Download presentation

Presentation is loading. Please wait.

Published byJose Carlson Modified over 3 years ago

1
Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson UC Berkeley IAS

2
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)

3
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

4
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, …

5
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

6
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?

7
+ |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)

8
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

9
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

10
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

11
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)

12
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?

13
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?

Similar presentations

OK

Sergey Bravyi, IBM Watson Center Robert Raussendorf, Perimeter Institute Perugia July 16, 2007 Exactly solvable models of statistical physics: applications.

Sergey Bravyi, IBM Watson Center Robert Raussendorf, Perimeter Institute Perugia July 16, 2007 Exactly solvable models of statistical physics: applications.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google