Jul' 24, 2002

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Quantum search by measurement
Andrew Landahl
QCMC 02
Institute for Quantum Information
http//www.iqi.caltech.edu
Andrew Childs, MITEnrico Deotto, MITEdward
Farhi, MITJeffrey Goldstone, MITSam Gutmann,
Northeastern
Based on quant-ph/0204013
Jul. 24, 2002
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Computation is Physical
Measurement circuit
Unitary circuit
Adiabatic
Topological field theory
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Adiabatic Theorem Born, Fock 28
Qualitative
System stays in ground state of if
changed slowly enough.
Quantitative
If , then the probability of
atransition
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Adiabatic algorithms Farhi et al. 00
Search problem
Find the minimum of the function
Adiabatic solution

• Encode in the Hamiltonian .

• Prepare the ground state of
  .

• Evolve adiabatically from to .

• Measure the final energy .

N.B. By Lloyd 96 this can be simulated by a
quantum circuit.
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Quantum measurement algorithms
Can we simulate adiabatic algorithms by the Zeno
effect? How well?
Two questions
How many measurements? How long does it take to
measure?
Perturbation theory
Phase estimation
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How many measurements?
Perturbation theory
So after M perfectly projective measurements, the
probability of staying in the ground state is
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The System-Meter Model von Neumann 55
Interaction
Evolution
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The System-Meter Model
Measurement complexity
Suppose we can resolve all displacementsto a
resolution
Then if
We can measure if
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The good, the bad, and the ugly
The good

• Measurement algorithm polynomially related to
  adiabatic algorithm

(Measurement)
(Adiabatic)
The bad

• Meter is continuous, not digital.

The ugly

• Measurements arent perfectly projective.

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Digital System-Meter Model Kitaev 95
Digitizing the system-meter model is phase
estimation!
Computational basis momentum eigenbasis
Pointer r qubits
Hamiltonian n qubits (digitizable a la Zalka
98)
Algorithm Phase estimation
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The Grover problem
Standard problem statement
Given a list of elements, all of which are
but one , find the by querying the list.
As a minimization problem
Minimize the function
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Grover oracles
Unitary
Hamiltonian
Measurement
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Adiabatic algorithm for Grover search
Features of the minimum gap
Features known independently of problem instance!
Naïve complexity
Actual complexity
Gap is only small for a time
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Grover in two measurements!
Another feature of the minimum gap
Naïve complexity
Two-measurement algorithm
Actual complexity

• Measure
• Measure

Success probability ½!
Time
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Summary

• Computation is Physical.
• The Zeno effect can efficiently simulate
  adiabatic evolution.
• Quantifying measurement complexity requires an
  explicit dynamical model.
• The digitized version of von Neumanns
  system-meter model is Kitaevs phase estimation
  algorithm.
• The Grover problem can be solved with two
  measurements, each taking time.

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Adiabatic algorithm for 3SAT
Instance
Set U of one-bit variables and collection C of
conjunctive clauses of at most three literals,
where a literal is a variable or a negated
variable in U.
Solution
A truth assignment for U.
Example
As a minimization problem
Minimize
, where
Adiabatic solution
, where
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Quantum Zeno Effect
Point-of-view 1
fixed projective measurements suppress
unitary evolution with probability .
Point-of-view 2
rotating projective measurements evolve
states by with probability .
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Quantum Zeno Effect
Pictorial analogy
Malus Law (Point-of-view 2)
http//230nsc1.phy-astr.gsu.edu/hbase/phyopt/polcr
oss.html
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Quantum Zeno Effect
Mathematical example
Unitary
Measurement
measurements every
Point-of-view 2
Point-of-view 1
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Effects of finite precision
Under the system-meter evolution
The reduced system density matrix becomes
Where is the Hadamard (elementwise) product,
and
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Effects of finite precision
Define
Then
If measurement were perfect, then we would have
. All we really need is
Which happens if