We investigate variational quantum eigensolver (VQE) ansatzes for simulating the ground state of the long-range XY model. We compare entanglers based on parametric controlled- \(R_{x}\) ( \(CR_{x}\) ), \(R_{xx}\) and combined \(R_{xx}\) and \(R_{yy}\) rotations ( \(R_{xx}R_{yy}\) ), considering both homogeneous and inhomogeneous parameter-sharing schemes. Numerical simulations show that the mean-field ansatz reproduces the mean-field energy but fails to capture quantum correlations. Introducing entanglers reveals distinct trade-offs between expressibility and trainability. The inhomogeneous \(CR_{x}\) ansatz yields accurate energies with shallow circuits, whereas the homogeneous \(R_{xx}\) ansatz benefits from symmetry and a smoother optimization landscape. Notably, the \(R_{xx}R_{yy}\) ansatz delivers the highest fidelity across the entire phase diagram, with the inhomogeneous version achieving high accuracy using only two layers, and the homogeneous one also exhibiting good performance with moderate depth. These results underline the importance of tailoring ansatz structure and parameter sharing to the underlying Hamiltonian to enable efficient and accurate quantum simulation on near-term devices.
We investigate variational quantum eigensolver (VQE) ansatzes for simulating the ground state of the long-range XY model. We compare entanglers based on parametric controlled- (R_{x}) ( (CR_{x}) ), (R_{xx}) and combined (R_{xx}) and (R_{yy}) rotations ( (R_{xx}R_{yy}) ), considering both homogeneous and inhomogeneous parameter-sharing schemes. Numerical simulations show that the mean-field ansatz reproduces the mean-field energy but fails to capture quantum correlations. Introducing entanglers reveals distinct trade-offs between expressibility and trainability. The inhomogeneous (CR_{x}) ansatz yields accurate energies with shallow circuits, whereas the homogeneous (R_{xx}) ansatz benefits from symmetry and a smoother optimization landscape. Notably, the (R_{xx}R_{yy}) ansatz delivers the highest fidelity across the entire phase diagram, with the inhomogeneous version achieving high accuracy using only two layers, and the homogeneous one also exhibiting good performance with moderate depth. These results underline the importance of tailoring ansatz structure and parameter sharing to the underlying Hamiltonian to enable efficient and accurate quantum simulation on near-term devices.
Finding the ground state energy and wavefunction of a quantum many-body system is a key problem in quantum physics and chemistry. We study this problem for the long-range XY model by using the variational quantum eigensolver (VQE) algorithm. We consider VQE ansatzes with full and linear entanglement structures consisting of different building gates: the CNOT gate, the controlled-rotation (CRX) gate, and the two-qubit rotation (TQR) gate. We find that the full-entanglement CRX and TQR ansatzes can sufficiently describe the ground state energy of the long-range XY model. In contrast, only the full-entanglement TQR ansatz can represent the ground state wavefunction with a fidelity close to one. In addition, we find that instead of using full-entanglement ansatzes, restricted-entanglement ansatzes where entangling gates are applied only between qubits that are a fixed distance from each other already suffice to give acceptable solutions. Using the entanglement entropy to characterize the expressive powers of the VQE ansatzes, we show that the full-entanglement TQR ansatz has the highest expressive power among them.
We propose an efficient circuit structure of variational quantum circuit Ans\"atze used for the variational quantum eigensolver (VQE) algorithm in calculating gapped topological phases on the currently feasible noisy intermediate-scale quantum computers. An efficient circuit Ansatz should include two layers: the initialization layer and the variational layer. In the initialization layer, a fixed-depth circuit state with a compatible entanglement structure to the target topological phase is constructed. The circuit state is further adjusted subsequently to capture the details of the local correlations, which is dictated with the Hamiltonian, in the parametrized variational layer. Based on this strategy, we design a circuit Ansatz to investigate the symmetry-protected topological Haldane phase in a nonexactly solvable alternating spin-21 Heisenberg chain by VQE calculations. Main characterizations of the Haldane phase, including the long-ranged string order, the fourfold nearly degenerate ground states associated with four different localized edge-mode patterns for the system with open boundaries, and the twofold degeneracy of the entanglement spectrum, are all observed for the optimized shallow circuit state with only one-depth variational layer both in numerical simulations and on real quantum computers. We further demonstrate that the computational capacity (i.e., expressibility) of this quantum circuit Ansatz is determined not by the system size but only by the intrinsic correlation length of the system, thus implying that the scalable VQE calculation is possible.
We explore the application of the Variational Quantum Eigensolver (VQE) to investigate the ground state properties, particularly the entanglement entropy, of the Transverse Field Ising Model (TFIM) in one, two, and three dimensions, considering systems of up to 27 spins. By benchmarking VQE results against exact diagonalization and analyzing the entanglement properties across different system sizes, we assess the algorithm's effectiveness in capturing faithful ground state. Using results of TFIM, we also investigate how VQE's expressivity and optimization influence the simulation of highly entangled quantum states. We employ different ansätze: the hardware-efficient EfficientSU2 from Qiskit, the physics-inspired Hamiltonian Variational ansätz (HVA) and HVA with symmetry breaking, and benchmark their performance using energy variance, entanglement entropy, spin correlations, and magnetization. We further discuss the implications for scaling these methods to larger quantum systems.
Quantum variational algorithms are one of the most promising applications of near-term quantum computers; however, recent studies have demonstrated that unless the variational quantum circuits are configured in a problem-specific manner, optimization of such circuits will most likely fail.In this paper, we focus on a special family of quantum circuits called the Hamiltonian variational ansatz (HVA), which its takes inspiration from the quantum approximate optimization algorithm and adiabatic quantum computation.Through the study of its entanglement spectrum and energy-gradient statistics, we find that the HVA exhibits favorable structural properties such as mild or entirely absent barren plateaus and a restricted state space that eases their optimization in comparison to the well-studied "hardware-efficient ansatz."We also numerically observe that the optimization landscape of the HVA becomes almost trap free, i.e., there are no suboptimal minima, when the ansatz is overparametrized.We observe a size-dependent "computational phase transition" as the number of layers in the HVA circuit is increased where the optimization crosses over from a hard to an easy region in terms of the quality of the approximations and the speed of convergence to a good solution.In contrast to the analogous transitions observed in the learning of random unitaries, which occur at a number of layers that grows exponentially with the number of qubits, our variational-quantum-eigensolver experiments suggest that the threshold to achieve the overparametrization phenomenon scales at most polynomially in the number of qubits for the transverse-field Ising and XXZ models.Lastly, as a demonstration of its entangling power and effectiveness, we show that the HVA can find accurate approximations to the ground states of a modified Haldane-Shastry Hamiltonian on a ring, which has long-range interactions and has a power-law entanglement scaling.
A variational quantum eigensolver (VQE) solves the ground state problem of a given Hamiltonian by finding the parameters of a quantum circuit Ansatz that minimizes the Hamiltonian expectation value. Among possible quantum circuit Ansätze, the Hamiltonian variational Ansatz (HVA) is widely studied for quantum many-body problems as the Ansatz with sufficiently large depth is theoretically guaranteed to express the ground state. However, since the HVA shares the same symmetry with the Hamiltonian, it is not necessarily good at finding symmetry-broken ground states that prevail in nature. In this paper, we systematically explore the limitations of the HVA for solving symmetry-broken systems and propose an alternative quantum circuit Ansatz with symmetry-breaking layers. With extensive numerical simulations, we show that the proposed Ansatz finds the ground state in depth significantly shorter than the bare HVA when the target Hamiltonian has symmetry-broken ground states.
Abstract Reducing circuit depth is essential for implementing quantum simulations of electronic structure on near-term quantum devices. In this work, we propose a variational quantum eigensolver (VQE) based perturbation theory (PT) algorithm to accurately simulate electron correlation of periodic materials with shallow ansatz circuits, which are generated from adaptive derivative-assembled pseudo-trotter or qubit-excitation-based VQE calculations using a loose convergence criteria. Here, the major part of the electron correlation is described using the VQE ansatz circuit and the remaining correlation energy is described by either multireference or similarity transformation-based PT. Numerical results demonstrate that the new algorithms are able to accurately describe electron correlation of the LiH crystal with only one circuit parameter, in contrast with ~30 parameters required in the adaptive VQE to achieve the same accuracy. Meanwhile, for fixed-depth ansätze, e.g. unitary coupled cluster, we demonstrate that the VQE-based PT provides an appealing scheme to improve their accuracy.