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Quantum Computing and Dynamical Quantum Models ( quant-ph/ ) Scott Aaronson, UC Berkeley QC Seminar May 14, 2002

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Talk Outline Why you should worry about quantum mechanics Dynamical models Schrödinger dynamics SZK DQP Search in N 1/3 queries (but not fewer)

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What we experience Quantum theory

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A Puzzle Let|O R = you seeing a red dot |O B = you seeing a blue dot What is the probability that you see the dot change color?

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Why Is This An Issue? Quantum theory says nothing about multiple-time or transition probabilities But then what is a prediction, or the output of a computation, or the utility of a decision? Reply: But we have no direct knowledge of the past anyway, just records

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When Does This Arise? When we consider ourselves as quantum systems Bohmian mechanics asserts an answer, but assumes a specific state space Not in explicit-collapse models

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Summary of Results (submitted to PRL, quant-ph/ ) What if you could examine an observers entire history? Defined class DQP SZK DQP. Combined with collision lower bound, implies oracle A for which BQP A DQP A Can search an N-element list in order N 1/3 steps, though not fewer

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Dynamical Model Given N N unitary U and state acted on, returns stochastic matrix S=D(,U) Must marginalize to single-time probabilities: diag( ) and diag(U U -1 ) Discrete time and state space Produces history for one N-outcome von Neumann observable (i.e. standard basis)

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Axiom: Symmetry D is invariant under relabeling of basis states: D(P P -1,QUP -1 ) = QD(,U)P -1

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Axiom: Locality 1 2 P 1 P 2 US Partition U into minimal blocks of nonzero entries Locality doesnt imply commutativity:

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Axiom: Robustness 1/poly(N) change to or U 1/poly(N) change to S

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Example 1: Product Dynamics Symmetric, robust, commutative, but not local

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Example 2: Dieks Dynamics Symmetric, commutative, local, but not robust

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Example 3: Schrödinger Dynamics

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Schrödinger Dynamics (cont) Theorem: Iterative process converges. (Uses max-flow-min-cut theorem.) Also symmetry and locality Commutativity for unentangled states only Theorem: Robustness holds.

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Computational Model Initial state: |0 n Apply poly-size quantum circuits U 1,…,U T Dynamical model D induces history v 1,…,v T v i : basis state of U i U 1 |0 n that youre in

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DQP (D): Oracle that returns sample v 1,…,v T, given U 1,…,U T as input (under model D) BQP DQP P #P DQP: Class of languages for which theres one BQP (D) algorithm that works for all symmetric local D

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BPP BQPSZK DQP

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SZK DQP Suffices to decide whether two distributions are close or far (Sahai and Vadhan 1997) Examples: graph isomorphism, collision-finding Two bitwise Fourier transforms

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Why This Works in any symmetric local model Let v 1 =|x, v 2 =|z. Then will v 3 =|y with high probability? Let F : |x 2 -n/2 w (-1) x w |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 (Z 2 ) n such that Mx=y, My=x, M T z=z Define permutations, by (x)=Mx and (z)=(M T ) -1 z; then (x) (z) x T M T (M T ) -1 z x z (mod 2) Implies that F is symmetric under application of to input basis states and -1 to output basis states

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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 (Z 2 ) 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, x z y z (mod 2) implies a T L T z = b T L T z So Q T (L T z) = L T z, implying M T z = z

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When Input Isnt Two-to-One Append hash register |h(x) on which Fourier transforms dont 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

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N 1/3 Search Algorithm N 1/3 Grover iterations t 2 /N = N -1/3 probability

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Concluding Remarks 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) N 1/3 bound is optimal: NP A DQP A for an oracle A Dynamical models: more reasonable?

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