Linear combination of unitaries with exponential convergence

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.