Quantum binary field multiplication with subquadratic Toffoli gate count and low space-time cost

• URL: http://arxiv.org/abs/2501.16136v1
• Date: Mon, 27 Jan 2025 15:26:11 GMT
• Title: Quantum binary field multiplication with subquadratic Toffoli gate count and low space-time cost
• Authors: Vivien Vandaele,
• Abstract summary: We present an algorithm for constructing quantum circuits that perform multiplication over $GF (2n)$ with $mathcalO(nlog_(n))bits.<n>For some primitives, such as trinomials, the multiplication can be done in logarithmic depth and with $mathcalO(nlog_(n))bits.
• Score: 3.129187821625805
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Multiplication over binary fields is a crucial operation in quantum algorithms designed to solve the discrete logarithm problem for elliptic curve defined over $GF(2^n)$. In this paper, we present an algorithm for constructing quantum circuits that perform multiplication over $GF(2^n)$ with $\mathcal{O}(n^{\log_2(3)})$ Toffoli gates. We propose a variant of our construction that achieves linear depth by using $\mathcal{O}(n\log_2(n))$ ancillary qubits. This approach provides the best known space-time trade-off for binary field multiplication with a subquadratic number of Toffoli gates. Additionally, we demonstrate that for some particular families of primitive polynomials, such as trinomials, the multiplication can be done in logarithmic depth and with $\mathcal{O}(n^{\log_2(3)})$ gates.

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