Quantum matrix arithmetics with Hamiltonian evolution

2025-10-07

AI Summary

This work addresses the efficient implementation of fundamental matrix arithmetic operations—multiplication, addition, inversion, Hermitian conjugation, scaling, and parity-preserving polynomial singular value transformations—in quantum computation. We propose a unified framework based on Hamiltonian block encoding, wherein the result of each operation is encoded in an off-diagonal block of a Hamiltonian, requiring at most two ancillary qubits (optimally just one), thereby significantly reducing hardware overhead. Key contributions include: (1) low-depth matrix multiplication via Lie group commutators and their higher-order generalizations; (2) controlled polynomial approximation enabling arbitrary parity-preserving polynomial singular value transformations; and (3) an overlap estimation algorithm that requires no additional qubits, ensuring efficient information extraction and composability. Applied to quantum chemical Hamiltonian simulation via two-factor decomposition and tensor compression, our method simultaneously optimizes both the number of evolution steps and the cost per step.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Sampling/Simulation-based Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

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 $ extit{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.

Problem

• Performing matrix arithmetic operations using Hamiltonian evolution of input operators
• Maintaining Hamiltonian block encoding for composing matrix operations with minimal ancilla qubits
• Developing methods for matrix multiplication, inversion, addition, and singular value transformation

Innovation

• Hamiltonian evolution encodes matrix operations
• Block encoding enables composable matrix arithmetic
• Ancilla-efficient quantum computation with minimal qubits