Fast Quantum Subroutines for the Simplex Method

We propose quantum subroutines for the simplex method that avoid classical computation of basis inverse. For an \(m \times n\) constraint matrix with at most \(d_c\) nonzero elements per column, d column or row basis, condition number \(\kappa \), and optimality tolerance \(\epsilon we show pricing can be performed in \(\tilde{O}(\frac{1}{\epsilon }\kappa \sqrt{n}(d_c n + m))\) time, where \(\tilde{O}\) notation hides polylogarithmic factors. If ratio n/m is larger than a certain threshold, running time subroutine reduced to d^{1.5} \sqrt{d_c} \sqrt{m})\). The steepest edge pivoting rule also admits implementation, increasing by factor ^2\). Classically, requires \(O(d_c^{0.7} m^{1.9} m^{2 o(1)} d_c n)\) worst case using fastest known algorithm sparse multiplication, m^2n)\) edge. Furthermore, test \(\tilde{O}(\frac{t}{\delta } \kappa d^2 m^{1.5})\) \(t, \delta \) determine feasibility tolerance; classically, this \(O(m^2)\) case. well-conditioned problems scale better m n, may therefore have worst-case asymptotic advantage. An important feature our paper speedup does not depend on data being available some “quantum form”: input natural description problem, output index variables should leave enter basis.