A new algorithm for directed quantum search Tathagat Tulsi, Lov Grover, Apoorva Patel Vassilina NIKOULINA, M2R III

Plan Introduction Direct quantum Search Algorithm Analysis Comparison

Introduction The 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 -> success  Otherwise -> error The goal :  Minimize error probability using smaller number of queries

Introduction f - sufficiently small ->  Optimal quantum search algorithm with f - large ->  Classical search algorithm may outperform quantum algorithm queries

Direct quantum search. Algorithm Iterating n times: Error probability: For ε>1/3 => better then Phase-π/3 Search For ε worse then Phase-π/3 Search 1>ε>1/2 -> probability decrease monotonically Set lower bound of ½ for ε  Set upper bound of ½ for f => extra ancilla |+> + controlled oracle query

Phase-π/3 Search Quantum search algorithm Phase-π/3 Search selective inversions Selective π/3-shift The best performance of Phase-π/3 Search ! Limitation : restricted number of oracle queries

Direct quantum search Algorithm Goal: Decrease probability of non-target state Initial state: Ancilla in initial state: Oracle query : flip ancilla Error probability : To decrease error -> apply Diffusion operator New state :

Direct quantum search Algorithm H U

Direct quantum search Analysis Initial state : Initial error probability:

Direct quantum search Analysis Non-target states: Target state: Joint search space of ancilla-1 and the register j

Direct quantum search Analysis

Error probability after 1 iteration Probability to find the register in non-target state: Error probability after q iteration Direct quantum search Analysis. Total error

Fixed point : ε=1 instead of ε=0 ->  Error probability (1- ε) 2q+1 Number of oracle queries  Directed quantum search to locate: ε=1-f  Thus q=O(1/f) < O(1/√f)

Advantages of Algorithm Real variables Allowed values of q ε ≤ ε th <1 Directed quantum search: q can take all odd positive numbers Phase-π/3 Search: q = (1,4,13,40,121,364,1093…) No. of ancilla states Directed quantum search: 2 Phase-π/3 Search: 6 (to obtain phase transformations from binary oracle) Improvement when ε has the lower bound Instead of |+> we can take initial state If r 1/2 then ε =(2r-1)/(2r+1) faster then Phase-π/3 Search

Conclusion Using irreversible measurement operators Superior to the Phase-π/3 Search Can be useful in other problems :  quantum error control

Questions