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Quantum Computing and Dynamical Quantum Models
(quant-ph/0205059)
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 N1/3 queries (but not fewer)
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What weexperience
Quantum theory
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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?
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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/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
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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)
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Axiom: Symmetry
D is invariant under relabeling of basis states:
D(PP-1,QUP-1) = QD(,U)P-1
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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
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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
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
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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
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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
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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.
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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
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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
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BPP
BQP SZK
DQP
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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
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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
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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 (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
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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
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N1/3 Search Algorithm
N1/3
Groveriterations
t2/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)
• N1/3 bound is optimal: NPA DQPA for an oracle A
• Dynamical models: more “reasonable”?