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)New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers( סקוט אהרונסון )Scott Aaronson( MIT עדויות חדשות שקשה לדמות את מכניקת הקוונטים עם מחשבים קלאסיים

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In 1994, something big happened in the foundations of computer science, whose meaning is still debated today… Why exactly was Shors algorithm important? Boosters: Because it means well build QCs! Skeptics: Because it means we wont build QCs! Me: For reasons having nothing to do with building QCs!

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Shors algorithm was a hardness result for one of the central computational problems of modern science: Q UANTUM S IMULATION Shors Theorem: Q UANTUM S IMULATION is not in probabilistic polynomial time, unless F ACTORING is also Use of DoE supercomputers by area (from a talk by Alán Aspuru-Guzik)

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Advantages of the new results: Based on generic complexity assumptions, rather than the classical hardness of F ACTORING Give evidence that QCs have capabilities outside the entire polynomial hierarchy Use only extremely weak kinds of QC (e.g. nonadaptive linear optics) testable before Im dead? Today: New kinds of hardness results for simulating quantum mechanics Disadvantages: Most apply to sampling problems (or problems with many possible valid outputs), rather than decision problems Harder to convince a skeptic that your QC is indeed solving the relevant hard problem Problems not useful (?)

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There exist black-box sampling and relational problems in BQP that are not in BPP PH Assuming the Generalized Linial-Nisan Conjecture, there exists a black-box decision problem in BQP but not in PH Original Linial-Nisan Conjecture was recently proved by Braverman, after being open for 20 years Results (from arXiv:0910.4698) Unconditionally, there exists a black-box decision problem that requires ( N) queries classically ( (N 1/4 ) even using postselection), but only O(1) queries quantumly

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Suppose the output distribution of any linear-optics circuit can be efficiently sampled classically (e.g., by Monte Carlo). Then P #P =BPP NP, and hence PH collapses. Indeed, even if such a distribution can be sampled in BPP PH, still PH collapses. Suppose the output distribution of any linear-optics circuit can even be approximately sampled in BPP. Then a BPP NP machine can approximate the permanent of a matrix of independent N(0,1) Gaussians. Conjecture: The above problem is #P-complete. Results (from recent joint work with Alex Arkhipov)

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BQP vs. PH: A Timeline Bernstein and Vazirani define BQP They construct an oracle problem, R ECURSIVE F OURIER S AMPLING, that has quantum query complexity n but classical query complexity n (log n) First example where quantum is superpolynomially better! A simple extension yields RFS MA Natural conjecture: RFS PH Alas, we cant even prove RFS AM! 19901995200020052010

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Fourier Sampling Problem Given oracle access to a random Boolean function The Task: Output strings z 1,…,z n, at least 75% of which satisfy and at least 25% of which satisfy where

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F OURIER S AMPLING Is In BQP Algorithm: H H H H H H f |0 Repeat n times; output whatever you see Distribution over Fourier coefficients Distribution over Fourier coefficients output by quantum algorithm

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F OURIER S AMPLING Is Not In PH Key Idea: Show that, if we had a constant-depth 2 poly(n) -size circuit C for F OURIER S AMPLING, then we could violate a known AC 0 lower bound, by sneaking a M AJORITY problem into the estimation of some random Fourier coefficient Obvious problem: How do we know C will output the particular s were interested in, thereby revealing anything about ? We dont! (Indeed, theres only a ~1/2 n chance it will) But we have a long time to wait, since our reduction can be nondeterministic! That just adds more layers to the AC 0 circuit

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Decision Version: F OURIER C HECKING Given oracle access to two Boolean functions Decide whether (i) f,g are drawn from the uniform distribution U, or (ii) f,g are drawn from the following forrelated distribution F: pick a random unit vector then let

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F OURIER C HECKING Is In BQP H H H H H H f |0 g H H H Probability of observing |0 n :

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Intuition: F OURIER C HECKING Shouldnt Be In PH Why? For any individual s, computing the Fourier coefficient is a #P-complete problem f and g being forrelated is an extremely global property: no polynomial number of f(x) and g(y) values should reveal much of anything about it But how to formalize and prove that?

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Key Definition: A distribution D is -almost k-wise independent if for all k-terms C, Theorem: For all k, the forrelated distribution F is O(k 2 /2 n/2 )-almost k-wise independent Proof: A few pages of Gaussian integrals, then a discretization step A k-term is a product of k literals of the form x i or 1-x i A distribution D over {0,1} N is k-wise independent if for all k-terms C, Approximation is multiplicative, not additive … thats important!

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Bazzi07 proved the depth-2 case Linial-Nisan Conjecture (1990) with weaker parameters that suffice for us : Let f:{0,1} n {0,1} be computed by a circuit of size and depth O(1). Then for all n (1) -wise independent distributions D, Generalized Linial-Nisan Conjecture: Let f be computed by a circuit of size and depth O(1). Then for all 1/n (1) -almost n (1) -wise independent distributions D, Razborov08 dramatically simplified Bazzis proofFinally, Braverman09 proved the whole thingAlas, we need the…

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Coming back to our result for relational problems: what was surprising was that we showed hardness of a BQP sampling problem, using a nondeterministic reduction from M AJORITY a #P problem! This raises a question: is something similar possible in the unrelativized (non-black-box) world? Indeed it is. Consider the following problem: QS AMPLING : Given a quantum circuit C, which acts on n qubits initialized to the all-0 state. Sample from Cs output distribution. Suppose QS AMPLING BPP. Then P #P =BPP NP (so in particular, PH collapses to the third level) Result/Observation:

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Why QS AMPLING Is Hard Let f:{0,1} n {-1,1} be any efficiently computable function. Suppose we apply the following quantum circuit: H H H H H H f |0 Then the probability of observing the all-0 string is

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Claim 1: p is #P-hard to estimate (up to a constant factor) Related to my result that PostBQP=PP Proof: If we can estimate p, then we can also compute x f(x) using binary search and padding Claim 2: Suppose QS AMPLING BPP. Then we could estimate p in BPP NP Proof: Let M be a classical algorithm for QS AMPLING, and let r be its randomness. Use approximate counting to estimate Conclusion: Suppose QS AMPLING BPP. Then P #P =BPP NP

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Ideally, we want a simple, explicit quantum system Q, such that any classical algorithm that even approximately simulates Q would have dramatic consequences for classical complexity theory We believe this is possible, using non-interacting bosons BOSONSFERMIONS There are two basic types of particle in the universe… Their transition amplitudes are given respectively by… All I can say is, the bosons got the harder job…

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U Our Result: Take a system of n identical photons, with m=O(n 2 ) modes (basis states) each. Put each photon in a known mode, then apply a random m m scattering matrix U: Let D be the distribution that results from measuring the photons. Suppose theres an efficient classical algorithm that samples any distribution even 1/poly(n)-close to D in variation distance. Then in BPP NP, one can estimate the permanent of a matrix X of i.i.d. N(0,1) complex Gaussians, to additive error with high probability over X. Conjecture: This problem is #P-complete

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The Permanent of Gaussians Conjecture (PGC) Given a matrix X of i.i.d, N(0,1) complex Gaussians, it is #P-complete to approximate Per(X) to within with 1-1/poly(n) probability over X But isnt the permanent easy to approximate, by Jerrum- Sinclair-Vigoda? Yesfor nonnegative matrices. For general matrices, can get huge cancellations between positive and negative terms, and indeed even approximating the permanent is #P-complete in the worst case Intuition for PGC: We know computing the permanent of a random matrix is #P-completeover finite fields. Merely need to extend that result to the reals or complex numbers! Basic difficulty: When doing LFKN-style interpolation, errors in the permanent estimates can blow up exponentially

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PGC Hardness of B OSON S AMPLING Idea: Given a Gaussian random matrix X, well smuggle X into the unitary transition matrix U for m=O(n 2 ) bosons Useful fact we rely on: given a Haar-random m m unitary matrix, an n n submatrix looks approximately Gaussian Neat Fact: The p S s sum to 1 where U S is an n n matrix containing s i copies of the i th row of U (first n columns only) Suppose that initially, modes 1,…,n contain one boson each while modes n+1,…,m are unoccupied. Then after applying U, we observe a configuration (list of occupation numbers) s 1,…,s m, with probability

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Problem: Bosons like to pile on top of each other! Call a configuration S=(s 1,…,s m ) good if every s i is 0 or 1 (i.e., there are no collisions between bosons), and bad otherwise If bad configurations dominated, then our sampling algorithm might work, without ever having to solve a hard P ERMANENT instance Furthermore, the bosonic birthday paradox is even worse than the classical one! rather than ½ as with classical particles Fortunately, we show that with n bosons and m kn 2 boxes, the probability of a collision is still at most (say) ½

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Experimental Prospects What would it take to implement the requisite experiment with photonics? Reliable phase-shifters and beamsplitters, to implement a Haar- random unitary on m photon modes Reliable single-photon sources Reliable photodetector arrays But crucially, no nonlinear optics or postselected measurements! Our Proposal: Concentrate on (say) n=30 photons, so that classical simulation is difficult but not impossible

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Prize Problems Prove the Generalized Linial-Nisan Conjecture! Yields an oracle A such that BQP A PH A Prove Generalized L-N even for the special case of DNFs. Yields an oracle A such that BQP A AM A Prove the Permanent of Gaussians Conjecture! Would imply that even approximate classical simulation of linear-optics circuits would collapse PH $100$200 NIS500

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More Open Problems (no prizes) Can we instantiate F OURIER C HECKING by an explicit (unrelativized) problem? Can we use B OSON S AMPLING to solve any decision problem outside BPP? Can you convince a skeptic (who isnt a BPP NP machine) that your QC is indeed doing B OSON S AMPLING ? Can we get unlikely classical complexity consequences from P=BQP or PromiseP=PromiseBQP?

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Summary I like to say that we have three choices: either (1)The Extended Church-Turing Thesis is false, (2)Textbook quantum mechanics is false, or (3)QCs can be efficiently simulated classically. For all intents and purposes?

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