Improving Ground State Accuracy of Variational Quantum Eigensolvers with Soft-coded Orthogonal Subspace Representations

TL;DR

The paper tackles the challenge of accurately estimating ground states with VQE under NISQ constraints by introducing soft-coded orthogonal subspace representations, where orthogonality among the subspace states is enforced via a penalty term rather than at the circuit level. After optimizing the subspace, the Hamiltonian restricted to the subspace is diagonalized (or solved via a subspace VQE), with off-diagonal elements and overlaps measured through Hadamard-test circuits to form a generalized eigenproblem. Benchmark results on a 3x3 transverse-field Ising model and a 4x4 Edwards–Anderson spin glass show that soft-coded orthogonality yields substantially higher final fidelities and robust convergence with shallower circuits compared to standard VQE and hard-orthogonal subspaces. The findings suggest that soft-coded subspace representations are particularly advantageous for NISQ-era quantum simulations and motivate future work on expressibility metrics, noise resilience, and adaptive subspace strategies.

Abstract

We propose a new approach to improve the accuracy of ground state estimates in Variational Quantum Eigensolver (VQE) algorithms by employing subspace representations with soft-coded orthogonality constraints. As in other subspace-based VQE methods, such as the Subspace-Search VQE (SSVQE) and Multistate Contracted VQE (MCVQE), once the parameters are optimized to maximize the subspace overlap with the low-energy sector of the Hamiltonian, one diagonalizes the Hamiltonian restricted to the subspace. Unlike these methods, where \emph{hard-coded} orthogonality constraints are enforced at the circuit level among the states spanning the subspace, we consider a subspace representation where orthogonality is \emph{soft-coded} via penalty terms in the cost function. We show that this representation allows for shallower quantum circuits while maintaining high fidelity when compared to single-state (standard VQE) and multi-state (SSVQE or MCVQE) representations, on two benchmark cases: a $3\times 3$ transverse-field Ising model and random realizations of the Edwards--Anderson spin-glass model on a $4\times 4$ lattice.

Figures (10)

• Figure 1: Generalized Hadamard test circuits for the estimate of real and imaginary parts of off-diagonal matrix elements between pure states $\ket{\psi_p}:=U_p \ket{0}$ and $\ket{\psi_q}:=U_q \ket{0}$. At the end of the circuit, the system register is either traced out (ignored) for the estimate of the overlap matrix elements $S_{pq}$, or measured to estimate of the Hamiltonian matrix elements $(H_{\rm trc})_{pq}$ (see text). The ancilla and system registers are labeled by '$a$' and '$s$' respectively.
• Figure 2: General structure of the parameterized circuit representing $U(\bm \theta)$ in all cases considered. The number of single-qubit independent rotation layers ( Rot) is $N_l\geq 1$. For $N_l\geq 2$, these are alternated with $N_l-1$ layers of entangling gates (e.g., CNOT, CZ).
• Figure 3: Subspace cost as function of the ratio between optimizer iterations and parameters for both hard-ortho (top row) and soft-ortho (bottom row) and for subspace dimension $K \in \{2,4,8\}$ for the 2D transverse-field Ising model discussed in the text. In each panel, the blue, orange and green curves (solid, dashed and dotted, respectively) correspond to ansätze for three selected depths: $N_l\in \{4,6,8\}$. The bottom lines indicate the best costs over 10 repeated independent runs with different random initializations, while the corresponding shaded bands indicate the spread of the cost values from the best run up to the 25th, 50th and 75th percentiles across runs. All runs are stopped at 1500 iterations.
• Figure 4: Behavior of the fidelity $\mathcal{F}_{\rm trc}$ between the exact ground state and the variational ground state candidate as a function of optimization steps for standard VQE ($K=1$), hard-ortho and soft-ortho subspace representations with subspace dimensions $K=2,4,8$ for the $3\times 3$ transverse-field Ising model. All ansätze have $N_l=4$ layers as described in the text. Solid lines show the best fidelity across 10 independent runs at each iteration, dashed lines indicate the median, and shaded bands indicate the spread of the fidelity values from the best run up to the 25th, 50th and 75th percentiles (darker to lighter).
• Figure 5: Scatter plot of the infidelity $1 - \mathcal{F}_{\rm trc}$ versus the normalized cost $\widetilde{C}_K$ (Eq. \ref{['eq:normalized-cost']}) for standard VQE (top left), hard-ortho (top center and right, $K=2,4$) and soft-ortho (bottom center and right, $K=2,4$) representations on the $3 \times 3$ transverse-field Ising model. All ansätze have 144 trainable parameters. Each point corresponds to one optimizer iteration step in one of the 10 independent runs; the color indicates the iteration number (see colorbar). For each run, we fit a linear relation $1 - \mathcal{F}_{\rm trc} \simeq m \widetilde{C} + q$, showing the average slope $m$ of the fits across all 10 runs as a dashed orange line. The bottom-left panel shows these averages $m$ (with standard error for the errorbars).