Quantum-Inspired Algorithms beyond Unitary Circuits: the Laplace Transform

Abstract:Quantum-inspired algorithms can deliver substantial speedups over classical state-of-the-art methods by executing quantum algorithms with tensor networks on conventional hardware. Unlike circuit models restricted to unitary gates, tensor networks naturally accommodate non-unitary maps. This flexibility lets us design quantum-inspired methods that start from a quantum algorithmic structure, yet go beyond unitarity to achieve speedups. Here we introduce a tensor-network approach to compute the discrete Laplace transform, a non-unitary, aperiodic transform (in contrast to the Fourier transform). We encode a length-$N$ signal on two paired $n$-qubit registers and decompose the overall map into a non-unitary exponential Damping Transform followed by a Quantum Fourier Transform, both compressed in a single matrix-product operator. This decomposition admits strong MPO compression to low bond dimension resulting in significant acceleration. We demonstrate simulations up to $N=2^{30}$ input data points, with up to $2^{60}$ output data points, and quantify how bond dimension controls runtime and accuracy, including precise and efficient pole identification.

Quantum Physics (quant-ph); Mathematical Physics (math-ph); Data Analysis, Statistics and Probability (physics.data-an)

arXiv:2601.17724 [quant-ph]

(or arXiv:2601.17724v3 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2601.17724 Focus to learn more arXiv-issued DOI via DataCite

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