Hamiltonian Simulation by Qubitization

We present the problem of approximating the time-evolution operator $e^{-i\hat{H}t}$ to error $\epsilon$, where the Hamiltonian $\hat{H}=(\langle G|\otimes\hat{\mathcal{I}})\hat{U}(|G\rangle\otimes\hat{\mathcal{I}})$ is the projection of a unitary oracle $\hat{U}$ onto the state $|G\rangle$ created by another unitary oracle. Our algorithm solves this with a query complexity $\mathcal{O}\big(t+\log({1/\epsilon})\big)$ to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are $d$-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where $\hat{H}$ is a density matrix. A key technical result is `qubitization', which uses the controlled version of these oracles to embed any $\hat{H}$ in an invariant $\text{SU}(2)$ subspace. A large class of operator functions of $\hat{H}$ can then be computed with optimal query complexity, of which $e^{-i\hat{H}t}$ is a special case.Comment: 23 pages, 1 figure; v2: updated notation; v3: accepted versio

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 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.Comment: 19 pages, 3 figures, minor correction

Hamiltonian Simulation Using Linear Combinations of Unitary Operations

We present a new approach to simulating Hamiltonian dynamics based on implementing linear combinations of unitary operations rather than products of unitary operations. The resulting algorithm has superior performance to existing simulation algorithms based on product formulas and, most notably, scales better with the simulation error than any known Hamiltonian simulation technique. Our main tool is a general method to nearly deterministically implement linear combinations of nearby unitary operations, which we show is optimal among a large class of methods.Comment: 18 pages, 3 figure