Quantum Computing and
Dynamical Quantum Models
(quant-ph/0205059)
Scott Aaronson, UC Berkeley
QC Seminar
May 14, 2002
Talk Outline
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Why you should worry about quantum
mechanics
Dynamical models
Schrödinger dynamics
SZK  DQP
Search in N1/3 queries (but not fewer)
Quantum
theory
What we
experience
A Puzzle
• Let |OR = you seeing a red dot
|OB = you seeing a blue dot
t1 :  R OR   B OB
( H )
t2 :  R OR   B OB
• 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 N1/3
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

P1P2
S
Partition U into minimal blocks of nonzero entries
Locality doesn’t imply commutativity:
D U A  ABU A1 ,U B  D   AB ,U A   D U B  ABU B1 ,U A  D   AB ,U B 
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

 


 4 / 5   4/5  4/5    3/ 5 





2
2


2
2
  3/ 5  3/5 3/5   4 / 5 
2
2
2
2
Symmetric, robust, commutative, but not local
Example 2: Dieks Dynamics
 4 / 5 0 1   3 / 5 
 3 / 5   1 0   4 / 5

 


 4 / 5   0




2
  3/ 5   1
2
1    3/ 5 


2
0   4 / 5 
2
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
.078 .130
.019
.360
.013 .410
.059
.640
.065
.922 .347
.640
.230
.461 .230
.360
.575
.461
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 U1,…,UT
• Dynamical model D induces history
v1,…,vT
• vi: basis state of UiU1|0n that “you’re” in
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
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
1
n/2
2

x f  x
x0,1
n

1
x  y
2
 f  x
Two bitwise Fourier
transforms
1
x  y
2
 f  x
Why This Works
in 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
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
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
t2/N = N-1/3 probability
N1/3
Grover
iterations
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”?