Improved parameter initialization for the (local) unitary cluster Jastrow ansatz

TL;DR

This work tackles the initialization challenge of the UCJ/LUCJ ansatz for quantum chemistry on near-term devices by introducing two complementary strategies: compressed double factorization of CCSD $t_2$ amplitudes to recover CCSD fidelity under truncation and locality constraints, and tensor-network–based surrogate optimization to refine sampling-based algorithm parameters. The methods are validated across extensive classical simulations and IBM quantum hardware, showing substantial improvements in QSCI and, in many cases, in VQE energies as well. The results demonstrate that carefully designed initialization and optimization pipelines can significantly boost the performance of variational quantum algorithms for chemistry under hardware limits, with practical benefits for both expectation-value and sampling-based approaches. The authors also provide open-source code and data to enable reproducibility and further development in this area.

Abstract

The unitary cluster Jastrow (UCJ) ansatz and its variant known as local UCJ (LUCJ) are promising choices for variational quantum algorithms for chemistry due to their combination of physical motivation and hardware efficiency. The parameters of these ansatzes can be initialized from the output of a coupled cluster, singles and doubles (CCSD) calculation performed on a classical computer. However, truncating the number of repetitions of the ansatz, as well as discarding interactions to accommodate the connectivity constraints of near-term quantum processors, degrade the approximation to CCSD and the resulting energy accuracy. In this work, we propose two methods to improve the parameter initialization. The first method, which is applicable to both expectation value- and sample-based algorithms, uses compressed double factorization of the CCSD amplitudes to improve or recover the CCSD approximation. The second method, which is applicable to sample-based algorithms, uses approximate tensor network simulation to improve the quality of samples produced by the ansatz circuit. We validate our methods using exact state vector simulation on systems of up to 52 qubits, as well as experiments on superconducting quantum processors using up to 65 qubits. Our results indicate that our methods can significantly improve the output of both expectation value- and sample-based quantum algorithms.

Figures (9)

• Figure 1: Potential energy curves and the effect of ansatz truncation. Data for N2 in a (10e, 16o) active space derived from the 6-31G basis set. The left column shows the "VQE energy", which is simply the expectation value of the Hamiltonian, and the right column shows the "QSCI energy" obtained by sampling configurations from the wavefunction and using them to run QSCI. For the relatively small N2/6-31G system, the variance of the QSCI energy was smaller than numerical precision with our experimental parameters (see Section \ref{['sec:computational_details']}), so error bars are not needed. Top row: Potential energy curves obtained by taking the CCSD $t$-amplitudes and plugging them into UCCSD and UCJ. Energy values for full configuration interaction (FCI), CCSD, and CISD are shown for reference. Middle row: Absolute energy error, computed as deviation from FCI. The horizontal line indicates 1.6 milliHartrees, a commonly cited accuracy target for chemistry. Bottom row: Energy error as a function of the number of repetitions retained in the ansatz, for bond length 1.2 Å. Data is shown for the UCJ ansatz, as well as variants of the LUCJ ansatz designed for square lattice and heavy-hex lattice qubit connectivity.
• Figure 2: Double-factorization loss function and multi-stage optimization. Data for the (10e, 16o) N2 system at bond length 1.2 Å. The left column shows data for the UCJ ansatz, and the right column shows data for the LUCJ ansatz with heavy-hex connectivity. Top row: Double-factorization final loss function value as a function of the number of ansatz repetitions, for naive truncation, compressed double-factorization, and compressed double-factorization without multi-stage optimization. Middle row: VQE energy error. Here, compressed double-factorization does not improve the VQE energy due to Trotter error, but this can be fixed by adding a regularization term to the loss function (see Fig. \ref{['fig:regularization']}). The horizontal line indicates the energy for the ansatz with the full untruncated number of repetitions. Bottom row: QSCI energy error. Compressed double-factorization does consistently improve the QSCI energy.
• Figure 3: Effect of adding an operator norm regularization term to the loss function. Data for the (10e, 16o) N2 system with the UCJ ansatz. The left column shows data for bond length 1.2 Å, and the right column shows data for bond length 2.4 Å. Top row: Double-factorization final loss function value as a function of the number of ansatz repetitions, for naive truncation, compressed double-factorization, and compressed double-factorization with regularization. Middle row: VQE energy error. With regularization, compressed double-factorization usually improves the VQE energy. The horizontal line indicates the energy for the ansatz with the full untruncated number of repetitions. Bottom row: QSCI energy error. For QSCI, regularization is not needed and can actually increase the energy error.
• Figure 4: The compressed operator wavefunctions are less concentrated and yield more unique configurations when sampled. Data for the (10e, 16o) N2 system at bond length 1.2 Å. The left column shows data for the UCJ ansatz, and the right column shows data for the LUCJ ansatz with heavy-hex connectivity. Top row: Entropy of the probability distribution associated with the state vector as a function of the number of ansatz repetitions, for naive truncation, compressed double-factorization, and compressed double-factorization with regularization. Middle row: Square root of the dimension of the subspace used for QSCI. The configurations of the subspace are formed by a Cartesian product, using the same single-spin configurations for both spin up and spin down. Generally, a higher-entropy distribution yields more unique configurations, and thus a larger subspace. The horizontal line indicates the entropy for the ansatz with the full untruncated number of repetitions. Bottom row: QSCI energy error. Generally, a higher subspace dimension yields better energy estimates. Compressed double-factorization can outperform even the full untruncated ansatz because the state vector can have higher entropy and produce more unique configurations when sampled.
• Figure 5: Compressed double factorization simulation results for N2 in a (10e, 26o) active space derived from the cc-pVDZ basis set. Data for UCJ (left column) and LUCJ heavy-hex (right column), and bond lengths 1.2 Å (top row) and 2.4 Å (bottom row). The top panel of each plot shows the QSCI energy error, and the bottom panel shows the square root of the dimension of the subspace used for QSCI. For each data point, we sampled 100,000 bitstrings from the state vector, and then subsampled 10 batches of 4,000 bitstrings uniformly at random from which to construct the QSCI subspace. The data point shows the average value from the 10 batches, with error bars indicating the minimum and maximum values (when not visible, error bars are smaller than symbol sizes. ).