Tensor-network (TN) states are efficient parametric representations of ground states of local quantum Hamiltonians extensively used in numerical simulations.Employing TN Ansatz states directly on a quantum simulator can potentially offer an exponential computational advantage over purely numerical simulation.We implement a quantum-encoded TN Ansatz state using a variational quantum eigensolver on an ion-trap quantum computer that approximates the ground states of the extended Su-Schrieffer-Heeger model.The generated states are characterized by estimating the topological invariants, verifying their topological order.Our TN encoding as a trapped-ion circuit employs only single-site optical pulses-the native operations naturally available on the platform.We reduce nearest-neighbor crosstalk by selecting different magnetic sublevels with well-separated transition frequencies to encode the qubits in neighboring ions.
Tensor-network (TN) states are efficient parametric representations of ground states of local quantum Hamiltonians extensively used in numerical simulations.Employing TN Ansatz states directly on a quantum simulator can potentially offer an exponential computational advantage over purely numerical simulation.We implement a quantum-encoded TN Ansatz state using a variational quantum eigensolver on an ion-trap quantum computer that approximates the ground states of the extended Su-Schrieffer-Heeger model.The generated states are characterized by estimating the topological invariants, verifying their topological order.Our TN encoding as a trapped-ion circuit employs only single-site optical pulses-the native operations naturally available on the platform.We reduce nearest-neighbor crosstalk by selecting different magnetic sublevels with well-separated transition frequencies to encode the qubits in neighboring ions.
Tensor-network (TN) states supply one of the most powerful variational tools to simulate quantum many-body systems. Though classical simulation of TN states (with approximation) is efficient, the required computational (classical) resources are still beyond our current capability when the size of the system and bond dimension becomes large (which is necessary for studying complicated quantum many-body systems). The TN contraction, which is the dominant cost in TN algorithms, can be replaced by measuring the corresponding physical observables directly in the experimentally prepared TN states. The computational cost is thus dramatically reduced. Here, we propose a scheme to generate TN states by combining multiple phonon modes and qubits in the ion trap platform. With the full connectivity and parallelism of the phonon modes operation, we can generate TN states with rough but complex entanglement structures [such as the multiscale entanglement renormalization Ansatz on the one-dimensional (1D) lattice and the projected entangled pair states on the kagome lattice] by shallow generation circuits. With the abundant local free parameters of the qubits operation, we can refine the local details of the generated TN states. We further benchmark the expressive power of the generated TN states by optimizing the ground energy, which is very close to the exact results of the 1D transverse-field Ising model near the critical point and the 1D Heisenberg model. Our method supplies a quantum-classical hybrid way to simulate the ground states of many-body systems and paves a promising way to demonstrate the advantage of quantum simulation.
Quantum computers are an ideal platform to study the ground state properties of strongly correlated systems due to the limitation of classical computing techniques particularly for systems exhibiting quantum phase transitions. While the error rates of Noisy Intermediate-Scale Quantum (NISQ) computers are still high, simulating strongly correlated systems on such devices and extracting information of possible phases may be within reach. The frustrated transverse-field Ising model (TFIM) is such a system with multiple ordered magnetic phases. In this study, we simulate a two-dimensional frustrated TFIM with next-nearest-neighbor spin-exchange interactions at zero temperature. The competition between the nearest-neighbor ferromagnetic and next-nearest-neighbor antiferromagnetic coupling gives rise to frustration in the system. Moreover, the presence of quantum fluctuations makes the ground-state phase profile even richer. We use the Variational Quantum Eigensolver (VQE) to compute the phases on a square lattice with periodic boundary conditions for a system of 16 sites (qubits). The trained VQE circuits are compared to exact diagonalization, allowing us to extract error measures of VQE. We focus on the ground-state phase transitions of this model, where VQE succeeds in finding the dominant magnetic phases. The optimized VQE circuits are then executed on the Quantinuum H1-1 trapped-ion quantum computer without using any error mitigation techniques. Our experiments show near perfect recovery of the magnetic phases of the frustrated model through ground-state energy, the energy derivative, and the spin correlation functions. Thus, we show that the trapped-ion quantum processor is able to achieve reliable simulations of a strongly correlated system within the limitations of the VQE approach.
In this paper, we introduce a tensor network (TN) scheme into the entanglement augmentation process of the synergistic optimization framework by Rudolph [] to build its process systematically for inhomogeneous systems. Our synergistic approach first embeds the variational optimal solution of the TN state with the entropic area law, which can be perfectly optimized in conventional (classical) computers, in a quantum variational circuit ansatz inspired by the TN state with the entropic volume law. Next, the framework performs a variational quantum eigensolver (VQE) process with embedded states as the initial state. We applied the synergistic to the ground-state analysis of the all-to-all coupled random transverse-field Ising, XYZ, Heisenberg model, employing the binary multiscale entanglement renormalization ansatz (MERA) state and branching MERA states as TN states with entropic area law and volume law, respectively. We then show that the synergistic accelerates VQE calculations in the three models without an initial parameter guess of the branching-MERA-inspired ansatz and can avoid a local solution trapped by a standard VQE with the ansatz in the Ising model. The improvement of optimizers for MERA in all-to-all coupled inhomogeneous systems, enhancement, and potential synergistic applications are also discussed. Published by the American Physical Society 2024
It appears that the tensor network ansatz, while not quite complete, is an efficient coordinate system for the tiny subset of a many-body Hilbert space which can be realized as a low energy state of a local Hamiltonian.However, we don't fully understand precisely which phases are captured by the tensor network ansatz, how to compute their physical observables (even numerically), or how to compute a tensor network representation for a ground state given a microscopic Hamiltonian. These questions are algorithmic in nature, but their resolution is intimately related to understanding the nature of quantum entanglement in many-body systems.For this reason it is useful to compute the tensor network representation of various `model' wavefunctions representative of different phases of matter; this allows us to understand how the entanglement properties of each phase are expressed in the tensor network ansatz, and can serve as test cases for algorithm development.Condensed matter physics has many illuminating model wavefunctions, such as Laughlin's celebrated wave function for the fractional quantum Hall effect, the Bardeen-Cooper-Schrieffer wave function for superconductivity, and Anderson's resonating valence bond ansatz for spin liquids.This thesis presents some results on exact tensor network representations of these model wavefunctions. In addition, a tensor network representation is given for the time evolution operator of a long-range one-dimensional Hamiltonian, which allows one to numerically simulate the time evolution of power-law interacting spin chains as well as two-dimensional strips and cylinders.
Quantum computers have long been anticipated to excel in simulating quantum many-body physics. In this work, we demonstrate the power of variational quantum circuits for resource-efficient simulations of dynamical and equilibrium physics in non-Hermitian systems. Using a variational quantum compilation scheme for fermionic systems, we reduce gate count, save qubits, and eliminate the need for postselection, a major challenge in simulating non-Hermitian dynamics via standard Trotterization. On the Quantinuum H1 trapped-ion processor, we experimentally observed a supersonic mode on an n = 18 fermionic chain after a non-Hermitian, nearest-neighbor interacting quench, which would otherwise be forbidden in a Hermitian system. Additionally, we investigate sequential quantum circuits generated by tensor networks for ground-state preparation using a variance minimization scheme, accurately capturing correlation functions and energies across an exceptional point on a dissipative spin chain up to length n = 20 using only 3 qubits. On the other hand, we provide an analytical example demonstrating that simulating single-qubit non-Hermitian dynamics for Θ(log(n)) time from certain initial states is exponentially hard on a quantum computer. Our work raises many intriguing questions about the intrinsic properties of non-Hermitian systems that permit efficient quantum simulation. Quantum simulation of non-Hermitian systems is non-trivial, as the use of Trotterization would incur into a huge sampling overhead. Here, the authors use a variational approach to simulate a quench in a 18-site fermionic chain on a trapped-ion quantum processor, seeing the signature of a supersonic mode in the connected density-density correlation function and providing insights on when quantum computers fail to simulate non-Hermitian dynamics.
The Su-Schrieffer-Heeger (SSH) chain provides a particularly stringent setting for variational quantum simulation because exact solvability, symmetry-protected topology, and boundary localized zero-mode physics coexist in a single analytically controlled lattice Hamiltonian. In this work, we use the open SSH chain as a one-particle benchmark for the Variational Quantum Eigensolver (VQE) and develop a two-part variational treatment that separately addresses its low-lying bulk spectrum and its topological near-zero sector. For the bulk benchmark, we restrict the SSH model to the one-particle sector, where the Hamiltonian reduces to an N times N hopping matrix that is encoded as a qubit operator for variational simulation. We compare three shallow variational families: a problem-aligned fermionic ansatz, a hardware-efficient ansatz, and a two-local baseline. Across the reported system sizes, the fermionic ansatz reproduces the benchmarked exact-diagonalization levels most faithfully, while the remaining circuits exhibit larger and more persistent deviations, particularly for the second reported level. These results demonstrate (in a controlled topological setting) that shallow variational accuracy depends strongly on whether the circuit manifold reflects the fermionic and lattice structure of the target Hamiltonian. We then turn to the central topological feature of the open SSH chain, namely the finite-size near-zero pair associated with exponentially localized edge states. Since these states are not extremal eigenvalues of the original Hamiltonian, they are not isolated naturally by VQE. To access this sector, we construct a folded-spectrum workflow based on the squared Hamiltonian, use variational minimization and deflation to identify the corresponding low-|E| subspace, and recover the signed near-zero energies by explicit orthonormalization and projection onto the original SSH Hamiltonian. The reconstructed pair agrees with the results of exact diagonalization and reproduces the expected exponential suppression of the finite-size splitting with increasing chain length. Taken together, the results establish a variational route to both the benchmarked bulk spectrum and the topological near-zero sector of a canonical one-dimensional topological Hamiltonian. More broadly, they show that in variational quantum simulation of topological systems, accuracy depends not only on ansatz compatibility with the Hamiltonian structure, but also on whether the chosen variational objective isolates the spectral feature of interest.