Quantum Computing and Dynamical Quantum Models ( quant-ph0205059)

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					 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)
Quantum    What we
 theory   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                                      P1P2
   U                                           S
Partition U into minimal blocks of nonzero entries

Locality doesn’t imply commutativity:
D U A  ABU A1 ,U B  D   AB ,U A   D U B  ABU B1 ,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

               2          2            
      3/ 5  3/5 3/5   4 / 5  
             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
                        1    3/ 5 
                                      2

                                  
      3/ 5   1     0   4 / 5  
              2                       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 .360 .640
            .019 .059
            .013 .065
            .130 .410
       .922 .640 .360
            .347 .575
            .461 .461
            .230 .230
 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
2n / 2
                  x f  x   
                                     1
                                        x  y    f  x
         x0,1
               n                      2
                   Two bitwise Fourier
                           transforms
                                    1
                                       x  y     f  x
                                     2
              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”?

				
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posted:8/31/2010
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