BLACK-BOX HAMILTONIAN SIMULATIONAND UNITARY IMPLEMENTATION

摘要

We present general methods for simulating black-box Hamiltonians using quantum walks.These techniques have two main applications: simulating sparse Hamiltonians and imple-menting black-box unitary operations. In particular, we give the best known simulationof sparse Hamiltonians with constant precision. Our method has complexity linear inboth the sparseness D (the maximum number of nonzero elements in a column) andthe evolution time t, whereas previous methods had complexity scaling as D~4 and weresuperlinear in t. We also consider the task of implementing an arbitrary unitary oper-ation given a black-box description of its matrix elements. Whereas standard methodsfor performing an explicitly specified N x N unitary operation use O(N~2)elementarygates, we show that a black-box unitary can be performed with bounded error usingO(N~(2/3)(log log N)~(4/3)) queries to its matrix elements. In fact, except for pathologicalcases, it appears that most unitaries can be performed with only O(1/2N) queries, whichis optimal.