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Quantum Computing and Dynamical Quantum Models ( quant-ph/0205059)

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 Schr ö dinger dynamics SZK  DQP Search in N 1/3 queries (but not fewer). Quantum theory.

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Quantum Computing and Dynamical Quantum Models ( quant-ph/0205059)

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  1. Quantum Computing and Dynamical Quantum Models(quant-ph/0205059) Scott Aaronson, UC Berkeley QC Seminar May 14, 2002

  2. Talk Outline • Why you should worry about quantum mechanics • Dynamical models • Schrödinger dynamics • SZK  DQP • Search in N1/3 queries (but not fewer)

  3. Quantum theory What we experience

  4. What is the probability that you see the dot change color? A Puzzle • Let |OR = you seeing a red dot • |OB = you seeing a blue dot

  5. 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”?

  6. 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

  7. 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

  8. 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) • Produces history for one N-outcome von Neumann observable (i.e. standard basis) • Discrete time and state space

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

  10. Axiom: Locality 12  P1P2 U S Partition U into minimal blocks of nonzero entries Locality doesn’t imply commutativity:

  11. Axiom: Robustness 1/poly(N) change to  or U  1/poly(N) change to S

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

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

  14. Example 3: Schrödinger Dynamics

  15. 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

  16. Computational Model • Initial state: |0n • Apply poly-size quantum circuits U1,…,UT • Dynamical model D induces history v1,…,vT • vi: basis state of UiU1|0n that “you’re” in

  17. 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

  18. DQP BQP SZK BPP

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

  20. Why This Worksin any symmetric local model Let v1=|x, v2=|z. Then will v3=|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 (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

  21. 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, xz  yz (mod 2) implies aTLTz = bTLTz So QT(LTz) = LTz, implying MTz = z

  22. 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

  23. t2/N = N-1/3 probability N1/3 Search Algorithm N1/3 Grover iterations

  24. 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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