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

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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 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 (?)

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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

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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!

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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

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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 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

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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

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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

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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

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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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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 )

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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:

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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) 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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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

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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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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

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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

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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 ) 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

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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

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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 ?

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