$σ$-VQE: Excited-state preparation of quantum many-body scars with shallow circuits

Abstract

We present and benchmark a type of variational quantum eigensolver (VQE), which we denote the $σ$-VQE. It is designed to target mid-spectrum eigenstates and prepare quantum many-body scar states. The approach leverages the fact that noisy intermediate-scale quantum devices are limited in their ability to generate generic highly-entangled states. This modified VQE pairs a low-depth circuit with an energy-selective objective that explicitly penalizes energy variance around a chosen target energy. The cost function exploits the limited expressibility of the shallow circuit as atypical low-entanglement eigenstates such as scar states are preferentially selected. We validate this mechanism across two complementary families of models that contain many-body scar states: the Shiraishi-Mori embedding approach, and the matrix-product state parent Hamiltonian construction. We define an unbiased estimation scheme for the nonlinear cost function that is compatible with qubit-wise commuting grouping and bitstring reuse. A proof-of-principle demonstration using a small-system instance was carried out on IBM Fez (Heron r2 QPU). These results motivate its use both as a practical "scar detector" and as a state-preparation primitive for initializing nonthermal eigenstate-supported dynamics.

Figures (8)

• Figure 1: Hardware-efficient ansatz used throughout this work. Each layer applies single-qubit rotations on all qubits followed by a pattern of two-qubit entangling gates (CZ) on a nearest-neighbor chain. The circuit depth denotes the number of such layers.
• Figure 2: We plot the bipartite entanglement entropy, $\mathcal{S}$ of eigenstates for our benchmark Hamiltonians. The scar states are circled in red. (a) The $N=9$ Shiraishi--Mori (SM) embedding yields a product-state scar at $E=0$. (b) The $N=8$ parent Hamiltonian construction without an embedded scar. (c)--(e) The $N=8$ parent Hamiltonian construction with an embedded MPS scar of bond dimension $\chi=1,2,3$ respectively. Increasing $\chi$ increases the scar entanglement and moves the scar state closer to the thermal band.
• Figure 3: $\sigma$-VQE performance when searching for the SM-embedded non-integrable $N=9$ open XXZ chain. The results shown are from setting the target energy of the cost function to 0, the energy of the scar state. We use ADAM with exact (statevector) evaluation of all expectation values and parameter-shift gradients. We simulate using 1 to 6 ansatz layers. We plot the infidelity ($1-\mathcal{F}$) of the circuit state with the scar state vs iteration of the algorithm. We observe that even a relatively shallow depth of $2$ is enough to achieve significant overlap with the scar state in this setting.
• Figure 4: We run the $\sigma$-VQE for the $N=9$ Shiraishi-Mori embedded chaotic XXZ model, now varying the target energy that is set in the cost function, Eq. \ref{['eq:cost']}. We run the $\sigma$-VQE routine for $300$ iterations with an ansatz depth of 3. We observe that the algorithm is only successful in reducing the cost function significantly when we specifically target the scar. Eigenstates at other target energies require more entanglement to be generated, and therefore are out of reach of the low-depth circuit. We additionally run the VQE for a control case of the chaotic XXZ model without an embedded scar. The shallow ansatz cannot significantly lower the cost function at all in the control case.
• Figure 5: We simulate the $\sigma$-VQE with a depth of $3$ for the $N=9$ open XXZ chain with a Shiraishi-Mori random-product QMBS state \ref{['eq:smxxz']} with the number of shots used per PSR optimization iteration being $S\in\{10^6,10^7,10^8\}$.