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Linear combination of unitaries with exponential convergence

Peter Brearley, Thomas Howarth

TL;DR

The paper addresses efficiently simulating non-unitary maps on quantum hardware by decomposing any operator into a linear combination of unitaries with exponential error decay. It introduces Fourier-LCU, which uses a smooth Fourier extension of $f(\tau)=\tau$ to construct a sine-series-based Hermitian/anti-Hermitian decomposition, yielding a sum of $4m$ unitaries with error shrinking as $m$ grows, and subnormalisation scaling only as $\alpha = O(\log\log(1/\epsilon))$. The authors present a concrete quantum circuit for block-encoding the target operator with complexity governed by $m$, $s$ (sparsity), and the norm $\|A\|_2$, and discuss coefficient-selection strategies that further minimize $\alpha$ via least-squares or regularised fitting, forming a Pareto front between fidelity and resource cost. Numerical simulations on a Lindblad open-system model substantiate exponential convergence and demonstrate practical subnormalisation reductions from ~5 to ~2 with regularised coefficients, while preserving fidelity. The framework offers a versatile and broadly applicable path to non-unitary quantum computing and to QSVT, contingent on implementing efficient underlying Hamiltonian simulations.

Abstract

We present a general method for decomposing non-unitary operators into a linear combination of unitary operators, where the approximation error decays exponentially. The decomposition is based on a smooth periodic extension of the identity map via the Fourier extension method, resulting in a sine series with exponentially decaying coefficients. Rewriting the sine series in terms of complex exponentials, then evaluating it on the Hermitian and anti-Hermitian parts of a non-unitary operator, yields its approximation by a linear combination of unitaries. When implemented in a quantum circuit, the subnormalisation of the resulting block encoding scales with the double logarithm of the inverse error, substantially improving over the polynomial relationship in existing methods. For hardware or applications with a fixed error budget, we discuss a strategy to minimise subnormalisation by exploiting the overcomplete nature of the Fourier extension basis. This regularisation procedure traces an error-subnormalisation Pareto front, identifying coefficients that maximise the subnormalisation at a fixed error budget. Fourier linear combinations of unitaries thus provides an accurate and versatile framework for non-unitary quantum computing.

Linear combination of unitaries with exponential convergence

TL;DR

The paper addresses efficiently simulating non-unitary maps on quantum hardware by decomposing any operator into a linear combination of unitaries with exponential error decay. It introduces Fourier-LCU, which uses a smooth Fourier extension of to construct a sine-series-based Hermitian/anti-Hermitian decomposition, yielding a sum of unitaries with error shrinking as grows, and subnormalisation scaling only as . The authors present a concrete quantum circuit for block-encoding the target operator with complexity governed by , (sparsity), and the norm , and discuss coefficient-selection strategies that further minimize via least-squares or regularised fitting, forming a Pareto front between fidelity and resource cost. Numerical simulations on a Lindblad open-system model substantiate exponential convergence and demonstrate practical subnormalisation reductions from ~5 to ~2 with regularised coefficients, while preserving fidelity. The framework offers a versatile and broadly applicable path to non-unitary quantum computing and to QSVT, contingent on implementing efficient underlying Hamiltonian simulations.

Abstract

We present a general method for decomposing non-unitary operators into a linear combination of unitary operators, where the approximation error decays exponentially. The decomposition is based on a smooth periodic extension of the identity map via the Fourier extension method, resulting in a sine series with exponentially decaying coefficients. Rewriting the sine series in terms of complex exponentials, then evaluating it on the Hermitian and anti-Hermitian parts of a non-unitary operator, yields its approximation by a linear combination of unitaries. When implemented in a quantum circuit, the subnormalisation of the resulting block encoding scales with the double logarithm of the inverse error, substantially improving over the polynomial relationship in existing methods. For hardware or applications with a fixed error budget, we discuss a strategy to minimise subnormalisation by exploiting the overcomplete nature of the Fourier extension basis. This regularisation procedure traces an error-subnormalisation Pareto front, identifying coefficients that maximise the subnormalisation at a fixed error budget. Fourier linear combinations of unitaries thus provides an accurate and versatile framework for non-unitary quantum computing.
Paper Structure (9 sections, 3 theorems, 28 equations, 4 figures, 2 tables)

This paper contains 9 sections, 3 theorems, 28 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Fourier-LCU Decomposition
  3. Quantum Circuit Implementation
  4. Coefficient Selection Strategy
  5. Exponential convergence via least squares
  6. Optimal subnormalisation via regularised fitting
  7. Resource Requirements
  8. Numerical Examples
  9. Conclusions

Key Result

Lemma 1

Let $J_{m}^{\star}(\lambda; \eta)$ be the minimum value of the cost function $J$ for a dictionary of size $m$, and let $\mathbf{a}_{m}^{\star}$ be the corresponding optimal coefficient vector. For a fixed regularisation parameter $\lambda > 0$ and extension parameter $\eta > 1$, the sequence of opti

Figures (4)

  • Figure 1: Quantum circuit for Fourier LCU, implementing the decomposition in Eq. \ref{['eq:fourier_lcu']}. The unitaries $V$ and $W$ are defined in Eqs. \ref{['eq:V_definition']} and \ref{['eq:W_definition']} respectively.
  • Figure 2: (a) Fourier extension sine series to approximate $f(\tau)=\tau$ by Eq. \ref{['eq:sine_series']} on the subinterval $\tau\in [-\pi/\eta,\pi/\eta]$, corresponding to $\eta=2+0.460m^{-0.319}$, using the coefficients given in Table \ref{['tab:exponential_coefficients']}. (b) Error $L^2$ norm on $[-\pi/\eta,\pi/\eta]$. (c) Block-encoding subnormalisation given by Eq. \ref{['eq:alpha']}. (d) Optimal extension factor $\eta$ to minimise $m\alpha$ given by Eq. \ref{['eq:eta_star']}.
  • Figure 3: Left and centre: The relationship between $\lambda$, $\epsilon$, and $\alpha$. As $\lambda \to 0$, the solver trades subnormalisation for higher precision. Right: The Pareto front. The horizontal lines show the cost of standard least-squares fits $\mathbf{a}^{\text{LS}}$. The vertical distance to the regularised curves represents the subnormalisation improvement, with the flatlining behaviour demonstrating convergence to the stable limit $\alpha^{\star}_{\infty}$.
  • Figure 4: Comparison of the coefficient selection strategies in Section \ref{['sec:coefficient_selection_strategy']} using statevector simulations. (a) Error $L^2$ norm $\| \ket{\rho(t)} - \vec{\rho}(t) \|_2$ between the computed statevector $\ket{\rho(t)}$ and the analytical solution $\vec{\rho}(t)$ rescaled to norm 1, against the number of terms in the Fourier series $m$, where the number of unitaries in the LCU is $4m$. (b) The subnormalisation $\alpha$ of the resulting block encoding, consistent with the definition in Eq. \ref{['eq:alpha']}. (c) LCU cost quantified by $\alpha m$.

Theorems & Definitions (7)

  • Definition 1: Block encoding
  • Lemma 1: Monotonicity of the joint functional $J$
  • proof
  • Corollary 1.1: Monotonicity of $\alpha^{\star}$ for a fixed error budget
  • proof
  • Corollary 1.2: Existence of $\alpha_{\infty}^{\star}$
  • proof