)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

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!

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)

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 (?)

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: ) 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

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)

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!

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

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

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

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

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

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?

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!

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…

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:

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

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

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…

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

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

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

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

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

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

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?

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?