Tailoring Fault-Tolerance to Trotter Circuits

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. 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 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\unicode{x27E6}n, n-2,2\unicode{x27E7}$</tex> error-detecting code family. Our analysis shows that this family implements Trotter circuits with optimal depth, 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. The solve-and-stitch algorithm has the potential to scale beyond this specific example and hence provide a principled approach to tailored fault-tolerance in quantum computing.