A Polylogarithmic-Depth Quantum Multiplier

TL;DR

This paper introduces a quantum multiplication algorithm with polylogarithmic depth and minimal T-depth, significantly improving the efficiency of quantum arithmetic operations.

Contribution

It presents the first quantum multiplier with both circuit depth and T-depth bounded by O(log^2 n), optimizing depth and fault-tolerance.

Findings

Achieves O(log^2 n) circuit and T-depth for quantum multiplication.

Uses O(n^2) gates and ancillary qubits for implementation.

Lowest T-depth among Clifford + T model multiplication algorithms.

Abstract

We present a quantum algorithm for multiplying two $n$-bit integers with overall circuit depth and $T$-depth both bounded by $O(\log^{2} n)$, while using $O(n^{2})$ gates and ancillary qubits. Our construction generates partial products via indicator-controlled copying and adds them using a binary adder tree, enabling parallel accumulation with logarithmic depth overhead per level. To the best of our knowledge, our design has the lowest $T$-depth among all multiplication algorithms using the Clifford + $T$ model. By optimizing both circuit depth and $T$-depth, our construction advances the practical feasibility of large-scale fault-tolerant quantum algorithms.