Quantum matrix arithmetics with Hamiltonian evolution

TL;DR

This work develops a comprehensive framework for matrix arithmetics on quantum computers by preserving a Hamiltonian block encoding $E_A=\exp(-i[0A^\dagger A0])$ of input matrices. By coherently maintaining block encodings, it enables a suite of primitives—matrix multiplication, addition, inversion, Hermitian conjugation, and fractional/complex-phase scaling—together with singular value transformation (QSVT) for both odd and even polynomials, all with at most two ancilla qubits and composable circuit structure. A dominated-polynomial approach to Hamiltonian QSVT is introduced to realize $f_{sv}(A)$ within a block-encoded framework, along with efficient overlap and Green’s-function estimation without extra qubits. The paper also demonstrates sum-of-squares Hamiltonian simulation, including a generic SOS scheme and a specialized SOS electronic-structure application (DFTHC), achieving commutator-scaling step costs and favorable ancilla usage. Collectively, these methods provide qubit-efficient, structurally transparent quantum-linear-algebra primitives with concrete applications to quantum chemistry and many-body physics, and suggest avenues for integrating with other quantum-simulation and linear-algebra techniques to further reduce resource overheads.

Abstract

The efficient implementation of matrix arithmetic operations underpins the speedups of many quantum algorithms. 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 $\textit{Hamiltonian block encoding}$, so that matrix operations can be composed one after another, and the entire quantum computation takes $\leq 2$ ancilla qubits. We achieve this for matrix multiplication, matrix addition, matrix inversion, Hermitian conjugation, fractional scaling, integer scaling, complex phase scaling, as well as 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 Hadmard test, at no extra cost of qubit. Our Hamiltonian matrix multiplication uses the Lie group commutator product formula and its higher-order generalizations due to Childs and Wiebe. 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. We describe a circuit for simulating a class of sum-of-squares Hamiltonians, attaining a commutator scaling in step count, while leveraging the power of matrix arithmetics to reduce the cost of each simulation step. In particular, we apply this to the 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 $1$ ancilla qubit.

Figures (13)

• Figure 1: Quantum circuit for Hamiltonian matrix multiplication. All circuit diagrams in the paper, including the present one, were prepared using Quantikz kay2018tutorial.
• Figure 2: Illustration of performing Hamiltonian QSVT for even functions via shifting and odd extension. Subfigure (a) illustrates some target function over a domain strictly enclosed by $[-\frac{\pi}{4}, \frac{\pi}{4}]$, with the desired approximation region shaded in blue. Subfigure (b) illustrates the constructed odd extension that approximates the initial even function on a shifted region.
• Figure 3: Quantum circuit for Hermitian conjugation.
• Figure 4: Quantum circuit for complex phase scaling ($R_z(\theta)=e^{-i\frac{\theta}{2}Z}$).
• Figure 5: Quantum circuit for integer scaling.

Theorems & Definitions (71)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3: Hermitian conjugation
• Proposition 4: Complex phase scaling
• Proposition 5: Integer scaling
• Proposition 6: Matrix addition
• Lemma 7: Perturbation of matrix exponential and matrix logarithm
• proof