* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Multilinear Formulas and Skepticism of Quantum
History of quantum field theory wikipedia , lookup
Compact operator on Hilbert space wikipedia , lookup
Hydrogen atom wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Double-slit experiment wikipedia , lookup
Particle in a box wikipedia , lookup
Quantum entanglement wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Theoretical and experimental justification for the Schrödinger equation wikipedia , lookup
Coherent states wikipedia , lookup
Noether's theorem wikipedia , lookup
EPR paradox wikipedia , lookup
Quantum machine learning wikipedia , lookup
Bell test experiments wikipedia , lookup
Quantum teleportation wikipedia , lookup
Quantum decoherence wikipedia , lookup
Quantum computing wikipedia , lookup
Quantum key distribution wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Quantum group wikipedia , lookup
Bell's theorem wikipedia , lookup
Canonical quantization wikipedia , lookup
Hidden variable theory wikipedia , lookup
Density matrix wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Multilinear Formulas and Skepticism of Quantum Computing / 2 Scott Aaronson, UC Berkeley http://www.cs.berkeley.edu/~aaronson Outline (1) (2) (3) (4) Four objections to quantum computing Sure/Shor separators Tree states Result: QECC states require n(log n) additions and tensor products (5) Experimental (!) proposal (6) Conclusions and open problems Four Objections Theoretical Practical Physical (A): QC’s can’t be built for fundamental reason (B): QC’s can’t be built for engineering reasons Algorithmic (C): Speedup is of limited theoretical interest (D): Speedup is of limited practical value (A): QC’s can’t be built for fundamental reason—Levin’s arguments (1) Analogy to unit-cost arithmetic model (2) Error-correction and fault-tolerance address only relative error in amplitudes, not absolute (3) “We have never seen a physical law valid to over a dozen decimals” (4) If a quantum computer failed, we couldn’t measure its state to prove a breakdown of QM—so no Nobel prize “The present attitude is analogous to, say, Maxwell selling the Daemon of his famous thought experiment as a path to cheaper electricity from heat” Responses (1) Continuity in amplitudes more benign than in measurable quantities—should we dismiss classical probabilities of order 10-1000? (2) How do we know QM’s successor won’t lift us to PSPACE, rather than knock us down to BPP? (3) To falsify QM, would suffice to show QC is in some state far from eiHt|. E.g. Fitch & Cronin won 1980 Physics Nobel “merely” for showing CP symmetry is violated Real Question: How far should we extrapolate from today’s experiments to where QM hasn’t been tested? How Good Is The Evidence for QM? (1) Interference: Stability of e- orbits, double-slit, etc. (2) Entanglement: Bell inequality, GHZ experiments (3) Schrödinger cats: C60 double-slit experiment, superconductivity, quantum Hall effect, etc. C60 Arndt et al., Nature 401:680-682 (1999) Alternatives to QM Roger Penrose Gerard ‘t Hooft (+ King of Sweden) Stephen Wolfram Exactly what property separates the Sure States we know we can create, from the Shor States that suffice for factoring? DIVIDING LINE My View: Any good argument for why quantum computing is impossible must answer this question—but I haven’t seen any that do What I’ll Do: - Initiate a complexity theory of (pure) quantum states, that studies possible Sure/Shor separators - Prove a superpolynomial lower bound on “tree size” of states arising in quantum error correction - Propose an NMR experiment to create states with large tree size AmpP Classes of Pure States H n 2 Circuit Tree P TSH OTree Vidal MOTree 2 2 1 1 Classical Tree size TS(|) = minimum number of unboundedfanin + and gates, |0’s, and |1’s needed in a tree representing |. Constants are free. Permutation order of qubits is irrelevant. Example: 00 2 01 10 11 / 7 + 2 7 3 7 1 2 |01 + 1 2 |11 1 2 |02 + 1 2 |12 |01 |12 TS 11 n H Tree States: Families n 2 n1 such that TS(|n)p(n) for some polynomial p Will abuse and refer to individual states Motivation: If we accept | and |, we almost have to accept || and |+|. But can only polynomially many “tensorings” and “summings” take place in the multiverse, because of decoherence? Example Tree State Pn i = equal superposition over n-bit strings of parity i Pn 0 Pn1 Pn i 1 Pn / 2 0 Pn / 2 0 Pn / 21 Pn / 21 2 1 Pn / 2 0 Pn / 21 Pn / 21 Pn / 2 0 2 TS Pn i n n 1 0 1 i 0 1 i 1 TS P n 2 2 2 O n2 O n Multilinear Formulas Trees involving +,, x1,…,xn, and complex constants, such that every vertex computes a multilinear polynomial (no xi multiplied by itself) Given f : 0,1 , let MFS(f) be minimum number of vertices in multilinear formula for f n + Theorem: If then x1 x2 -3i x1 TS x0,1 f x x , n MFS f Depth Reduction Theorem: Any tree state has a tree of polynomial size and logarithmic depth Proof Idea: Follows Brent’s Theorem (1974), that any function with a poly-size arithmetic formula has a formula of polynomial size and logarithmic depth H 2n is an orthogonal tree state if it has a polynomial-size tree that only adds orthogonal states Theorem: Any orthogonal tree state can be prepared by a poly-size quantum circuit Proof Idea: If we can prepare | and |, clearly can prepare ||. To prepare |+| where |=0: let U|0n=|, V|0n=|. Then 0 0 n 1 U V 0 1 0 n 0 0 n 0 U V 0 1 n Add OR of 2nd register to 1st register Theorem: If H 2n is chosen uniformly under the Haar measure, then with 1-o(1) probability, no state | with TS(|)=2o(n) satisfies |||215/16 Why It’s Not Obvious: : TS 2 o n Proof Idea: Use Warren’s Theorem from real algebraic geometry H n 2 TreeBQP Class of problems solvable by a quantum computer whose state at every time is a tree state. (1-qubit intermediate measurements are allowed.) BPP TreeBQP BQP Theorem: TreeBQP P 3 P 3 Proof Idea: Guess and verify trees; use GoldwasserSipser approximate counting Evidence that TreeBQP BQP? QECC States Let C be a coset in n 2 ; then C 1 C x xC Codewords of stabilizer codes (Gottesman, CSS) Later we’ll add phases to reduce codeword size Take the following distribution over cosets: choose A Z 2k n , b Z 2k u.a.r. (where k=n1/3), then let C x Z : Ax b n 2 Result: log n 1 Pr TS C n C Raz’s Breakthrough Given coset C, let 1 if x C f x 0 otherwise Need to lower-bound multilinear formula size MFS(f) Until June, superpolynomial lower bounds on MFS didn’t exist Raz: n(log n) MFS lower bounds for Permanent and Determinant of nn matrix (Exponential bounds conjectured, but n(log n) is the best Raz’s method can show) Idea of Raz’s Method Given f : 0,1 , choose 2k input bits u.a.r. Label them y1,…,yk, z1,…,zk n Randomly restrict remaining bits to 0 or 1 u.a.r. Yields a new function f R y, z : 0,1 0,1 k k Let MR = fR(y,z) z{0,1}k y{0,1}k Show MR has large rank with high probability over choice of fR Intuition: Multilinear formulas can compute functions with huge rank, i.e. IP x, w x1w1 xn wn mod 2 But once we restrict everything except y1,…,yk, z1,…,zk, with high probability rank becomes small x1 x2 w w 2 1 x3 w3 x4 w4 x5 w5 x6 w6 x7 w7 x8 y2 w8 0 1 z3 z1 1 1 1 0 z2 y3 0 1 y1 0 1 Theorem (Raz): Pr rank M R c2k 1 MFS f n log n Lower Bound for Coset States 0 x 0 1 1 0 0 0 1 0 A 1 0 1 1 0 1 1 1 y1 0 0 0 1 0 1 0 0 y2 1 b y 3 1 1 0 If these two kk matrices are z1 invertible (which they are with z 2 2 probability > 0.288 ), then MR is z3 a permutation of the identity matrix, so rank(MR)=2k Corollary First superpolynomial gap between multilinear and general formula size of functions • f(x) is trivially NC1—just check whether Ax=b • Determinant not known to be NC1—best formulas known are nO(log n) Still open: Is there a polynomial with a poly-size formula but no poly-size multilinear formula? Inapproximability of Coset States Fact: For an NN complex matrix M=(mij), rank M N mij ij 2 ij (Follows from Hoffman-Wielandt inequality) Corollary: With (1) probability over coset C, no state | with TS(|)=no(log n) has ||C|20.98 Shor States Superpositions over binary integers in arithmetic progression: letting w = (2n-a-1)/p, a p 1 w a pi w i 0 (= 1st register of Shor’s alg after 2nd register is measured) Conjecture: Let S be a set of integers with |S|=32t and |x|exp((log t)c) for all xS and some c>0. Let Sp={x mod p : xS}. For sufficiently large t, if we choose a prime p uniformly at random from [t,5t/4], then |Sp|3t/4 with probability at least 3/4 Theorem: Assuming the conjecture, there exist p,a for which TS(|pZ+a)=n(log n) Challenge for NMR Experimenters • Create a uniform superposition over a n “generic” coset of 2 (n9) or even better, Clifford group state • Worthwhile even if you don’t demonstrate error correction • We’ll overlook that it’s really (1-10-5)I/512 + 10-5|CC| New test of QM: are all states tree states? What’s been done: 5-qubit codeword in liquid NMR (Knill, Laflamme, Martinez, Negrevergne, quant-ph/0101034) 1 00000 10010 01001 10100 01010 11011 00110 11000 4 11101 00011 11110 01111 10001 01100 10111 00101 TS(|) 69 Tree Size Upper Bounds for Coset States log2(# of nonzero amplitudes) 0 1 n 1 1 3 2 3 7 7 3 4 9 17 10 4 5 11 21 27 13 5 6 13 25 49 33 16 6 7 15 29 57 77 39 19 7 8 17 33 65 121 89 45 8 9 19 37 73 145 185 101 51 9 10 21 41 81 161 305 225 113 57 10 11 23 45 89 177 353 433 249 125 63 11 12 25 49 97 193 385 705 545 273 137 69 12 13 27 53 105 209 417 833 993 593 297 149 75 # o f q u b i t s 2 3 4 5 6 7 8 9 10 11 12 “Hardest” cases (to left, use naïve strategy; to right, Fourier strategy) 22 25 28 31 34 37 For Clifford Group States log2(# of nonzero amplitudes) 0 1 n 1 1 3 2 3 7 11 3 4 9 17 25 4 5 11 21 41 53 5 6 13 25 49 89 6 7 15 29 57 113 153 133 7 8 17 33 65 129 225 233 189 8 9 19 37 73 145 289 369 345 301 9 10 21 41 81 161 321 545 561 537 413 10 11 23 45 89 177 353 705 865 817 793 541 11 12 25 49 97 193 385 769 1281 1313 1265 1177 733 12 13 27 53 105 209 417 833 1665 1985 1889 1841 1689 # o f q u b i t s 2 3 4 5 6 7 8 9 10 11 12 85 957 Open Problems • Exponential tree-size lower bounds • Lower bound for Shor states • Explicit codes (i.e. Reed-Solomon) • Concrete lower bounds for (say) n=9 Important for experiments • Extension to mixed states • Separate tree states and orthogonal tree states • PAC-learn multilinear formulas? TreeBQP=BPP? • Non-tree states already created in solid state? Conclusions • Complexity theory is relevant for experimental QIP • Complexity of quantum states deserves further attention • QC skeptics can strengthen their case (and help us) by proposing Sure/Shor separators • QC experiments will test quantum mechanics itself in a fundamentally new way