noisy quantum computers naturally arose. This also motivates AA problem 2 which assumes that quantum computers can sample quantum distributions up to some small error in the total variation distance between distributions.

Suppose that you have a noisy quantum memory of n qubits at a certain state $\rho$. We can think about noise where every qubit is corrupted with a small probability $\eta$. The probability distribution represented by the noisy state is something like this: We start with the required distribution of 0-1 strings and we flip a bit or set a bit to 0 with small probability $\eta$.  This notion of approximation is quite damaging to the probability distribution with which we started. The whole point with fault tolerance is to represent a “protected state” on m qubits by a “physical state” on more qubits. However, fault tolerance as is does not allow us to sample or even to approximately sample the protected state. (If you measure noisily the state of a quantum n-qubit memory then the distribution you obtain  is similar to the noise we talk about in the context of noise sensitivity. Roughly, each 0-1 string is replaced by a random 0-1 string in a large Hamming ball around it.)

Question 4: Do noisy quantum computers allow QSAMPLE (on the nose)? (Answer: yes)

Here we assume that noiseless classical computing is possible. Let me present an approach towards a positive answer. At first I thought that this will require a new type of quantum error correcting codes that I was not sure if they can be constructed but after a few conversations with FT experts  I realized that the answer is ‘Yes’ by rather routine FT methods.

## 5) Is there a quantum analog to Aaronson and Arkhipov’s result?

AA’s result deals with the quantum complexity class QSAMPLE, but besides that it refers to the classical polynomial hierarchy.

Maybe one can say something under the weaker assumption that QSAMPLE belongs to BQP?

Question 5: If QSAMPLE is in BQP, does this imply unreasonable computational complexity consequences?

What I mean is this: Suppose you are given a classical computer with BQP subroutine. So every decision problem in BQP the subroutine can solve in polynomial time. How strong is this computer? Lets call it a BQP machine. Can this computer create (or well approximate) probability distributions in QSAMPLE? If the answer is yes then already P=BQP collapses the hierarchy by AA’s result. But it is possible that if a BQP machine can approximate QSAMPLE then this already will have unreasonable computational complexity consequences.

The direct quantum Analogs of AA’s result will ask: If BQP machines can QSAMPLE maybe  #P-hard problems occurs in a low level of the quantum polynomial hierarchy? Does it imply that that #P-hard problems are in $BQP^{QMA}$? (Maybe even in $BQP^{NP}$?) These analogs are naive since there is no satisfactory analog of the quantum polynomial hierarchy.  At least for some such notions  it may be the case that #P problems belong unconditionally to some “low level” quantum complexity class.

But again what I ask about is this. Perhaps QSAMPLE represents much stronger computational power compared to BQP, so that AA’s theorem (perhaps even their proof) can be extended to show that simulating QSAMPLE by BQP-machines already has surprising computational-complexity consequences.

## 6) Three Two competing scenarios

So let me draw two three competing scenarios:

a) QSAMPLE and BQP both represent the computational complexity of a quantum computer (and both can be simulated for a noisy quantum computer). The is perhaps the common sense scenario.

b) QSAMPLE is considerably more powerful than BQP while still can be achieved by quantum fault tolerance. (This would be my fantasy.)

c) QSAMPLE is considerably more powerful than BQP and reflects a genuine difference between what quantum computers can do and what noisy quantum computers can do.

These possibilities are especially interesting because I expect that it will be possible to decide between them. So this seems an interesting and feasible research question.

## 7) AA’s photon machines and BOSONSAMPLING.

AA move one step further (This description was based on the hearing the lecture, now that the paper is out, perhaps I should say several further steps) and present a very simple type of quantum computers which already enable creating probability distributions which (under reasonable computational complexity assumptions) cannot be created classically.  This is very nice, and trying to test the hypothesis of superior computational power on small quantum devices is an inportant direction. Indeed, the paper proposes that there might be a simple linear optics experiment that “overthrows” the “polynomial Church-Turing thesis”.

The analog for QSAMPLE for these simple quantum computers is called BOSONSAMPLING. It is not known if BOSONSAMPLING represents the full power of quantum computers (quite likely it is weaker) but  it already suffices to imply the results mentioned above.

There is one practical as well as conceptual concern though. It seems that even if you try to experiment on a small fragment of quantum computers like the photon machines that AA consider, quantum fault tolerance is likely to require the full monty, namely, universal quantum computer capabilities. (But this itself is an interesting question.)

Question 6: Does fault tolerance on AA’s photon machines requires universal quantum computing?

We can ask a question of a more general nature. There are various canditates for intermidiate computational classes between BPP and BQP. (Like when you allow log depth quantum circuits). Maybe fault tolerance leaves you just with BPP and BQP and nothing in between?

Question 7: Is it the case(under the standard assumptions regarding noise) that there is no intermediate computational class between BPP and BQP described by noisy quantum devices.

I see now some problems with Question 7 (and perhaps also Question 6) when you understand “full monty”  as a computational complexity notion. (For example, applying current fault tolerance methods on log depth quantum circuit will still requires circuits of rather small depth.) We can ask less formally if fault tolerance for any noisy quantum device which may represent a lower complexity power than universal noisy quantum computers still requires the full apparatus of current FT techniques.