Following are some preliminary observations connecting BosonSampling, an interesting computational task that quantum computers can perform (that we discussed in this post), and noise-sensitivity in the sense of Benjamini, Schramm, and myself (that we discussed here and here.)
BosonSampling and computational-complexity hierarchy-collapse
Suppose that you start with n bosons each can have m locations. The i-th boson is in superposition and occupies the j-th location with complex weight . The bosons are indistinguishable which makes the weight for a certain occupation pattern proportional to the permanent of a certain n by n submatrix of the n by m matrix of weights.
Boson Sampling is a task that a quantum computer can perform. As a matter of fact, it only requires a “boson machine” which represents only a fragment of quantum computation. A boson machine is a quantum computer which only manipulates indistinguishable bosons with gated described by phaseshifters and beamsplitters.
BosonSampling and boson machines were studied in a recent paper The Computational Complexity of Linear Optics of Scott Aaronson and Alex Arkhipov (AA). They proved (Theorem 1 in the paper) that if (exact) BosonSampling can be performed by a classical computer then this implies a collapse of the computational-complexity polynomial hierarchy (PH, for short). This result adds to a similar result achieved independently by Michael J. Bremner, Richard Jozsa, and Dan J. Shepherd (BJS) in the paper entitled: “Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy,” and to older results by Barbara Terhal and David DiVincenzo (TD) in the paper Adaptive quantum computation, constant depth quantum circuits and Arthur-Merlin games, Quant. Inf. Comp. 4, 134-145 (2004).
Since universal quantum computers can achieve BosonSampling (and the other related computational tasks considered by TD and BJS), this is a very strong indication for the computational complexity advantage of quantum computers which arguably brings us with quantum computers to the “cozy neighborhood” of NP-hardness.
Noisy quantum computers with quantum fault-tolerance are also capable of exact BosonSampling and this strong computational-complexity quantum-superiority applies to them as well.
Realistic BosonSampling and Gaussian Permanent Estimation (GPE)
Aaronson an Arkhipov considered the following question that they referred to as Gaussian Permanent Approximation.
Problem (Problem 2 from AA’s paper): (): Given as imput a matrix of i.i.d Gaussians,together with error bounds ε, δ > o, estimate to within additive error with probability at leat 1-δ over X, in time.
They conjectured that such Gaussian Permanent Approximation is computationally hard and showed (Theorem 3) that this would imply that sampling w.r.t. states achievable by boson machines cannot even be approximated by classical computing (unless PH collapses). They regarded questions about approximation more realistic in the context of decoherence where we cannot expect exact sampling.
Scott Aaronson also expressed guarded optimism that even without quantum fault-tolerance BosonSampling can be demonstrated by boson machines for 20-30 bosons, leading to strong experimental evidence for computational advantage of quantum computers (or, if you wish, against the efficient Church-Turing thesis).
Is it so?
More realistic BosonSampling and Noisy Gaussian Permanent Estimation (NGPE)
Let us consider the following variation that we refer to as Noisy Gaussian Permanent Estimation:
Problem 2′: (): Given as imput a matrix of i.i.d Gaussians, and a parameter t>0 let NPER (M), be the expected value of the permanent for where E= . Given the input matrix M together with error bounds ε, δ > o, estimate NPER(M) to within additive error with probability at leat 1-δ over X, in time.
Problem 2′ seems more relevant for noisy boson machines (without fault-tolerance). The noisy state of the computer is based on every single boson being slightly mixed, and the permanent computation will average these individual mixtures. We can consider the relevant value for t to be a small constant. Can we expect Problem 2′ to be hard?
The answer for Question 2′ is surprising. I expect that even when is very very tiny, namely for , the expected value of NPER(M) (essentially) does not depend at all on M!
Noise Sensitivity a la Benjamini, Kalai and Schramm
Noise sensitivity for the sense described here for Boolean functions was studied in a paper by Benjamini Schramm and me in 1999. (A related notion was studied by Tsirelson and Vershik.) Lectures on noise sensitivity and percolation is a new beautiful monograph by Christophe Garban and Jeff Steif which contains a description of noise sensitivity. The setting extends easily to the Gaussian case. See this paper by Kindler and O’donnell for the Gaussian case. In 2007, Ofer Zeituni and I studied the noise sensitivity in the Gaussian model of the maximal eigenvalue of random Gaussian matrices (but did not write it up).
Noise sensitivity depends on the degree of the support of the Fourier expansion. For determinants or permanents of an n by n matrices the basic formulas as sums of generalized diagonals describe the Fourier expansion, so the Fourier coefficients are supported on degree-n monomials. This implies that the determinant and the permanent are very noise sensitive.
Noisy Gaussian Permanent Estimation is easy
Noisy Gaussian Permanent Estimation is easy, even for very small amount of noise, because the outcome does not depend at all on the input. It is an interesting question what is the hardness of NGPE is when the noise is below the level of noise sensitivity.
Update (March, 2014) Exploring the connection between BosonSampling and BKS-noise sensitivity shows that the picture drawn here is incorrect. Indeed, the square of the permanent is not noise stable even when the noise is fairly small and this shows that the noisy distribution does not approximate the noiseless distribution. However the noisy distribution does depend on the input!
AA’s protocol and experimental BosonSampling
Scott and Alex proposed a simple experiment described as follows : “An important motivation for our results is that they immediately suggest a linear-optics experiment, which would use simple optical elements (beamsplitters and phaseshifters) to induce a Haar-random unitary transformation U on an input state of n photons, and would then check that the probabilities of various final states of the photons correspond to the permanents of submatrices, as predicted by quantum mechanics.”
Recently, four groups carried out interesting BosonSampling experiments with 3 bosons (thus for permanents of 3 by 3 matrices.) (See this post on Scott’s blog.)
BKS-noise sensitivity is giving simple predictions on how things will behave as a function of the number of bosons and this can be tested even with experiments with very small number of bosons. When you increase the number of bosons the distribution will quickly become uniform for the various final states. The correlation between the probabilities and the value corresponding to permanents will rapidly go to zero.
Some follow-up questions
Here are a few interesting questions that deserve further study.
1) Does problem 2′ capture the general behavior of noisy boson machines? To what generality noise sensitivity applies for general functions described by Boson sampling distributions?
(There are several versions for photons-based quantum computers including even an important model by Knill, Laflamme, and Milburn that support universal quantum computation. The relevance of BKS noise-sensitivity should be explored carefully for the various versions.)
2) Is the connection with noise sensitivity relevant to the possibility to have boson machines with fault tolerance?
3) What is the Gaussian-quantum analog for BKS’s theorem asserting that noise sensitivity is the law unless we have substantial correlation with the majority function?
4) What can be said about noise-sensitivity of measurements for other quantum codes?
A few more remarks:
More regarding noisy boson machines and quantum fault tolerance
Noisy boson machines and BosonSampling are related to various other issues regarding quantum fault-tolerance. See my added recent remarks (about the issue of synchronization, and possible modeling using smoothed Lindblad evolutions) to my old post on AA’s work.
Noise sensitivity and the special role of the majority function
The main result of Itai, Oded, and me was that a Boolean function which is not noise sensitive must have a substantial correlation with the majority function. Noise sensitivity and the special role of majority for it gave me some motivation to look at quantum fault-tolerance in 2005 (see also this post,) and this is briefly discussed in my first paper on the subject, but until now I didn’t find an actual connection between quantum fault-tolerance and BKS-noise-sensitivity.
It is an interesting question which bosonic states are realistic, and it came up in some of my papers and in the discussion with Aram Harrow and Steve Flammia (and their paper on my “Conjecture C”).
A sort of conclusion
BosonSampling was offered as a way to prove that quantum mechanics allows a computational advantage without using the computational advantage for actual algorithmic purpose. If you wish, the AA’s protocol is offered as a sort of zero-knowledge proof that the efficient Church-Turing thesis is false. It is a beautiful idea that attracted interest and good subsequent work from theoreticians and experimentalists. If the ideas described here are correct, BosonSampling and boson machines may give a clear understanding based on BKS noise-sensitivity for why quantum mechanics does not offer computational superiority (at least not without the magic of quantum fault-tolerance).
Avi’s joke and common sense
Here is a quote from AA referring to a joke by Avi Wigderson: “Besides bosons, the other basic particles in the universe are fermions; these include matter particles such as quarks and electrons. Remarkably, the amplitudes for n-fermion processes are given not by permanents but by determinants of n×n matrices. Despite the similarity of their definitions, it is well-known that the permanent and determinant differ dramatically in their computational properties; the former is #P-complete while the latter is in P. In a lecture in 2000, Wigderson called attention to this striking connection between the boson and fermion dichotomy of physics and the permanent-determinant dichotomy of computer science. He joked that, between bosons and fermions, ‘the bosons got the harder job.’ One could view this paper as a formalization of Wigderson’s joke.”
While sharing the admiration to Avi in general and Avi’s jokes in particular, if we do want to take Avi’s joke seriously (as we always should), then the common-sense approach would be first to try to understand why is it that nature treats bosons and fermions quite equally and the dramatic computational distinction is not manifested at all. The answer is that a crucial ingredient for a computational model is the modeling of noise/errors, and that noise-sensitivity makes bosons and fermions quite similar physically and computationally.
Eigenvalues, determinants, and Yuval Filmus
It is an interesting question (that I asked over Mathoverflow) to understand the Fourier expansion of the set of eigenvalues, the maximum eigenvalue and related functions. At a later point, last May, I was curious about the Fourier expansion of the determinant, and for the Boolean case I noticed remarkable properties of its Fourier expansion. So I decided to ask Yuval Filmus about it:
Yuval’s answer came a couple of hours later like a cold shower:
In other words, the usual formula for the determinant *is* its Fourier transform
This reminded me of a lovely story of how I introduced Moni Naor to himself that I should tell sometime.
What else can a quantum computer sample?
The ability of quantum computers to sample (exactly) random complex Gaussian matrices according to the value of their permanents is truly amazing! If you are not too impressed by efficient factoring but still do not believe that QC can reach the neighborhood of NP-hard problems you may find this possibility too good to be true.
I am curious if sharp P reductions give us further results of this nature. For example, can a QC sample random 3-SAT formulas (by a uniform distribution or by a certain other distribution coming from sharp-P reductions) according to the number of their satisfying assignments?
Can QC sample integer polytopes by their volume or by the number of integer points in them? Graphs by the number of 4-colorings?