Fast quantum subroutines for the simplex method
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We propose quantum subroutines for the simplex method that avoid classical computation of the basis inverse. For a well-conditioned m × n constraint matrix with at most d_c nonzero elements per column, at most d nonzero elements per column or row of the basis, and optimality tolerance ϵ, we show that pricing can be performed in time Õ(1/ϵ√(n)(d_c n + d^2 m)), where the Õ notation hides polylogarithmic factors. If the ratio n/m is larger than a certain threshold, the running time of the quantum subroutine can be reduced to Õ(1/ϵd √(d_c) n √(m)). Classically, pricing would require O(d_c^0.7 m^1.9 + m^2 + o(1) + d_c n) in the worst case using the fastest known algorithm for sparse matrix multiplication. We also show that the ratio test can be performed in time Õ(t/δ d^2 m^1.5), where t, δ determine a feasibility tolerance; classically, this requires O(m^2) in the worst case. For well-conditioned sparse problems the quantum subroutines scale better in m and n, and may therefore have a worst-case asymptotic advantage. An important feature of our paper is that this asymptotic speedup does not depend on the data being available in some "quantum form": the input of our quantum subroutines is the natural classical description of the problem, and the output is the index of the variables that should leave or enter the basis.
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