Fixed-Point Adiabatic Quantum Search

TL;DR

This paper investigates whether adiabatic quantum search algorithms can possess the fixed-point property similar to gate-model algorithms, demonstrating that it depends on the interpolation schedule used and providing bounds and simulations.

Contribution

It shows that fixed-point property in adiabatic quantum search depends on the interpolation schedule and provides a rigorous analysis with bounds and simulation results.

Findings

Fixed-point adiabatic search depends on the interpolation schedule.

Explicit upper bounds on adiabatic approximation error are derived.

Fixed-point adiabatic search can be simulated in the gate model without losing quadratic speedup.

Abstract

Fixed-point quantum search algorithms succeed at finding one of $M$ target items among $N$ total items even when the run time of the algorithm is longer than necessary. While the famous Grover's algorithm can search quadratically faster than a classical computer, it lacks the fixed-point property --- the fraction of target items must be known precisely to know when to terminate the algorithm. Recently, Yoder, Low, and Chuang gave an optimal gate-model search algorithm with the fixed-point property. Meanwhile, it is known that an adiabatic quantum algorithm, operating by continuously varying a Hamiltonian, can reproduce the quadratic speedup of gate-model Grover search. We ask, can an adiabatic algorithm also reproduce the fixed-point property? We show that the answer depends on what interpolation schedule is used, so as in the gate model, there are both fixed-point and non-fixed-point versions of adiabatic search, only some of which attain the quadratic quantum speedup. Guided by geometric intuition on the Bloch sphere, we rigorously justify our claims with an explicit upper bound on the error in the adiabatic approximation. We also show that the fixed-point adiabatic search algorithm can be simulated in the gate model with neither loss of the quadratic Grover speedup nor of the fixed-point property. Finally, we discuss natural uses of fixed-point algorithms such as preparation of a relatively prime state and oblivious amplitude amplification.