arXiv subjects

Peter Brearley

Publications and source records attributed to Peter Brearley.

5 recordsLinked to original sources

arXiv2026

Fourier extensions for matrix-function block encodings with error-independent subnormalization bounds

Block encodings of non-unitary matrix functions are central to quantum numerical linear algebra. Hamiltonian simulation is a natural input model for Hermitian matrices, but accurate block encodings often incur large subnormalization. We decouple the accuracy from the subnormalization by formulating the matrix-function block encoding as a Fourier-extension approximation problem, yielding a linear combination of unitaries for Hermitian matrix inputs. Fourier extensions approximate non-periodic functions by a Fourier series on a larger periodic domain, creating redundant coefficients that can be optimized for their absolute sum, and hence subnormalization. The coefficients may be chosen for optimal subnormalization with algebraic convergence, by tuning the subnormalization bound for increasing rates of exponential convergence, or by Sobolev-regularized fitting to accommodate more general spectral sets. Fourier-extension block encodings apply to eigenvalue transforms of Hermitian matrices, or to odd singular-value transforms of general matrices, including as a quantum linear systems algorithm.

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arXiv2025

High-order splitting of non-unitary operators on quantum computers

Dissipation and irreversibility are central to many physical systems, yet they lead to non-unitary dynamics that are challenging to realise on quantum processors. High-order operator splitting is an attractive approach for simulating unitary dynamics, yet conventional product formulas introduce negative time steps at high orders that are ill-conditioned for dissipative dynamics. We show how block encodings of complex-coefficient product formulas can be constructed by a sequence of simple Hamiltonian evolutions in real and imaginary time with high-order accuracy. The unitary substages use positive real coefficients, while the dissipative substages use complex coefficients with positive real parts; the real parts preserve the contractive evolution, and the imaginary parts are additional unitary evolutions. We use known formulas of orders 4 and 6 and give a new symmetric formula of order 8 in this restricted class. We apply the approach to the classical problem of lossy mechanical wave propagation in statevector simulations and on a trapped-ion quantum processor. Orders 4 and 6 require comparable CNOT gates to orders 1 and 2 at large error tolerances, but quickly become more efficient as the tolerance is reduced. On the quantum hardware, a single step of order 4 achieves lower error than order 1 and comparable error to order 2, while order 6 is noise-limited. Our results show that operator splitting can combine practical circuit constructions with high-order convergence for non-unitary dissipative dynamics.

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arXiv2025

Spectral quantum algorithm for passive scalar transport in shear flows

The mixing of scalar substances in fluid flows by stirring and diffusion is ubiquitous in natural flows, chemical engineering, and microfluidic drug delivery. Here, we present a spectral quantum algorithm for scalar mixing by solving the advection-diffusion equation in a quantum computational fluid dynamics framework. The exact gate decompositions of the advection and diffusion operators in spectral space are derived. For all but the simplest one-dimensional flows, these operators do not commute. Therefore, we use operator splitting to construct quantum circuits capable of simulating arbitrary polynomial velocity profiles in multiple dimensions, such as the Blasius profile of a laminar boundary layer. Periodic, Neumann, and Dirichlet boundary conditions can be imposed with the appropriate quantum spectral transform. We evaluate the approach in statevector simulations of a Couette flow, plane Poiseuille flow, and a polynomial Blasius profile approximation. For an advection-diffusion problem in one dimension, we compare the time evolution of an ideal quantum simulation with those of real quantum computers with superconducting and trapped-ion qubits. The required number of two-qubit gates grows with the logarithm of the number of grid points raised to one higher power than the order of the polynomial velocity profile.

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arXiv2024

Quantum Algorithm for the Advection-Diffusion Equation by Direct Block Encoding of the Time-Marching Operator

A quantum algorithm for simulating multidimensional scalar transport problems using a time-marching strategy is presented. A direct unitary block encoding of the explicit time-marching operator is constructed, resulting in the intrinsic success probability of the squared solution norm without the need for amplitude amplification, thereby retaining a linear dependence on the simulation time. The algorithm separates the explicit time-marching operator into an advection-like component and a corrective shift operator. The advection-like component is mapped to a Hamiltonian simulation and combined with the shift operator through the linear combination of unitaries algorithm. State-vector simulations of a scalar transported in a steady two-dimensional Taylor-Green vortex support the theoretical findings.

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arXiv2023

Quantum Algorithm for Solving the Advection Equation using Hamiltonian Simulation

A quantum algorithm for solving the advection equation by embedding the discrete time-marching operator into Hamiltonian simulations is presented. One-dimensional advection can be simulated directly since the central finite difference operator for first-order derivatives is anti-Hermitian. Here, this is extended to industrially relevant, multi-dimensional flows with realistic boundary conditions and arbitrary finite difference stencils. A single copy of the initial quantum state is required and the circuit depth grows linearly with the required number of time steps, the sparsity of the time-marching operator and the inverse of the allowable error. Statevector simulations of a scalar transported in a two-dimensional channel flow and lid-driven cavity configuration are presented as a proof of concept of the proposed approach.

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