* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download The Learnability of Quantum States
Scalar field theory wikipedia , lookup
Particle in a box wikipedia , lookup
Renormalization wikipedia , lookup
Wave–particle duality wikipedia , lookup
Quantum decoherence wikipedia , lookup
Hydrogen atom wikipedia , lookup
Quantum dot wikipedia , lookup
X-ray fluorescence wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Quantum field theory wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Path integral formulation wikipedia , lookup
Quantum fiction wikipedia , lookup
Bell's theorem wikipedia , lookup
Wheeler's delayed choice experiment wikipedia , lookup
Quantum entanglement wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Bell test experiments wikipedia , lookup
Density matrix wikipedia , lookup
Coherent states wikipedia , lookup
EPR paradox wikipedia , lookup
Double-slit experiment wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
History of quantum field theory wikipedia , lookup
Quantum teleportation wikipedia , lookup
Hidden variable theory wikipedia , lookup
Quantum group wikipedia , lookup
Probability amplitude wikipedia , lookup
Quantum state wikipedia , lookup
Quantum computing wikipedia , lookup
Quantum machine learning wikipedia , lookup
Delayed choice quantum eraser wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Canonical quantization wikipedia , lookup
New Computational Insights from Quantum Optics Scott Aaronson Based on joint work with Alex Arkhipov The Extended ChurchTuring Thesis (ECT) Everything feasibly computable in the physical world is feasibly computable by a (probabilistic) Turing machine But building a QC able to factor n>>15 is damn hard! Can’t CS “meet physics halfway” on this one? I.e., show computational hardness in more easily-accessible quantum systems? Also, factoring is a “special” problem Our Starting Point Det A 1 sgn S n n a i, i 1 i Per A n a S n i 1 i, i All I can say is, the bosons In #P-complete [Valiant] gotPthe harder job FERMIONS BOSONS So if n-boson amplitudes correspond to permanents… Can We Use Bosons to Calculate the Permanent? That sounds way too good to be true—it would let us solve NP-complete problems and more using QC! Explanation: Amplitudes aren’t directly observable. To get a reasonable estimate of Per(A), you might need to repeat the experiment exponentially many times Basic Result: Suppose there were a polynomial-time classical randomized algorithm that took as input a description of a noninteracting-boson experiment, and that output a sample from the correct final distribution over n-boson states. Then P#P=BPPNP and the polynomial hierarchy collapses. Motivation: Compared to (say) Shor’s algorithm, we get “stronger” evidence that a “weaker” system can do interesting quantum computations Crucial step we take: switching attention to sampling problems P#P SampBQP PERMANENT BQP PH X Y… BPPNP FACTORING BPP 3SAT SampP Related Work Valiant 2001, Terhal-DiVincenzo 2002, “folklore”: A QC built of noninteracting fermions can be efficiently simulated by a classical computer Knill, Laflamme, Milburn 2001: Noninteracting bosons plus adaptive measurements yield universal QC Jerrum-Sinclair-Vigoda 2001: Fast classical randomized algorithm to approximate Per(A) for nonnegative A A. 2011: Generalization of Gurvits’s algorithm that gives better approximation if A has repeated rows or columns The Quantum Optics Model A rudimentary subset of quantum computing, involving only non-interacting bosons, and not based on qubits Classical counterpart: Galton’s Board, on display at the Boston Museum of Science Using only pegs and noninteracting balls, you probably can’t build a universal computer— but you can do some interesting computations, like generating the binomial distribution! The Quantum Version Let’s replace the balls by identical single photons, and the pegs by beamsplitters Then we see strange things like the Hong-Ou-Mandel dip 1 2 1 2 1 2 1 2 The two photons are now correlated, even though they never interacted! Explanation involves destructive 1 1 1 1 0 interference of amplitudes: 2 2 2 2 Final amplitude of non-collision is Getting Formal The basis states have the form |S=|s1,…,sm, where si is the number of photons in the ith “mode” We’ll never create or destroy photons. So s1+…+sm=n is constant. For us, m=poly(n) Initial state: |I=|1,…,1,0,……,0 U You get to apply any mm unitary matrix U If n=1 (i.e., there’s only one photon, in a superposition over the m modes), U acts on that photon in the obvious way m n 1 M : n In general, there are ways to distribute n identical photons into m modes U induces an MM unitary (U) on the n-photon states as follows: Per U U S ,T S ,T s1! sm !t1! t m ! Here US,T is an nn submatrix of U (possibly with repeated rows and columns), obtained by taking si copies of the ith row of U and tj copies of the jth column for all i,j Beautiful Alternate Perspective The “state” of our computer, at any time, is a degree-n polynomial over the variables x=(x1,…,xm) (n<<m) Initial state: p(x) := x1xn We can apply any mm unitary transformation U to x, to obtain a new degree-n polynomial p' x pUx x S s1 ,, sm s1 S 1 sm m x s1 1 Then on “measuring,” we see the monomial x x 2 with probability S s1! sm ! sm m OK, so why is it hard to sample the distribution over photon numbers classically? Given any matrix ACnn, we can construct an mm unitary U (where m2n) as follows: A B U C D Suppose we start with |I=|1,…,1,0,…,0 (one photon in each of the first n modes), apply U, and measure. Then the probability of observing |I again is p : I U I 2 2n Per A 2 Claim 1: p is #P-complete to estimate (up to a constant factor) Idea: Valiant proved that the PERMANENT is #P-complete. Can use a classical reduction to go from a multiplicative approximation of |Per(A)|2 to Per(A) itself. Claim 2: Suppose we had a fast classical algorithm for boson sampling. Then we could estimate p in BPPNP Idea: Let M be our classical sampling algorithm, and let r be its randomness. Use approximate counting to estimate Pr M r outputs I 2 r Conclusion: p : Suppose I U weI had a fast classical Per Aalgorithm for boson sampling. Then P#P=BPPNP. 2n 2 The Elephant in the Room The previous result hinged on the difficulty of estimating a single, exponentially-small probability p—but what about noise and error? The “right” question: can a classical computer efficiently sample a distribution with 1/poly(n) variation distance from the boson distribution? Our Main Result: Suppose it can. Then there’s a BPPNP algorithm to estimate |Per(A)|2, with high probability nn over a Gaussian matrix C A ~ N 0 ,1 Our Main Conjecture Estimating |Per(A)|2, for most Gaussian matrices A, is a #P-hard problem If the conjecture holds, then even a noisy n-photon experiment could falsify the Extended Church Thesis, assuming P#PBPPNP! Most of our paper is devoted to giving evidence for this conjecture We can prove it if you replace “estimating” by “calculating,” or “most” by “all” First step: understand the distribution of |Per(A)|2 for random A Conjecture: There exist constants C,D and >0 such that for all n and >0, Pr X ~ N 0,1Cnn PerX n! Cn D Empirically true! Also, we can prove it with determinant in place of permanent The Equivalence of Sampling and Searching [A., CSR 2011] [A.-Arkhipov] gave a “sampling problem” solvable using quantum optics that seems hard classically—but does that imply anything about more traditional problems? Recently, I found a way to convert any sampling problem into a search problem of “equivalent difficulty” Basic Idea: Given a distribution D, the search problem is to find a string x in the support of D with large Kolmogorov complexity Using Quantum Optics to Prove that the Permanent is #P-Hard [A., Proc. Roy. Soc. 2011] Valiant famously showed that the permanent is #P-hard— but his proof required strange, custom-made gadgets We gave a new, more transparent proof by combining three facts: (1) n-photon amplitudes correspond to nn permanents (2) Postselected quantum optics can simulate universal quantum computation [Knill-Laflamme-Milburn 2001] (3) Quantum computations can encode #P-hard quantities in their amplitudes Experimental Issues Basically, we’re asking for the n-photon generalization of the Hong-Ou-Mandel dip, where n = big as possible Our results suggest that such a generalized HOM dip would be evidence against the Extended Church-Turing Thesis Physicists: “Sounds hard! But not as hard as full QC” Remark: If n is too large, a classical computer couldn’t even verify the answers! Not a problem yet though… Several experimental groups (Imperial College, U. of Queensland) are currently working to do our experiment with 3 photons. Largest challenge they face: reliable single-photon sources Summary Thinking about quantum optics led to: - A new experimental quantum computing proposal - New evidence that QCs are hard to simulate classically - A new classical randomized algorithm for estimating permanents - A new proof of Valiant’s result that the permanent is #P-hard - (Indirectly) A new connection between sampling and searching Open Problems Prove our main conjecture: that Per(A) is #P-hard to approximate for a matrix A of i.i.d. Gaussians ($1,000)! Can our model solve classically-intractable decision problems? Fault-tolerance in the quantum optics model? Find more ways for quantum complexity theory to “meet the experimentalists halfway” Bremner, Jozsa, Shepherd 2011: QC with commuting Hamiltonians can sample hard distributions