Multi-product Zeno effect with higher order convergence rates

TL;DR

This paper introduces a multi-product formula to enhance the quantum Zeno effect, achieving higher order convergence rates for approximating projected quantum dynamics, with applications to quantum error correction and system decoupling.

Contribution

It develops a novel multi-product approach that significantly improves convergence rates of the quantum Zeno effect beyond the traditional order 1/n, using advanced mathematical tools.

Findings

Achieves convergence rate of 1/n^{K+1} with the multi-product formula

Demonstrates the scheme using bosonic cat code example

Applicable to system decoupling via the 'Bang-Bang' method

Abstract

To implement the dynamics of a projected Hamiltonian or Lindbladian, the quantum Zeno effect is a fundamental quantum phenomenon that approximates the effective dynamic by intersecting the Hamiltonian or Lindblad evolution by any quantum operation that converges to the desired projected subspace. Unlike the related Trotter product formula, the best-known convergence rate of the quantum Zeno effect is limited to the order $1/n$. In this work, we improve the convergence rate using a multi-product formula to achieve any power of $1/n^{K+1}$, employing a modified Chernoff Lemma, a modified Dunford-Segal approximation, and the holomorphic functional calculus. We briefly illustrate this scheme using the bosonic cat code as an example, along with a broader class of cases governed by the `Bang-Bang' method for decoupling systems from their environment.