Quantum SDP-Solvers: Better upper and lower bounds

Abstract

Brandão and Svore \cite{brandao2016QSDPSpeedup} recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension n of the problem and the number m of constraints, but worse in terms of various other parameters. In this paper we improve their algorithms in several ways, getting better dependence on those other parameters. To this end we develop new techniques for quantum algorithms, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure.We also show limits on this approach to quantum SDP-solvers, for instance for combinatorial optimization problems that have a lot of symmetry. Finally, we prove some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver (and hence also SDP-solver) has to scale linearly with mn when m≈n, which is the same as classical.

Warm-starting quantum optimization

“…A classical laptop can solve instances of (SDP) relaxations of QUBO, where Σ has 10 13 entries [51]. Furthermore, quantum computers offer the prospect of some speed-ups in solving SDPs [66,67], although recent quantum-inspired algorithms for SDPs may reduce the potential speedup [68].…”

“…While more elaborate representations of the matrix have been proposed[66,67], it is not yet clear how to implement the related oracles in practice.Accepted in Quantum 2021-06-15, click title to verify. Published under CC-BY 4.0.…”

Quantum machine learning: a classical perspective

“…Based on a classical algorithm to solve SDPs by Arora & Kale [ 108 ], which has a runtime of , where r is an upper bound on the sum of the entries of the optimal solution to the dual problem, in 2016 Brandão & Svore [ 109 ] developed a quantum algorithm for SDPs that is quadratically faster in m and n . The dependence on the error parameters of this result has been improved in [ 110 ]. In this work, the authors obtain a final scaling of .…”

“…The main problem of these quantum algorithms is that the dependence on R , r , s and 1/ ϵ is considerably worse than in [ 108 ]. This quantum algorithm thus provides a speed-up only in situations where R , r , s ,1/ ϵ are fairly small compared with mn and, to date, it is unclear whether there are interesting examples of SDPs with these features (for more details, see [ 110 ]).…”

Quantum-accelerated constraint programming

“…Similarly, there have been efforts to develop quantum algorithms for constraint satisfaction and search. Quantum search has been extended to problems within mathematical programming, such as semidefinite programming [43,44] and the acceleration of the simplex method [45]. The latter, in a similar fashion to this work, uses quantum search to accelerate the subroutines of the simplex method (e.g., variable pricing).…”