{"id":430905,"date":"2018-11-06T16:45:33","date_gmt":"2018-11-07T00:45:33","guid":{"rendered":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/?post_type=msr-research-item&p=430905"},"modified":"2018-11-06T16:45:33","modified_gmt":"2018-11-07T00:45:33","slug":"quantum-speed-ups-semidefinite-programming","status":"publish","type":"msr-research-item","link":"https:\/\/cm-edgetun.pages.dev\/en-us\/research\/publication\/quantum-speed-ups-semidefinite-programming\/","title":{"rendered":"Quantum Speed-ups for Semidefinite Programming"},"content":{"rendered":"

We give a quantum algorithm for solving semidefinite programs (SDPs). It has worst-case running time n^<\/span>1\/<\/span>2 <\/span><\/span><\/span><\/span><\/span>m^<\/span>1\/<\/span>2 <\/span><\/span><\/span><\/span><\/span>s^<\/span>2 <\/span><\/span>poly<\/span>(<\/span>log<\/span><\/span>(<\/span>n<\/span>)<\/span>,<\/span>log<\/span><\/span>(<\/span>m<\/span>)<\/span>,<\/span>R<\/span>,<\/span>r<\/span>,<\/span>1<\/span>\/<\/span><\/span><\/span>\u03b4<\/span>)<\/span><\/span><\/span><\/span> , with n<\/span><\/span><\/span><\/span> and s<\/span><\/span><\/span><\/span> the dimension and row-sparsity of the input matrices, respectively, m<\/span><\/span><\/span><\/span> the number of constraints, \u03b4<\/span><\/span><\/span><\/span> the accuracy of the solution, and R<\/span>,<\/span>r<\/span><\/span><\/span><\/span> a upper bounds on the size of the optimal primal and dual solutions. This gives a square-root unconditional speed-up over any classical method for solving SDPs both in n<\/span><\/span><\/span><\/span> and m<\/span><\/span><\/span><\/span> . We prove the algorithm cannot be substantially improved (in terms of n<\/span><\/span><\/span><\/span> and m<\/span><\/span><\/span><\/span> ) giving a \u03a9<\/span>(<\/span>n^<\/span>1\/<\/span>2<\/span><\/span><\/span><\/span><\/span>+<\/span>m^<\/span>1\/<\/span>2<\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span> quantum lower bound for solving semidefinite programs with constant s<\/span>,<\/span>R<\/span>,<\/span>r<\/span><\/span><\/span><\/span> and \u03b4<\/span><\/span><\/span><\/span> . The quantum algorithm is constructed by a combination of quantum Gibbs sampling and the multiplicative weight method. In particular it is based on a classical algorithm of Arora and Kale for approximately solving SDPs. We present a modification of their algorithm to eliminate the need for solving an inner linear program which may be of independent interest.<\/p>\n","protected":false},"excerpt":{"rendered":"

We give a quantum algorithm for solving semidefinite programs (SDPs). It has worst-case running time n^1\/2 m^1\/2 s^2 poly(log(n),log(m),R,r,1\/\u03b4) , with n and s the dimension and row-sparsity of the input matrices, respectively, m the number of constraints, \u03b4 the accuracy of the solution, and R,r a upper bounds on the size of the optimal 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