Asymptotically Faster Quantum Algorithms to Solve Multivariate Quadratic Equations

This paper designs and analyzes a quantum algorithm to solve a system of m quadratic equations in n variables over a finite field Fq. In the case m = n and q = 2, under standard assumptions, the algorithm takes time 2 on a mesh-connected computer of area 2, where t ≈ 0.45743 and a ≈ 0.01467. The area-time product has asymptotic exponent t+ a ≈ 0.47210. For comparison, the area-time product of Grover’s algorithm has asymptotic exponent 0.50000. Parallelizing Grover’s algorithm to reach asymptotic time exponent 0.45743 requires asymptotic area exponent 0.08514, much larger than 0.01467.