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Gradient methods for variational optimization of projected entangled-pair states

Laurens Vanderstraeten, Jutho Haegeman, Philippe Corboz, Frank Verstraete

arXiv:1606.09170v2cond-mat.str-elquant-ph

TL;DR

Finding optimal states in the PEPS manifold is challenging because evaluating the global energy gradient is difficult. The paper uses a conjugate-gradient variational method with effective corner environments, and reports lower energies and better order parameters than imaginary-time methods.

  • Problem

    For translation-invariant PEPS, evaluating the global energy gradient requires accounting for local and nonlocal contributions and is highly nontrivial.

    Source: §III · p. 7

  • Method

    The method optimizes the global PEPS energy with conjugate gradients, using effective corner environments to evaluate the gradient.

    Source: §I · p. 2, §III · p. 7

  • Results

    The variational search systematically finds lower energies than imaginary-time algorithms with local truncations, and at the same bond dimension obtains lower energies and better order parameters for the XY and isotropic Heisenberg models.

    Source: §IV · p. 11, §V · p. 13

  • Takeaways & Limitations

    The framework also supports global symmetries, structure factors, and energy variance as an unbiased measure of variational error.

    Source: §V · p. 13

    The PEPS ansatz is not guaranteed a priori to provide an accurate parametrization.

    Source: B. The energy variance · p. 8

Abstract

from arXiv · show

We present a conjugate-gradient method for the ground-state optimization of projected entangled-pair states (PEPS) in the thermodynamic limit, as a direct implementation of the variational principle within the PEPS manifold. Our optimization is based on an efficient and accurate evaluation of the gradient of the global energy functional by using effective corner environments, and is robust with respect to the initial starting points. It has the additional advantage that physical and virtual symmetries can be straightforwardly implemented. We provide the tools to compute static structure factors directly in momentum space, as well as the variance of the Hamiltonian. We benchmark our method on Ising and Heisenberg models, and show a significant improvement on the energies and order parameters as compared to algorithms based on imaginary-time evolution.

I. INTRODUCTION

Simulating two-dimensional quantum systems is difficult, while infinite-PEPS optimization poses a challenging nonlinear energy-minimization problem. This paper presents a gradient-based variational PEPS algorithm as an alternative to imaginary-time evolution.

I. INTRODUCTION

  • Strong quantum correlations make low-dimensional systems difficult to simulate, and perturbation theory often fails in these regimes.The introduction emphasizes two-dimensional systems as a particularly important setting because many quantum phenomena there remain unexplored.Source: §I · p. 1
  • Variational methods are motivated by the limitations of Monte Carlo sampling and exact diagonalization, while PEPS offer a candidate ansatz for two-dimensional ground states.PEPS can be formulated directly in the thermodynamic limit, focusing on bulk physics without finite-size or boundary effects.Source: §I · p. 1
  • Optimizing infinite PEPS requires minimizing a highly nonlinear energy functional whose gradient is difficult to evaluate.Existing imaginary-time algorithms evolve a trial state with e^-τH, use a Trotter-Suzuki decomposition, and truncate the bond dimension after each step.Source: §I · p. 1
  • Imaginary-time truncation methods do not guarantee the globally optimal PEPS, and simple-update neglect of the environment can be poor for strongly correlated systems.Full-update methods account for the wave function but require inversion of an effective environment that can be badly conditioned; their local truncations are not guaranteed to maximize global overlap.Source: §I · pp. 1–2
  • The paper optimizes the global PEPS energy with conjugate gradients and a corner-environment contraction scheme, providing a clear convergence criterion.The approach also facilitates physical symmetries and computes energy variance, two-point correlations, and static structure factors.Source: §I · p. 2
  • The paper benchmarks its method on the transverse Ising, XY, and isotropic Heisenberg models.The introduction also previews possible extensions and applications.Source: §I · p. 2

II. EFFECTIVE ENVIRONMENTS AND TWO-POINT CORRELATION FUNCTIONS

A PEPS assigns a tensor to each site of an infinite square lattice, contracts virtual indices, and leaves physical indices open to encode spin configurations. Its norm and observables require contraction of an infinite tensor network, generally approximately.

II. EFFECTIVE ENVIRONMENTS AND TWO-POINT CORRELATION FUNCTIONS

  • A PEPS associates a five-legged tensor A with every site, contracts the virtual indices, and leaves physical indices open for spin configurations.The virtual degrees of freedom carry quantum correlations, while their dimension D tunes the variational class.Source: §II · p. 2
  • The PEPS norm is represented by an infinite double-layer tensor network, whose contraction is generally approximate.In the double-layer representation, each block contracts A with its conjugate over the physical index, and ket and bra virtual indices are grouped.Source: §II · pp. 2–3
  • Contracting the infinite network enables evaluation of the PEPS norm, expectation values, and correlation functions.The contraction is the evaluation of the infinite tensor network denoted by C2(. . . ).Source: §II · pp. 2–3

A. The linear transfer matrix

The linear transfer matrix reduces contraction of the two-dimensional PEPS network to a one-dimensional fixed-point problem. An MPS approximation to that fixed point, with tunable bond dimension, supports norm and local-observable calculations.

A. The linear transfer matrix

  • Successive applications of the linear transfer matrix reduce the infinite PEPS norm contraction to its leading eigenvalue and associated fixed point.The fixed point contains information about correlations in a half-infinite part of the lattice.Source: A. The linear transfer matrix · p. 3
  • The transfer matrix carries PEPS correlations from one row to the next, and its effective one-dimensional network can be evaluated through leading left and right fixed points.The normalization of the fixed-point MPS tensors determines the eigenvalue scale, which can be set to one.Source: A. The linear transfer matrix · pp. 3–4
  • The fixed point is generally approximated by an MPS with tunable bond dimension χ, allowing contraction accuracy to be improved.An exact fixed-point representation is possible only in special cases.Source: A. The linear transfer matrix · p. 3
  • Rescaling the PEPS tensor so the leading transfer-matrix eigenvalue is unity normalizes the PEPS.The MPS fixed point then supports computation of a local operator expectation value.Source: A. The linear transfer matrix · p. 3

B. The corner transfer matrix

The corner transfer matrix method compresses regions of the infinite network into tensors that form an effective one-site environment. Extending the environment allows evaluation of nonlocal operators, while a corner-MPS ansatz reduces its fixed-point problem to an efficient eigenvalue equation.

B. The corner transfer matrix

  • Corner transfer matrices compress network corners, while row tensors capture the infinite rows and together form an effective one-site environment.This environment supports computation of the PEPS norm or local expectation values.Source: B. The corner transfer matrix · p. 4
  • Leaving the region above the corner uncompressed extends the environment to evaluate two-point correlations and operators at arbitrary lattice locations.The scheme extends the corner-environment construction introduced for nonlocal expectations.Source: B. The corner transfer matrix · p. 4
  • The corner environment is found from a fixed-point equation, approximating the full fixed point with an MPS and an extra corner tensor.The resulting linear equation for the corner matrix is a simple eigenvalue problem that can be solved efficiently.Source: B. The corner transfer matrix · p. 4
  • Far from the corner, the fixed-point equation reduces to that of the linear transfer matrix, motivating the MPS approximation.The corner matrix captures the effect of the corner shape, and the approach uses MPS tensors from the linear-transfer-matrix fixed points.Source: B. The corner transfer matrix · p. 4

C. Channel environments

Channel environments use transfer-matrix fixed points and corner tensors to contract PEPS norms, expectation values, and correlations. They also provide a route to arbitrary two-point and three-point correlation functions.

C. Channel environments

  • With fixed points in all directions and four corner tensors, the network can be contracted for the norm, local expectation values, and correlation functions.The transfer-matrix eigenvalue is set to unity; corner tensors can then be rescaled to normalize the state.Source: C. Channel environments · pp. 4–5
  • Channel fixed points are largest-eigenvalue eigenvectors, with the left-right fixed-point inner product set to one and the leading eigenvector projected out for later use.The fixed points are defined for the top channel and the other direction.Source: C. Channel environments · p. 5
  • Arbitrary two-point correlation functions at generic lattice positions can be computed straightforwardly with the channel environment.The construction also evaluates three-point functions by orienting the corners appropriately.Source: C. Channel environments · p. 5
  • Finding transfer-matrix fixed points is the hardest step and scales as O(χ^3D^4 + χ^2D^6); corner and channel fixed points have similar scaling but are computed only once.D is the PEPS bond dimension and χ the fixed-point bond dimension.Source: C. Channel environments · p. 5

D. Static structure factor

The corner environment computes the static structure factor directly in momentum space by summing connected correlations across relative operator positions. Channel-wise geometric-series resummation handles the infinite contributions, alongside a same-site term.

D. Static structure factor

  • The static structure factor s(q) is computed directly in momentum space from connected correlations, with operators defined to have zero ground-state expectation value.The full expression also includes the contribution where both operators act at the same site.Source: D. Static structure factor · pp. 5–6
  • Moving the operators independently through the channels accounts for all relative positions, and the resulting infinite sums are resummed as geometric series.The momentum-resolved operator captures the superposition of positions within a channel.Source: D. Static structure factor · pp. 5–6
  • The channel inverse is well defined because the component along the channel fixed point is zero after removing the disconnected correlation contribution.The geometric series converges for every momentum.Source: D. Static structure factor · p. 6
  • Structure-factor evaluation scales as O(χ^3D^4 + χ^2D^6), with the infinite channel sum solved by an iterative linear solver.The same-site operator contribution is included in the full expression.Source: D. Static structure factor · p. 6

III. VARIATIONAL CONJUGATE-GRADIENT METHOD

The method minimizes the PEPS energy over a nonlinear variational class in the thermodynamic limit. It uses the energy gradient in a conjugate-gradient search, with a gradient-norm convergence criterion and the option to restrict the search to symmetry subspaces.

III. VARIATIONAL CONJUGATE-GRADIENT METHOD

  • For translation-invariant PEPS, optimizing the ground-state approximation is a highly nonlinear minimization of the energy functional.The discussion assumes nearest-neighbor interactions and a square lattice, while noting extensions to other lattices, larger unit cells, or longer-range Hamiltonians.Source: §III · pp. 6–7
  • The method evaluates the gradient by summing local and non-local energy changes, using the channel environment to compute this zero-momentum-like quantity efficiently.The gradient is a nontrivial object for a translation-invariant PEPS.Source: §III · p. 7
  • Nonlinear conjugate gradient combines the current gradient with the previous search direction, and this work uses the Fletcher-Reeves prescription for its parameter.The step size is selected by a line search using bisection with an Armijo condition.Source: §III · p. 7
  • A sufficiently small gradient norm supplies a convergence criterion indicating that the energy cannot be further optimized.Source: §III · p. 7
  • The search can be confined to a variational subspace, including one imposing symmetries, and the reported results start from a random tensor.The number of variational parameters can also be controlled.Source: §III · p. 7

A. Computing the gradient

The energy gradient is obtained by differentiating the normalized PEPS energy with respect to bra tensors and summing their contributions. Channel contractions collect the infinite sum over Hamiltonian placements and open tensor positions.

A. Computing the gradient

  • The objective is a real function of complex-valued A, treated as independent variables A and Ā; the implementation normalizes the PEPS and shifts each Hamiltonian term by its expectation value.These choices simplify the gradient expression.Source: A. Computing the gradient · p. 7
  • Differentiating with respect to each bra tensor and summing over open tensor positions yields the gradient, with Hamiltonian terms and open spots propagated independently through the channels.The construction parallels the channel summation used for the structure factor.Source: A. Computing the gradient · pp. 7–8
  • A channel tensor sums Hamiltonian insertions across relative positions, and its momentum-zero channel inverse is well defined because the acted-on vector has no component along the fixed point.The Hamiltonian is redefined by subtracting its expectation value.Source: A. Computing the gradient · pp. 7–8
  • The differentiated Hamiltonian terms include distinct orientations because each interaction acts on two sites.Source: A. Computing the gradient · p. 8
  • Gradient evaluation scales as O(χ^3D^4 + χ^2D^6), and rotational invariance of A makes the gradient rotationally invariant as well.The hardest step is computing the infinite channel sum with an iterative linear solver.Source: A. Computing the gradient · p. 8

B. The energy variance

The section presents energy variance as an unbiased diagnostic for PEPS ground-state accuracy and describes its evaluation from nearest-neighbor terms. Rotational invariance simplifies the calculation, whose stated complexity exceeds that of the gradient.

B. The energy variance

  • Energy variance provides an unbiased extrapolation quantity, while increasing PEPS bond dimension is proposed as a reliability test because energy and order-parameter behavior with D is unclear.The variance measures how closely a variational wave function approximates a ground state or, more generally, an eigenstate.Source: B. The energy variance · p. 9
  • The energy variance is v = ⟨Ψ(A)| (H − e)^2 |Ψ(A)⟩, where e is the energy expectation value.It measures the extent to which the variational wave function approximates a ground state or other Hamiltonian eigenstate.Source: B. The energy variance · p. 9
  • Because the variance is a zero-momentum structure factor of the Hamiltonian, its evaluation uses geometric series, with normalized two-site-channel fixed points and a renormalized Hamiltonian.Renormalization makes disconnected contributions drop out and the stated inverse well defined; the blue tensor with χ^2D^4 elements is the most costly step.Source: B. The energy variance · p. 9
  • The energy variance per site is obtained by summing variance terms associated with the Hamiltonian’s horizontal and vertical nearest-neighbor terms.The construction first associates a variance term with each nearest-neighbor Hamiltonian term.Source: B. The energy variance · p. 9
  • For rotationally invariant PEPS tensors, horizontal and vertical contributions are equal; the variance evaluation scales as O(χ^3D^6), more than gradient evaluation.Without rotational invariance, the other diagrams can be obtained by rotating the displayed ones.Source: B. The energy variance · p. 10

IV. BENCHMARKS

The benchmarks show lower energies and better order parameters than imaginary-time evolution for the XY and isotropic Heisenberg models. They also assess Ising transition capture, optimization convergence, zero-variance energy extrapolation, and momentum-space structure factors.

IV. BENCHMARKS

  • At λ = 3.04, the variational search finds lower energies and better order parameters than full-update imaginary-time evolution, even as its Trotter error vanishes.The comparison uses a rotationally symmetric one-site PEPS for the variational method and a rotationally asymmetric two-site unit cell for imaginary-time evolution.Source: §IV · p. 10, Figure 2 · p. 11
  • λc ≈3.09 for D = 2 and λc ≈3.054 for D = 3; the Ising transition is captured accurately by D = 3, with only slight further accuracy gains expected from increasing D.The estimates locate the critical point at the maximum slope of the magnetization curve.Source: Figure 1 · p. 10, §IV · p. 10
  • For D = 2, the criterion ∥g∥≤10^-5 coincides with clear convergence of the energy and magnetization; near convergence, D = 3 requires χ ≈100 for gradient tolerance.Large corner-environment bond dimensions are needed close to convergence because many gradient terms undergo subtle cancellations.Source: Figure 3 · p. 11, §IV · p. 11
  • For the XY and isotropic Heisenberg models, the variational method reaches lower energies and better order parameters than imaginary-time evolution at the same bond dimension.The comparisons are reported after convergence and cover XY (Jz = 0) and Heisenberg (Jz = 1).Source: §IV · p. 11
  • The XY and Heisenberg benchmarks plot relative energy error and staggered magnetization against Monte Carlo results as bond dimension varies.The models are XY (Jz = 0) and the Heisenberg antiferromagnet (Jz = 1).Source: Figure 4 · p. 12, Figure 5 · p. 12
  • For the isotropic Heisenberg model, zero-variance extrapolation from D = 4, 5 improves the energy estimate by a factor of two.The variance is used to assess how well the variational states approximate the true ground state.Source: §IV · pp. 11–12, Figure 6 · p. 12
  • The Heisenberg structure factor’s regular parts are reproduced even at low bond dimensions, while its divergence and zero are better captured as D increases, though not smoothly.The divergence is around the X point and the zero is at the Γ point.Source: §IV · p. 12, Figure 7 · p. 12

V. CONCLUSIONS

The paper presents a direct variational optimization method for PEPS ground states and reports systematically lower energies than imaginary-time methods with local truncations. It also discusses symmetry handling, additional observables, and possible extensions of the framework.

V. CONCLUSIONS

  • Starting from a random PEPS tensor, the algorithm finds a variational minimum at fixed bond dimension using the energy gradient and a clear convergence criterion.The method is presented as a direct implementation of the variational principle.Source: §V · p. 13
  • The variational search systematically finds lower energies than algorithms based on imaginary-time evolution and local truncations.The authors describe their approach as complementary to other PEPS algorithms.Source: §V · p. 13
  • Global symmetries can be exploited, including virtual matrix product operator symmetries for topological order and restrictions to PEPS subclasses [14, 66–68].The authors state that symmetry exploitation should lead to more efficient simulations.Source: §V · p. 13
  • Using the same effective environment for energy and normalization could permit consistent regularization and efficient variational optimization on finite PEPS [7, 34, 69–73].This is proposed as a finite-lattice extension of the corner-environment method.Source: §V · p. 13
  • The framework computes experimentally relevant structure factors and energy variance, an unbiased measure of PEPS variational error.Variance extrapolations are not straightforward, but may contribute to improved energy extrapolations across bond dimensions.Source: §V · p. 13
  • The effective environment could also support Hessian evaluation for Newton or quasi-Newton optimization, while gauge fixing might improve efficiency [74].The paper presents conjugate-gradient search as a first step toward these more advanced schemes.Source: §V · p. 13
  • The authors suggest tangent-space PEPS methods may support real-time evolution and low-energy spectrum calculations for generic two-dimensional spin models [38, 39, 75, 76].These are presented as prospects rather than results demonstrated in this paper.Source: §V · p. 13

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