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arXiv:2604.09847

A Logarithmic-Depth Quantum Multiplier

Fred Sun, Anton Borissov

quant-phSubmitted Apr 10, 2026• Announced Apr 14, 2026• 0 citations• Revised Oct 19, 2026 (v2)
v1 · Apr 10, 2026v2 · Oct 19, 2026
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Abstract

We present a quantum algorithm for multiplying two $n$-bit integers with overall circuit depth and $T$-depth both bounded by $O(\log n)$, while using $O(n^2)$ gates and ancillary qubits. Our construction generates partial products via indicator-controlled copying and adds them using a tree of reversible $3$-to-$2$ adders followed by one quantum carry-lookahead adder. Building on the carry-save adders of Kim et al., we combine full input fanout, workspace reuse, and phase-corrected gate implementations to reduce $T$-depth. We provide explicit circuit constructions, proofs of correctness, and concrete closed-form resource bounds.

Comments
Replaced the binary adder tree in V1 with a tree of carry-save adders and updated the resource bounds

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