Black-box Hamiltonian simulation and unitary implementation

TL;DR

This paper introduces efficient quantum algorithms for simulating sparse Hamiltonians and implementing black-box unitaries, significantly improving complexity bounds over previous methods and approaching theoretical optimality.

Contribution

It presents the first linear-in-time quantum walk-based simulation for sparse Hamiltonians and a sub-quadratic query complexity method for black-box unitary implementation.

Findings

Sparse Hamiltonian simulation with linear complexity in sparseness D and time t.

Black-box unitary implementation with O(N^{2/3} (log log N)^{4/3}) queries.

Most unitaries can be implemented with O(√N) queries, near optimal.

Abstract

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 O(N^2) elementary gates, we show that a black-box unitary can be performed with bounded error using O(N^{2/3} (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(sqrt{N}) queries, which is optimal.