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arXiv · 2312.10054

Efficient Quantum Oracle for Solving Bilinear Diophantine Equations on Digital Quantum Computers

S. Whitlock·T. D. Kieu
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Abstract

We present a concrete oracle construction for bilinear Diophantine equations of the form $f(x,y) = Axy + Bx + Cy + D$, together with its application as a scalable, hardware-agnostic benchmark for digital quantum computers. The oracle can be used in a Grover search algorithm in two variants suitable for both noisy-intermediate scale quantum devices and early fault-tolerant quantum processors. Applied to integer factoring via a residue-class encoding, the circuit requires $2n-5$ qubits or fewer to factor an $n$-bit biprime $N = pq$; for $N = 143$ requiring as few as 7 qubits and 135 two-qubit gates compared to 19 qubits and 51,048 two-qubit gates for a qubit-efficient variant of Shor's algorithm. Large-scale simulations confirm a success probability approaching 100\% for $>$800 randomly selected biprimes with $5 \leq n \leq 35$. The circuit family provides a scalable, deterministically convergent and easily verifiable benchmark in a range accessible to near term quantum hardware.

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S. Whitlock, T. D. Kieu. 2023-12-03. Efficient Quantum Oracle for Solving Bilinear Diophantine Equations on Digital Quantum Computers. https://arxiv.org/abs/2312.10054

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