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Self-Verifying Variational Quantum Simulation of the Lattice Schwinger Model

Christian Kokail, Christine Maier, Rick van Bijnen, Tiff Brydges, Manoj K. Joshi, Petar Jurcevic, Christine A. Muschik, Pietro Silvi, Rainer Blatt, Christian F. Roos, Peter Zoller

arXiv:1810.03421v2quant-phcond-mat.quant-gas

TL;DR

The paper addresses how to variationally simulate lattice gauge theories without directly realizing their target Hamiltonians and how to verify approximate quantum-simulation energies. It demonstrates a trapped-ion VQS of the lattice Schwinger model, obtaining 20-ion ground-state results with self-verification from Hamiltonian variances. The final 20-ion simulated ground-state fidelity approaches 0.8.

  • Problem

    Direct analog implementation of the lattice Schwinger model is challenging, while variational methods face scaling, parameter-growth, and measurement-cost limitations.

  • Method

    The experiment uses a classical-quantum variational feedback loop with a programmable trapped-ion simulator, symmetry-preserving trial states, global DIRECT optimisation, and a central data repository for post-processing Hamiltonian evaluations.

  • Results

    The experiment simulates the lattice Schwinger model for up to 20 qubits, and the final 20-ion quantum many-body ground-state fidelity approaches 0.8.

  • Takeaways & Limitations

    Measuring the target-Hamiltonian variance provides algorithmic error bars for approximate energies directly on the quantum device.

Abstract

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Hybrid classical-quantum algorithms aim at variationally solving optimisation problems, using a feedback loop between a classical computer and a quantum co-processor, while benefitting from quantum resources. Here we present experiments demonstrating self-verifying, hybrid, variational quantum simulation of lattice models in condensed matter and high-energy physics. Contrary to analog quantum simulation, this approach forgoes the requirement of realising the targeted Hamiltonian directly in the laboratory, thus allowing the study of a wide variety of previously intractable target models. Here, we focus on the Lattice Schwinger model, a gauge theory of 1D quantum electrodynamics. Our quantum co-processor is a programmable, trapped-ion analog quantum simulator with up to 20 qubits, capable of generating families of entangled trial states respecting symmetries of the target Hamiltonian. We determine ground states, energy gaps and, by measuring variances of the Schwinger Hamiltonian, we provide algorithmic error bars for energies, thus addressing the long-standing challenge of verifying quantum simulation.

I. VARIATIONAL QUANTUM SIMULATION OF THE SCHWINGER MODEL WITH TRAPPED-IONS

The lattice Schwinger model is simulated variationally using a programmable trapped-ion platform whose resource Hamiltonians generate symmetry-preserving trial states. The target Hamiltonian is evaluated through measured spin correlations rather than directly implemented.

  • Target – lattice Schwinger model: The lattice Schwinger model represents 1D quantum electrodynamics as an Abelian lattice gauge theory encoded on a spin-1/2 lattice.The Kogut-Susskind encoding maps fermionic configurations to alternating positron and electron spin occupations.
  • Target – lattice Schwinger model: Gauss’ law eliminates the electric-field degrees of freedom, producing an effective spin-1/2 Hamiltonian with long-range interactions.The long-range terms originate from squaring the electric-field operator expressed through Pauli operators.
  • Quantum resource – trapped-ion analog quantum simulator: A 20-ion trapped-ion analog simulator supplies entangling XY dynamics and local z rotations as quantum resources for variational-state preparation.The ions encode qubits, while the resource Hamiltonians generate highly entangled states and local spin rotations.
  • Quantum resource – trapped-ion analog quantum simulator: Trial states begin from Néel-ordered product states and are generated by alternating entangling and local light-shift layers.The initial Néel state represents the bare vacuum, and the circuit parameters specify the selected resource operations.
  • Symmetry-protecting variational circuits: Symmetry constraints keep the variational states in the relevant charge-conjugation and parity sector while reducing single-qubit-layer parameters to N/2.Approximate bulk translational symmetry can reduce parameters further through θ(j) = θ(j+2).
  • Measurement and verification: The target energy and its variance are reconstructed from projective measurements after local basis rotations, requiring 3 bases for ⟨HT⟩θ and 3N bases for ⟨H2T⟩θ.The same circuit operations preserve a decoherence-free subspace with respect to major experimental decoherence sources.

II. RESULTS

Experiments use variational quantum simulation to determine Schwinger-model ground states, verify energies through Hamiltonian variances, and probe a quantum phase transition with trapped-ion systems.

  • Optimisation procedure: The DIRECT global optimiser samples parameter-space cells, updates an internal energy-landscape model, and allocates up to 10^5 quantum-simulator calls with adaptive measurements.This search can continue exploring new regions even after finding an initial local minimum.
  • Ground-state determination: 20-ion variational states converged with six circuit layers and 15 parameters, while the final many-body ground-state fidelity approached 0.8.A 16-ion run reached fidelity approaching 0.9.
  • Ground-state determination: The 8-ion experiment reached E(θopt) = −3.24 ± 0.36 with fidelity approaching 0.95, corresponding to (11±18)% of the experimentally measured energy gap.The measured gap was ∆exp = 2.11 ± 0.24.
  • Self-verification: Hamiltonian-variance measurements provide algorithmic error bars that decrease toward eigenstate convergence, reaching E/∆exp = 0.64 ± 0.20 for the 8-ion run.The variance bound estimates the distance to the nearest exact eigenstate energy, while statistical error arises mainly from shot noise.
  • Quantum phase transition: The 8-ion phase-transition study varied the bare mass and detected the critical point near mc ∼−0.7 using the order parameter and second-order Rényi entropy.Convergence at the critical point was achieved with a five-layer circuit.

III. SUMMARY AND OUTLOOK

The authors demonstrate self-verifying variational quantum simulation of the lattice Schwinger model on up to 20 qubits. They argue that symmetry-aware circuits and approximate bulk translational invariance support scaling to larger lattice systems.

  • Self-verifying VQS was experimentally demonstrated for the lattice Schwinger model using a programmable trapped-ion analog quantum simulator.
  • The experiments considered systems of up to 20 qubits and globally optimised up to 15 parameters within a budget of up to 10^5 quantum-co-processor calls.
  • Symmetry-preserving circuits reduce variational parameters, while approximate bulk translational invariance offers a route toward scaling to large and infinite systems.

Appendix A: Global search algorithm with noisy cost function

The optimisation problem is a noisy, high-dimensional search over an energy landscape with multiple local minima. A modified DIRECT algorithm balances exploration and exploitation using hypercells, while a Gaussian-process metamodel represents the landscape within a finite measurement budget.

  • The variational optimiser searches a high-dimensional energy landscape revealed only through noisy quantum-observable measurements and containing multiple local minima.
  • DIRECT divides parameter space into hypercells and prioritises low-energy and large cells to balance exploitation with exploration.
  • The algorithm’s exploration stages ensure that it is guaranteed to ultimately find the global minimum, and it solves VQS optimisation efficiently up to dimension 20.
  • A Gaussian-process metamodel maintains an internal representation of the energy landscape using a squared-exponential covariance kernel and fitted correlation lengths.
  • The optimiser uses a finite measurement budget, beginning with relatively few samples at unexplored points before refining promising regions.
  • In the reported procedure, optimisation continues until the allocated resource budget is exhausted, rather than stopping at an energy-precision threshold or stationarity criterion.

Appendix B: Protecting symmetries of the Schwinger model

The Schwinger model’s charge and CP-related symmetries define the target subspace for the variational simulation. Resource Hamiltonians and single-qubit rotations are tailored to preserve these symmetries while reducing control parameters.

  • The spin-encoded Schwinger model retains global charge conservation and a CP-related symmetry inherited from the lattice gauge formulation.
  • The zero-excess-charge sector contains the Néel state representing the bare vacuum and is the symmetry sector used to target the quantum phase transition.
  • For even lattice length, CP combines spatial reflection with charge conjugation and spin flips, forming a Z2 transformation.
  • The target Hamiltonian is block-diagonal in charge sectors, and the zero-magnetisation sector splits into CP=+1 and CP=−1 blocks; simulations use the 0,+1 sector.
  • The native entangling resources approximately protect z-magnetisation when B≫max{|Jij|}, while reflection-symmetric couplings support CP protection.
  • Z-axis single-qubit rotations preserve magnetisation, and paired opposite-site angles preserve CP while reducing controls from 3N to N/2 for single-qubit layers.

Appendix C: Measurement Scheme

The measurement scheme evaluates the Schwinger energy and its variance by decomposing them into Pauli components measured in product bases. Symmetry can reduce the required bases, although the experiment’s approximate magnetisation protection increases the practical cost.

  • The closed-loop protocol requires measurements of both ⟨HT⟩ and ⟨H^2_T⟩ to optimise the energy and estimate its variance.
  • The three components of ⟨HT⟩ are measured in x, y, and z bases, requiring three bases independently of system size.
  • Evaluating ⟨H^2_T⟩ requires measurements of long-range, short-range, and anticommutator correlators assembled from product-basis projective measurements.
  • For anticommutator terms, rotating sites j and j+1 before measurement yields correlators for each j, requiring N−1 bases for the relevant family.
  • With perfect magnetisation protection, symmetry relations reduce the energy evaluation to 2 bases and the complete variance evaluation to 2N bases.
  • Because the trapped-ion platform protects magnetisation only approximately and input preparation can break it, the experiment uses 3 bases for ⟨HT⟩ and 3N for ⟨H^2_T⟩.

Appendix D: Parallelisation via Central Data Repository

The Central Data Repository stores measured correlations with their variational parameters, allowing previously collected data to be reused for different Hamiltonian parameters. This parallelisation provides a rough energy landscape that accelerates subsequent optimisation runs.

  • The Central Data Repository permanently associates correlation measurements with variational control parameters θ for later classical post-processing.Hamiltonian parameters enter only during post-processing, so the same prepared-state data can evaluate any Hamiltonian measurable in the available bases.
  • Changing the mass parameter m allows stored correlations to be re-evaluated without repeating the original measurements.The procedure fixes m, stores measured correlations, then reuses the repository for the next m value.
  • The reused data supplies a rough energy landscape before each new optimisation, locating the next global minimum much faster.

Appendix E: Energy gap estimation via quantum subspace expansion

Quantum subspace expansion estimates low-energy levels from measurements on an optimised variational ground state and a symmetry-preserving excitation subspace. The resulting generalised eigenproblem improves the ground-state estimate and yields an energy-gap estimator.

  • Quantum subspace expansion estimates inter-sector energy gaps using data acquired solely from the quantum processor.The method is adopted in the context of variational quantum eigensolvers.
  • The method constructs a symmetry-preserving subspace above the optimised variational ground state and measures effective-Hamiltonian and overlap matrices.The matrices are Heff_q,q′ = ⟨Ψ(θopt)|Ôq ĤT Ôq′|Ψ(θopt)⟩ and M_q,q′ = ⟨Ψ(θopt)|Ôq Ôq′|Ψ(θopt)⟩.
  • The generalised eigenproblem produces ordered eigenvalues λ^(k), with λ^(0) improving the ground-state energy and λ^(1) − λ^(0) estimating the energy gap.The quality depends on the chosen quantum subspace, with low-energy excitations typically giving substantial corrections near the lower spectrum.
  • The excitation operators include single nearest-neighbour electron-positron pair creation or annihilation processes that preserve global charge and are tailored to preserve CP symmetry.
  • The qubit-chain encoding illustrates long-range spin-spin interactions through curves whose thickness represents the coupling strengths c_i,j.The encoding maps particles into spins for an eight-site product state.
  • Reconstructing the full effective and overlap matrices requires measurement bases scaling as ∝N^2, while the generalized eigenproblem has dimension N/2 + 1.

Appendix F: Mapping of the Lattice Schwinger Model

The lattice Schwinger model maps one-dimensional U(1) quantum electrodynamics onto a spin lattice. Gauge-field degrees of freedom are eliminated using Gauss’ law, producing nearest-neighbour hopping and long-range density interactions before the Jordan-Wigner spin mapping.

  • The Schwinger model is a one-dimensional gauge field theory coupling a flavourless fermion field to an Abelian U(1)-symmetric gauge field.
  • The Kogut-Susskind formulation places gauge fields on lattice bonds and spinless fermion fields on lattice sites.Even and odd sites encode electron and positron fields, respectively.
  • Gauss’ law relates neighbouring electric fields to the fermion occupation and, with the left-boundary field fixed, determines the entire gauge-field configuration.
  • Gauge fields can therefore be eliminated as independent dynamical degrees of freedom and represented as collective operators on matter degrees of freedom.
  • After gauge transformation, fermionic hopping remains nearest-neighbour while density-density interactions become long range.
  • A Jordan-Wigner transformation maps the fermionic model into a spin Hamiltonian with the form used in the main text.

Appendix G: Variational quantum simulator workflow

The variational quantum simulator alternates classical parameter selection, quantum state preparation, measurement in multiple bases, and classical optimisation. Correlation samples are collected until every target-Hamiltonian component is covered, then noisy cost estimates guide the next iteration.

  • The target Hamiltonian is decomposed into Pauli operator strings, and the variational cost function is their weighted sum of expectation values.The prepared state is |Ψ(θ)⟩ = exp(−iθ_k Ĥ_R^(i_k))|Ψ0⟩ for a fixed-depth resource circuit.
  • A classical computer passes control variables θ to the controllable quantum device, which initializes |Ψ0⟩ and prepares |Ψ(θ)⟩ through programmable dynamics.
  • The prepared state is measured in projective bases, producing samples that simultaneously provide compatible correlator components.
  • Measurements are repeated across bases until one sample exists for every component q of the target Hamiltonian.
  • The procedure estimates the cost function F(θ) and its statistical error bar ΔF, with additional sampling directed toward components contributing most to the variance.
  • A global optimiser uses noisy cost values and their error bars to choose new controls, optionally revisiting earlier points to reduce uncertainty until the measurement budget is exhausted.

Appendix H: On Scalability

Numerical and analytical studies examine VQS scalability, showing how circuit depth depends on system size and proximity to the critical point while resource symmetries constrain targetability and can reduce variational complexity.

  • Numerical scalability: Infidelity is evaluated against exact ground states while varying qubit number, circuit depth, and distance from the critical point.For 5% infidelity or less, the required circuit depth nreq can be identified from these simulations.
  • Numerical scalability: Near the critical point, the required circuit depth grows approximately linearly with system size, whereas away from criticality it tends to saturate.The numerical study considers systems up to 12 sites and finds smooth dependence on N and δm.
  • Numerical scalability: The observed scaling suggests polynomial rather than exponential variational complexity at finite algorithmic error.This conclusion assumes a polynomially scaling search algorithm and is consistent with one-dimensional entanglement-area-law behavior.
  • Controllability and symmetries: Resource Hamiltonians generate a Lie algebra through real linear combinations and commutators, with Suzuki–Trotter and Baker–Campbell–Hausdorff constructions becoming exact with unlimited resources.The available dynamics are characterized by the closure of the resource Hamiltonians under these operations.
  • Controllability and symmetries: Variational preparation preserves resource symmetries, so target states must share the relevant symmetry content for controllability.Additional target symmetries can be incorporated into the resources, reducing the number of variational parameters per circuit layer.

Appendix J: Experimental details

The experiment uses a linear chain of 40Ca+ ions, with cooling and state preparation followed by layered single-qubit and entangling operations, automated optimisation, and basis-resolved fluorescence measurements.

  • Experimental system: Each calcium ion encodes a qubit in two long-lived electronic levels coupled through a 729 nm optical quadrupole transition.The ions are confined in a linear Paul trap.
  • Initialisation: Initialisation combines Doppler cooling, optical pumping, sideband cooling, and laser-induced light shifts to prepare a Néel state.Sideband cooling lasts 6 ms for eight ions and 11 ms for twenty ions.
  • Trial state preparation: Trial states are prepared with local z-axis rotations and bichromatic-laser entangling operations based on an XY Hamiltonian.The measured coupling-range exponent is α = 1.34 for eight ions and α = 0.98 for a larger chain.
  • Measurement: Energy and variance estimation use projective measurements in 3 and 24 bases, respectively, followed by spatially resolved fluorescence readout.For 20-ion measurements, post-selection retains measurements within the zero-magnetisation subspace.
  • Feedback loop: The automated classical–quantum feedback loop repeats the experimental sequence for 30 to 200 iterations before averaging and returning data to the classical optimiser.The classical computer generates the experimental control script after calibration.

Appendix K: Figures: Algorithmic error analysis and 16 ion optimisation

Appendix K examines algorithmic error through circuit-depth and mass dependence, and presents a 16-ion optimisation run in which the DIRECT algorithm navigates an energy landscape with several local minima.

  • 16-ion optimisation: A 16-ion optimisation run is shown using a large budget and the DIRECT algorithm.The energy trajectory is plotted against iteration number during a single optimisation run.
  • 16-ion optimisation: The optimisation encountered several local minima in the energy landscape.This highlights the nontrivial search landscape encountered during the 16-ion run.
  • Algorithmic error analysis: Algorithmic error is analysed as a function of circuit depth and bare mass around the critical point.The analysis is based on numerically simulated ground-state optimisation for eight ions.
  • Algorithmic error analysis: Increasing circuit depth decreases the standard deviation of the target Hamiltonian, especially near the critical point.Peaks near criticality indicate more entangled target states that require deeper circuits for a specified precision.
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