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Efficient tensor network simulation of IBM's Eagle kicked Ising experiment
Joseph Tindall, Matt Fishman, Miles Stoudenmire, Dries Sels
TL;DR
The paper addresses whether the kicked Ising dynamics on IBM’s heavy-hex lattice can be simulated accurately and efficiently classically. It uses a connectivity-matched tensor network contracted with belief propagation, achieving highly accurate results for the 127-qubit system and enabling long-time thermodynamic-limit simulations. The approach is most applicable when correlations are locally tree-like and loop effects are limited.
Problem
The work examines accurate classical simulation of noisy-processor kicked Ising dynamics where existing quantum and classical approaches provide an important comparison point.
Method
The authors use a heavy-hex-connectivity tensor-network ansatz, evolve it with bond-dimension truncation, and approximately contract it using belief propagation.
Results
The method gives ∼10^-14 accuracy for average magnetization on 127 qubits in under 10 seconds and remains within 10^-4 of exact results for a weight-17 observable after six Trotter steps.
Takeaways & Limitations
Tree-like correlations and increasing edge-environment separability make this approach effective for heavy-hex dynamics and support simulations of the infinite lattice to long times.
Takeaways & Limitations
The belief-propagation approximation assumes locally tree-like structure, while finite loops can still contribute errors and the error generally decreases with increasing loop size under gapped loop correlations.
Abstract
from arXiv · showhide
We report an accurate and efficient classical simulation of a kicked Ising quantum system on the heavy-hexagon lattice. A simulation of this system was recently performed on a 127 qubit quantum processor using noise mitigation techniques to enhance accuracy (Nature volume 618, p.~500-505 (2023)). Here we show that, by adopting a tensor network approach that reflects the geometry of the lattice and is approximately contracted using belief propagation, we can perform a classical simulation that is significantly more accurate and precise than the results obtained from the quantum processor and many other classical methods. We quantify the tree-like correlations of the wavefunction in order to explain the accuracy of our belief propagation-based approach. We also show how our method allows us to perform simulations of the system to long times in the thermodynamic limit, corresponding to a quantum computer with an infinite number of qubits. Our tensor network approach has broader applications for simulating the dynamics of quantum systems with tree-like correlations.
MAIN
The paper develops a belief-propagation tensor-network simulation tailored to heavy-hex connectivity, showing high accuracy for the 127-qubit Eagle problem and extending the approach to infinite lattices. Its accuracy is linked to increasingly tree-like correlations and separable edge environments as system size grows.
- Method: The tensor-network ansatz mirrors heavy-hex connectivity and uses belief propagation for approximate gate evolution and contraction.The approach limits entanglement through bond dimension χ and is designed for locally tree-like correlations.
- Small-system validation: For circuit depths n < 6, the method agrees perfectly with exact simulation because the circuit light cone does not reach around lattice loops.At larger depths, deviations are characterized by reduced separability of edge environments.
- Correlation structure: Increasing the number of heavy-hexagons improves belief-propagation accuracy at fixed χ, with highly separable edge environments persisting to 20 Trotter steps and larger systems.The authors connect this trend to increasing system size and possible interference effects between lattice loops.
- 127-qubit results: For the 127-qubit system, average magnetization reaches ∼10^-14 accuracy in under 10 seconds on a laptop, while the state requires 0.3GB of memory.The simulations reproduce highly accurate expectation values for higher-weight observables as well.
- Deeper circuits: At 6 Trotter steps, the χ = 500 result for a weight-17 observable is within 10^-4 of exact values across all plotted θh.At 20 steps, the authors extrapolate finite-χ results toward infinite bond dimension because exact data is unavailable.
- Infinite lattice: The method extends to the thermodynamic limit through a 5-site periodic unit cell, with close agreement between infinite- and finite-lattice magnetization indicating minimal boundary effects.The authors also observe rapid initial entanglement growth followed by substantially slower growth and possible long-time saturation.
- Scope and implications: The method scales as O(Lχ^4) per Trotter step and is expected to work best for systems with locally tree-like correlations and limited entanglement.The authors identify lattice topology as an exploitable resource beyond low entanglement and shallow circuits.
The Vidal gauge
The Vidal gauge imposes isometric constraints on grouped local tensors, with bond tensors living on network edges. These constraints support accurate gate application and expectation-value calculations.
- The tensor network state uses local tensors Γv on sites and bond tensors Λe on edges.
- In the Vidal gauge, suitable groupings of local tensors around each edge form isometric tensors.
- The isometric condition has the identity matrix as its right-hand side for each site and neighboring-edge grouping.
- Maintaining these gauge conditions is important for accuracy when applying two-site gates and computing expectation values.
- The same operations can be performed in an arbitrary gauge using fixed-point belief-propagation message tensors, so the Vidal gauge is not strictly necessary.
The simple update procedure
The simple update applies a two-site gate by combining neighboring tensors and bonds, decomposing the composite with an SVD, truncating the updated bond, and restoring the local tensor structure.
- The simple update combines two neighboring site tensors, their bond tensors, and the gate into a composite tensor Θv,w.
Measuring expectation values within the Vidal
Single-site expectation values are computed by absorbing neighboring bonds into the site tensor and contracting with the observable, while belief propagation approximates the surrounding environment.
- To measure ⟨Ov⟩, neighboring bond tensors are absorbed into the on-site tensor Γv.
- The resulting tensor is contracted with its conjugate while inserting Ov along the contracted physical index.
- The expectation-value approximation treats the site environment as a tensor product of environments supplied by neighboring sites.
- In the Vidal gauge, this approximation is equivalent to using fixed-point belief-propagation messages as environment approximations in an arbitrary gauge.
- For locally tree-like networks, the environment approximation can provide very good local-observable estimates.
Belief propagation on a tensor network state
Belief propagation approximates tensor-network environments with directed message tensors, supports gauge transformations into the Vidal gauge, and is used repeatedly during simulation to maintain accuracy.
- Belief propagation approximates graphical-model marginals and has been adapted to tensor-network contraction.
- For BP on a Vidal-gauge state, square roots of bond tensors are absorbed into the local tensors before forming the norm network.
- Directed message tensors Mv,vi are defined on network edges, with directions distinguishing messages between neighboring tensors.
- Self-consistent message equations are iterated until convergence using contractions over all neighboring tensors except the recipient.
- Converged messages provide rank-one environment approximations for expectation values and can define gauge transformations into the Vidal gauge.
- The simulations perform BP gauging after every Trotter step to maintain accuracy, while shorter circuits may not require it for comparable accuracy.
Estimating the error of BP for general tensor
The paper estimates belief-propagation error by measuring how separable an edge environment is after cutting and contracting the tensor-network norm network. For gapped loop correlations, the error is expected to decrease exponentially with the smallest loop size.
- Edge-environment construction: The edge environment is obtained by cutting one network edge, contracting all other indices, and forming a χ^2 × χ^2 matrix.This construction supports singular-value analysis of the cut norm network.
- BP approximation: Belief propagation assumes the edge environment is rank 1, with singular-value spectrum {1, 0, ..., 0}.Deviation from this spectrum quantifies the failure of the rank-1 approximation.
- Error measure: The BP error along an edge is estimated from the index of separability, computed from the singular values in a fixed symmetric gauge.The edge choice generally does not qualitatively change the observed results for this model.
- Loop-size scaling: For a translationally invariant periodic ring with gapped transfer-matrix spectrum, BP error scales as O(exp(−cl)) with loop size l.The same exponential decrease is expected generically when loop correlations are gapped and the smallest loop size is l.
Computing resources and software packages
The numerical work uses publicly available Julia tensor-network software, with tensor-network diagrams generated separately. Reported 127-qubit data are available online, while Fig. 3 resource measurements use a MacBook M1 Pro.
- Software: The numerical results were produced with ITensorNetworks.jl, a publicly available Julia package for manipulating tensor-network states of arbitrary geometry.The package supports gauging, contracting, partitioning, and evolving tensor networks.
- Software: ITensorNetworks.jl is built on ITensors.jl, which provides the basic tensor operations.
- Visualization: Tensor-network diagrams were produced with the publicly available Graph-Tikz.jl Julia package for graph visualization.
- Resources: Fig. 3 timings and memory usage were measured on a MacBook M1 Pro, while the code was also run on a Rusty computing cluster.Data for the 127-qubit simulations are available online.
I. APPENDIX A: COMPARISON OF METHODS FOR CALCULATING Z62
The appendix compares BP-evolved tensor-network results for ⟨Z62⟩ after 20 Trotter steps with quantum and classical approaches. Bond-dimension extrapolation brings the BP results into close agreement with other accurate methods, while one-dimensional ansätze require larger bond dimensions.
- Comparison setup: BP-evolved TNS results at χ = 200 and χ = 500 are compared with an extrapolation to χ →∞ for ⟨Z62⟩ after 20 Trotter steps.The shaded region represents the difference between χ = 500 data and the infinite-bond-dimension extrapolation.
- BP-TNS behavior: Finite-bond-dimension BP-TNS results overshoot the inferred true result for π/4 ≲ θh ≲ 7π/16, an effect resolved by bond-dimension extrapolation.
- Method comparison: Extrapolated BP-evolved TNS results closely agree with a mixed BP-evolved TNS-TNO method, while truncated Pauli-string results provide another approximation to the Heisenberg evolution.
- One-dimensional methods: MPS and MPO methods based on one-dimensional ansätze require much larger bond dimensions than more general tensor-network approaches for this problem.The appendix highlights this limitation when comparing MPS data and MPO-based simulations.
- Contraction accuracy: BP contraction is considered sufficiently accurate for this model based on comparisons with MPS and analyses of the BP approximation using exact and boundary-MPS contractions.
- Compared approaches: Figure 8 compares the Eagle processor with MPS, isoTNS, BP-evolved TNS, mixed TNS-TNO, sparse Pauli dynamics, MPO, 31-qubit simulation, and TNO methods.
- Additional baselines: A 31-qubit full-state calculation and dissipative mean-field results are included as additional comparison points, with the latter reported as relatively accurate at approximately 5 Trotter steps.
APPENDIX B: MPS CALCULATIONS
The appendix benchmarks MPS simulations using a heavy-hexagon site ordering, MPO evolution, truncation, and light-cone depth reduction. It estimates accumulated truncation error from discarded singular values across MPO-MPS applications.
- MPS ordering: The MPS benchmark maps the 127 Eagle qubits directly to MPS sites using an ordering distinct from the one in Ref..The alternative ordering is illustrated in Fig. 9 alongside the reference ordering.
- MPS evolution: The MPS is evolved by successive MPO applications, truncating after each application to a maximum bond dimension χ.Single-site gates are applied exactly at the start of each Trotter step.
- Light-cone reduction: Light-cone depth reduction omits MPO two-site gates outside the remaining measurement light cone.This reduces the gates included when only n′ Trotter steps remain before measurement.
- Error estimate: The approximate simulation error sums discarded singular values ϵi from each MPO applied during the n-step simulation.The number of MPO-MPS applications can be less than 5n because of light-cone depth reduction.
- Error estimate: The per-gate error estimate is approximately the overlap error between exact gate application and the truncated state.
APPENDIX C: BOUNDARY MPS FOR THE HEAVY-HEX LATTICE
The appendix uses a generalized boundary-MPS contraction to approximate edge environments and assess belief-propagation errors on large heavy-hex lattices. Comparisons of magnetization from BP and boundary MPS show differences of about 10^-4 for the largest lattices, decreasing with system size.
- Edge-environment approximation: A boundary-MPS-like method generalized to arbitrary network structures approximately contracts the norm network to compute edge environments for lattices too large for exact contraction.The method contracts pairs of lattice rows as MPSs and reduces the calculation to MPSs incident to the chosen row.
- Numerical comparison: For the largest lattices, BP and boundary-MPS magnetization values differ by approximately 10^-4.This comparison quantifies the discrepancy between the two approximate evaluation methods for the parameters studied.
- Numerical comparison: The difference between BP and boundary-MPS evaluations generally decreases as system size increases.This trend is observed in the magnetization comparisons presented for the considered parameters.
- Numerical comparison: Figure 11 compares single-site magnetization from BP and boundary-MPS evaluations of the BP-evolved tensor network across heavy-hex lattice sizes.The figure also reports relative differences and a BP error estimate based on Eq. (12).