A quantum eigenvalue solver based on tensor networks

TL;DR

The paper addresses the challenge of accurately computing electronic ground-state energies for strongly correlated systems, which is difficult for classical methods due to extensive configuration spaces. It introduces the tensor network quantum eigensolver (TNQE), a hybrid quantum-classical approach that forms a multi-reference wavefunction as a sum of matrix product states (MPS) in rotated orbital bases, and optimizes it via a gradient-free quantum subspace diagonalization framework with a generalized sweep. The key results show chemical accuracy for stretched H$_2$O and H$_6$ in STO-3G using modest bond dimensions and subspace sizes, with substantially lower quantum-resource estimates and strong tolerance to shot noise compared to VQE-UCCSD; this points to a practical near-term quantum advantage for certain strongly correlated systems and geometries. The work highlights a scalable route to extend tensor-network based chemistry to higher connectivities and complex geometries, leveraging compact, linear-depth quantum circuits and a flexible multi-reference ansatz, while identifying open questions about scaling, encoding costs, and error mitigation essential for large-scale implementation.

Abstract

Electronic ground states are of central importance in chemical simulations, but have remained beyond the reach of efficient classical algorithms except in cases of weak electron correlation or one-dimensional spatial geometry. We introduce a hybrid quantum-classical eigenvalue solver that constructs a wavefunction ansatz from a linear combination of matrix product states in rotated orbital bases, enabling the characterization of strongly correlated ground states with arbitrary spatial geometry. The energy is converged via a gradient-free generalized sweep algorithm based on quantum subspace diagonalization, with a potentially exponential speedup in the off-diagonal matrix element contractions upon translation into compact quantum circuits of linear depth in the number of qubits. Chemical accuracy is attained in numerical experiments for both a stretched water molecule and an octahedral arrangement of hydrogen atoms, achieving substantially better correlation energies compared to a unitary coupled-cluster benchmark, with orders of magnitude reductions in quantum resource estimates and a surprisingly high tolerance to shot noise. This proof-of-concept study suggests a promising new avenue for scaling up simulations of strongly correlated chemical systems on near-term quantum hardware.

Figures (15)

• Figure 1: A tensor network diagram depicting the tensor-train (TT) factorization of the FCI coefficient tensor in Eq. \ref{['eq:fci']}, resulting in a matrix product state defined in Eq. \ref{['eq:mps']}.
• Figure 2: Tensor network diagrams depicting a matrix product state in canonical form. The black tensor denotes the canonical center in each case. The arrows indicate contraction (a.k.a. 'bubbling') of the highlighted tensors. (a) The canonical center (diamond-shaped tensor) is a diagonal matrix of singular values corresponding to the Schmidt decomposition over partitions $A$ (top) and $B$ (bottom) as in Eq. \ref{['eq:schmidt']}. (b) The canonical center is contracted into the single-site tensor at site $p$ (in $A$ for this example). (c) The canonical center is contracted into the two-site tensor at sites $p$ and $p+1$.
• Figure 3: A molecular orbital graph indicating all possible tensor network bond indices between $N=6$ spin-orbital sites. The solid lines denote the bond indices of a matrix product state as defined in Eq. \ref{['eq:mps']}. The restricted Hartree-Fock orbitals of the H$_3^+$ cation in the STO-3G basis set are provided as a minimal example of a closed-shell molecule with six spin-orbital sites.
• Figure 4: Graph representations of matrix product states. (a) MPSs $\ket{\phi_j}$ and $\ket{\phi_i}$ with different site orderings over the same set of $N=6$ molecular orbitals, and the superposition state $c_i\ket{\phi_i}+c_j\ket{\phi_j}$. (b) A sequence of FSWAP operations $\hat{F}_{ij}$ transforms the MPS $\ket{\phi_i}$ into the orbital ordering of $\ket{\phi_j}$, giving rise to the tensor network in Fig. \ref{['fig:mat_el']}a.
• Figure 5: Tensor networks to compute the off-diagonal matrix element $\braket{\phi_i|\hat{H}|\phi_j}$ between matrix product states with permuted or rotated orbitals, given an efficient MPO representation of $\hat{H}$. (a) The MPSs are related by a permutation over the same set of orbitals. The FSWAPs, denoted by pairs of crossed tensors, are selected by a bubble sort comparator network. Each comparator applies an FSWAP to rearrange neighboring orbitals if their order does not match the order in which they appear on the right. The dashed outlines indicate the positions of non-swapping comparators. (b) The MPSs are related by an arbitrary orbital rotation $\hat{G}_{ij}$, which is decomposed into a pyramidal structure of Givens rotation tensors (see Eq. \ref{['eq:givens_decomp']}).