Quantum Computing and Dynamical Quantum Models quantph0205059 Scott

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 N 1/3 queries (but not fewer)

Quantum theory What we experience

A Puzzle • Let |OR = you seeing a red dot |OB = you seeing a blue dot • 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 N 1/3 steps, though not fewer

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

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

Axiom: Locality 1 2 U P 1 P 2 S Partition U into minimal blocks of nonzero entries Locality doesn’t imply commutativity:

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

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

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

Example 3: Schrödinger Dynamics

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: |0 n Apply poly-size quantum circuits U 1, …, UT • Dynamical model D induces history v 1, …, v. T • vi: basis state of Ui U 1|0 n that “you’re” in

DQP • (D): Oracle that returns sample v 1, …, v. T, given U 1, …, 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

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

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, MTz=z Define permutations , by (x)=Mx and (z)=(MT)-1 z; then (x) (z) x. TMT(MT)-1 z 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 (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. TLTz = b. TLTz 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

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

Concluding Remarks • N 1/3 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”?