The efficient implementation of matrix arithmetic operations underpins the speedups of many quantum algorithms. Standard circuit constructions rely on ancilla qubits and multi-qubit controlled gates, which do not align well with the capabilities of quantum devices expected in the foreseeable future.

We develop a suite of methods to perform matrix arithmetics—with the result encoded in the off-diagonal blocks of a Hamiltonian—using Hamiltonian evolutions of input operators. We show how to maintain this Hamiltonian block encoding after specifying all its entries, so that matrix operations can be composed one after another and the entire quantum computation takes at most two ancilla qubits. We achieve this for matrix multiplication, matrix addition, matrix inversion, Hermitian conjugation, fractional scaling, integer scaling, complex phase scaling, and singular value transformation for both odd and even polynomials. We also present an overlap estimation algorithm to extract classical properties of Hamiltonian-block-encoded operators, analogous to the well-known Hadamard test, at no extra qubit cost. Our Hamiltonian matrix multiplication uses the Lie group commutator product formula and its higher-order general¬izations due to Childs and Wiebe. We prove a concrete error bound exactly matching the Baker–Campbell–Hausdorff series to third order, which is provably tight up to a single application of the triangle inequality. Our Hamiltonian singular value transformation employs a dominated polynomial approximation, where the approximation holds within the domain

of interest while the constructed polynomial is upper bounded by the target function over the entire unit interval.

When applied to quantum simulation, our methods inherit the commutator scaling of conventional product formulas and leverage the power of matrix arithmetics to reduce the cost of each simulation step. To illustrate this feature, we describe a circuit for simulating a class of sum-of-squares Hamiltonians, attaining a commutator scaling in step count while the gate cost per step remains comparable to that of more advanced algorithms. In particular, we apply this to doubly factorized tensor-hypercontracted Hamiltonians from recent studies of quantum chemistry, obtaining further improvements for initial states with a fixed number of particles. We achieve this with one ancilla qubit. This is joint work with Yuan Su.