Quantum Lattice Systems Research Articles

An introduction to infinite projected entangled-pair state methods for variational ground state simulations using automatic differentiation

Tensor networks capture large classes of ground states of phases of quantum matter faithfully and efficiently. Their manipulation and contraction has remained a challenge over the years, however. For most of the history, ground state simulations of two-dimensional quantum lattice systems using (infinite) projected entangled pair states have relied on what is called a time-evolving block decimation. In recent years, multiple proposals for the variational optimization of the quantum state have been put forward, overcoming accuracy and convergence problems of previously known methods. The incorporation of automatic differentiation in tensor networks algorithms has ultimately enabled a new, flexible way for variational simulation of ground states and excited states. In this work we review the state-of-the-art of the variational iPEPS framework, providing a detailed introduction to automatic differentiation, a description of a general foundation into which various two-dimensional lattices can be conveniently incorporated, and demonstrative benchmarking results.

Continuous extension of the discrete shift translations on one-dimensional quantum lattice systems

We study the continuous extension of discrete shift translations on one-dimensional quantum lattice systems. We explore a specific example provided by a quasi-free C ∗ {C}^{\ast } -flow on the one-dimensional fermion lattice system, comparing it with its original discrete shift-translations. Specifically, we demonstrate its explicit dynamical formula, which reveals violations of causality and locality. Furthermore, we prove that this quasi-free C ∗ {C}^{\ast } -flow on the one-dimensional fermion lattice system cannot be extended to the one-dimensional quantum spin lattice system via the Jordan-Wigner transformation.

Decoding topological XYZ 2 codes with reinforcement learning based on attention mechanisms

Quantum error correction, a technique that relies on the principle of redundancy to encode logical information into additional qubits to better protect the system from noise, is necessary to design a viable quantum computer. For this new topological stabilizer code-XYZ 2 code defined on the cellular lattice, it is implemented on a hexagonal lattice of qubits and it encodes the logical qubits with the help of stabilizer measurements of weight six and weight two. However topological stabilizer codes in cellular lattice quantum systems suffer from the detrimental effects of noise due to interaction with the environment. Several decoding approaches have been proposed to address this problem. Here, we propose the use of a state-attention based reinforcement learning decoder to decode XYZ 2 codes, which enables the decoder to more accurately focus on the information related to the current decoding position, and the error correction accuracy of our reinforcement learning decoder model under the optimisation conditions can reach 83.27% under the depolarizing noise model, and we have measured thresholds of 0.18856 and 0.19043 for XYZ 2 codes at code spacing of 3–7 and 7–11, respectively. our study provides directions and ideas for applications of decoding schemes combining reinforcement learning attention mechanisms to other topological quantum error-correcting codes.

Stability of the bulk gap for frustration-free topologically ordered quantum lattice systems

We prove that uniformly small short-range perturbations do not close the bulk gap above the ground state of frustration-free quantum spin systems that satisfy a standard local topological quantum order condition. In contrast with earlier results, we do not require a positive lower bound for finite-system spectral gaps uniform in the system size. To obtain this result, we extend the Bravyi–Hastings–Michalakis strategy so it can be applied to perturbations of the GNS Hamiltonian of the infinite-system ground state.

Correction: Large Deviations in Quantum Lattice Systems: One-Phase Region

Correction: Large Deviations in Quantum Lattice Systems: One-Phase Region

Dissipative quantum many-body dynamics in (1+1)D quantum cellular automata and quantum neural networks

Classical artificial neural networks, built from elementary units, possess enormous expressive power. Here we investigate a quantum neural network (QNN) architecture, which follows a similar paradigm. It is structurally equivalent to so-called (1+1)D quantum cellular automata, which are two-dimensional quantum lattice systems on which dynamics takes place in discrete time. Information transfer between consecutive time slices—or adjacent network layers—is governed by local quantum gates, which can be regarded as the quantum counterpart of the classical elementary units. Along the time-direction an effective dissipative evolution emerges on the level of the reduced state, and the nature of this dynamics is dictated by the structure of the elementary gates. We show how to construct the local unitary gates to yield a desired many-body dynamics, which in certain parameter regimes is governed by a Lindblad master equation. We study this for small system sizes through numerical simulations and demonstrate how collective effects within the quantum cellular automaton can be controlled parametrically. Our study constitutes a step towards the utilization of large-scale emergent phenomena in large QNNs for machine learning purposes.

Interscale entanglement production in a quantum system simulating classical chaos

It is a fundamental problem how the universal concept of classical chaos emerges from the microscopic description of quantum mechanics. We here study standard classical chaos in a framework of quantum mechanics. In particular, we design a quantum lattice system that exactly simulates classical chaos after an appropriate continuum limit, which is called the ‘Hamiltonian equation limit’. The key concept of our analysis is an entanglement entropy defined by dividing the lattice into many blocks of equal size and tracing out the degrees of freedom within each block. We refer to this entropy as the ‘interscale entanglement entropy (IEE)’ because it measures the amount of entanglement between the microscopic degrees of freedom within each block and the macroscopic degrees of freedom that define the large-scale structure of the wave function. By numerically simulating a quantum lattice system corresponding to the Hamiltonian of the kicked rotor, we find that the long-time average of the IEE becomes positive only when chaos emerges in the Hamiltonian equation limit, and the growth rate of the entropy in the initial stage is proportional to that of the coarse-grained Gibbs entropy of the corresponding classical system.

Block-diagonalization of infinite-volume lattice Hamiltonians with unbounded interactions

In this paper we extend the local iterative Lie-Schwinger block-diagonalization method – introduced in [8] for quantum lattice systems with bounded interactions in arbitrary dimension– to systems with unbounded interactions, i.e., systems of bosons. We study Hamiltonians that can be written as the sum of a gapped operator consisting of a sum of on-site terms and a perturbation given by relatively bounded (but unbounded) interaction potentials of short range multiplied by a real coupling constant t. For sufficiently small values of |t|independent of the size of the lattice, we prove that the spectral gap above the ground-state energy of such Hamiltonians remains strictly positive.As in [8], we iteratively construct a sequence of local block-diagonalization steps based on unitary conjugations of the original Hamiltonian and inspired by the Lie-Schwinger procedure. To control the supports of the effective potentials generated in the course of our block-diagonalization steps, we use methods introduced in [8] for Hamiltonians with bounded interactions potentials. However, due to the unboundedness of the interaction potentials, weighted operator norms must be introduced, and some of the steps of the inductive proof by which we control the weighted norms of the effective potentials require special care to cope with matrix elements of unbounded operators.We stress that no “large-field problems” appear in our construction. In this respect our operator methods turn out to be an efficient tool to separate the low-energy spectral region of the Hamiltonian from other spectral regions, where the unbounded nature of the interaction potentials would become manifest.

The second law of thermodynamics as a deterministic theorem for quantum spin systems

We review our approach to the second law of thermodynamics as a theorem asserting the growth of the mean (Gibbs–von Neumann) entropy of quantum spin systems undergoing automorphic (unitary) adiabatic transformations. Non-automorphic interactions with the environment, although known to produce on the average a strict reduction of the entropy of systems with finite number of degrees of freedom, are proved to conserve the mean entropy on the average. The results depend crucially on two properties of the mean entropy, proved by Robinson and Ruelle for classical systems and Lanford and Robinson for quantum lattice systems: upper semicontinuity and affinity.

An optimized infinite time-evolving block decimation algorithm for lattice systems

The infinite time-evolving block decimation algorithm (iTEBD) provides an efficient way to determine the ground state and dynamics of the quantum lattice systems in the thermodynamic limit. In this paper we suggest an optimized way to take the iTEBD calculation, which takes advantage of additional reduced decompositions to speed up the calculation. The numerical calculations show that for a comparable computation time our method provides more accurate results than the traditional iTEBD, especially for lattice systems with large on-site degrees of freedom.

Tensor-network approach to work statistics for one-dimensional quantum lattice systems

We introduce a tensor-network approach to calculate the statistics of work done on one-dimensional quantum lattice systems initially prepared in thermal equilibrium states. In this approach, the dynamics is simulated with time-evolving block decimation (TEBD), and the initial thermal equilibrium state is prepared either directly with TEBD or with minimally entangled typical thermal states, which generates a set of typical states representing the Gibbs canonical ensemble. As an illustrative example, we apply this approach to the Ising chain in mixed transverse and longitudinal fields. Under a prescribed protocol, the moment generating function for work distribution can be calculated, from which the quantum Jarzynski equality and the generalized quantum work relation involving a functional of an arbitrary observable are tested.

Local Noether theorem for quantum lattice systems and topological invariants of gapped states

We study generalizations of the Berry phase for quantum lattice systems in arbitrary dimensions. For a smooth family of gapped ground states in d dimensions, we define a closed d + 2-form on the parameter space, which generalizes the curvature of the Berry connection. Its cohomology class is a topological invariant of the family. When the family is equivariant under the action of a compact Lie group G, topological invariants take values in the equivariant cohomology of the parameter space. These invariants unify and generalize the Hall conductance and the Thouless pump. A key role in these constructions is played by a certain differential graded Fréchet–Lie algebra attached to any quantum lattice system. As a by-product, we describe ambiguities in charge densities and conserved currents for arbitrary lattice systems with rapidly decaying interactions.

Time evolution of an infinite projected entangled pair state: A gradient tensor update in the tangent space

Time evolution of an infinite 2D many body quantum lattice system can be described by the Suzuki-Trotter decomposition applied to the infinite projected entangled pair state (iPEPS). Each Trotter gate increases the bond dimension of the tensor network, $D$, that has to be truncated back in a way that minimizes a suitable error measure. This paper goes beyond simplified error measures -- like the one used in the full update (FU), the simple update (SU), and their intermediate neighborhood tensor update (NTU) -- and directly maximizes an overlap between the exact iPEPS with the increased bond dimension and the new iPEPS with the truncated one. The optimization is performed in a tangent space of the iPEPS variational manifold. This gradient tensor update (GTU) is benchmarked by a simulation of a sudden quench of a transverse field in the 2D quantum Ising model and the quantum Kibble-Zurek mechanism in the same 2D system.

Trotter Product Formulae for *-Automorphisms of Quantum Lattice Systems

We consider the dynamics tmapsto tau _t of an infinite quantum lattice system that is generated by a local interaction. If the interaction decomposes into a finite number of terms that are themselves local interactions, we show that tau _t can be efficiently approximated by a product of n automorphisms, each of them being an alternating product generated by the individual terms. For any integer m, we construct a product formula (in the spirit of Trotter) such that the approximation error scales as n^{-m}. Our bounds hold in norm, pointwise for algebra elements that are sufficiently well approximated by finite volume observables.

Local iterative block-diagonalization of gapped Hamiltonians: A new tool in singular perturbation theory

In this paper, the local iterative Lie–Schwinger block-diagonalization method, introduced and developed in our previous work for quantum chains, is extended to higher-dimensional quantum lattice systems with Hamiltonians that can be written as the sum of an unperturbed gapped operator, consisting of a sum of on-site terms, and a perturbation, consisting of bounded interaction potentials of short range multiplied by a real coupling constant t. Our goal is to prove that the spectral gap above the ground-state energy of such Hamiltonians persists for sufficiently small values of |t|, independently of the size of the lattice. New ideas and concepts are necessary to extend our method to systems in dimension d > 1: As in our earlier work, a sequence of local block-diagonalization steps based on judiciously chosen unitary conjugations of the original Hamiltonian is introduced. The supports of effective interaction potentials generated in the course of these block-diagonalization steps can be identified with what we call minimal rectangles contained in the lattice, a concept that serves to tackle combinatorial problems that arise in the course of iterating the block-diagonalization steps. For a given minimal rectangle, control of the effective interaction potentials generated in each block-diagonalization step with support in the given rectangle is achieved by exploiting a variety of rather subtle mechanisms, which include, for example, the use of weighted sums of paths consisting of overlapping rectangles and of large denominators, expressed in terms of sums of orthogonal projections, which serve to control analogous sums of projections in the numerators resulting from the unitary conjugations of the interaction potential terms involved in the local block-diagonalization step.

TeNeS: Tensor network solver for quantum lattice systems

TeNeS (Tensor Network Solver) [1,2] is a free/libre open-source software program package for calculating two-dimensional many-body quantum states based on the tensor network method and the corner transfer matrix renormalization group (CTMRG) method. This package calculates ground-state wavefunctions for user-defined Hamiltonians and evaluates user-defined physical quantities such as magnetization and correlation functions. For certain predefined models and lattices, there is a tool that makes it easy to generate input files. TeNeS uses an OpenMP/MPI hybrid parallelized tensor operation library and thus can perform large-scale calculations using massively parallel machines. Program summaryProgram Title: TeNeSCPC Library link to program files:https://doi.org/10.17632/psm26xxbvd.1Code Ocean capsule:https://codeocean.com/capsule/1510058Licensing provisions: GNU General Public License version 3Programming language:C++ and python3External routines/libraries: mptensorNature of problem: TeNeS calculates the approximate ground states and their properties of user-defined two-dimensional quantum lattice models using user-friendly input files. Numerically exact solutions of such tasks generally require an exponentially diverging computational time, whereas the error in the output of TeNeS is well controlled and can be reduced with a polynomial cost.Solution method: TeNeS implements the tensor networks method based on a tensor-product-state (TPS) wavefunction and the corner transfer matrix renormalization group method. TeNeS also supports massively parallel computing using the library mptensor, which implements OpenMP/MPI hybrid parallelized tensor operations.

Many-body localization in a quantum gas with long-range interactions and linear external potential

We study theoretically transitions between the localized and chaotic many-body regimes in one-dimensional quantum lattice systems with long-range couplings between particles and linear external potential. In terms of established criteria characterizing localization, we construct effective phase diagrams for several types of lattice systems with variable amplitude of the external linear tilt and interaction strength. By means of exact diagonalization and time-dependent variational principle numerical approaches we analyze system dynamics after quenches. Our results reveal that the Stark localization without any artificial source of disorder remains stable upon inclusion of long-range interactions.

Generalized quantum measurements with matrix product states: Entanglement phase transition and clusterization

We propose a method, based on matrix product states, for studying the time evolution of many-body quantum lattice systems under continuous and site-resolved measurement. Both the frequency and the strength of generalized measurements can be varied within our scheme, thus allowing us to explore the corresponding two-dimensional phase diagram. The method is applied to one-dimensional chains of nearest-neighbor interacting hard-core bosons. A transition from an entangling to a disentangling (area-law) phase is found. However, by resolving time-dependent density correlations in the monitored system, we find important differences between different regions at the phase boundary. In particular, we observe a peculiar phenomenon of measurement-induced particle clusterization that takes place only for frequent moderately strong measurements, but not for strong infrequent measurements.

Universal order-parameter and quantum phase transition for two-dimensional q-state quantum Potts model

We investigate quantum phase transitions for q-state quantum Potts models (q = 2,3,4) on a square lattice and for the Ising model on a honeycomb lattice by using the infinite projected entangled-pair state algorithm with a simplified updating scheme. We extend the universal order parameter to a two-dimensional lattice system, which allows us to explore quantum phase transitions with symmetry-broken order for any translation-invariant quantum lattice system of the symmetry group G. The universal order parameter is zero in the symmetric phase, and it ranges from zero to unity in the symmetry-broken phase. The ground-state fidelity per lattice site is computed, and a pinch point is identified on the fidelity surface near the critical point. The results offer another example highlighting the connection between (i) critical points for a quantum many-body system undergoing a quantum phase-transition and (ii) pinch points on a fidelity surface. In addition, we discuss three quantum coherence measures: the quantum Jensen–Shannon divergence, the relative entropy of coherence, and the l1 norm of coherence, which are singular at the critical point, thereby identifying quantum phase transitions.

On the equivalence of the KMS condition and the variational principle for quantum lattice systems with mean-field interactions

On the equivalence of the KMS condition and the variational principle for quantum lattice systems with mean-field interactions