New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers

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 Shor’s algorithm important? Boosters: Because it means we’ll build QCs! Skeptics: Because it means we won’t build QCs! Me: For reasons having nothing to do with building QCs!

Shor’s algorithm was a hardness resultfor one of the central computational problems of modern science: Quantum Simulation Use of DoE supercomputers by area (from a talk by Alán Aspuru-Guzik) Shor’s Theorem: Quantum Simulation is not in probabilistic polynomial time, unless Factoring is also

Today: New kinds of hardness results for simulating quantum mechanics Advantages of the new results: Based on more “generic” complexity assumptions than classical hardness of Factoring Give evidence that QCs have capabilities outside the entire polynomial hierarchy Use only extremely weak kinds of QC (e.g. nonadaptive linear optics) 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” (?)

What Is The Polynomial Hierarchy? 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?” “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 1990 1995 2000 2005 2010 Bernstein and Vazirani define BQP They construct an oracle problem, Recursive Fourier Sampling, 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 can’t even prove RFSAM!

Results In The Oracle WorldFrom arXiv:0910.4698 There exist oracle sampling and relational problems in BQP that are not in BPPPH • 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 Unconditionally, there exists an oracle decision problem that requires (N1/4) queries classically (even using postselection), but only 1 query quantumly

Results In The “Real” WorldFrom not-yet-arXived joint work with Alex Arkhipov • Suppose the output distribution of any linear-optics circuit can be efficiently sampled classically (e.g., by Monte Carlo). Then P#P=BPPNP, and hence PH collapses. • Indeed, even if such a distribution can be sampled in BPPPH, still PH collapses. • Suppose the output distribution of any linear-optics circuit can even be approximately sampled in BPP. Then a BPPNP 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.

Fourier Fishing Problem Given oracle access to a random Boolean function The Task: Output strings z1,…,zn, at least 75% of which satisfy and at least 25% of which satisfy where

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

Fourier Fishing 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 AC0(constant-depth, unbounded-fanin, quasipolynomial-size) circuit for an “exponentially-scaled down” version of the problem And AC0 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 AC0 circuit for Majority(or for computing a Fourier coefficient) must have exponential size Problem: In our case, the AC0 circuit C doesn’t have to compute the Fourier coefficients—it 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 AC0 circuit for Majority

Decision Version: Fourier Checking Given oracle access to two Boolean functions • Decide whether • f,g are drawn from the uniform distribution U, or • f,g are drawn from the following “forrelated” distribution F: pick a random unit vector then let

Fourier Checking Is In BQP |0 H H H |0 H f H g H |0 H H H Probability of observing |0n:

Evidence That Fourier Checking 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 Conjecture—the same statement, but without the “almost”—was proved last year by Braverman, in a major breakthrough in complexity theory (indirectly inspired by this work )

Coming back to the first result, what’s surprising is that we showed hardness of a BQP sampling problem, by using a nondeterministic reduction from Majority—a “#P” problem! This raises a question: is something similar possible in the unrelativized (non-black-box) world? Result/Observation: Suppose QSamplingBPP. Then P#P=BPPNP (so in particular, PH collapses to the third level) Indeed it is. Consider the following problem: QSampling: Given a quantum circuit C, which acts on n qubits initialized to the all-0 state. Sample from C’s output distribution.

WhyQSampling Is Hard Let f:{0,1}n{-1,1} be any efficiently computable function. Suppose we apply the following quantum circuit: |0 H H |0 H f H |0 H H 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 xf(x) using binary search and padding Claim 2: Suppose QSamplingBPP. Then we could estimate p in BPPNP Proof: Let M be a classical algorithm for QSampling, and let r be its randomness. Use approximate counting to estimate Conclusion: Suppose QSamplingBPP. Then P#P=BPPNP

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 asimple, 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 There are two basic types of particle in the universe… All I can say is, the bosons got the harder job… BOSONS FERMIONS Their transition amplitudes are given respectively by…

Linear Optics for Dummies (or computer scientists) Computational basis states have the form |S=|s1,…,sm, where s1,…,sm are nonnegative integers such that s1+…+sm=n n = # of photons m = # of “modes” (boxes) that each photon can be in Theorem (Lloyd 1996 et al.):BosonPBQP Proof Idea: Decompose U into a product of O(m2) “elementary linear-optics gates” (beamsplitters and phase-shifters), then simulate each gate using standard qubit gates 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 Theorem (Knill, Laflamme, Milburn 2001): Linear optics with adaptive measurements can do all of BQP By contrast, we’ll use just a single (nonadaptive) measurement of the photon numbers at the end where US,T is an nn submatrix of U indexed by S,T (containing an sitj block of uij’s for each i,j) Then you get to measure V| in the computational basis

Our Result: Take a system of n identical photons with m=O(n2) modes. Put each photon in a known mode, then apply a Haar-random mm unitary transformation U: U Permanent-of-Gaussians Conjecture: This problem is #P-complete Let D be the distribution that results from measuring the photons. Suppose there’s a BPP algorithm that takes U as input, and samples any distribution even1/poly(n)-close to D in variation distance. Then in BPPNP, 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.

PGCHardness of BosonSampling • Idea:Given a Gaussian random matrix X, we’ll “smuggle” X into the unitary transition matrix U for m=O(n2) photons—in 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 about—so 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 BPPNP, and thereby solve #P Difficulty: The “bosonic birthday paradox”! Identical bosons like to pile on top of each other, and that’s 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 sourcesFock 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 BQPAPHA Prove the Permanent of Gaussians Conjecture!Would imply that even approximate classical simulation of linear-optics circuits would collapsePH 140Fr $200 Do a linear optics experiment that overthrows the Polynomial-Time Church-Turing Thesis ?

New Evidence That Quantum Mechanics Is Hard to Simulate on Classical Computers