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Multilinear Formulas and Skepticism of Quantum Computing / 2 Scott Aaronson, UC Berkeley “The Proving Of” Documentary Trailers for Future Talks Spanish Version Announcements Start Talk Multilinear Formulas and Skepticism of Quantum Computing / 2 Scott Aaronson, UC Berkeley “The Proving Of” Documentary Trailers for Future Talks Spanish Version Announcements Start Talk Multilinear Formulas and Skepticism of Quantum Computing / 2 Scott Aaronson, UC Berkeley “The Proving Of” Documentary Trailers for Future Talks Spanish Version Announcements Start Talk Multilinear Formulas and Skepticism of Quantum Computing / 2 Scott Aaronson, UC Berkeley “The Proving Of” Documentary Trailers for Future Talks Spanish Version Announcements Start Talk Live Coverage of QIP’2004 http://fortnow.com/lance/complog Multilinear Formulas and Skepticism of Quantum Computing / 2 Scott Aaronson, UC Berkeley “The Proving Of” Documentary Trailers for Future Talks Spanish Version Announcements Start Talk Four Objections to Quantum Computing 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 AmpP I hereby propose a complexity theory of pure quantum states n Circuit Tree P TSH OTree H2 Vidal MOTree one of whose goals is to study possible Sure/Shor separators. 2 2 1 1 Strict containment Containment Non-containment Classical Boring Bonus Feature: Relations Between Computational and Quantum State Complexity Questions BQP = P#P implies AmpP P AmpP P implies NP BQP/poly P = P#P implies P AmpP P AmpP implies BQP P/poly Tree size TS(|) = minimum number of unbounded-fanin + and gates, |0’s, and |1’s needed in a tree representing |. Constants are free. Permutation order of qubits is irrelevant. Tree states are states with polynomially-bounded TS 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 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 Grab Bag of Theorems Theorem 1: Any tree state has a tree of polynomial size and logarithmic depth Theorem 2: Any orthogonal tree state (where all additions are of orthogonal states) can be prepared by a polynomial-size quantum circuit Theorem 3: Most quantum states can’t even be approximated by a state with subexponential tree size Theorem 4: A quantum computer whose state is always a tree state can be simulated in the 3rd level of the classical polynomial-time hierarchy. Yields weak evidence that TreeBQP BQP Coset States Let C be a coset in n 2 ; then C 1 C x xC Codewords of stabilizer codes (Gottesman, Calderbank-Shor-Steane) Take the following distribution over cosets: choose A Z 2k n , b Z 2k uniformly at random (where k=n1/3), then let C x Z n : Ax b 2 Lower Bound To Be Proven: 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) Cartoon 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 Let MR = fR(y,z) z{0,1}k k ALL QUESTIONS WILL BE ANSWERED BY THE NEXT TALK y{0,1}k Theorem: Pr rank M R c2 1 MFS f n k 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 If these two kk matrices are 0 invertible (which they are with z1 probability > 0.2882), then MR is z 2 a permutation of the identity z3 matrix, so rank(M )=2k R Non-Quantum Corollary: First superpolynomial gap between general and multilinear formula size of functions 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) Theorem: Assuming a number-theoretic conjecture, there exist p,a for which TS(|pZ+a)=n(log n) Bonus Feature: My original conjecture has been falsified by Carl Pomerance Revised Conjecture (Not Yet Falsified & Obviously True) Let A consist of 5+log(n1/3) subsets of {20,…,2n-1} chosen uniformly at random. For all 32n1/3 subsets B of A, let S contain the sum of the elements of B. Let S mod p = {x mod p : xS}. If p is chosen uniformly at random from [n1/3,1.1n1.3], then Prp [|S mod p| 3n1/3/4] 3/4 Theorem: Assuming this conjecture, quantum states that arise in Shor’s algorithm have tree size n(log n) Partial results toward proving the revised conjecture by Don Coppersmith Bonus Feature: Cluster States Equal superposition over all settings of qubits in a nn lattice, with phase=(-1)m where m is the number of pairs of neighboring ‘1’ qubits 0 0 1 0 1 0 1 1 1 0 0 0 1 1 1 1 0 1 0 0 1 0 0 1 1 Conjecture: Cluster states have superpolynomial tree size Bonus Feature: Terhal States Given an nn unitary matrix U and string x1…xn with Hamming weight k, let Ux be the kk submatrix of U formed by the first k rows and the columns corresponding to xi=1. Then a Terhal state is x0,1 : x k det U x x n (Amazingly, these are always normalized) Conjecture: Terhal states have superpolynomial tree size Challenge for 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?