A new algorithm for directed quantum search Tathagat Tulsi, Lov Grover, Apoorva Patel

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A new algorithm for directed quantum
searchTathagat Tulsi, Lov Grover, Apoorva Patel

• Vassilina NIKOULINA, M2R III

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Plan

• Introduction
• Direct quantum Search Algorithm
• Analysis
• Comparison

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IntroductionThe problem of search

• Database with fraction f of marked items, we have
  no precise knowledge of f
• Algorithm returns 1 item from database
• Marked item -gt success
• Otherwise -gt error
• The goal
• Minimize error probability using smaller number
  of queries

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Introduction

• f - sufficiently small -gt
• Optimal quantum search algorithm with
• f - large -gt
• Classical search algorithm may outperform quantum
  algorithm

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Direct quantum search.Algorithm
Iterating n times
Error probability
For egt1/3 gt better then Phase-p/3 Search For
elt1/3 gt worse then Phase-p/3 Search 1gtegt1/2
-gt probability decrease monotonically Set lower
bound of ½ for e ? Set upper bound of ½ for f
gt extra ancilla gt controlled oracle query
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Phase-p/3 Search
Quantum search algorithm Phase-p/3
Search selective inversions Selective
p/3-shift
The best performance of Phase-p/3 Search
! Limitation restricted number of oracle queries
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Direct quantum searchAlgorithm
Goal Decrease probability of non-target state
Initial state
Ancilla in initial state
Error probability
flip ancilla
Oracle query
To decrease error -gt apply Diffusion operator
New state
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Direct quantum searchAlgorithm
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Direct quantum searchAnalysis
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Direct quantum searchAnalysis
Joint search space of ancilla-1 and the register j
Non-target states
Target state
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Direct quantum searchAnalysis
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Direct quantum searchAnalysis. Total error
Error probability after 1 iteration
Error probability after q iteration
Probability to find the register in non-target
state
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Direct quantum searchAnalysis. Total error

• Fixed point e1 instead of e0 -gt
• Error probability (1- e)2q1
• Number of oracle queries
• Directed quantum search to locate e1-f
• Thus qO(1/f) lt O(1/vf)

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Advantages of Algorithm

• Real variables
• Allowed values of q
• e eth lt1
• Directed quantum search q can take all odd
  positive numbers
• Phase-p/3 Search q (1,4,13,40,121,364,1093)
• No. of ancilla states
• Directed quantum search 2
• Phase-p/3 Search 6 (to obtain phase
  transformations from binary oracle)
• Improvement when e has the lower bound
• Instead of gt we can take initial state
• If rlt1/2 then e 1-1/2r If rgt1/2 then e
  (2r-1)/(2r1)
• faster then Phase-p/3 Search

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Conclusion

• Using irreversible measurement operators
• Superior to the Phase-p/3 Search
• Can be useful in other problems
• quantum error control

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Questions