Fast quantum integer multiplication with zero ancillas

TL;DR

This paper introduces a novel quantum multiplication algorithm that operates in sub-quadratic time with zero ancilla qubits, significantly reducing gate count and qubit requirements for quantum factoring algorithms.

Contribution

The authors develop a zero-ancilla, sub-quadratic-time quantum multiplication method, improving efficiency and qubit usage in quantum algorithms like Shor's and Regev's.

Findings

Achieves $ ilde{O}(n^{1+ ext{epsilon}})$ gate complexity

Yields the smallest known circuits for classically-verifiable quantum advantage

Reduces qubit count to $2n + O( ext{log} n)$ in factoring circuits

Abstract

The multiplication of superpositions of numbers is a core operation in many quantum algorithms. The standard method for multiplication (both classical and quantum) has a runtime quadratic in the size of the inputs. Quantum circuits with asymptotically fewer gates have been developed, but generally exhibit large overheads, especially in the number of ancilla qubits. In this work, we introduce a new paradigm for sub-quadratic-time quantum multiplication with zero ancilla qubits -- the only qubits involved are the input and output registers themselves. Our algorithm achieves an asymptotic gate count of $\mathcal{O}(n^{1+\epsilon})$ for any $\epsilon > 0$; with practical choices of parameters, we expect scalings as low as $\mathcal{O}(n^{1.3})$. Used as a subroutine in Shor's algorithm, our technique immediately yields a factoring circuit with $\mathcal{O}(n^{2+\epsilon})$ gates and only $2n + \mathcal{O}(\log n)$ qubits; to our knowledge, this is by far the best qubit count of any factoring circuit with a sub-cubic number of gates. Used in Regev's recent factoring algorithm, the gate count is $\mathcal{O}(n^{1.5+\epsilon})$. Finally, we demonstrate that our algorithm has the potential to outperform previous proposals at problem sizes relevant in practice, including yielding the smallest circuits we know of for classically-verifiable quantum advantage.