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 \({\mathbf {F}}_q\). In the case \(m=n\) and \(q=2\), under standard assumptions, the algorithm takes time \(2^{(t+o(1))n}\) on a mesh-connected computer of area \(2^{(a+o(1))n}\), where \(t\approx 0.45743\) and \(a\approx 0.01467\). The area-time product has asymptotic exponent \(t+a\approx 0.47210\).