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Algorithms for finite Projected Entangled Pair States

Michael Lubasch, J. Ignacio Cirac, Mari-Carmen Bañuls

arXiv:1405.3259v2quant-phcond-mat.str-el

TL;DR

Finite PEPS can model two-dimensional systems, but their computational cost limits feasible simulations and convergence. This paper analyzes environment and update strategies and benchmarks finite-PEPS ground states, reporting results on lattices up to 21 × 21 sites.

  • Problem

    Finite PEPS can address non-translationally invariant settings, but their higher computational cost restricts feasible simulations and their convergence.

    Source: §I · p. 1

  • Method

    The paper analyzes cluster-environment accuracy and purification MPOs, develops gauge-fixed tensor updates, and benchmarks finite-PEPS calculations.

    Source: §1 · pp. 3–4, §2 · p. 9, §V · p. 13

  • Results

    Full Update results show energy and order parameter converging fast with increasing bond dimension, and Heisenberg correlator precisions at D = 6 and 7 previously attained with D = 16.

    Source: §V · p. 13

  • Takeaways & Limitations

    Finite PEPS at sizes used for finite-size scaling offer a feasible, unbiased alternative to iPEPS, while the methods can also be applied to iPEPS.

    Source: §V · p. 14

    Simple Update and small-cluster Cluster Update are less costly but do not give the best ground-state approximation at fixed D and yield limited correlation lengths.

    Source: §V · p. 13

Abstract

from arXiv · show

Projected Entangled Pair States (PEPS) are a promising ansatz for the study of strongly correlated quantum many-body systems in two dimensions. But due to their high computational cost, developing and improving PEPS algorithms is necessary to make the ansatz widely usable in practice. Here we analyze several algorithmic aspects of the method. On the one hand, we quantify the connection between the correlation length of the PEPS and the accuracy of its approximate contraction, and discuss how purifications can be used in the latter. On the other, we present algorithmic improvements for the update of the tensor that introduce drastic gains in the numerical conditioning and the efficiency of the algorithms. Finally, the state-of-the-art general PEPS code is benchmarked with the Heisenberg and quantum Ising models on lattices of up to $21 \times 21$ sites.

I. INTRODUCTION

Finite PEPS can represent non-translationally invariant systems and support finite-size scaling, but their computational cost has limited feasible tensor dimensions and convergence. This paper examines environment approximation and tensor updates, then benchmarks improved finite-PEPS algorithms.

I. INTRODUCTION

  • PEPS algorithms remain computationally costly relative to MPS, restricting feasible tensor dimensions and leaving results less converged than in one dimension.Finite PEPS avoid a predefined unit cell, but their implementation is more involved and running times grow with system size.Source: §I · p. 1
  • Finite PEPS allow a less biased finite-size-scaling approach and suit intrinsically non-translationally invariant problems such as optical-lattice experiments in inhomogeneous traps.The paper benchmarks state-of-the-art finite-PEPS algorithms on the Heisenberg and quantum Ising models.Source: §I · pp. 1–2
  • The paper analyzes environment approximation and tensor updates, relating cluster-contraction convergence to correlation length and proposing purification-based positive environments and stabilizing gauge choices.It also studies reduced tensors to speed finite-case computations and extends gauge choices to full-tensor updates.Source: §I · pp. 1–2

II. NOTATION AND PRELIMINARY CONCEPTS

  • A PEPS represents a many-body state by contracting site tensors, whose physical indices encode local degrees of freedom and virtual indices connect neighboring sites.For the square-lattice geometry considered, virtual-index dimension D limits the maximum block entropy according to an area law.Source: §II · p. 2
  • Ground-state algorithms use alternating least squares to update tensors site by site, while imaginary-time evolution minimizes the distance between the current PEPS and its Trotter-evolved target.The environment approximation supplies the effective matrix and vector for each local tensor equation; the update solves that equation and changes the PEPS.Source: §II · p. 2
  • Simple Update assumes a separable environment and uses simple SVDs, whereas Full Update uses a more accurate norm-environment approximation for sequential gate application.Simple Update is more efficient and stable but produces limited accuracy for finite PEPS.Source: §II · p. 2

III. ALGORITHMIC ASPECTS

The section examines finite-PEPS environment approximation and tensor-update strategies, including their physical implications and implementation trade-offs.

III. ALGORITHMIC ASPECTS

  • Cluster-contraction accuracy is connected to the state's correlation length, and purification MPOs provide a way to maintain a positive environment approximation.The section also considers the feasibility and cost of explicitly retaining that positivity.Source: §III · p. 3
  • For tensor updates, the paper proposes gauge choices for different schemes to improve numerical stability and uses reduced tensors to speed computations.Tensor normalization is another factor that can improve stability.Source: §III · p. 3
  • The algorithmic considerations are relevant to implementing finite-PEPS methods and have physical implications, particularly for environment contraction..Source: §III · p. 3

A. Environment approximation

The section describes boundary-MPO and cluster approximations for PEPS environments, relating cluster-contraction accuracy to the state’s correlation length. Cluster size and boundary bond dimension provide an accuracy–efficiency trade-off.

A. Environment approximation

  • Applying Trotter gates one at a time requires approximating the norm environment alone; the vector for each update uses an exact single-gate contraction.The original approach approximates separate environments for the norm and the full set of gates.Source: A. Environment approximation · p. 3
  • The original contraction approximates each bulk-row product with a boundary MPO of unchanged bond dimension D′, at cost O(dD6D′2) + O(D4D′3).For typical precision, D′ scales as D2, giving an overall contraction cost O(D10).Source: Figure 2 · p. 3, A. Environment approximation · p. 3

1. Accuracy of the cluster contraction

  • A cluster treats the considered row and neighboring rows more precisely than the outside environment, trading cluster size and D′ against cost and accuracy.For δ = 1, the outside separable positive boundary MPO costs O(dD5), while the cluster contraction costs O(dD5D′2); δ = 0 gives an SU-equivalent environment.Source: §1 · p. 3, Figure 3 · p. 4
  • The characteristic cluster size δ0 is approximately the correlation length ζ, and contraction error decays more steeply when correlations decay faster.The error and correlation decay were fit exponentially; contractions used D′ = 100 so the error was independent of D′.Source: §1 · pp. 3–4

2. Positive environment

Exact norm environments are positive Hermitian, but common approximate contractions need not preserve positivity. The section examines purification-based contractions, finding improved accuracy at increased computational cost.

2. Positive environment

  • The SL method preserves positivity and stability, but its environment error can exceed that of the original contraction by several orders of magnitude and depend strongly on system size.The proposed purification results suggest that SL’s error mainly reflects its restricted class of attainable purifications, rather than a small D′′ alone.Source: §2 · pp. 4–5
  • A good positive MPO approximation with moderate D′ does not guarantee that a small D′′ suffices, and single-layer operations do not find the most general purification at fixed D′′.The authors conclude that nonlinear optimization improves SL significantly but is too costly to use for their cluster and full contractions.Source: §2 · pp. 4–5
  • A double-layer purification fit replaces linear local equations with quartic cost functions and iteratively solved nonlinear equations.The method optimizes the purification tensors while retaining a boundary purification with virtual bond D′′ and purification bond d′.Source: §2 · p. 5, Figure 6 · p. 5
  • At fixed purification bond d′, relative norm error decreases fast with D′′; comparing 11 × 11 and 21 × 21 lattices shows no strong system-size dependence, while increasing d′ makes curves converge to original-contraction error.Source: §2 · p. 5

B. Tensor update

The tensor-update strategies reduce computational demands and improve numerical stability. Reduced-tensor updates can reach larger bond dimensions and produced the lowest energies reported in this work.

B. Tensor update

  • Sequentially applying Trotter gates while changing only the acted-on tensors lets all pair updates use a fixed environment.The neighboring-tensor environment is represented by a periodic boundary MPO.Source: B. Tensor update · p. 5, Figure 8 · p. 6

1. Reduced tensor

  • Reduced-tensor updates minimize the full-tensor cost function exactly only with a separable environment; they can enable larger bond dimensions, potentially compensating for fewer variational parameters.Forming Nred costs O(d4D4D′2) + O(d2D6D′2) + O(d2D4D′3), and its eigendecomposition costs O(d6D6).Source: B. Tensor update · pp. 5–6, §1 · p. 7

B. Tensor update

  • A proper gauge choice and tensor normalization drastically improve update stability, although PEPS cannot gauge the effective norm matrix to identity as open-boundary MPS can.The effective norm matrix is N_l.Source: B. Tensor update · p. 6

1. Reduced tensor

  • QR decomposition forms a reduced tensor containing the physical index and gated link; only this tensor changes during the update, reducing parameters from dD4 to d2D2.The unchanged tensor parts are contracted with a periodic MPO to form the reduced-pair environment, whose positive approximant is constructed from its Hermitian part and nonnegative eigenvalues.Source: §1 · p. 6, Figure 9 · p. 6
  • The less costly reduced-tensor update enabled larger bond dimensions and ultimately yielded the lowest energies, so it was used for the paper’s results.For D = 2 and 3, full-tensor updates gave better energies; at D = 4, the approaches’ energies were already very similar.Source: §1 · p. 7

2. Gauge fixing

The authors use environment-derived gauge transformations to improve PEPS tensor-update conditioning, then combine them with SVD initialization and ALS sweeping. They report better-conditioned norm matrices and faster ALS convergence, and extend the gauge strategy to full tensors and long-range interactions.

2. Gauge fixing

  • For separable environments, square roots of six positive semidefinite environment matrices are absorbed into the tensor pair before the SVD update.The stated leading update cost is O(d^6D^3) + O(d^2D^5), assuming d ≤ D^2.Source: §2 · p. 7
  • For non-separable pair environments, independent QR and LQ decompositions of the same environment tensor produce gauge transformations for both virtual bonds.The transformations improve conditioning for both reduced tensors; applying them to the tensors preserves the state.Source: §2 · p. 7, Figure 11 · p. 7
  • The reduced-tensor update initializes the pair by SVD, optimizes it with ALS and tensor gauge fixing, then gauges the converged pair's internal link.SVD initialization keeps the D largest singular values; the final gauge choice puts the two tensors on an equal footing.Source: §2 · pp. 7–8, Figure 12 · p. 8
  • Gauge fixing improves norm-matrix condition numbers by several orders of magnitude in all studied cases, and no instabilities were encountered after gauge fixing.The condition number gives only an upper bound on solution error: a large value does not imply low accuracy, but a small value implies high accuracy.Source: §2 · p. 8
  • Gauge fixing accelerates ALS convergence: the initial SVD alone reduces the cost function more than one ungauged sweep, and total reduction is also greater.For the Ising model, gauge fixing plus SVD yields a lower cost than ten ungauged sweeps, although further sweeping can help, especially for Heisenberg.Source: Figure 13 · p. 8, §2 · p. 8
  • Local gauge choices avoid diagonalizing the large norm matrix for non-adjacent tensors and are expected to extend the update to long-range interactions.The local norm matrix has size d^2D^2 × d^2D^2; simulations found comparable condition numbers to pair-environment gauging.Source: §2 · pp. 8–9
  • The gauge-fixing approach also improves conditioning and ALS convergence for full-tensor updates, and may apply to PEPO-based time evolution.The transformations are derived from the norm tensor alone and do not explicitly depend on the approximated operator.Source: §2 · p. 9

3. Stability issues

Rough environment approximations can make tensor updates less stable, precise, and efficient. The authors recommend positive-semidefinite norm approximants, pseudoinverses on a cutoff subspace, and consistent normalization and scaling.

3. Stability issues

  • Replace each non-Hermitian, non-positive-semidefinite norm matrix with its closest Hermitian approximant, then remove negative eigenvalues to obtain a positive-semidefinite approximant.This is especially relevant when the environment approximation is rough, such as with small clusters or boundary bond dimensions.Source: §3 · p. 9
  • Because norm matrices can have zero eigenvalues and an ill-conditioned positive subspace, construct their pseudoinverses using only eigenvalues above a cutoff.The cutoff defines which part of the norm-matrix spectrum contributes to the inverse.Source: §3 · p. 9
  • Normalize the PEPS after each set of Trotter gates and scale all tensors to share the same largest absolute element.Imaginary-time evolution changes the state norm; common tensor scaling also avoids very small or very large tensors.Source: §3 · p. 9

IV. PERFORMANCE OF FINITE PEPS

The benchmark simulations use the full-update algorithm with sequential Trotter gates and general boundary MPO contractions, reaching lattices up to 21 × 21. The convergence procedure uses successively smaller time steps and increasing bond dimension, with converged simple-update states serving as useful full-update starting points.

IV. PERFORMANCE OF FINITE PEPS

  • The best benchmark results use full updates of reduced tensors, sequential Trotter gates, and general boundary MPO contractions on lattices up to 21 × 21.The study also ran simple updates on the same systems.Source: §IV · p. 10

A. Convergence procedure

  • Imaginary-time evolution starts from a D = 2 PEPS, converges the energy at τ = 0.01 and then smaller time steps, and increases bond dimension from a previously converged state.The initial PEPS embeds a separable state and replaces zero entries with small random numbers.Source: A. Convergence procedure · p. 10
  • A converged simple-update PEPS is a useful full-update starting state, often requiring only a few further steps and smaller D′ during continued evolution.This holds when continuing with the full update at the same bond dimension.Source: A. Convergence procedure · p. 10
  • Reported energies and correlators use the smallest-time-step PEPS, with observable error estimated from the previous time step; contractions use D′ = 100, checked against D′ = 200.The authors considered D′ = 100 large enough to neglect contraction errors.Source: A. Convergence procedure · p. 10

B. Heisenberg model

For the Heisenberg model, full-update energies improve rapidly with bond dimension while simple-update energies saturate, and larger lattices yield better energies. Spin correlators converge toward the reference values as bond dimension increases, with correlation behavior also varying with cluster size and lattice size.

B. Heisenberg model

  • Full-update energy error decreases rapidly with D, while simple-update energies saturate; both methods give better energies on larger lattices.For D up to 6, the lowest simple-update energies fall between full-update values at D = 3 and 4.Source: Figure 14 · p. 10, B. Heisenberg model · p. 10
  • A small staggered field, switched off during evolution, improves convergence and lets the simple update avoid local minima and reach lower energies, especially on 20 × 20.For the full update it also permits lower D′ while low energies are reached with the field still on.Source: B. Heisenberg model · p. 10
  • −0.62637(2) per site is the reported 10 × 10, D = 4 energy, below the earlier −0.62515 result; at D = 6 the energy is −0.62774(1).The D = 4 result also beats the cited finite-system wave-function ansatz energies, while the best DMRG results remain more accurate.Source: B. Heisenberg model · p. 11
  • The spin-spin correlator is used to assess non-local accuracy, and its diagonal and vertical values were quantitatively similar in the converged PEPS.The authors note that this direction-independence follows from the PEPS ansatz and would be harder to reproduce with MPS in two dimensions.Source: B. Heisenberg model · p. 11
  • At fixed separation, correlator error decreases quickly with bond dimension, while at fixed D it grows with distance.At x = L/2, reported relative errors are approximately 0.003 for D = 6 on 10 × 10 and 0.01 for D = 5 on 14 × 14.Source: Figure 15 · p. 11, B. Heisenberg model · p. 11
  • The study reaches correlator precisions reported for D = 16 Monte Carlo results on 8 × 8 and 16 × 16 lattices using D = 6 and 7 on 10 × 10 and 14 × 14 lattices.The cited D = 16 errors are 0.003(2) and 0.013(2), respectively.Source: B. Heisenberg model · p. 11
  • Larger update clusters produce more slowly decaying correlation functions in the CUδ comparison.The authors relate this to the cluster-contraction error behaving like the PEPS correlation function.Source: Figure 16 · p. 12, B. Heisenberg model · p. 12
  • On the studied open-boundary finite lattices, correlations decay faster and energy accuracy improves on the 14 × 14 lattice than on 10 × 10.The authors interpret this as a slight decrease in the Heisenberg model's true correlation length with increasing lattice size.Source: B. Heisenberg model · p. 12

C. Quantum Ising model

The finite PEPS Ising benchmark estimates critical behavior on a 21 × 21 lattice and compares correlation lengths from Full and Simple Update. Its results are close to iPEPS, although the finite-size analysis is limited.

C. Quantum Ising model

  • The 21 × 21 lattice yields Bc ≈ 3.0 and β ≈ 0.34 without finite-size scaling, close to iPEPS estimates.The authors present the Ising calculations primarily as benchmarks, using 11 × 11 and 21 × 21 lattices at fields around B = 3.Source: C. Quantum Ising model · p. 12
  • For the 21 × 21 lattice at B = 3.0, the FU D = 4 correlation length is ≈ 2.6, versus ≈ 1.2 for SU D = 7.The 11 × 11 FU D = 4 correlation length reaches ≈ 4.3; the authors note they did not perform the extensive D′ convergence analysis used for Heisenberg.Source: C. Quantum Ising model · p. 13
  • The 21 × 21 FU correlation functions agree with the best iPEPS results, while the SU performs poorly where the correlation length should be large.The plotted correlations compare FU and SU bond dimensions across the 21 × 21 lattice at B = 3.0 and an 11 × 11 lattice at B = 2.8.Source: C. Quantum Ising model · pp. 12–13
  • General boundary MPOs can contract large clusters with high accuracy using a boundary bond dimension much smaller than the exact-contraction requirement D′ = D^2δ.The paper identifies long-correlation-length states as a setting where large clusters can be analyzed this way.Source: C. Quantum Ising model · p. 13

V. CONCLUSIONS

The paper combines environment and tensor-update improvements in finite PEPS algorithms and benchmarks them on large Heisenberg and Ising systems. Full Update results show strong accuracy at moderate bond dimensions, while small-cluster updates can limit correlation lengths.

V. CONCLUSIONS

  • The cluster strategy trades environment precision against computational cost, and its accuracy is fundamentally connected to the PEPS correlation length.Purification MPOs provide a positive environment approximation and overcome limitations of the Single-Layer algorithm; these techniques also apply to Cluster and Full Update and other positive-MPO settings.Source: §V · p. 13
  • The proposed update restricts variational parameters to reduced tensors for speed and uses gauge fixing to improve conditioning for both reduced and full tensors.Gauge fixing combined with a cheap SVD forms a simplified, fast update procedure.Source: §V · p. 13
  • The implementation combines sequential Trotter gates, general boundary MPO contraction, and reduced-tensor updates for finite PEPS imaginary-time evolution.The code was applied to Heisenberg and transverse-field quantum Ising ground-state searches.Source: §V · p. 13
  • Full Update calculations reach lattices up to 21 × 21 and bond dimensions up to D ≈ 7, 8, with fast energy and order-parameter convergence as bond dimension increases.For Heisenberg spin-spin correlator precision, D = 6 and 7 match what Ref.13 previously attained only at D = 16; 21 × 21 Ising estimates also agree well with iPEPS without finite-size scaling.Source: §V · p. 13
  • Small-cluster Simple and Cluster Updates are cheaper and support larger bond dimensions, but do not give the best fixed-D ground states and yield limited correlation lengths.This limitation matters especially for strongly correlated settings such as the Ising model near criticality; maximum bond dimension alone is not a sufficient measure of algorithmic power.Source: §V · pp. 13–14
  • Finite PEPS on finite-size-scaling lattices offer an unbiased alternative to iPEPS, and the methods may also help analyze unit-cell bias in iPEPS.The analysis used a generic implementation; adapting it to specific problem properties is expected to further enhance performance.Source: §V · p. 14

Appendix A: Purification approximations

The appendix compares linearization, conjugate gradient, and Newton approaches for optimizing purification MPO tensors. Newton performed best in the reported benchmark, and the authors propose a two-step strategy that uses it to refine an initial purification.

Appendix A: Purification approximations

  • Linearization solves equations for AlBl while treating Al and A*l independently, then damps each tensor change with a weight α.α must be large enough to avoid unnecessarily slow convergence but small enough to ensure decreasing cost; solving the linear equations involves a pseudoinverse.Source: Appendix A · p. 14
  • Conjugate gradient uses cost and gradient evaluations, while Newton additionally requires the cost-function Hessian and its pseudoinverse.The Newton step width is naturally given, unlike the heuristic α in linearization or the line search in conjugate gradient.Source: Appendix A · p. 14
  • Newton converged reliably for every tested d′ with few local updates per tensor, outperforming linearization and conjugate gradient in the purification benchmark.The benchmark fixed D′′ = 2, varied d′, and used a reference purification with D′′ = 4 and d′ = 4 from Ising SU states on 11 × 11 lattices.Source: Appendix A · p. 14
  • The proposed practical strategy first finds an optimal purification with the SL algorithm, then further optimizes it with Newton.The authors note that all methods benefit from initial variables close to the final solution.Source: Appendix A · pp. 14–15

Appendix B: Gauge fixing for the full tensors

For full-tensor updates with a non-separable environment, the appendix constructs gauge transformations from positive approximants of the left and right environments. It also reports condition-number comparisons and collects Heisenberg energy values alongside quantum Monte Carlo references.

Appendix B: Gauge fixing for the full tensors

  • For non-separable environments, the method independently computes positive approximants of the left and right D^4 × D^4 environments, then uses their square roots to construct bond gauge transformations.The inserted R−1R factors are absorbed into the full tensors and environment; the update initializes tensors before ALS sweeping.Source: Appendix B · p. 15, Figure 20 · p. 15
  • Gauge fixing drastically improves the full-update norm-matrix condition number compared with no fixing or using only the positive approximant.Table II reports mean condition numbers and standard deviations for D = 2 under the setting of Table I.Source: Appendix B · p. 15, Table II · p. 16

Appendix C: Finite PEPS energies

  • Heisenberg PEPS energies are compared with quantum Monte Carlo values, and the reported T = 10^-4 values and errors agree with T = 10^-3 within error bars.The reference quantum Monte Carlo energies are summarized in Table III.Source: Appendix C · p. 16, Table III · p. 16

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