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Variational Hamiltonian simulation for translational invariant systems via classical pre-processing

Refik Mansuroglu, Timo Eckstein, Ludwig Nützel, Samuel A. Wilkinson, Michael J. Hartmann

arXiv:2106.03680v5quant-ph

TL;DR

Large quantum-system time evolution is generally classically intractable, and NISQ implementations must control gate counts because gate imperfections reduce output fidelity. The method classically optimizes a variational time-evolution circuit on a small system and reuses its parameters for larger translationally invariant systems, including open-boundary variants. On small systems, the variational approach improves Trotter-Suzuki accuracy by up to 3 orders of magnitude in the cost function while reducing gate count by a factor of 2.

  • Problem

    Large quantum-system time evolution is generally classically intractable, and NISQ implementations must control gate counts because gate imperfections reduce output fidelity.

    Source: §1 · p. 2

  • Method

    The method classically optimizes a variational time-evolution circuit on a small system and reuses its parameters for larger translationally invariant systems, including open-boundary variants.

    Source: §2 · p. 3, §4 · p. 5, §4.1 · p. 5

  • Results

    On small systems, the variational approach improves Trotter-Suzuki accuracy by up to 3 orders of magnitude in the cost function while reducing gate count by a factor of 2.

    Source: Model · p. 3

  • Takeaways & Limitations

    The method provides a classical-assisted route to lower-gate-count quantum time evolution whose advantage persists for open boundaries and larger systems.

    Source: §6 · p. 11

    The operator-norm performance bound can become loose by a factor κ = 2^n for a pathological error concentrated in one poorly simulated state.

    Source: §3 · p. 5

Abstract

from arXiv · show

The simulation of time evolution of large quantum systems is a classically challenging and in general intractable task, making it a promising application for quantum computation. A Trotter-Suzuki approximation yields an implementation thereof, where a higher approximation accuracy can be traded for an increased gate count. In this work, we introduce a variational algorithm which uses solutions of classical optimizations to predict efficient quantum circuits for time evolution of translationally invariant quantum systems. Our strategy can improve upon the Trotter-Suzuki accuracy by several orders of magnitude. It translates into a reduction in gate count and hence gain in overall fidelity at the same algorithmic accuracy. This is important in NISQ-applications where the fidelity of the output state decays exponentially with the number of gates. The performance advantage of our classical assisted strategy can be extended to open boundaries with translational symmetry in the bulk. We can extrapolate our method to beyond classically simulatable system sizes, maintaining its total fidelity advantage over a Trotter-Suzuki approximation making it an interesting candidate for beyond classical time evolution.

1. Introduction

Hamiltonian time evolution is difficult to simulate classically because many-body Hilbert spaces grow exponentially, motivating quantum simulation. The paper addresses NISQ gate costs and state dependence by using classically optimized variational circuits for translationally invariant systems.

1. Introduction

  • Exponential Hilbert-space growth makes large-system time evolution generally intractable classically, motivating quantum computation for non-equilibrium dynamics.Source: §1 · p. 2
  • NISQ gate imperfections make reducing circuit depth important because output-state fidelity decreases exponentially with gate count.Source: §1 · p. 2
  • Higher-order Trotter-Suzuki formulas improve approximation accuracy but incur a constant factor 5^2q that grows with order q, disadvantaging moderate NISQ simulations.Source: §1 · p. 2
  • The proposed approach moves costly variational optimization to a classical pre-processor for translationally invariant Hamiltonians, avoiding repeated quantum sampling.Source: §1 · p. 2
  • Unlike methods tied to particular initial states, the method is state agnostic and can reuse the same circuit on arbitrary initial and intermediate states.Source: §1 · pp. 2–3

2. Main Results

The paper uses classical optimization on small systems to construct variational circuits for larger translationally invariant systems. The method also adapts to open boundaries by combining boundary and bulk parameter sets, while numerical tests show substantially improved accuracy and reduced gate counts.

2. Main Results

  • Classical pre-processing optimizes a small, solvable system and reuses its parameters to generate circuits for larger translationally invariant systems.Source: §2 · p. 3
  • For open boundaries, separate classical optimizations provide boundary and bulk parameters that are combined during upscaling, with two-dimensional extensions following the same procedure.Source: §2 · p. 3
  • The variational approach improves Trotter-Suzuki accuracy by up to 3 orders of magnitude in the cost function while halving the gate count on small systems.Source: Model · p. 3

Model.

Under NISQ conditions, the method is especially advantageous at larger simulation times for fixed circuit depth. In a 50-qubit example, it achieves substantially lower preparation infidelity than the best Trotter-Suzuki ansatz.

Model.

  • For fixed circuit depth, these improvements become decisive when the simulation time τ is large under imperfect-gate NISQ conditions.Source: Model · p. 3
  • At 50 qubits with 0.1% single-gate error and τ = 2.5, the algorithm achieves approximately 29% preparation infidelity, only 7% above the gate-error minimum.Source: Model · p. 3
  • The 29% preparation infidelity compares with approximately 57% for the best Trotter-Suzuki ansatz in the same numerical experiment.Source: Model · p. 3

3. Variational Optimization

The method replaces fixed Trotter coefficients with variational parameters optimized classically against exact time evolution, while retaining a circuit structure compatible with Hamiltonian terms. This optimization is state-independent but requires classical access to the full evolution operator, so its quantum advantage comes from transferring trained parameters to larger systems.

Trotter benchmark

  • First-order Trotter-Suzuki is used as the starting circuit structure, while higher-order formulas improve leading-order error at the cost of more gates.The paper uses first-order formulas as the main benchmark because higher-order approaches use exponentially more gates in q and do not improve total fidelity in the reported analysis.Source: §3 · pp. 3–4

Ansatz construction

  • The variational ansatz replaces Trotter coefficients with A·m trainable parameters θr,a across m layers and A interaction types.The gates are generated by Hamiltonian terms chosen for direct implementation on the target hardware, subject to any additional decomposition required by locality and architecture.Source: §3 · pp. 3–4

Optimization objective

  • The classical optimization minimizes a Frobenius-norm distance between the variational circuit and exact time evolution, normalized to remove system-size scaling.Because the benchmark uses the full evolution operator, the objective is agnostic to the initial state and is computed classically.Source: §3 · p. 5

Error interpretation

  • The cost function controls average full-Hilbert-space evolution error and can bound observable errors, although the bound depends on how errors are distributed across states.For concentrated errors κ=1, whereas a pathological single-state error can make the bound loose by κ=2^n; the paper notes concentrated errors are typical for Trotter algorithms.Source: §3 · p. 5

Classical pre-processing

  • Classical optimization alone provides no quantum advantage because it requires calculating the full exact evolution operator, but trained parameters can be applied to larger unseen systems.This transfer to larger systems is the NISQ-oriented use of the method.Source: §3 · p. 5

4. Upscaling System Size

The upscaling strategy exploits local translational invariance by copying optimized gates from a small system onto newly added qubits. For open boundaries, separately optimized boundary and periodic-bulk circuits are glued together, neglecting bulk boundary effects as an approximation.

4. Upscaling System Size

  • Translational invariance lets the method generalize a small-system variational solution to larger systems by reusing the same local gate parameters.The procedure is model-insensitive when the Hamiltonian remains translationally invariant.Source: §4 · p. 5, §4.1 · p. 6

4.1. Translationally Invariant Systems

  • For periodic boundaries, qubits are added successively by copying gates and reconnecting the former boundary interactions to the new qubits, thereby creating entanglement rather than tensoring independent systems.The construction can be repeated to extend an initially small circuit by K qubits.Source: §4.1 · pp. 5–6
  • The gluing procedure is equivalent to a coarser-level Trotterization and reduces exactly to ordinary Trotter-Suzuki decomposition when variational parameters are reset to Trotter parameters.This establishes how the variational construction relates to the underlying product-formula circuit.Source: §4.1 · p. 6
  • The expected upscaling accuracy degradation is linear in system size for local Hamiltonians because the leading approximation error is governed by commutators between Hamiltonian terms.The paper reports that numerical experiments confirm this expectation.Source: §4.1 · p. 6

4.2. Open Boundary Conditions

  • For open boundaries, a small open system supplies boundary parameters while a periodic system supplies bulk parameters, and the two circuits are joined by inserting periodic qubits into the open system’s middle.The approach is motivated by short-depth circuits, for which bulk qubits are approximately unaffected by the boundary choice.Source: §4.2 · p. 6
  • The open-boundary construction approximates dynamics by neglecting open-boundary effects on bulk qubits, with separate optimizations for the boundary and periodic bulk.Numerical experiments for one- and two-dimensional Transverse Field Ising Models indicate that only a few dedicated boundary gates can suffice.Source: §4.2 · p. 6

5. Transverse Field Ising Model

The TFIM experiments evaluate variational circuits optimized on small systems, their scaling to larger one- and two-dimensional lattices, repeated time steps, and observable dynamics against Trotter-Suzuki benchmarks.

5. Transverse Field Ising Model

  • The TFIM study uses a variational ansatz initialized from first-order Trotter-Suzuki structure and tests one- and two-dimensional periodic or open-boundary systems with gate imperfections.The cost function is evaluated exactly up to 12 qubits and approximated by random-input sampling for larger systems, reaching simulations up to 24 qubits.Source: §5 · p. 7

5.1. Single Time Step

  • More than 1000-fold one-dimensional and 100-fold two-dimensional cost improvements are maintained over larger extrapolated systems while using half as many gates.Linear fits indicate these improvement factors remain approximately constant beyond classically simulable system sizes.Source: §5.1 · p. 8
  • The variational single-step circuit delays algorithmic-error dominance to τ = 2.5 in one dimension, compared with earlier Trotter-Suzuki turning points.In two dimensions, the variational turning point occurs around τ ≈ 1.5.Source: Figure 2 · p. 8, §5.1 · p. 9
  • At 50 qubits with periodic one-dimensional boundaries, variational evolution increases NISQ infidelity by only 7% due to approximation infidelity, versus about 57% for the best Trotter-Suzuki ansatz.Source: §5.1 · p. 9

5.2. Multiple Time Steps

  • The state-agnostic variational sequence remains better than Trotter-Suzuki after 100 repetitions for most Jz and hx values, but Trotter-Suzuki can overtake when either coefficient is small.When Jz or hx approaches zero, Trotter-Suzuki becomes exact; in extreme cases it overtakes after about 50 repetitions, when gate infidelity dominates total error.Source: §5.2 · p. 9, Figure 3 · p. 10

5.3. Precision of Time-Evolved Observables

  • For the initial product state |+⟩^⊗N, the variational circuit predicts neighboring-spin two-point correlations closer to exact dynamics than all tested Trotter-Suzuki benchmarks.A three-layer variational circuit and six-layer Trotter-Suzuki circuit can have similar approximation error, but the latter uses twice as many gates and therefore suffers squared gate fidelity.Source: §5.3 · p. 9
  • Optimizing a larger time step 2τ = 5.0 with m = 6 layers reduces infidelity to 18.8%, compared with 44.5% for the best Trotter-Suzuki decomposition.Longer times and more parameters do not further improve long-time performance beyond this setting, while half-open boundaries can yield smaller infidelities because fewer gates contribute errors.Source: §5.3 · p. 11

6. Conclusion

The paper uses classical pre-processing to optimize variational time-evolution circuits on small systems and reuse them on larger translationally invariant systems. The method improves approximation fidelity and remains applicable to open boundaries and NISQ settings, while its gluing scheme and suitable optimization sizes require further study.

6. Conclusion

  • Classically optimized variational circuits can be reused on larger translationally invariant systems by gluing gate sequences together.The optimization is performed on a small tractable system before scaling the resulting parameters to larger systems.Source: §6 · p. 11
  • The method avoids quantum-hardware sampling costs and can achieve the same approximation fidelity as Trotter-Suzuki with fewer gates.The paper reports higher approximation fidelity at equal gate count, enabling reduced gate counts at comparable accuracy.Source: §6 · p. 11
  • The approach extends to open boundaries with translational symmetry in the bulk, where its observed advantages persist and can increase.Such systems are relevant to NISQ hardware because open boundaries are easier to realize.Source: §6 · p. 11
  • The gluing scheme may be limited by the relationship between locally optimized dynamics, system size, and the range of correlations.The authors identify further investigation of suitable optimization sizes and correlation ranges as an open direction.Source: §6 · p. 11
  • Quantum-optimized alternatives and randomized ansätze remain directions for comparison or extension of the classical pre-processing strategy.The paper notes that suitable optimization system sizes and the use of randomized ansätze require further investigation.Source: §6 · p. 11

Appendix A. Higher Order Trotter-Suzuki Decomposition

Higher-order Trotter-Suzuki formulas improve approximation accuracy but require substantially more gates, making gate imperfections decisive in NISQ comparisons. In the presented simulation, the variational algorithm has lower total infidelity than the higher-order benchmarks once gate fidelity is included.

Higher Order Trotter-Suzuki Decomposition

  • For the 3 × 6 Transverse Field Ising system, the variational algorithm beats the fourth-order product formula at m=1 using only 30% of its gates.The sixth- and eighth-order formulas improve approximation accuracy but require 50- and 250-fold more gates, respectively.Source: Figure A1 · p. 13
  • Higher-order Trotter-Suzuki formulas recursively improve the approximation and can be repeated with m>1, but repetition increases gate count by a factor of m.The qth-order construction is recursively defined, while increasing the Trotter number subdivides and repeats the sequence.Source: Appendix A · p. 13
  • The appendix restricts the main comparison to m∈{1,2} because higher-order formulas require gate counts that are unfavorable for NISQ-focused optimization.The relevant limitation is the growth of gate-related error with product-formula order.Source: Appendix A · pp. 13–14
  • With pg=99.9% gate fidelity, gate errors dominate after one time step, and the variational algorithm outruns all higher-order product formulas in total infidelity.Total infidelity combines imperfect gate fidelity with the approximate algorithmic fidelity.Source: Figure A1 · p. 13, Appendix A · p. 14

Appendix B. Smallest Two-dimensional Transverse Field Ising Model with Fully Open Boundary Conditions

The open-boundary construction combines optimized boundary and bulk blocks to scale a two-dimensional system while preserving the required parameter structure. Its minimal fully open glued configuration is 5 × 7, but its full cost function is beyond the stated numerical feasibility limit.

Smallest Two-dimensional Transverse Field Ising Model with Fully Open Boundary Conditions

  • The construction copies optimized parameters across boundary, half-open, and periodic bulk blocks to assemble larger open systems.Two-dimensional open systems require cuts along both dimensions and separate treatment of boundary and bulk regions.Source: Appendix B · p. 14
  • Periodic, half-open, and fully open optimization blocks have minimal sizes 3 × 3, 3 × 4, and 4 × 4, respectively.At least three qubits are needed along a periodic direction so the left and right neighbors remain distinct.Source: Appendix B · pp. 14–15
  • The smallest fully open glued configuration is 5 × 7 and consists of three optimized blocks plus a central row of three qubits.The construction combines half-open and fully open blocks with periodic bulk qubits.Source: Appendix B · p. 15
  • Although the individual blocks can be optimized classically, the 5 × 7 cost function cannot be evaluated because it exceeds numerical feasibility.Half-open systems of size 3 × (4 + K) remain numerically evaluable examples of the gluing procedure.Source: Appendix B · p. 15

Appendix C. The XY Model with a Transverse Field

The XY model with a transverse field provides a more non-commuting test of the variational method, showing favorable cost scaling but more volatile observable errors. Larger time steps and deeper variational circuits extend the useful simulation-time range at the cost of more complex classical optimization.

The XY Model with a Transverse Field

  • The XY model adds a YY interaction, increasing non-commuting terms and providing a test expected to have larger Trotter error.The appendix compares variational and Trotter-Suzuki cost values, time scaling, and observable errors.Source: Appendix C · p. 15
  • For 50 qubits, the variational cost is 10^-5 in one dimension versus approximately 0.01 for Trotter m=3 and 5 · 10^-3 for Trotter m=6.In two dimensions with 50 qubits, the corresponding cost values are of order 10^-3 for variational m=3, 10^-1 for Trotter m=3, and 10^-2 for Trotter m=6.Source: Appendix C · p. 15
  • The m=3 variational sequence remains ahead of Trotter-Suzuki for about 40 repetitions in one dimension and 20 in two dimensions.Optimizing a doubled time step τ=0.6 with m=6 layers can extend this limit, at the cost of a more complex classical optimization.Source: Appendix C · p. 15
  • Observable errors show no universal winner: m=6 Trotter-Suzuki sometimes performs better in two dimensions, while the variational sequence wins across the tested one-dimensional repetitions.The comparison is for the two-point correlation ⟨Z8Z9⟩ over 50 repetitions.Source: Appendix C · p. 15
  • Figure C1 compares cost scaling with system size and time, then compares relative two-point-correlation errors for one- and two-dimensional periodic systems.Panels a–b show cost versus system size, c–d show cost over repeated time steps, and e–f show observable error.Source: Figure C1 · p. 16
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