IACR News item: 26 April 2024
Seyoon Ragavan
ePrint Report
In this note, we improve the space-efficient variant of Regev's quantum factoring algorithm [Reg23] proposed by Ragavan and Vaikuntanathan [RV24] by constant factors in space and/or size. This allows us to bridge the significant gaps in concrete efficiency between the circuits by [Reg23] and [RV24]; [Reg23] uses far fewer gates, while [RV24] uses far fewer qubits.
The main observation is that the space-efficient quantum modular exponentiation technique by [RV24] can be modified to work with more general sequences of integers than the Fibonacci numbers. We parametrize this in terms of a linear recurrence relation, and through this formulation construct three different circuits for quantum factoring:
- A circuit that uses $\approx 12.4n$ qubits and $\approx 54.9n^{1/2}$ multiplications of $n$-bit integers.
- A circuit that uses $(9+\epsilon)n$ qubits and $O_\epsilon(n^{1/2})$ multiplications of $n$-bit integers, for any $\epsilon > 0$.
- A circuit that uses $(24+\epsilon)n^{1/2}$ multiplications of $n$-bit integers, and $O_\epsilon(n)$ qubits, for any $\epsilon > 0$.
In comparison, the original circuit by [Reg23] uses at least $\approx 3n^{3/2}$ qubits and $\approx 6n^{1/2}$ multiplications of $n$-bit integers, while the space-efficient variant by [RV24] uses $\approx 10.32n$ qubits and $\approx 138.3n^{1/2}$ multiplications of $n$-bit integers (although a very simple modification of their Fibonacci-based circuit uses $\approx 11.32n$ qubits and only $\approx 103.7n^{1/2}$ multiplications of $n$-bit integers). The improvements proposed in this note take effect for sufficiently large values of $n$; it remains to be seen whether they can also provide benefits for practical problem sizes.
In comparison, the original circuit by [Reg23] uses at least $\approx 3n^{3/2}$ qubits and $\approx 6n^{1/2}$ multiplications of $n$-bit integers, while the space-efficient variant by [RV24] uses $\approx 10.32n$ qubits and $\approx 138.3n^{1/2}$ multiplications of $n$-bit integers (although a very simple modification of their Fibonacci-based circuit uses $\approx 11.32n$ qubits and only $\approx 103.7n^{1/2}$ multiplications of $n$-bit integers). The improvements proposed in this note take effect for sufficiently large values of $n$; it remains to be seen whether they can also provide benefits for practical problem sizes.
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