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Quantum Computing and Dynamical Quantum Models (quant-ph/0205059) Scott Aaronson, UC Berkeley QC Seminar May 14, 2002 Talk Outline • Why you should worry about quantum mechanics • Dynamical models • Schrödinger dynamics • SZK DQP • Search in N1/3 queries (but not fewer) Quantum What we theory experience A Puzzle • Let |OR = you seeing a red dot |OB = you seeing a blue dot t1 : R OR B OB ( H ) t2 : R OR B OB • What is the probability that you see the dot change color? Why Is This An Issue? • Quantum theory says nothing about multiple-time or transition probabilities • Reply: “But we have no direct knowledge of the past anyway, just records” • But then what is a “prediction,” or the “output of a computation,” or the “utility of a decision”? When Does This Arise? • When we consider ourselves as quantum systems • Not in “explicit-collapse” models • Bohmian mechanics asserts an answer, but assumes a specific state space Summary of Results (submitted to PRL, quant-ph/0205059) • What if you could examine an observer’s entire history? Defined class DQP • SZK DQP. Combined with collision lower bound, implies oracle A for which BQPA DQPA • Can search an N-element list in order N1/3 steps, though not fewer Dynamical Model • Given NN unitary U and state acted on, returns stochastic matrix S=D(,U) • Must marginalize to single-time probabilities: diag() and diag(UU-1) • Produces history for one N-outcome von Neumann observable (i.e. standard basis) • Discrete time and state space Axiom: Symmetry D is invariant under relabeling of basis states: D(PP-1,QUP-1) = QD(,U)P-1 Axiom: Locality 12 P1P2 U S Partition U into minimal blocks of nonzero entries Locality doesn’t imply commutativity: D U A ABU A1 ,U B D AB ,U A D U B ABU B1 ,U A D AB ,U B Axiom: Robustness 1/poly(N) change to or U 1/poly(N) change to S Example 1: Product Dynamics 4 / 5 0 1 3/ 5 3/ 5 1 0 4 / 5 4 / 5 4/5 4/5 3/ 5 2 2 2 2 2 2 3/ 5 3/5 3/5 4 / 5 2 2 Symmetric, robust, commutative, but not local Example 2: Dieks Dynamics 4 / 5 0 1 3/ 5 3/ 5 1 0 4 / 5 4 / 5 0 2 1 3/ 5 2 3/ 5 1 0 4 / 5 2 2 Symmetric, commutative, local, but not robust Example 3: Schrödinger Dynamics 7 / 25 3/ 5 4 / 5 3/ 5 24 / 25 4 / 5 3/ 5 4 / 5 .360 .640 .078 .360 .640 .019 .059 .013 .065 .130 .410 .922 .640 .360 .347 .575 .461 .461 .230 .230 Schrödinger Dynamics (con’t) • Theorem: Iterative process converges. (Uses max-flow-min-cut theorem.) • Theorem: Robustness holds. • Also symmetry and locality Commutativity for unentangled states only Computational Model • Initial state: |0n Apply poly-size quantum circuits U1,…,UT • Dynamical model D induces history v1,…,vT • vi: basis state of UiU1|0n that “you’re” in DQP • (D): Oracle that returns sample v1,…,vT, given U1,…,UT as input (under model D) • DQP: Class of languages for which there’s one BQP(D) algorithm that works for all symmetric local D • BQP DQP P#P DQP BQP SZK BPP SZKDQP • Suffices to decide whether two distributions are close or far (Sahai and Vadhan 1997) Examples: graph isomorphism, collision-finding 1 2n / 2 x f x 1 x y f x x0,1 n 2 Two bitwise Fourier transforms 1 x y f x 2 Why This Works in any symmetric local model Let v1=|x, v2=|z. Then will v3=|y with high probability? Let F : |x 2-n/2 w (-1)xw|w be Fourier transform Observation: x z y z (mod 2) Need to show F is symmetric under some permutation of basis states that swaps |x and |y while leaving |z fixed Suppose we had an invertible matrix M over (Z2)n such that Mx=y, My=x, MTz=z Define permutations , by (x)=Mx and (z)=(MT)-1z; then (x) (z) xTMT(MT)-1z x z (mod 2) Implies that F is symmetric under application of to input basis states and -1 to output basis states Why M Exists Assume x and y are nonzero (they almost certainly are) Let a,b be unit vectors, and let L be an invertible matrix over (Z2)n such that La=x and Lb=y Let Q be the permutation matrix that interchanges a and b while leaving all other unit vectors fixed Set M := LQL-1 Then Mx=y, My=x Also, xz yz (mod 2) implies aTLTz = bTLTz So QT(LTz) = LTz, implying MTz = z When Input Isn’t Two-to-One • Append hash register |h(x) on which Fourier transforms don’t act • Choose h uniformly from all functions {0,1}n {1,…,K} • Take K=1 initially, then repeatedly double K and recompute |h(x) • For some K, reduces to two-to-one case with high probability N1/3 Search Algorithm t2/N = N-1/3 probability N1/3 Grover iterations Concluding Remarks • N1/3 bound is optimal: NPA DQPA for an oracle A • With direct access to the past, you could decide graph isomorphism in polytime, but probably not SAT • Contrast: Nonlinear quantum theories could decide NP and even #P in polytime (Abrams and Lloyd 1998) • Dynamical models: more “reasonable”?

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posted: | 8/31/2010 |

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