Quantum Search Algorithms for Multiple Solution Problems

1
Quantum Search Algorithms for Multiple Solution
Problems

• EECS 598 Class Presentation
• Manoj Rajagopalan

2
Outline

• Recap of Grovers algorithm for the unique
  solution case
• Grovers algorithm for multiple solutions
  multiplicity known
• Quantum search algorithm for multiple solutions
  multiplicity unknown
• Quantum counting to determine multiplicity

3
References

• Quantum Computing and Quantum Information
  textbook
• A fast quantum mechanical algorithm for database
  search, LK Grover, 1996
• Tight bounds on quantum searching, M Boyer, G
  Brassard, P Hoyer, 1996
• Quantum counting, G Brassard, P Hoyer, A Tapp,
  1998

4
Notation

• n qubits in the system
• N of possible values of n qubits 2n
• M multiplicity of solution
• k probability amplitude of system in solution
  state
• l probability amplitude of system in
  non-solution state
• A set of indices that denote solutions (good
  states)
• B set of indices denoting bad states
• ? rotation angle corresponding to Grover
  operator

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Grovers Algorithm for Unique Solution Case

• Given F0,1n ? 0,1, find i0 ? F(i0)1 and ? i
  ? i0 F(i)0
• Set up initial state 0? ?n
• Apply the Hadamard transform
• H?n 0? ?? ??
• Let i0 be the solution ?? k i0?
• Grover operator made of 4 steps
• Apply the oracle
• Apply H?n
• Conditional phase shift
• Apply H?n

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Unique Solution Case Recap (contd)

• Apply the Grover operator. After j iterations,

• Need bound on the number of iterations

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Unique Solution Case Recap (contd)
Let sin2? 0 lt ? ?
For km 1, (2m1)? ? /2 gt For
large N, ? ? sin ? ? m ?
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Multiple Solutions Multiplicity known
Given F0,1n ?? 0,1, find all i?0,1n ?
F(i)1 M number of solutions gt 1 Define good
states A i F(i) 1 A M bad states
B j F(j) 0 B N - M Suffices to
tackle good and bad states as groups k
probability amplitude of each solution (element
of set A) l probability amplitude of each
element of set B Mk2 (N-M)l2 1
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Multiple Solutions Multiplicity known

• Grovers algorithm for the multiple solution case
• Structurally the same as that in the case of
  unique solution
• Set up initial state 0? ?n
• Apply the Hadamard transform
• 3. Apply Grover operator repeatedly
• Apply the oracle
• Apply H?n
• Conditional phase shift
• Apply H?n
• Differs in the oracle implementation Oracle
  lends a relative phase shift of 1 to all
  solutions

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Multiple Solutions Multiplicity known
Define
After j iterations
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Multiple Solutions Multiplicity known
Let m upper bound on number of iterations We
want lm 0 cos ((2m1)?) gt

• cos(2m1)? ? sin ?
• Probability of failure after exactly m iterations
• (N-M) lm2 cos2((2m1)?) ? sin2?

Negligible for M ltlt N
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Multiple Solutions Multiplicity known
For M ltlt N, ? ? sin ?
Knowing M, we can predetermine the upper bound on
the number of iterations, m. Unique solution
problem is a special case of this for M1.
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Multiple Solutions unknown Multiplicity
Number of iterations required to obtain a
solution with significant confidence depends on
the solutions multiplicity. If M is not known,
then there is no way of telling how many
iterations will suffice. Take m to be on
the safe side? (max iterations) No!
Probability of success minuscule when M 4a2
where a is a small integer.
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Multiple Solutions unknown Multiplicity

• Modified procedure for unknown M
• Initialize m 1 and ? 8/7 (actually 1 lt ? lt
  4/3)
• Choose integer j such that 0 ? j ? m
• Apply j iterations of Grovers algorithm
• Measure and let outcome be i
• If F(i) 1 then solution found exit program
• Else m min(?m, ) goto step 2
• Theorem This algorithm finds a solution in O(
  )

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Multiple Solutions unknown Multiplicity
For M gt 3N/4 constant expected time by classical
sampling For 0 lt M ? 3N/4, runtime O( ) For
M ltlt N, runtime lt 6 times runtime_if_M_were_known
Knowing the number of solutions helps in reducing
runtime. This motivates quantum counting
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Quantum Counting

• Aim To determine the number of solutions M to an
  N item unstructured search problem
• Classical computing consults the oracle ?(N)
  times to determine M
• Quantum computing can combine Grovers algorithm
  and phase estimation to determine M much faster!
• Why count?
• Fast estimation of M gt rapid solution
  detection
• Is there a solution at all? NP-Complete
  problems

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Quantum Counting
Recall The computational bases can be
partitioned into two subsets, the good states
set A containing all the solutions, and
Letting
we get
in the basis.
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Quantum Counting
Eigenvalues of G are ei2? and ei(2?-2?) The value
of ? can be determined by phase estimation From
?, the value of M can be calculated PHASE
ESTIMATION Given a unitary operator U and one of
its eigenvectors, the phase ? of its
corresponding eigenvalue ei2?? is determined
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Quantum Counting
Complexity of phase estimation algorithms