New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers Scott Aaronson Parts based on joint work with Alex Arkhipov

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 more generic complexity assumptions than 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) 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 (?)

Example of a PH problem: For all n-bit strings x, does there exist an n-bit string y such that for all n-bit strings z, (x,y,z) holds? What Is The Polynomial Hierarchy? such-and-such is true PH collapses to a finite level is complexity-ese for such-and-such would be almost as insane as P=NP

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!

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

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 additively approximate Per(X), with high probability over a matrix X of i.i.d. N(0,1) Gaussians. Permanent-of-Gaussians Conjecture: The above problem is #P-complete. Results In The Real World From not-yet-arXived joint work with Alex Arkhipov

F OURIER F ISHING 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 F ISHING 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 F ISHING Is Not In PH Basic Strategy: Suppose an oracle problem is in PH. Then by reinterpreting every quantifier as an OR gate, and every quantifier as an AND gate, we can get an AC 0 (constant-depth, unbounded-fanin, quasipolynomial-size) circuit for an exponentially- scaled down version of the problem And AC 0 circuits are one of the few things in complexity theory that we can actually lower-bound! In particular, it was proved in the 1980s that any AC 0 circuit for M AJORITY (or for computing a Fourier coefficient) must have exponential size Problem: In our case, the AC 0 circuit C doesnt have to compute the Fourier coefficientsit just has to sample from some probability distribution defined in terms of them! To deal with that, we use a nondeterministic reduction (which adds more layers to the circuit), to show that C would nevertheless lead to an AC 0 circuit for M AJORITY

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 :

Evidence That F OURIER C HECKING PH We can prove that, even after you condition on any setting for any polynomial number of f(x)s and g(y)s, you still have almost no information about whether f and g are independent or forrelated We conjecture that this property, by itself, is enough to imply an oracle problem is not in PH. We call this the Generalized Linial-Nisan Conjecture The original Linial-Nisan Conjecturethe same statement, but without the almostwas proved last year by Braverman, in a major breakthrough in complexity theory (indirectly inspired by this work )

Coming back to the first result, whats surprising is that we showed hardness of a BQP sampling problem, by 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) 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

Related Results A. 2004: PostBQP=PP Bremner, Jozsa, Shepherd (poster #1): PostIQP=PP, hence efficient simulation of IQP collapses PH Fenner, Green, Homer, Pruim 1999: Determining whether a quantum circuit accepts with nonzero probability is hard for PH

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…

Starting from a fixed basis state (like | =|1,…,1,0,…0 ), you get to choose an arbitrary m m unitary U to apply U induces an unitary V on n-photon states, defined by Linear Optics for Dummies (or computer scientists) Computational basis states have the form |S =|s 1,…,s m, where s 1,…,s m are nonnegative integers such that s 1 +…+s m =n n = # of photons m = # of modes (boxes) that each photon can be in Then you get to measure V| in the computational basis where U S,T is an n n submatrix of U indexed by S,T (containing an s i t j block of u ij s for each i,j) Theorem (Lloyd 1996 et al.): BosonP BQP Proof Idea: Decompose U into a product of O(m 2 ) elementary linear-optics gates (beamsplitters and phase-shifters), then simulate each gate using standard qubit gates Theorem (Knill, Laflamme, Milburn 2001): Linear optics with adaptive measurements can do all of BQP By contrast, well use just a single (nonadaptive) measurement of the photon numbers at the end

U Our Result: Take a system of n identical photons with m=O(n 2 ) modes. Put each photon in a known mode, then apply a Haar-random m m unitary transformation U: Let D be the distribution that results from measuring the photons. Suppose theres a BPP algorithm that takes U as input, and 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. Permanent-of-Gaussians Conjecture: This problem is #P-complete

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 ) photonsin such a way that S|V| =Per( X), for some basis state |S Useful fact we rely on: given a Haar-random m m unitary matrix, an n n submatrix looks approximately Gaussian Then the sampler has no way of knowing which submatrix of U we care aboutso even if it has 1/poly(n) error, with high probability it will return |S with probability |Per( X)| 2 Then, just like before, we can use approximate counting to estimate Pr[|S ] |Per( X)| 2 in BPP NP, and thereby solve #P Difficulty: The bosonic birthday paradox! Identical bosons like to pile on top of each other, and thats bad for us SO WE DEAL WITH IT

Experimental Prospects What would it take to implement the requisite experiment? Reliable phase-shifters and beamsplitters, to implement an arbitrary unitary on m photon modes Reliable single-photon sources Fock states, not coherent states 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 the Permanent of Gaussians Conjecture! Would imply that even approximate classical simulation of linear-optics circuits would collapse PH $ Fr Do a linear optics experiment that overthrows the Polynomial-Time Church-Turing Thesis ?