Efficient quantum state preparation of multivariate functions using tensor networks

TL;DR

It is shown that paradigmatic multivariate functions can be accurately prepared such as a 17-dimensional Gaussian encoded in the state of 102 qubits and, through experiments, a 9-dimensional Gaussian realized using 54 qubits on Quantinuum's H2 quantum processor.

Abstract

For the preparation of high-dimensional functions on quantum computers, we introduce tensor network algorithms that are efficient with regard to dimensionality, optimize circuits composed of hardware-native gates and take gate errors into account during the optimization. To avoid the notorious barren plateau problem of vanishing gradients in the circuit optimization, we smoothly transform the circuit from an easy-to-prepare initial function into the desired target function. We show that paradigmatic multivariate functions can be accurately prepared such as, by numerical simulations, a 17-dimensional Gaussian encoded in the state of 102 qubits and, through experiments, a 9-dimensional Gaussian realized using 54 qubits on Quantinuum's H2 quantum processor.

Figures (6)

• Figure 1: The proposed IQSP algorithm. To prepare a desired target function $F(\bm{x})$ on a quantum computer, IQSP uses a sequence of smoothly connected functions $f(\bm{x}, \lambda_{k})$ where $k \in \{0, 1, \dots, K\}$ and $f(\bm{x}, \lambda_{0})$ is easy to prepare and $f(\bm{x}, \lambda_{K}) = F(\bm{x})$. For each value of $\lambda_{k}$, IQSP first creates a TN representation of $f(\bm{x}, \lambda_{k})$ using TCI and then optimizes a PQC to maximize the fidelity with that TN. The IQSP algorithm warm-starts the training for each $k > 0$ using the optimal parameters from the preceding step $k-1$, which can ensure large gradients throughout the circuit optimization. Shown are a comb TN for $d$ variables as well as a PQC for $d = 2$, as they are used in the paper. Here $\ket{+} = (\ket{0}+\ket{1}) / \sqrt{2}$ and the green circles/boxes represent variational tensors/unitaries.
• Figure 2: Gradient magnitudes in IQSP versus random parameter initialization. (a) Average gradient magnitude \ref{['eq:gradient']} versus overlap at the beginning of the optimization. Each data point represents a single realization and each dashed line the average over 100 random initializations. (b) Average gradient magnitude \ref{['eq:gradient']} as a function of the qubit count $n$, whereby the error bars range from the minimum to the maximum gradient magnitude of 100 random parameter choices.
• Figure 3: Noise-aware IQSP optimization. Infidelity \ref{['eq:infid']} after noise-unaware (NU) and noise-aware (NA) optimization, evaluated with (w/) and without (w/o) noise, for a four-dimensional Gaussian realized using 24 qubits.
• Figure 4: Covariances \ref{['eq:covs']} measured on H2-2. We create the 9-dimensional Gaussian with covariance matrix \ref{['eq:gaus_quadr']} using $54$ qubits. The PQC is obtained by noise-aware (NA) optimization and is simulated without (w/o) noise and run on H2-2. The error bars represent 2 standard deviations. The noiseless results (empty circles) are estimated using $n_{\text{shots}} = 10^4$ and the experimental results on H2 (crosses) using $n_{\text{shots}} = 1024$.
• Figure 5: Error \ref{['eq:err_max']} as a function of two-qubit gate count. (a) We consider the Ricker wavelet and compare to the Chebyshev polynomial approach of rosenkranz_2025. (b) We consider the Student's t-distribution and compare to the Fourier series approach of rosenkranz_2025.