Measurement-based uncomputation of quantum circuits for modular arithmetic

TL;DR

This paper formalizes measurement-based uncomputation (MBU) for quantum circuits, especially in modular arithmetic, reducing gate counts and depths, and enhancing efficiency for quantum cryptography applications.

Contribution

It introduces a formal framework for MBU in single-qubit registers and applies it to optimize various modular arithmetic circuits, filling gaps in existing literature.

Findings

Reduced Toffoli count and depth by 10-15% for certain architectures

Achieved nearly 25% reduction for other modular adder architectures

Potential improvements for modular multiplication and exponentiation circuits

Abstract

Measurement-based uncomputation (MBU) is a technique used to perform probabilistic uncomputation of quantum circuits. We formalize this technique for the case of single-qubit registers, and we show applications to modular arithmetic. First, we present formal statements for several variations of quantum circuits performing non-modular addition: controlled addition, addition by a constant, and controlled addition by a constant. We do the same for subtraction and comparison circuits. This addresses gaps in the current literature, where some of these variants were previously unexplored. Then, we shift our attention to modular arithmetic, where again we present formal statements for modular addition, controlled modular addition, modular addition by a constant, and controlled modular addition by a constant, using different kinds of plain adders and combinations thereof. We introduce and prove a "MBU lemma" in the context of single-qubit registers, which we apply to all aforementioned modular arithmetic circuits. Using MBU, we reduce the Toffoli count and depth by $10\%$ to $15\%$ for modular adders based on the architecture of [VBE96], and by almost $25\%$ for modular adders based on the architecture of [Bea02]. Our results have the potential to improve other circuits for modular arithmetic, such as modular multiplication and modular exponentiation, and can find applications in quantum cryptanalysis.