Simultaneous determination of multiple low-lying energy levels on a superconducting quantum processor

TL;DR

This work demonstrates the experimental realization of ancilla-entangled VQE (AEVQE) to simultaneously determine multiple low-lying eigenenergies and eigenstates on a superconducting quantum processor. By encoding $K=2^{N_a}$ target energies into maximally entangled ancilla–qubit pairs and optimizing with SPSA, the authors solve the ground and excited states of H$_2$ and transverse-field Ising models, obtaining energy curves and phase-transition indicators, while employing symmetry verification and readout mitigation to improve accuracy. The study analyzes how system size, optimizer choice, and hyperparameters affect optimization performance, and compares AEVQE to ancilla-free methods (weighted SSVQE and MCVQE) in terms of shot budgeting and diagonalization requirements, highlighting both efficiency gains and scalability challenges. The results establish experimental feasibility for AEVQE on public quantum platforms and offer guidance for applying VQE approaches to realistic problems, while noting bottlenecks such as barren plateaus and the need for improved architectures or approximate diagonalization strategies.

Abstract

Determining the ground and low-lying excited states is critical in numerous scenarios. Recent work has proposed the ancilla-entangled variational quantum eigensolver (AEVQE) that utilizes entanglement between ancilla and physical qubits to simultaneously tagert multiple low-lying energy levels. In this work, we report the experimental implementation of the AEVQE on a superconducting quantum cloud platform, demonstrating the full procedure of solving the low-lying energy levels of the H$_2$ molecule and the transverse-field Ising models (TFIMs). We obtain the potential energy curves of H$_2$ and show an indication of the ferromagnetic to paramagnetic phase transition in the TFIMs from the average absolute magnetization. Moreover, we investigate multiple factors that affect the algorithmic performance and provide a comparison with ancilla-free VQE algorithms. Our work demonstrates the experimental feasibility of the AEVQE algorithm and offers a guidance for the VQE approach in solving realistic problems on publicly-accessible quantum platforms.

Figures (7)

• Figure 1: Schematic diagram of AEVQE. The quantum circuit consists of ancilla qubits $a_i$ and physical qubits $p_i$. First, the ancilla and physical qubits are initialized to entangled states. In the optimization iterations, the variational circuits with parameters $\bm{\theta}$ are executed on the quantum processor. The measured results of the physical qubits (noted as bitstrings) are fed into the classical computer for searching $\bm{\theta}_{\text{opt}} = \text{arg}\mathop{\text{min}}\limits_{\bm{\theta}}\mathcal{L}(\bm{\theta})$. After optimization, $\bm{\theta}_{\text{opt}}$ is obtained and applied to calculate $H_{\text{sub}}$. Finally, unitary transformation $T$ is applied to diagonalize $H_{\text{sub}}$ and determine the eigenenergies and eigenstates.
• Figure 2: Calculation of the ground and first excited state energies $E_0$ and $E_1$ of the $\text{H}_{2}$ molecule. (a) The circuit schematic. The gray and orange areas represent the initialization and variational circuits, respectively. $H$ represents the Hadamard gate. $R_x$ and $R_z$ are the rotations along the $x$ and $z$ axes of the Bloch sphere, with the rotation angles to be optimized. The unitary transformation $T$ is applied to diagonalize $H_\text{sub}$ after the loss function convergence. (b) The optimization of the loss function at H-H bond distance $d = 0.6$ angstrom. The inset shows the result of the final ten iterations. (c) The experimental energy potentials $E_0$ (blue dots) and $E_1$ (orange dots) as a function of the H-H bond distance. The lines are the corresponding theoretical results. The error bars represent the standard error of the average energy. All energies are in the unit of Hartree.
• Figure 3: Calculation of the ground and first excited state energies $E_0$ and $E_1$ of the transverse field Ising models with one ancilla qubit. (a) The circuit schematic for the five-spin system. Decompositions of $U$, $W$, and $T$ are shown on the right. $S = R_z(\pi/2)$ is the phase gate and $\sqrt{-Y} = R_y(-\pi/2)$. (b) The optimization of the loss function for $h/J = 0.4$. The blue and orange lines represent the results of the three- and five-spin TFIMs, respectively. (c) The ground and first excited state energies ($E_0$ , $E_1$) of the three- and five-spin TFIMs obtained in experiment. The open circles and solid circles represent the raw energies and the symmetry-verified energies,respectively. The error bars represent the standard error of the average energy. (d) The experimental (dots) and theoretical (lines) average absolute magnetization $m_{\text{abs}}$ of the three- and five-spin TFIMs.
• Figure 4: Calculation of the lowest four eigenenergies $E_0$, $E_1$, $E_2$, and $E_3$ of the transverse field Ising model with two ancilla qubits. (a) The circuit schematic for the three-spin system. The transformation matrix $T(4\times4)$ is decomposed into single qubit rotations and CZ gates, as shown at the bottom. The unitary operators $U$, $W$, and $T(2\times2)$ are the same as those in Fig. \ref{['Fig3']}(a). (b) The optimization of the loss function for $h/J = 0.5$. (c) The experimental eigenenergies $E_0$, $E_1$, $E_2$, and $E_3$ of the three-spin TFIM with $h/J = 0.5$. The dashed-line rectangles are the corresponding theoretical results. The error bars represent the standard errors of the average energy.
• Figure 5: (a) The AEVQE circuit for simulating the $N_p$-spin TFIMs, consisting of $N_p$ physical qubits and $N_a$ ancilla qubits. (b) The number of iterations as a function of the number of physical qubits, with fixed one ancilla qubit. (c) The number of iterations in simulating the five-spin TFIM with different ancilla qubit numbers. The error bars represent the standard errors of the average iterations.