Hamiltonian simulation with nearly optimal dependence on spectral norm

Abstract

We present a quantum algorithm for approximating the real time evolution e−iHt of an arbitrary d-sparse Hamiltonian to error є, given black-box access to the positions and b-bit values of its non-zero matrix entries. The query complexity of our algorithm is O((t√d||H||1 → 2)1+o(1)/єo(1)) with respect to the largest Euclidean row norm ||H||1 → 2, which is shown to be optimal up to subpolynomial factors through a matching lower bound, and it uses a factor O(b) more gates. This provides a polynomial speedup in sparsity for the common case where the spectral norm is known, and generalizes previous approaches which achieve optimal scaling, but with respect to more restrictive parameters. By exploiting knowledge of the spectral norm, our algorithm solves the black-box unitary implementation problem – O(d1/2+o(1)) queries suffice to approximate any d-sparse unitary in the black-box setting, which matches the quantum search lower bound of Ω(√d) queries and improves upon prior art [Berry and Childs, QIP 2010] of Õ(d2/3) queries. Combined with known techniques, we also solve systems of sparse linear equations with condition number κ using O((κ √d)1+o(1)/єo(1)) queries, which is a quadratic improvement in sparsity.