Searching

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Quantum Searching Related AlgorithmsLov K.
Grover, Bell Labs, Alcatel-Lucent

• Searching quantum classical
• Quantum Searching
• Fixed Point Searching
• The search algorithm combines the two main
  building blocks for quantum algorithms---fast
  transforms and amplitude amplification---and is
  deceptively simple. - David Meyer (Three
  views of the search algorithm)

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Classical Searching out of 5 items
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Quantum Mechanical Search
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Search Quantum Classical
In amplitude amplification, amplitude in target
state is amplified. (after h iterations, the
probability of success is sin(2hUts)2) .
In classical searching probabilities in
non-target states is reduced (e.g. after h
iterations, the probability of success is 1-
(1-Uts2)h?).
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Quantum Search Algorithm

• Encode N states with log2N qubits.
• Start with all qubits in 0 state.
• Apply the following operations

Observe the state.
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Optimality of quantum search algorithm
We are allowed to hook up O(log N)
hardware. Problem - find the single point at
which f(x) ? 0.

• Classically we need N steps.
• Quantum mechanically, we need only vN steps.

Quantum search algorithm is best possible
algorithm for exhaustive searching. - Chris
Zalka, Phys. Rev. A, 1999
However, only optimal for exhaustive search of 1
in N items.
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Quantum searching amidst uncertainty

• Quantum search algorithm is optimal only if
  number of solutions is known.
• Puzzle - Find a solution if the number of
  solutions is either 1 or 2 with equal
  probability.
• (Only one observation allowed)

½½(1-(½)pt/4)
½(sin2(t)sin2(2t))
Fixed point searching converges to 1.
Maximum success probability 3/4
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Fixed Point Quantum Searching

• Fixed point point of monotonic convergence (no
  overshoot).

• Iterative quantum procedures cannot have fixed
  points(Reason Unitary transformations have
  eigenvalues of modulus unity
  so inherently periodic).

• Fixed points achieved by 1. Using
  measurements2. Iterating with slightly different
  unitary operations in different iterations.

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Slightly different operations in different
iterations

• If Vts2 1-d, denote p/3 phase shift of t s
  state by Rt Rs.
• VRsV RtVts2 1-d3
• V(RsVRtV)(RsVRtV )(RsVRtV)(RsVRtV)ts2
  1-d9
• Non-periodic sequence and can hence have
  fixed-points

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Error correction - idea

• U takes us to within e of the target state.
  lttUsgt2 1- e
• then URsURtU takes us to within e3 of
  target lttURsURtUsgt21-e3
• Can cancel errors in any unitary U by URsURtU
• - need to run U twice and U once, with same
  errors.- need to be able to do Rs Rt

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Quantum search

• Database search function inversion
• Scheduling Problems
• Collision problem Element Distinctness
• Precision Measurements
• Pendulum Modes
• Moving Particles in a Harmonic oscillator
• Confocal Resonator Design.

A good idea finds application in contexts beyond
where it was originally conceived.