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 N 1/3 queries (but not fewer)

3. What we experience Quantum theory

4. 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?

5. 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

6. 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

7. Summary of Results (submitted to PRL, quant-ph/0205059) 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

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

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 P 1 P 2 US Partition U into minimal blocks of nonzero entries Locality doesnt 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 (cont) Theorem: Iterative process converges. (Uses max-flow-min-cut theorem.) Also symmetry and locality Commutativity for unentangled states only Theorem: Robustness holds.

16. 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

17. 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

18. BPP BQPSZK DQP

19. 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

20. 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

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 (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

22. 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

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

24. 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?