Tailoring Fault-Tolerance to Quantum Algorithms

TL;DR

This paper introduces a method for customizing quantum error correction to specific algorithms, demonstrated with Trotter circuits, to reduce overhead and improve fault-tolerance in quantum simulation.

Contribution

It presents a systematic approach for tailoring fault-tolerance to specific quantum algorithms, exemplified by Trotter circuits, using the solve-and-stitch algorithm and flag gadgets.

Findings

Achieves near-optimal circuit depth with tailored error correction.

Demonstrates fault-tolerance for Trotter circuits with minimal overhead.

Generalizes approach to larger logical Clifford circuits.

Abstract

The standard approach to universal fault-tolerant quantum computing is to develop a general purpose quantum error correction mechanism that can implement a universal set of logical gates fault-tolerantly. Given such a scheme, any quantum algorithm can be realized fault-tolerantly by composing the relevant logical gates from this set. However, we know that quantum computers provide a significant quantum advantage only for specific quantum algorithms. Hence, a universal quantum computer can likely gain from compiling such specific algorithms using tailored quantum error correction schemes. In this work, we take the first steps towards such algorithm-tailored quantum fault-tolerance. We consider Trotter circuits in quantum simulation, which is an important application of quantum computing. We develop a solve-and-stitch algorithm to systematically synthesize physical realizations of Clifford Trotter circuits on the well-known $\llbr n,n-2,2 \rrbr$ error-detecting code family. Our analysis shows that this family implements Trotter circuits with essentially optimal depth under reasonable assumptions, thereby serving as an illuminating example of tailored quantum error correction. We achieve fault-tolerance for these circuits using flag gadgets, which add minimal overhead. Importantly, the solve-and-stitch algorithm has the potential to scale beyond this specific example, as illustrated by a generalization to the four-qubit logical Clifford Trotter circuit on the $\llbr 20,4,2 \rrbr$ hypergraph product code, thereby providing a principled approach to tailored fault-tolerance in quantum computing.