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arXiv:2604.09847

A Polylogarithmic-Depth Quantum Multiplier

Fred Sun, Anton Borissov

quant-phSubmitted Apr 10, 2026• Announced Apr 14, 2026• 0 citations
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 $O(\log^{2} n)$, 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.

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We present a quantum algorithm for multiplying two nnn-bit integers with overall circuit depth and TTT-depth both bounded by O(log⁡2n)O(\log^{2} n)O(log2n), while using O(n2)O(n^{2})O(n2) 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 TTT-depth among all multiplication algorithms using the Clifford + TTT model. By optimizing both circuit depth and TTT-depth, our construction advances the practical feasibility of large-scale fault-tolerant quantum algorithms.
Quantum multiplication is a fundamental operation in quantum computing. It is important to have a quantum multiplier with low complexity. In this paper, we propose the Quantum Multiplier Based on Exponent Adder (QMbead), a new approach that requires just log⁡2(n)\log_2(n)log2​(n) qubits to multiply two nnn-bit integer numbers, in addition to O(n)O(n)O(n) ancillary qubits used for quantum state preparation. The QMbead uses a so-called exponent encoding to respectively represent two multiplicands as two superposition states which are prepared by a quantum state preparation method, then employs a quantum adder to obtain the sum of these two superposition states, and subsequently measures the outputs of the quantum adder to calculate the product of the multiplicands. Different quantum adders can be used in the QMbead. The circuit depth and time complexity of the QMbead, using a logarithmic-depth quantum carry lookahead adder (QCLA) as adder, are O(log⁡n)O(\log n)O(logn) and O(nlog⁡n)O(n \log n)O(nlogn), respectively. The gate complexity of the QMbead is O(n)O(n)O(n). The circuit depth and gate complexity of the QMbead is better than existing quantum multipliers such as the quantum Karatsuba multiplier and the QFT based multiplier. The time complexity of the QMbead is identical to that of the fastest classical multiplication algorithm, Harvey-Hoeven algorithm. Interestingly, the QMbead maintains an advantage over the Harvey-Hoeven algorithm, given that the latter is only suitable for excessively large numbers, whereas the QMbead is valid for both small and large numbers. The multiplicand can be either an integer or a decimal number. The QMbead has been implemented on quantum simulators to compute products with a bit length of up to 273 bits using only 17 qubits, excluding the ancillary qubits used for quantum state preparation. This establishes QMbead as an efficient solution for multiplying large integer or decimal numbers with many bits.
We construct an exact nnn-bit quantum adder with O(n)O(n)O(n) gates and O(log⁡n)O(\log n)O(logn) depth that requires no ancillary qubits. The circuit uses only XXX, CNOT, and Toffoli gates under all-to-all connectivity and is classically reversible. Our construction accelerates the controlled ladder underlying addition. Grouping the data into blocks and use existing control qubits as conditional workspace gives a recursive construction with O(nlog⁡log⁡n)O(n\log\log n)O(nloglogn) gates and O(log⁡n)O(\log n)O(logn) depth. We then coordinate inter-block propagation via two short-path forest operations to cut the per-block workspace requirement, yielding a nonrecursive implementation with O(n)O(n)O(n) gates. Both the gate count and circuit depth are asymptotically optimal.
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Quantum squaring circuits have been used as helpful arithmetic modules in various quantum algorithms for calculating series expansions or distances of vectors, etc. Quantum multipliers can replace quantum squaring circuits, but squaring with quantum multipliers is inefficient because it involves using quantum gates for unnecessary bitwise multiplication. In this paper, we propose a depth-optimized quantum circuit dedicated to squaring by eliminating these unnecessary quantum gates and implementing quantum gates in parallel. We also discuss the optimal distribution of the partial products to reduce further the gate cost of the quantum adder used for the sum of the partial products. The proposed partial product distribution method lowers the quantum adder’s number of gates and depth by half. Our quantum squaring circuit is the most efficient, with an average improvement of 68% and 79.7% in T-count and T-depth, respectively, compared to existing quantum squaring circuits. Despite the increased qubit counts caused by the depth optimization, we demonstrate that the proposed circuit has the smallest KQT\hbox {KQ}_\textrm{T}KQT​ .