Halving the cost of quantum addition

TL;DR

This paper introduces a new method to reduce the T gate count in quantum addition circuits by half, using a temporary logical-AND technique that is broadly applicable to various quantum algorithms, significantly lowering quantum computational costs.

Contribution

The paper presents a novel temporary logical-AND construction that halves the T gate count for quantum addition and demonstrates its wide applicability across multiple quantum algorithms.

Findings

Reduced T gates for n-bit adder to 4n + O(1)

Developed an n-bit controlled adder with T-count 8n + O(1)

Enhanced quantum circuit efficiency with temporary logical-ANDs

Abstract

We improve the number of T gates needed to perform an n-bit adder from 8n + O(1) to 4n + O(1). We do so via a "temporary logical-AND" construction which uses four T gates to store the logical-AND of two qubits into an ancilla and zero T gates to later erase the ancilla. This construction is equivalent to one by Jones, except that our framing makes it clear that the technique is far more widely applicable than previously realized. Temporary logical-ANDs can be applied to integer arithmetic, modular arithmetic, rotation synthesis, the quantum Fourier transform, Shor's algorithm, Grover oracles, and many other circuits. Because T gates dominate the cost of quantum computation based on the surface code, and temporary logical-ANDs are widely applicable, this represents a significant reduction in projected costs of quantum computation. In addition to our n-bit adder, we present an n-bit controlled adder circuit with T-count of 8n + O(1), a temporary adder that can be computed for the same cost as the normal adder but whose result can be kept until it is later uncomputed without using T gates, and discuss some other constructions whose T-count is improved by the temporary logical-AND.