Phases and phase transition in Grover's algorithm with systematic noise

TL;DR

This paper analyzes how systematic noise affects Grover's quantum search algorithm, revealing phase transitions between ergodic and non-ergodic regimes and identifying thresholds where computational power is lost, using random matrix theory.

Contribution

It introduces a random matrix theory framework to analytically study phase transitions in Grover's algorithm under systematic noise, highlighting two distinct critical points.

Findings

Identification of an ergodicity breaking transition at _{gap}  L^{-1}

Discovery of a computational power loss at _{comp}  L^{-1/2}2^{-L/2}

Relevance to various quantum computing platforms and noise types

Abstract

While limitations on quantum computation by Markovian environmental noise are well-understood in generality, their behavior for different quantum circuits and noise realizations can be less universal. Here we consider a canonical quantum algorithm - Grover's algorithm for unordered search on $L$ qubits - in the presence of systematic noise. This allows us to write the behavior as a random Floquet unitary, which we show is well-characterized by random matrix theory (RMT). The RMT analysis enables analytical predictions for phases and phase transitions of the many-body dynamics. We find two separate transitions. At moderate disorder $\delta_{c,\mathrm{gap}}\sim L^{-1}$, there is a ergodicity breaking transition such that a finite-dimensional manifold remains non-ergodic for $\delta < \delta_{c,\mathrm{gap}}$. Computational power is lost at a much smaller disorder, $\delta_{c,\mathrm{comp}} \sim L^{-1/2}2^{-L/2}$. We comment on relevance to non-systematic noise in realistic quantum computers, including cold atom, trapped ion, and superconducting platforms.