Researchers Achieve Simultaneous Determination of Low-Lying Energy Levels with AEVQE

Scientists are continually striving to accurately determine ground and low-lying excited states for a range of complex systems. Huili Zhang, Yibin Guo, and Guanglei Xu, from the Beijing Key Laboratory of Fault-Tolerant Quantum Computing, alongside Yulong Feng, Jingning Zhang, and Hai-feng Yu et al., have now experimentally implemented a novel approach , the ancilla-entangled variational quantum eigensolver (AEVQE) , on a publicly accessible quantum cloud platform. This research demonstrates the full procedure for solving low-lying energy levels in both the H molecule and transverse-field Ising models (TFIMs), revealing potential energy curves and indications of phase transitions. Significantly, this work proves the experimental feasibility of AEVQE and provides valuable guidance for applying variational quantum eigensolver techniques to tackle realistic problems on current quantum hardware.

Scientists are continually striving to accurately determine ground and low-lying excited states for a range of complex systems.

AEVQE Demonstrates Simultaneous Eigenstate Determination with high fidelity

The study unveils a novel approach to quantum computation, employing a set of ancilla qubits to construct maximally Entangled states with physical qubits, enabling the simultaneous extraction of multiple final eigenstates from different ancillary states. Measurements of the physical qubits then feed into a classical computer, which iteratively optimises the circuit parameters θ to minimise a carefully constructed loss function, ultimately yielding the desired eigenenergies and eigenstates. For the three-spin TFIM system, the team calculated four eigenenergies, while for the five-spin system, they determined two eigenenergies, with average differences of 0.029 and 0.099 for the higher-lying excited states, respectively. Experiments show that the AEVQE algorithm’s performance is influenced by several factors, including system size, the choice of classical optimiser, and hyperparameter settings, all of which were thoroughly investigated. Furthermore, the research establishes a direct comparison between AEVQE and other established algorithms like weighted subspace search VQE (SSVQE) and multistate contracted VQE (MCVQE), with a particular focus on analysing the required shot budget during the optimisation stage. This detailed analysis provides valuable guidance for implementing VQE approaches on publicly-accessible quantum platforms, paving the way for solving increasingly realistic and complex problems.

AEVQE Implementation for Molecular and Magnetic Simulations

This work details the full procedure, showcasing a significant step towards utilising near-term quantum devices for complex simulations. To implement AEVQE, the study pioneered a specific experimental setup utilising the superconducting quantum processor Baihua on the Quafu SQC platform. Researchers began by preparing a maximally entangled state between Na ancilla qubits and corresponding physical qubits, constructing the initial state |Ψinit⟩= 1/√K ΣK−1m=0 |m⟩a|ψm⟩p, where K represents the number of target eigenstates and {|m⟩} denotes the computational basis. The team then applied a unitary operation, Ia ⊗U(θ), acting on the physical qubits with variational parameters θ, initiating the iterative optimisation process.

Crucially,. This iterative process, preparing entangled states, applying the unitary, measuring, and optimising, was repeated until convergence. Following optimisation, the team calculated Hsub, with matrix elements defined as Hsub[m, n] = ⟨Ψopt|(|m⟩⟨n|a ⊗H)|Ψopt⟩, utilising the optimal final state |Ψopt⟩= Ia ⊗U(θopt)|Ψinit⟩. Diagonalisation of Hsub, achieved via a unitary transformation T, then revealed the eigenenergies and eigenstates, allowing for the preparation of individual eigenstates through projective measurement of the ancilla qubits. For the H2 molecule, the study obtained the H-H bond distance dependence of two eigenenergies, achieving an average energy difference of 0.027 Hartree for the first excited state. In TFIM simulations, four eigenenergies were calculated for a three-spin system and two for a five-spin system, with average differences of 0.029 and 0.099 for the high-lying excited states, respectively.

AEVQE Solves Molecular and Magnetism Problems efficiently

The work required Na ancilla qubits, with K = 2Na, to determine K eigenenergies embedded within Np physical qubits. Applying a unitary operation Ia ⊗U(θ) to this initial state, where Ia is the identity operator on the ancilla qubits and U(θ) is the parameterized circuit acting on the physical qubits, allowed for optimisation of the loss function L(θ). Data shows the loss function, L(θ) = 1/K ΣKm=0 ⟨ψm|pU†(θ)HU(θ)|ψm⟩p, provides an upper bound on the average of the K low-lying eigenenergies of H, and its gradient was fed to a classical optimizer to refine the variational parameters θ. Diagonalization of Hsub, yielding D = diag(E0. EK−1), allowed determination of the eigenenergies and eigenstates. Specifically, for the H2 molecule simulation, the team employed two physical qubits and a UCCGSD ansatz, with the Hamiltonian defined as H = c0 + c1Z0 + c2X0 + c3Z0Z1 + c4X0X1. The experimental energy potentials E0 and E1 were obtained as a function of the H-H bond distance, achieving results consistent with theoretical calculations, and error bars representing the standard error of the average energy were recorded. The simultaneous perturbation stochastic approximation (SPSA) optimizer was used, with a perturbation strength of ε = 0.1 and a learning rate of η = 0.2, and measurements were repeated 15 × 1024times per iteration to reduce sampling errors.

AEVQE Demonstrates Molecular and Magnetic State Solutions

The authors also investigated factors influencing algorithmic performance and compared AEVQE with ancilla-free VQE algorithms, revealing potential benefits in accuracy when decoherence is a primary error source. However, they acknowledge limitations related to ancilla-physical entanglement becoming more challenging with increasing numbers of ancilla qubits, particularly in one-dimensional qubit arrangements. Future research should focus on employing ladder-shaped qubit architectures to alleviate entanglement difficulties and exploring approximate diagonalization techniques for targeting a larger number of low-lying energy levels. Addressing the barren plateau phenomenon, which becomes more severe as system size increases, also requires further investigation to improve optimization efficiency and expand the applicability of VQAs like AEVQE. The deposited data in the zenodo database will allow for further validation and expansion of these findings.