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Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley Trailers for Future Talks The Proving Of DocumentarySpanish Version AnnouncementsStart Talk

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Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley Trailers for Future Talks The Proving Of DocumentarySpanish Version AnnouncementsStart Talk

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Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley Trailers for Future Talks The Proving Of DocumentarySpanish Version AnnouncementsStart Talk

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Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley Trailers for Future Talks The Proving Of DocumentarySpanish Version AnnouncementsStart Talk

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Live Coverage of QIP2004 http://fortnow.com/lance/complog

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Multilinear Formulas and Skepticism of Quantum Computing Scott Aaronson, UC Berkeley Trailers for Future Talks The Proving Of DocumentarySpanish Version AnnouncementsStart Talk

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Four Objections to Quantum Computing TheoreticalPractical Physical(A): QCs cant be built for fundamental reason (B): QCs cant be built for engineering reasons Algorithmic(C): Speedup is of limited theoretical interest (D): Speedup is of limited practical value

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(A): QCs cant be built for fundamental reasonLevins 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 couldnt measure its state to prove a breakdown of QMso 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

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Responses (1) Continuity in amplitudes more benign than in measurable quantitiesshould we dismiss classical probabilities of order 10 -1000 ? (2) How do we know QMs successor wont 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 e iHt |. E.g. Fitch & Cronin won 1980 Physics Nobel merely for showing CP symmetry is violated Real Question: How far should we extrapolate from todays experiments to where QM hasnt been tested?

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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: C 60 double-slit experiment, superconductivity, quantum Hall effect, etc. C 60 Arndt et al., Nature 401:680-682 (1999)

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Alternatives to QM Roger PenroseGerard t Hooft (+ King of Sweden) Stephen Wolfram

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Exactly what property separates the Sure States we know we can create, from the Shor States that suffice for factoring? DIVIDING LINE

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I hereby propose a complexity theory of pure quantum states one of whose goals is to study possible Sure/Shor separators. Classical Vidal Circuit AmpP MOTree OTree TSH Tree P 1 2 1 2 Strict containment Containment Non-containment

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Boring Bonus Feature: Relations Between Computational and Quantum State Complexity Questions BQP = P #P impliesAmpP P AmpP PimpliesNP BQP/poly P = P #P implies P AmpP P AmpPimpliesBQP P/poly

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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: + |0 1 |1 2 ++ |0 1 |1 1 |0 2 |1 2

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Trees involving +,, x 1,…,x n, and complex constants, such that every vertex computes a multilinear polynomial (no x i multiplied by itself) Given let MFS(f) be minimum number of vertices in multilinear formula for f Multilinear Formulas + -3ix1x1 x1x1 x2x2 Theorem: If then

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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 cant 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 3 rd level of the classical polynomial-time hierarchy. Yields weak evidence that TreeBQP BQP Grab Bag of Theorems

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Coset States Let C be a coset in then Codewords of stabilizer codes (Gottesman, Calderbank-Shor-Steane) Take the following distribution over cosets: choose uniformly at random (where k=n 1/3 ), then let Lower Bound To Be Proven:

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Razs Breakthrough Given coset C, let Need to lower-bound multilinear formula size MFS(f) LOOKS HARD Until June, superpolynomial lower bounds on MFS didnt exist Raz: n (log n) MFS lower bounds for Permanent and Determinant of n n matrix (Exponential bounds conjectured, but n (log n) is the best Razs method can show)

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Cartoon of Razs Method Given choose 2k input bits u.a.r. Label them y 1,…,y k, z 1,…,z k Randomly restrict remaining bits to 0 or 1 u.a.r. Yields a new function Let Theorem: f R (y,z)M R = y {0,1} k z {0,1} k ALL QUESTIONS WILL BE ANSWERED BY THE NEXT TALK

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Lower Bound for Coset States b x A If these two k k matrices are invertible (which they are with probability > 0.288 2 ), then M R is a permutation of the identity matrix, so rank(M R )=2 k Non-Quantum Corollary: First superpolynomial gap between general and multilinear formula size of functions

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Inapproximability of Coset States Fact: For an N N complex matrix M=(m ij ), (Follows from Hoffman-Wielandt inequality) Corollary: With (1) probability over coset C, no state | with TS(| )=n o(log n) has | |C | 2 0.98

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Superpositions over binary integers in arithmetic progression: letting w = (2 n -a-1)/p, (= 1 st register of Shors alg after 2 nd register is measured) Theorem: Assuming a number-theoretic conjecture, there exist p,a for which TS(|pZ+a )=n (log n) Shor States Bonus Feature: My original conjecture has been falsified by Carl Pomerance

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Revised Conjecture (Not Yet Falsified & Obviously True) Let A consist of 5+log(n 1/3 ) subsets of {2 0,…,2 n-1 } chosen uniformly at random. For all 32n 1/3 subsets B of A, let S contain the sum of the elements of B. Let S mod p = {x mod p : x S}. If p is chosen uniformly at random from [n 1/3,1.1n 1.3 ], then Pr p [|S mod p| 3n 1/3 /4] 3/4 Theorem: Assuming this conjecture, quantum states that arise in Shors algorithm have tree size n (log n) Partial results toward proving the revised conjecture by Don Coppersmith

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Bonus Feature: Cluster States Equal superposition over all settings of qubits in a n n lattice, with phase=(-1) m where m is the number of pairs of neighboring 1 qubits Conjecture: Cluster states have superpolynomial tree size 00101 01110 00111 10100 10011

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Given an n n unitary matrix U and string x 1 …x n with Hamming weight k, let U x be the k k submatrix of U formed by the first k rows and the columns corresponding to x i =1. Then a Terhal state is ( Amazingly, these are always normalized) Conjecture: Terhal states have superpolynomial tree size Bonus Feature: Terhal States

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Challenge for Experimenters Create a uniform superposition over a generic coset of (n 9) or even better, Clifford group state Worthwhile even if you dont demonstrate error correction Well overlook that its really (1-10 -5 )I/512 + 10 -5 |C C| New test of QM: are all states tree states? Whats been done: 5-qubit codeword in liquid NMR (Knill, Laflamme, Martinez, Negrevergne, quant-ph/0101034) TS(| ) 69

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Tree Size Upper Bounds for Coset States 0123456789101112 113 2377 3491710 4511212713 56 25493316 67152957773919 78173365121894522 891937731451851015125 9102141811613052251135728 10112345891773534332491256331 11122549971933857055452731376934 121327531052094178339935932971497537 log 2 (# of nonzero amplitudes) n#ofqubitsn#ofqubits Hardest cases (to left, use naïve strategy; to right, Fourier strategy)

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For Clifford Group States 0123456789101112 113 23711 3491725 4511214153 561325498985 67152957113153133 78173365129225233189 89193773145289369345301 910214181161321545561537413 1011234589177353705865817793541 1112254997193385769 1281131312651177 733 12132753105209417833 16651985188918411689957 log 2 (# of nonzero amplitudes) n#ofqubitsn#ofqubits

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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 Extension to mixed states Separate tree states and orthogonal tree states PAC-learn multilinear formulas? TreeBQP=BPP? Non-tree states already created? Important for experiments

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