Unconditional correctness of recent quantum algorithms for factoring and computing discrete logarithms

In 1994, Shor introduced his famous quantum algorithm to factor integers and compute discrete logarithms in polynomial time. In 2023, Regev proposed a multi-dimensional version of Shor’s algorithm that requires far fewer quantum gates. His algorithm relies on a number-theoretic conjecture on the elements in $(\mathbb{Z}/N\mathbb{Z})^{\times}$ that can be written as short products of very small prime numbers. We prove a version of this conjecture using tools from analytic number theory such as zero-density estimates. As a result, we obtain an unconditional proof of correctness of this improved quantum algorithm and of subsequent variants.

💡 Research Summary

The paper addresses a long‑standing gap in the theoretical foundation of Regev’s multidimensional quantum algorithms for integer factorisation and discrete logarithms. While Shor’s original algorithm (1994) solves both problems in polynomial time using a quantum circuit of size $O(n^{2}\log n)$ (with $n=\lceil\log_{2}N\rceil$) and $O(n\log n)$ qubits, Regev (2023) proposed a higher‑dimensional variant that dramatically reduces the gate count to $O(n^{3/2}\log n)$ at the cost of invoking the quantum sub‑circuit $O(\sqrt n)$ times. The speed‑up hinges on choosing a set of small integers $b_{1},\dots,b_{d}$ (with $d\approx\sqrt{\log N}$) such that a short exponent vector $(e_{1},\dots,e_{d})$ exists with
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