Black-box hamiltonian simulation and unitary implementation

We present general methods for simulating black-box Hamiltonians using quantum walks. These techniques have two main applications: simulating sparse Hamiltonians and implementing black-box unitary operations. In particular, we give the best known simulation of sparse Hamiltonians with constant precision. Our method has complexity linear in both the sparseness D (the maximum number of nonzero elements in a column) and the evolution time t, whereas previous methods had complexity scaling as D ⁴ and were superlinear in t. We also consider the task of implementing an arbitrary unitary operation given a black-box description of its matrix elements. Whereas standard methods for performing an explicitly specified N X N unitary operation use Õ(N ²) elementary gates, we show that a black-box unitary can be performed with bounded error using O(N ^2/3(log logN) ^4/3) queries to its matrix elements. In fact, except for pathological cases, it appears that most unitaries can be performed with only Õ (√N) queries, which is optimal.