Fast quantum subroutines for the simplex method

Abstract

We propose quantum subroutines for the simplex method that avoid classical computation of the basis inverse. We show how to quantize all steps of the simplex algorithm, including checking optimality, unboundedness, and identifying a pivot (i.e., pricing the columns and performing the ratio test) according to Dantzig's rule or the steepest edge rule. The quantized subroutines obtain a polynomial speedup in the dimension of the problem, but have worse dependence on other numerical parameters. For example, for a problem with $m$ constraints, $n$ variables, at most $d_c$ nonzero elements per column of the costraint matrix, at most $d$ nonzero elements per column or row of the basis, basis condition number $\kappa$, and optimality tolerance $\epsilon$, pricing can be performed in $\tilde{O}(\frac{1}{\epsilon}\kappa d \sqrt{n}(d_c n + d m))$ time, where the $\tilde{O}$ notation hides polylogarithmic factors; classically, pricing requires $O(d_c^{0.7} m^{1.9} + m^{2 + o(1)} + d_c n)$ time in the worst case using the fastest known algorithm for sparse matrix multiplication. For well-conditioned sparse problems the quantum subroutines scale better in $m$ and $n$, and may therefore have an advantage for very large problems. The running time of the quantum subroutines can be improved if the constraint matrix admits an efficient algorithmic description, or if quantum RAM is available.