Halving the cost of quantum addition

We improve the number of T gates needed to perform an n-bit adder from 8n + O(1) [1, 6, 8] 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. Temporary logical-ANDs are a generally useful tool when optimizing T-counts. They 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, our constructions represent a significant reduction in projected costs of quantum computation. We also 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.