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)

What weexperience

Quantum theory

A Puzzle• Let |OR = you seeing a red dot

|OB = you seeing a blue dot

1

( )

2

:

:

R R B B

H

R R B B

t O O

t O O

• 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

• 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”

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

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)

• Discrete time and state space

• Produces history for one N-outcome von Neumann observable (i.e. standard basis)

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:

1 1, , , ,A AB A B AB A B AB B A AB BD U U U D U D U U U D U

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

2 2

2 2

4 / 5 3/ 5

3/ 5 4 / 5

2 2

2 2

4 / 5 4 / 5

3/ 5 3/ 5

Symmetric, robust, commutative, but not local

Example 2: Dieks Dynamics

4 / 5 0 1 3/ 5

3/ 5 1 0 4 / 5

2 2

2 2

4 / 5 3/ 5

3/ 5 4 / 5

0 1

1 0

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

.640 .360

.360 .640

.078

.922

.130 .410

.230 .230

.019 .059

.461 .461

.013 .065

.347 .575

Schrödinger Dynamics (con’t)

• Theorem: Iterative process converges. (Uses max-flow-min-cut theorem.)

• Also symmetry and locality

Commutativity for unentangled states only

• Theorem: Robustness holds.

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)

• BQP DQP P#P

• DQP: Class of languages for which there’s one BQP(D) algorithm that works for all symmetric local D

BPP

BQP SZK

DQP

SZKDQP• Suffices to decide whether two distributions are close or far (Sahai and Vadhan 1997)

Examples: graph isomorphism, collision-finding

/ 20,1

1 1

2 2nn

x

x f x x y f x

1

2x y f x

Two bitwise Fourier transforms

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)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

N1/3

Groveriterations

t2/N = N-1/3 probability

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)

• N1/3 bound is optimal: NPA DQPA for an oracle A

• Dynamical models: more “reasonable”?