Matrix Product State Methods Research Articles

Nuclear norm regularized loop optimization for tensor network

We propose a loop optimization algorithm based on nuclear norm regularization for the tensor network. The key ingredient of this scheme is to introduce a rank penalty term proposed in the context of data processing. Compared to the standard variational periodic matrix product states method, this algorithm can circumvent the local minima related to short-ranged correlation in a simpler fashion. We demonstrate its performance when used as a part of the tensor network renormalization algorithms [S. Yang, Z.-C. Gu, and X.-G. Wen, ] for the critical two-dimensional Ising model. The scale invariance of the renormalized tensors is attained with higher accuracy while the higher parts of the scaling dimension spectrum are obtained in a more stable fashion. Published by the American Physical Society 2024

Elucidating Dynamical Behaviors in Kinetically Constrained Models via Energy-Activity Double-Biased Matrix Product State Analysis.

We investigate dynamical phase transitions in two representative kinetically constrained models: the 1D Fredrickson-Andersen and East Models. A recently developed energy-activity double-bias approach utilizing both s and g fields, conjugated with dynamical activity and trajectory energy, is combined with matrix product state (MPS) methods. It is demonstrated that MPS methods facilitate the numerical approximation of large-deviation statistics of dynamics by determining the eigenvalues of tilted dynamical generators under the influence of double-biasing fields. Specifically, by focusing on the g-field, a nearly "half-filled" state at moderate negative g values is identified, indicating the potential existence of an anomalous phase. Additionally, dynamical quantities under various s, g, and T conditions are obtained via tensor networks, showing good qualitative consistency with mean-field results and our previous extensive numerical simulation results obtained by the path sampling method. Our study introduces novel methodologies for examining decoupled dynamical behaviors that, although energetically active, remain dynamically inactive within the system. This approach offers a fresh perspective on the theoretical framework and computational strategies for studying dynamical phase transitions in kinetically constrained systems.

Matrix product study of spin fractionalization in the one-dimensional Kondo insulator

The Kondo lattice is one of the classic examples of strongly correlated electronic systems. We conduct a controlled study of the Kondo lattice in one dimension, highlighting the role of excitations created by the composite fermion operator. Using time-dependent matrix product state methods, we compute various correlation functions and contrast them with both large-N mean-field theory and the strong-coupling expansion. We show that the composite fermion operator creates long-lived, charge-e and spin-1/2 excitations, which cover the low-lying single-particle excitation spectrum of the system. Furthermore, spin excitations can be thought to be composed of such fractionalized quasiparticles with a residual interaction which tend to disappear at weak Kondo coupling. Published by the American Physical Society 2024

Work statistics for quantum spin chains: Characterizing quantum phase transitions, benchmarking time evolution, and examining passivity of quantum states

We study three aspects of work statistics in the context of the fluctuation theorem for quantum spin chains up to 1024 sites by numerical methods based on matrix-product states (MPSs). First, we use our numerical method to evaluate the moments/cumulants of work done by the sudden quench process on the Ising or Haldane spin chains, and we study their behaviors across the quantum phase transitions. Our results show that, up to the fourth cumulant, the work statistics can indicate the quantum phase transition characterized by the local order parameters but barely for purely topological phase transitions. Second, we propose to use the fluctuation theorem, such as Jarzynski's equality, which relates the real-time correlator to the ratio of the thermal partition functions, as a benchmark indicator for the numerical real-time evolving methods. Third, we study the passivity of ground and thermal states of quantum spin chains under some cyclic impulse processes. We show that the passivity of thermal states and ground states under the Hermitian actions is ensured by the second laws and variational principles, respectively, and we also verify this by numerical calculations. In addition, we also consider the passivity of ground states under non-Hermitian actions, for which the variational principle cannot be applied. Despite that, we find no violation of passivity from our numerical results for all the cases considered in the Ising and Haldane chains. Overall, we demonstrate that the work statistics for the sudden quench and impulse processes can be evaluated precisely by the numerical MPS method to characterize quantum phase transitions and examine the passivity of quantum states. We also propose to exploit the universality of the fluctuation theorem to benchmark the numerical real-time evolutions in an algorithmic and model-independent way. Published by the American Physical Society 2024

Exact solution to quantum dynamical activity.

The quantum dynamical activity constitutes a thermodynamic cost in trade-off relations such as the quantum speed limit and the quantum thermodynamic uncertainty relation. However, calculating the quantum dynamical activity has been a challenge. In this paper, we present the exact solution for the quantum dynamical activity by deploying the continuous matrix product state method. Moreover, using the derived exact solution, we determine the upper bound of the dynamical activity, which comprises the standard deviation of the system Hamiltonian and jump operators. We confirm the exact solution and the upper bound by performing numerical simulations.

Heisenberg-limited metrology with perturbing interactions

We show that it is possible to perform Heisenberg-limited metrology on GHZ-like states, in the presence of generic spatially local, possibly strong interactions during the measurement process. An explicit protocol, which relies on single-qubit measurements and feedback based on polynomial-time classical computation, achieves the Heisenberg limit. In one dimension, matrix product state methods can be used to perform this classical calculation, while in higher dimensions the cluster expansion underlies the efficient calculations. The latter approach is based on an efficient classical sampling algorithm for short-time quantum dynamics, which may be of independent interest.

Two-dimensional excitation information by a matrix product state method on infinite helices

Two-dimensional excitation information by a matrix product state method on infinite helices

Quasi-fractionalization of edge spin in chirality-assisted cluster-based Haldane state on triangular spin tube

Fractionalization of quantum degrees of freedom holds the key to finding new phenomena in physics, e.g., the quark model in hadron physics and the spin-charge separation in strongly-correlated electron systems. A typical example of the fractionalization in quantum spin systems is the spin-1 Haldane state, whose intriguing characteristics are well described by fractionalized virtual spins, delivering two individual spin-1/2 degrees of freedom as edge states. Here we theoretically propose an exotic extension of the Haldane state to the pseudo spin-1 model consisting of the mixture of real spin and spin chirality, resulting in quasi-fractionalization of spin-1/2 magnetization, i.e., an approximately-1/4 spin. Existence of the edge state is confirmed both analytically and numerically in a triangular spin tube, combining a low-energy perturbation theory and variational matrix-product state method. Our study not only proposes an unconventional quantum spin object but paves a way to chop the elementary quantities further.

MPS-VQE: A variational quantum computational chemistry simulator with matrix product states

Quantum computing is attracting more and more attention in material and biological science. In the noisy intermediate-scale quantum (NISQ) computing era, the variational quantum eigensolver (VQE) is expected as an effective method to solve quantum chemistry problems which has potential quantum advantage. Classically simulating quantum algorithms plays a critical role in algorithmic development and the validation of noisy quantum devices before large-scale, fully fault-tolerant quantum computers are developed. However, most existing simulators are based on the state vector or density matrix representation of the quantum state which is faced with the memory bottleneck due to the exponential growth of memory requirement as the number of qubits increases. In this work, we present a VQE simulator based on the matrix product states (MPS) method from quantum many-body physics, and a detailed study of our MPS-VQE simulator in quantum chemistry applications. The memory requirement of MPS-VQE grows only polynomially and we demonstrate that accurate results can be obtained despite the truncation of the smallest singular values during the algorithm. A distributed parallelization scheme is also presented for massively parallel computer systems, and different implementations of the parallelization scheme based on the Julia language are benchmarked which demonstrate the efficiency and scalability of our method. Our method could open a new avenue towards large-scale classical simulation of quantum computational chemistry.

Matrix product states and numerical mode decomposition for the analysis of gauge-invariant cavity quantum electrodynamics

There has been a problem of gauge ambiguities with the Rabi Hamiltonian due to the fact that it can be derived from two formally different but physically equivalent fundamental Hamiltonians. This problem has recently been resolved for models with a single quantized electromagnetic mode. In this paper, we mathematically and numerically verify this for multimode models. With this established, we combine the numerical methods, matrix product states (MPS) and numerical mode decomposition (NMD), for analyzing cavity QED systems. The MPS method is used to efficiently represent and time evolve a quantum state. However, since the coupling structure of the Rabi Hamiltonian is incompatible with MPS, it is numerically transformed into an equivalent Hamiltonian that has a chain coupling structure, which allows efficient application of MPS. The technique of NMD is used to extract the numerical electromagnetic modes of an arbitrary environment. As a proof of concept, this combined approach is demonstrated by analyzing one-dimensional cavity QED systems in various settings.

Two-dimensional isometric tensor networks on an infinite strip

The exact contraction of a generic two-dimensional (2D) tensor network state (TNS) is known to be exponentially hard, making simulation of 2D systems difficult. The recently introduced class of isometric TNS (isoTNS) represents a subset of TNS that allows for efficient simulation of such systems on finite square lattices. The isoTNS ansatz requires the identification of an "orthogonality column" of tensors, within which one-dimensional matrix product state (MPS) methods can be used for calculation of observables and optimization of tensors. Here we extend isoTNS to infinitely long strip geometries and introduce an infinite version of the Moses Move algorithm for moving the orthogonality column around the network. Using this algorithm, we iteratively transform an infinite MPS representation of a 2D quantum state into a strip isoTNS and investigate the entanglement properties of the resulting state. In addition, we demonstrate that the local observables can be evaluated efficiently. Finally, we introduce an infinite time-evolving block decimation algorithm (iTEBD\textsuperscript{2}) and use it to approximate the ground state of the 2D transverse field Ising model on lattices of infinite strip geometry.

Bridging quantum criticality via many-body scarring

Quantum dynamics in certain kinetically-constrained systems can display a strong sensitivity to the initial condition, wherein some initial states give rise to persistent quantum revivals -- a type of weak ergodicity breaking known as `quantum many-body scarring' (QMBS). Recent work [Phys.Rev.B 105, 125123 (2022)] pointed out that QMBS gets destroyed by tuning the system to a quantum critical point, echoing the disappearance of long-range order in the system's ground state at equilibrium. Here we show that this picture can be much richer in systems that display QMBS dynamics from a continuous family of initial conditions: as the system is tuned across the critical point while at the same time deforming the initial state, the dynamical signatures of QMBS at intermediate times can undergo an apparently smooth evolution across the equilibrium phase transition point. We demonstrate this using the PXP model -- a paradigmatic model of QMBS that has recently been realized in Rydberg atom arrays as well as ultracold bosonic atoms in a tilted optical lattice. Using exact diagonalization and matrix product state methods, we map out the dynamical phase diagram of the PXP model with the quenched chemical potential. We demonstrate the existence of a continuous family of initial states that give rise to QMBS and formulate a ramping protocol that can be used to prepare such states in experiment. Our results show the ubiquity of scarring in the PXP model and highlight its intriguing interplay with quantum criticality.

Two-stage melting of an intercomponent Potts long-range order in two dimensions

The interplay of topology and competing interactions can induce enriched phases and phase transitions at finite temperatures. We consider a weakly coupled two-dimensional hexatic-nematic XY model with a relative ${Z}_{3}$ Potts degrees of freedom, and apply the matrix product state method to solve this model rigorously. Since the partition function is expressed as a product of two-legged one-dimensional transfer matrix operator, an entanglement entropy of the eigenstate corresponding to the maximal eigenvalue of this transfer operator can be used as a stringent criterion to determine various phase transitions precisely. At low temperatures, the intercomponent ${Z}_{3}$ Potts long-range order (LRO) exists, indicating that the hexatic and nematic fields are locked together and their respective vortices exhibit quasi-LRO. In the hexatic regime, below the BKT transition of the hexatic vortices, the intercomponent ${Z}_{3}$ Potts LRO appears, accompanying the binding of nematic vortices. In the nematic regime, however, the intercomponent ${Z}_{3}$ Potts LRO undergoes a two-stage melting process. An intermediate Potts liquid phase emerges between the Potts ordered and disordered phases, characterized by an algebraic correlation with formation of charge-neutral pairs of both hexatic and nematic vortices. These two-stage phase transitions are associated with the proliferation of the domain walls and vortices of the relative ${Z}_{3}$ Potts variable, respectively. Our results thus provide a prototype example of two-stage melting of a two-dimensional LRO driven by multiple topological defects.

Partially disordered Heisenberg antiferromagnet with short-range stripe correlations

Zero-point quantum fluctuations of a N\'eel order can produce effective interactions between quasi-orphan spins weakly coupled to the lattice. On the $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$-distorted triangular lattice, this phenomenon leads to a correlated partially disordered phase. In this article, we use matrix product state methods to study a similar model: the $S=1/2$ stuffed square lattice. Tuning the exchange amplitudes we go from a square lattice plus orphan central spins at ${J}^{\ensuremath{'}}/J=0$, to the union jack lattice at ${J}^{\ensuremath{'}}/J=1$, and a square lattice including all spins at $J/{J}^{\ensuremath{'}}=0$. We calculate the complete antiferromagnetic phase diagram, dominated by ferrimagnetic and N\'eel orders, and compare it with existing results. Most importantly, we find a partially disordered phase in the weakly frustrated regime. In this phase, the N\'eel order from the square lattice is unaffected, while the central spins form a collective state with exponentially decaying double-striped correlations. We also study the role of quantum fluctuations by introducing an ordering staggered magnetic field on the square sublattice and find that the central spins order ferromagnetically when fluctuations from the N\'eel order are suppressed.

Nonlinear spectroscopy of bound states in perturbed Ising spin chains

We study the nonlinear response of non-integrable 1D spin models using infinite matrix-product state techniques. As a benchmark and demonstration of the method, we first calculate the 2D coherent spectroscopy for the exactly soluble ferromagnetic transverse field Ising model, where excitations are freely moving domain-walls. We then investigate the distinct signatures of confined bound states by introducing a longitudinal field and observe the emergence of strong non-rephasing like signals. To interpret the observed phenomena, we use a two-kink approximation to perturbatively compute the 2D spectra. We find good agreement in comparison with the exact results of the infinite matrix-product state method in the strongly confined regime. We discuss the relevance of our results for quasi$-$1D Ising spin chain materials such as $\mathrm{CoNb}_2\mathrm{O}_6$.

Chiral Spin-Chain Interfaces Exhibiting Event-Horizon Physics.

The interface between different quantum phases of matter can give rise to novel physics, such as exotic topological phases or nonunitary conformal field theories. Here we investigate the interface between two spin chains in different chiral phases. Surprisingly, the mean field theory approximation of this interacting composite system is given in terms of Dirac fermions in a curved space-time geometry. In particular, the interface between the two phases represents a black hole horizon. We demonstrate that this representation is faithful both analytically, by employing bosonization to obtain a Luttinger liquid model, and numerically, by employing matrix product state methods. A striking prediction from the black hole equivalence emerges when a quench, at one side of the interface between two opposite chiralities, causes the other side to thermalize with the Hawking temperature for a wide range of parameters and initial conditions.

Dynamical conductivity of disordered quantum chains

We study the transport properties of a one-dimensional quantum system with disorder. We numerically compute the frequency dependence of the conductivity of a fermionic chain with nearest-neighbor interaction and a random chemical potential by using the Chebyshev matrix product state (CheMPS) method. As a benchmark, we investigate the noninteracting case first. Comparison with exact diagonalization and analytical solutions demonstrates that the results of CheMPS are reliable over a wide range of frequencies. We then calculate the dynamical conductivity spectra of the interacting system for various values of the interaction and disorder strengths. In the high-frequency regime, the conductivity decays as a power law, with an interaction-dependent exponent. This behavior is qualitatively consistent with the bosonized field theory predictions, although the numerical evaluation of the exponent shows deviations from the analytically expected values. We also compute the characteristic pinning frequency at which a peak in the conductivity appears. We confirm that it is directly related to the inverse of the localization length, even in the interacting case. We demonstrate that the localization length follows a power law of the disorder strength with an exponent dependent on the interaction, and find good quantitative agreement with the field theory predictions. In the low-frequency regime, we find a behavior consistent with the one of the noninteracting system omega ^{2}(ln omega )^{2} independently of the interaction. We discuss the consequences of our finding for experiments in cold atomic gases.Graphical abstract

Entanglement scaling for λϕ24

We study the $\ensuremath{\lambda}{\ensuremath{\phi}}^{4}$ model in $0+2$ dimensions at criticality, focusing on the scaling properties originating from the UV and IR physics. We demonstrate that the entanglement entropy, the correlation length $\ensuremath{\xi}$ and order parameters $\ensuremath{\phi}$ and ${\ensuremath{\phi}}^{3}$ exhibit distinctive double scaling properties that prove a powerful tool in the data analysis. The calculations are performed with boundary matrix product state methods on tensor network representations of the partition function to which the entanglement scaling hypothesis is applied, though the technique is equally applicable outside the realm of tensor networks. We find the value ${\ensuremath{\alpha}}_{c}=11.09698(31)$ for the critical point, improving on previous results.

Quantum-inspired method for solving the Vlasov-Poisson equations.

Kinetic simulations of collisionless (or weakly collisional) plasmas using the Vlasov equationare often infeasible due to high-resolution requirements and the exponential scaling of computational cost with respect to dimension. Recently, it has been proposed that matrix product state (MPS) methods, a quantum-inspired but classical algorithm, can be used to solve partial differential equationswith exponential speed-up, provided that the solution can be compressed and efficiently represented as a MPS within some tolerable error threshold. In this work, we explore the practicality of MPS methods for solving the Vlasov-Poisson equationsfor systems with one coordinate in space and one coordinate in velocity, and find that important features of linear and nonlinear dynamics, such as damping or growth rates and saturation amplitudes, can be captured while compressing the solution significantly. Furthermore, by comparing the performance of different mappings of the distribution functions onto the MPS, we develop an intuition of the MPS representation and its behavior in the context of solving the Vlasov-Poisson equations, which will be useful for extending these methods to higher-dimensional problems.

Intertwined string orders of topologically trivial and nontrivial phases in an interacting Kitaev chain with spatially varying potentials

By using the variational matrix product state method, we numerically study the interacting Kitaev chain with spatially varying periodic and quasiperiodic potentials, and the latter follows the Fibonacci sequence. The edge correlation functions of Majorana fermions and low-lying ground states are computed to explore the robustness of topological superconducting phase. It is found that the original topologically nontrivial phase is separated into two branches by an emergent topologically trivial phase, as a result of the competition among spatially varying potential, electronic, Coulomb interaction, and chemical potential. The analysis of energy gap, occupation number, and intertwined string order parameters together suggests that the lift of degeneracy in the topologically trivial phase is enabled by a potential-induced fracton mechanism, namely, the pairing of four Majorana fermions. The evolution from the emergent fractal structure of population to the original structure of charge density wave is investigated as well.