Quantum Computing and Dynamical Quantum Models (quant-ph/0205059)

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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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(No Transcript)
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A Puzzle

• Let OR? you seeing a red dot
• OB? you seeing a blue dot

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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?

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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

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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 BQPA ? DQPA

• Can search an N-element list in order N1/3
  steps, though not fewer

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Dynamical Model

• Given N?N unitary U and state ? acted on,
  returns stochastic matrix SD(?,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

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Axiom Symmetry
D is invariant under relabeling of basis
states D(P?P-1,QUP-1) QD(?,U)P-1
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Axiom Locality
?1??2 ? P1?P2
U
S
Partition U into minimal blocks of nonzero entries
Locality doesnt imply commutativity
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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
Symmetric, robust, commutative, but not local
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Example 2 Dieks Dynamics
Symmetric, commutative, local, but not robust
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Example 3 Schrödinger Dynamics
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Schrödinger Dynamics (cont)

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

• Theorem Robustness holds.

• Also symmetry and locality
• Commutativity for unentangled states only

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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?U10??n that youre in

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DQP

• ?(D) Oracle that returns sample v1,,vT, given
  U1,,UT as input (under model D)

• DQP Class of languages for which theres one
  BQP?(D) algorithm that works for all symmetric
  local D

• BQP ? DQP ? PP

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DQP
BQP
SZK
BPP
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SZK?DQP

• Suffices to decide whether two distributions are
  close or far (Sahai and Vadhan 1997)
• Examples graph isomorphism, collision-finding

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Why This Worksin any symmetric local model
Let v1x?, v2z?. Then will v3y? with high
probability? Let F x? ? 2-n/2 ?w (-1)x?ww? 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 Mxy,
Myx, MTzz 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 Lax and
Lby Let Q be the permutation matrix that
interchanges a and b while leaving all other unit
vectors fixed Set M LQL-1 Then Mxy,
Myx Also, x?z ? y?z (mod 2) implies aTLTz
bTLTz So QT(LTz) LTz, implying MTz z
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When Input Isnt Two-to-One

• Append hash register h(x)? on which Fourier
  transforms dont act

• Choose h uniformly from all functions
• 0,1n ? 1,,K

• Take K1 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 Grover iterations




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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?