Black-box Hamiltonian simulation and unitary implementation (pp0029-0062)
          Andrew M. Childs and Dominic W. Berry
          doi: https://doi.org/10.26421/QIC12.1-2-4
Abstracts: 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 D4 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 × N unitary operation use O˜(N2 ) elementary gates, we show that a black-box unitary can be performed with bounded error using O(N2/3 (log log N) 4/3 ) queries to its matrix elements. In fact, except for pathological cases, it appears that most unitaries can be performed with only O˜( √ N) queries, which is optimal.
Key words: Quantum computation, quantum query complexity, Hamiltonian simulation, quantum walk