Unifying Projected Entangled Pair States contractions

• Exact PEPS norm and expectation-value contractions have exponential cost in system size, limiting feasible simulations to relatively small tensor dimensions.The environment surrounding a tensor is also needed for local updates during optimization.Source: §1 · p. 2
• Different contraction assumptions can limit ground-state accuracy even when their computational cost permits large bond dimensions.The paper uses the environment representation as a unifying perspective for analyzing these limitations.Source: §1 · p. 2
• The authors introduce a cluster contraction scheme that interpolates between the cheapest, least precise method and the costliest, most precise method.Their finite-PEPS results also apply to iPEPS, whose repeated unit cell can affect the observed order.Source: §1 · p. 3
• The analysis finds Simple Update can yield less accurate ground-state energies when its assumed environment differs from the true one, while Single-Layer error has strong system-size dependence.The cluster strategy is presented for tensor updates and expectation values, with discussion of parallelization and iPEPS.Source: §1 · p. 3

• PEPS assign one tensor to each physical site, connect neighboring tensors by virtual indices of bond dimension D, and contract those indices to form state coefficients.For a regular subsystem of side length ℓ, the entanglement entropy is bounded by a quantity proportional to ℓ^(dim−1) log(D).Source: §2 · pp. 3–4
• The sandwich method approximates a two-dimensional norm or expectation-value network by successive one-dimensional MPS contractions.It contracts rows inward from opposite edges, truncating each MPS-MPO product to a fixed boundary bond dimension D′.Source: §2 · p. 4, Figure 2 · p. 5
• The boundary MPS acts as an MPO mapping virtual indices between ket and bra, and the approximate norm contraction costs O(D^4D′^3) + O(dD^6D′^2).In practice, D′ typically scales as D^2 independent of system size, giving total cost O(D^10).Source: §2 · pp. 4–5
• The boundary MPO can be interpreted as a virtual-boundary thermal state, and locality of its Hamiltonian would support efficient MPO approximation for gapped models.The text notes that such locality is suggested by evidence, but efficient expectation-value computation with PEPS has not been proven.Source: §2 · p. 5
• During imaginary-time optimization, alternating least squares updates one tensor at a time by solving N_l A_l = b_l, which requires contracting its environment approximately.The environment is the open network left after removing the tensor and its adjoint from the norm; its approximation affects algorithm accuracy and cost.Source: §2 · p. 6

3. Simple Update

• SU generalizes one-dimensional TEBD using Γ tensors and diagonal positive λ matrices; its update costs O(D5) but assumes a local, separable environment.This approximation can be rough when the true environment is not local and separable.Source: §3 · pp. 6–7
• The finite-PEPS tests use imaginary-time evolution, decreasing the time step τ successively until convergence.The tested Hamiltonians are the transverse-field Ising and Heisenberg models.Source: §3 · p. 7
• Below 10^-5 relative energy error at D=3 for Ising B=1.0, 3.0, and 4.0, but SU stalls for B=2.0 and Heisenberg; original PEPS beats SU for 4×4 Heisenberg.At D=4, the original method reaches energy per site −0.5739 versus exact −0.5743.Source: Figure 4 · p. 7, §3 · p. 8

4. Single-Layer

• SL contracts one sandwich layer using successive MPO-MPS operations and represents the contracted boundary as a positive purification MPO with bond dimension D′′ and purification bond d′.The purification boundary can capture nonlocal, non-separable correlations beyond SU's local environment.Source: §4 · p. 8
• Even at maximum purification bond, SL contraction error remains orders of magnitude above sandwich contraction at the same D′, and its cost can be higher for large D′ = D′′ ≫ D.For fixed d′, the SL error quickly converges with D′′ to a value determined by d′.Source: §4 · p. 9
• For D=4, d′=8, and D′′=10, SL norm error rises from 0.06 on 11×11 to 0.6 on 21×21, versus 10^-5 for sandwich contraction with D′=10 on both sizes.Figure 6 identifies the lattice sizes and SL truncation curves used in this comparison.Source: §4 · p. 9, Figure 6 · p. 10
• Setting d′ = D = D′′ gives O(D7) cost but is generally too restrictive for sandwich-comparable precision, and the required truncations depend strongly on system size.Unlike SL, the sandwich method's cost-controlling D′ can typically scale as D² with a prefactor dependent on the state rather than system size.Source: §4 · p. 10
• SL optimization allows only local isometries, not the most general boundary purification; optimizing the general form would require quartic costs and nonlinear equations.The authors therefore favor a general boundary MPO without explicitly imposing positivity for the problems considered.Source: §4 · pp. 10–11

5. Clusters

• CUδ uses a separable environment outside the cluster and a more precise contraction inside, interpolating from CU0, closely related to SU, to the original algorithm at δ = L−1.The cluster includes rows or columns within distance δ of the one being updated.Source: §5 · p. 11

5.1. Cluster size δ = 0: a generalized Simple Update

• CU0 optimizes a positive separable boundary MPO, generalizing SU; its final energies are very similar to SU, though it depends less on the initial state.SU applies a mean λSU across gate sets, while CU0 optimizes the separable environment for each update.Source: §5.1 · pp. 12, 14

5.2. From Simple to Full Update

• CU1 improves SU energies, while the Full Update (FU) lowers them further for D ≥ 4; the gains depend on system size and bond dimension.The FU sequentially approximates individual Trotter gates and is not identical to the original PEPS algorithm.Source: §5.2 · pp. 15, 17, Figure 10 · p. 16
• For fixed cluster size, increasing D′ consistently lowers the energy; D = 2 reaches its lowest energy with CU1, whereas D = 4 improves with larger clusters and reaches the FU value with CU4.The D′ value at which energy converges depends on both PEPS bond dimension D and cluster size.Source: §5.2 · p. 17, Figure 11 · p. 17
• For sufficiently large fixed D′, the distance between the cluster boundary MPO and the reference drops exponentially with cluster size until full-contraction accuracy is reached.D = 4 requires larger clusters and boundary bond dimensions than D = 2 to attain the same distance.Source: §5.2 · p. 18, Figure 12 · p. 18

5.3. Computation of expectation values

• For local expectation values, energy error likewise drops exponentially with cluster size at sufficiently large fixed D′, while larger PEPS bond dimensions require larger clusters and D′.The tested clusters surround local Hamiltonian terms on 20 × 20 PEPS.Source: §5.3 · pp. 18–19, Figure 13 · p. 19

5.4. Applicability to a parallel PEPS code and to iPEPS

• For finite PEPS, clusters beyond δ = 1 do not lower sequential cost, but parallel CU is estimated to yield L/δ speed-up with similar imaginary-time convergence.The estimate neglects sub-leading computations and processor communication.Source: §5.4 · pp. 19–20
• CUδ can also be applied to iPEPS, with cluster size and D′ controlling the cost and precision of tensor updates and expectation values.The authors also describe cluster-based Monte Carlo sampling of observables and energy gradients in iPEPS.Source: §5.4 · p. 21

6. Conclusions

• CUδ systematically increases environment precision from the Simple Update’s simplest representation to the full contraction’s most accurate environment.The paper identifies environment approximation as key to precision in PEPS contractions for both norms and expectation values.Source: §6 · p. 22
• Increasing cluster size improves environment-approximation precision exponentially, and clusters can accelerate local-observable computations.CU0 has the same computational cost as Simple Update while slightly improving it for the models analyzed; CU1 gives significantly better ground-state energies than SU at lower cost than FU.Source: §6 · p. 22
• For sequential finite-PEPS updates, every cluster size δ > 1 costs more than FU, though CUδ may support parallel PEPS algorithms and is argued to apply to iPEPS.FU can reuse intermediate calculations more efficiently in the sequential setting.Source: §6 · p. 22
• Environment approximation is a key determinant of precision for PEPS contractions of both norms and expectation values.The CUδ strategy provides a means to control that approximation accuracy across contractions.Source: §6 · p. 22

Appendix A

• The exact Heisenberg ground-state energy per site is −0.57432544 on 4 × 4 and −0.628655(2) on 10 × 10.The 4 × 4 value comes from exact diagonalization; the 10 × 10 value comes from Quantum Monte Carlo at T = 0.0001.Source: Appendix A · p. 23
• Tables A1–A4 report Heisenberg energy per site on 4 × 4 and 10 × 10 lattices for SU, CU0, CU1, and FU.Table A3 uses CU1 with D′ = D2.Source: Table A1 · p. 23, Table A2 · p. 23, Table A3 · p. 23
• Tables A5–A6 report Heisenberg energy per site on a 10 × 10 lattice for D = 2 and D = 4, respectively.Both tables refer to results presented in Figure 11.Source: Table A5 · p. 24, Table A6 · p. 24

Appendix B

• Imaginary-time evolution starts from a noisy separable D = 2 PEPS, proceeds through specified step-and-time-step schedules, and can grow D recursively by one.For Cluster and Full Update, D′ is typically 1, 2, or related to D as D, D2, and so on; the final PEPS at one D′ initializes increased D with that D′.Source: Appendix B · p. 25
• Energy evaluations use D′ = 100 on the final PEPS at the smallest time step; energy error compares it with an intermediate state.The intermediate state is either the half-evolution PEPS or the final PEPS at the immediately larger time step, depending on whether that larger step was run.Source: Appendix B · p. 25
• Figure 4’s schedule uses 1000 steps at τ = 0.1, 2000 at τ = 0.01, and 8000 at τ = 0.001 before its stated configuration.The configuration then specifies bond-dimension growth with 2000 steps at τ = 0.01 and 8000 at τ = 0.001.Source: Appendix B · p. 25
• Figure 6’s schedule runs 10000 steps at each of τ = 0.1, 0.01, and 0.001, then 20000 at τ = 0.0001 and 50000 at τ = 0.00001.The D = 3 and 4 results extend the final D = 2 PEPS from τ = 0.0001 with 20000 steps at that time step.Source: Appendix B · p. 25
• Table 1 evolves an 11 × 11 Ising PEPS with SU and then CU0; the displayed numbers are converged to machine precision.Each method uses 10000 steps at τ = 0.1 followed by 10000 at τ = 0.01.Source: Appendix B · p. 25
• For FU with D = 7 and D′ = 130, the state is propagated for 100 steps at τ = 0.01.The passage also gives a schedule fragment of 500 steps at τ = 0.01 and 1000 at τ = 0.001.Source: Appendix B · p. 26
• For Figure 11, CU1 at D = 4 uses 1000 steps at τ = 0.01 and 2000 at τ = 0.001; CU2 uses 1000 steps at each.CU3, CU4, and FU at D = 4 use 500 steps at τ = 0.01 and 1000 at τ = 0.001.Source: Appendix B · p. 26