Quantum-Inspired Algorithms beyond Unitary Circuits: the Laplace Transform

TL;DR

This work introduces a tensor-network approach to compute the discrete Laplace transform, a non-unitary, aperiodic transform (in contrast to the Fourier transform) and quantifies how bond dimension controls runtime and accuracy, including precise and efficient pole identification.

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.

Figures (6)

• Figure 1: (a) Decomposition of the z-transform into the damping transform $\hat{\mathrm{DT}}$, and the quantum Fourier transform $\hat{\mathrm{QFT}}$, shown in the paired-register layout $\ket{j}\ket{j'}$. The $\hat{\mathrm{DT}}$ consists of damping operations, defined in (c), separated into two steps. The solid blue box are damping operations acting on register $\ket{j}$, while the dashed purple box acts on both $\ket{j}\ket{j'}$, using register $\ket{j'}$ as controls. The $\hat{\mathrm{QFT}}$ acts only on register $\ket{j'}$ and uses unitary gates shown in (b). The controlled operations can be understood as sequential applications of controlled quantum gates, or as a joint MPO where identity operations are applied on wire crossings.
• Figure 2: Numerical results for the MPO representation of the $z$-Transform. a) Maximum MPO bond dimension $D$ for $\hat{\mathrm{DT}}$, $\hat{\mathrm{QFT}}$, and the composite $\hat{\mathrm{zT}}$ versus the number of output qubits $m = 2n=2\log_2(N)$ at cutoff $10^{-15}$ and $(\omega_r,\omega_i)=(2\pi,2\pi)$, indicating strong MPO compressibility. $N$ is the size of input data and $n$ is the number of input qubits. b) Error versus the inverse square root of the SVD cutoff, $1/\sqrt{\tau}$, on log–log axes for $m=20$ with a Gaussian random input $x_j \sim \mathcal{N}(0,1)$. Max and mean absolute errors, $\Delta_M$ and $\bar{\Delta}$ respectively, exhibit $\sqrt{\tau}$ scaling as confirmed by linear fits (dashed/dot–dashed). c) End-to-end runtime (t) of the MPO $z$-transform ($\tau = 10^{-15}$) versus system size $n$ (log-scale $y$-axis). “Full” includes the time required for encoding the signal into the initial MPS; “Core” does not include that cost. Curves for several input types are shown (see legend and Appendix \ref{['app:functions']}). For the random input, a tail fit of the "Full" timings (solid black) indicates runtime growing approximately linearly with $N=2^{n}$; the fitted relation is $t\sim 1.3\times 10^{-3} N + 10^{-3}$.
• Figure 3: Pole identification from the compressed MPS at $n=20$, accessing $2^{40}$ candidate samples without materializing the full grid. Color encodes normalized $|\chi|$ ($|\chi|/\max|\chi|$). Each sample is computed on a $(k,\ell)$ grid point and plotted at the corresponding $z$-plane location $z_{k,\ell}=e^{-s_{k,\ell}}$. (a) $|\chi(z)|$ on a coarse $(k\ell)$ grid (step $2^{12}$; $256\times256$ samples) reveals a peak near $z\approx 1$. (b) A fine scan centered at $z=1$ resolves two peaks at $z \approx 0.99999+0.00409\,{\rm i}$ and $z \approx 0.99996-0.00817\,{\rm i}$. Across the plotted samples, the max absolute error, $\Delta_M\lesssim 10^{-7}$.
• Figure C1: Circuit diagram of the damping transform $\hat{\mathrm{DT}}$, which forms the non-unitary part of the $z$-transform $\hat{\mathrm{zT}}$. Each two-qubit controlled gate and the "damping-Hadamard" in this circuit will be mapped to a local tensor to obtain an MPO.
• Figure C2: Singular-value spectra across all internal bonds for $n=30$. Panels (a), (b), and (c) show $\sigma_k$ versus index $k$ for $\hat{\mathrm{DT}}$, $\hat{\mathrm{QFT}}$, and $\hat{\mathrm{zT}}$, respectively, on a $\log_{10}$ y-axis with relative cutoff $\tau=10^{-15}$ and $(\omega_r,\omega_i)=(2\pi,2\pi)$. Each curve corresponds to the spectrum at a single bond.