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Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson UC Berkeley IAS

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Is Quantum Computing Fundamentally Impossible? It violates the extended Church-Turing thesis Analogy to analog computing and unit-cost arithmetic We have never seen a physical law valid to over a dozen decimals (Leonid Levin) Even if QM is right, maybe our complexity model is wrong (Oded Goldreich)

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Question: If quantum computing is impossible, then what criterion separates the quantum states that suffice for factoring from the states weve already seen? DIVIDING LINE Sure/Shor Separator

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Goal of Paper: Provide a formal framework for studying possible Sure/Shor separators The simplest candidates for such a criterion are nonstarters… Exponentially small amplitudes? Thousands of coherent qubits? YAWN 10000 polarized photons Buckyballs, superconducting coils, …

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Key Idea: Complexity classification of pure quantum states Classical Vidal Circuit AmpP MOTree OTree TSH Tree P 1 2 1 2 Strict containment Containment Non-containment

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Intuitively, once we admit | and | into our set of possible states, were almost forced to admit | | and | + | as well! But what if we take the closure under a polynomial number of those operations?

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+ |1 1 |1 2 ++ |0 1 |1 1 |0 2 |1 2 Tree size of | : Minimum number of vertices in such a tree representing | Tree states: Infinite families of states {| n } n 1 for which TS(| n )=n O(1)

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Many interesting states have polynomial TS 1D spin chain, superposition over multiples of 3, … Basic Facts About Tree Size A random state cant even be approximated by states with subexponential TS If then TS(| ) equals the multilinear formula size of f to within a constant factor Minimum size of an arithmetic formula for f(x 1,…,x n ) involving +,, and complex constants, every node of which computes a multilinear polynomial in x 1,…,x n

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Let C = {x | Ax b (mod 2)}, where A is chosen uniformly at random from Main Result: Tree Size Lower Bound Then w.h.p. over A, the coset state has tree size n (log n) Want to know the proof technique? Go to my talk RAN RAZ

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Significance: Coset states underlie quantum error correction. Our result gives a second reason to prepare them: refuting the hypothesis that all states in Nature are tree states Purely classical corollary: First superpolynomial gap between general and multilinear formula size of functions Extensions: Tree size n (log n) is needed even to approximate coset states Can derandomize, to get an n (log n) lower bound for an explicit coset state

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But what about states arising in Shors algorithm? Conjecture: Let A consist of 5+log(n 1/3 ) uniform random elements of {2 0,…,2 n-1 }. Let S consist of all 32n 1/3 sums of subsets of A. If a prime p is drawn uniformly from [n 1/3,1.1n 1.3 ], then |S p | 3n 1/3 /4 with probability at least ¾, where S p = {x mod p : x S} Theorem: Assuming the conjecture, Shor states have tree size n (log n)

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Experimental Situation To interpret experiments, would be nice to have explicit tree size lower bounds for small n Whats been done in liquid NMR: (Knill et al. 2001) Non-tree states might already have been observed in solid state! 2D or 3D spin lattices with nearest-neighbor Hamiltonianssimilar to cluster states 11001 00101 11100 01110 10100 Key question: What kind of evidence is needed to prove a states existence?

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Open Problems Can we show exponential tree size lower bounds? If a quantum computer is in a tree state at every time step, can it then be simulated classically? Current result: Can be simulated in Do all codeword states have superpolynomial tree size? Relationship between tree size and persistence of entanglement?

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