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Scalable Circuits for Preparing Ground States on Digital Quantum Computers: The Schwinger Model Vacuum on 100 Qubits

Roland C. Farrell, Marc Illa, Anthony N. Ciavarella, Martin J. Savage

arXiv:2308.04481v3quant-phhep-lathep-phnucl-th

TL;DR

Preparing useful initial states for large quantum simulations is challenging. This paper introduces SC-ADAPT-VQE, which constructs scalable circuits classically and applies them to the Schwinger-model vacuum on up to 100 qubits, with quantum-computer observables agreeing with Matrix Product State results.

  • Problem

    Preparing initial states for quantum simulations of physically relevant systems remains a major challenge, especially when large spacetime volumes are needed for quantitative results.

  • Method

    SC-ADAPT-VQE determines circuits on modest lattices using classical computers, then exploits translation invariance and finite correlation length to scale them to larger systems.

  • Results

    The method prepared the Schwinger-model vacuum on up to 100 qubits, while energy density and chiral condensate converged exponentially with circuit depth and quantum-computer observables agreed with Matrix Product State results.

  • Takeaways & Limitations

    SC-ADAPT-VQE provides a route to preparing large-system lattice quantum field theory states without optimizing the circuits directly on a quantum computer.

  • Takeaways & Limitations

    The volume-extrapolation comparison remains limited by the small number of points in the plateau region, with larger-L results needed for a more complete comparison.

Abstract

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The vacuum of the lattice Schwinger model is prepared on up to 100 qubits of IBM's Eagle-processor quantum computers. A new algorithm to prepare the ground state of a gapped translationally-invariant system on a quantum computer is presented, which we call Scalable Circuits ADAPT-VQE (SC-ADAPT-VQE). This algorithm uses the exponential decay of correlations between distant regions of the ground state, together with ADAPT-VQE, to construct quantum circuits for state preparation that can be scaled to arbitrarily large systems. These scalable circuits can be determined using classical computers, avoiding the challenging task of optimizing parameterized circuits on a quantum computer. SC-ADAPT-VQE is applied to the Schwinger model, and shown to be systematically improvable, with an accuracy that converges exponentially with circuit depth. Both the structure of the circuits and the deviations of prepared wavefunctions are found to become independent of the number of spatial sites, $L$. This allows for a controlled extrapolation of the circuits, determined using small or modest-sized systems, to arbitrarily large $L$. The circuits for the Schwinger model are determined on lattices up to $L=14$ (28 qubits) with the qiskit classical simulator, and subsequently scaled up to prepare the $L=50$ (100 qubits) vacuum on IBM's 127 superconducting-qubit quantum computers ibm_brisbane and ibm_cusco. After introducing an improved error-mitigation technique, which we call Operator Decoherence Renormalization, the chiral condensate and charge-charge correlators obtained from the quantum computers are found to be in good agreement with classical Matrix Product State simulations.

I. INTRODUCTION

SC-ADAPT-VQE addresses the difficulty of preparing nontrivial ground states at scale by determining scalable circuits classically and extrapolating them to larger lattices. Applied to the Schwinger model, it achieves exponentially improving accuracy and supports preparation up to 100 qubits.

  • Ground-state preparation remains limited for modest and large quantum many-body simulations, despite progress on preparing nontrivial initial states.
  • A mass gap produces a correlation length and exponential decay of distant ground-state correlations, motivating scalable preparation for translationally invariant systems.
  • SC-ADAPT-VQE determines circuits on modest lattices using classical computers, then uses translation invariance to scale them to arbitrarily large systems.
  • The method avoids quantum optimization involving statistical uncertainty and device noise, while the prepared-state quality becomes independent of L up to O(e^-ξ/L) corrections.
  • Classical simulations of the Schwinger model reached L = 14 (28 qubits), while the resulting circuits were used to prepare vacua on up to 100 qubits.

II. THE LATTICE SCHWINGER MODEL

The Schwinger model is formulated as a lattice gauge theory whose gauge field is constrained by charge distribution. In axial gauge with open boundaries, fermions are discretized and mapped to spins, leaving nonlocal Coulomb interactions in the Hamiltonian.

  • The Schwinger model is a continuum theory described by a Lagrange density involving charged fermions and an electromagnetic gauge field.
  • In 1 + 1 dimensions, Gauss’s law completely constrains the gauge field through the distribution of fermion charges.
  • In axial gauge, the spatial gauge field is absent and temporal-gauge effects appear as nonlocal Coulomb interactions.
  • With open boundary conditions, staggered fermions are discretized and mapped to spins using the Jordan-Wigner transformation.
  • The lattice contains L spatial sites corresponding to 2L staggered fermion sites, with m and g denoting the bare electron mass and charge.

A. Infinite Volume Extrapolations of Local Observables

Exponential decay of correlations makes local ground-state observables converge rapidly away from boundaries, enabling infinite-volume extrapolations. Boundary perturbations instead produce O(ξ/L) corrections in volume averages.

  • The ground-state correlation length is ξ = 1/mhadron, and correlations between regions separated beyond ξ are exponentially suppressed.
  • Local observables converge to their infinite-volume values with O(e^-ξ/L) corrections, while boundary effects produce O(ξ/L) deviations in volume averages.
  • The vacuum energy density ε and chiral condensate χ are extrapolated from finite-L exact-diagonalization and DMRG results to L →∞.
  • Linear and quadratic 1/L extrapolations overlap, with their infinite-volume difference used to estimate fitting uncertainties.
  • The chiral condensate is an order parameter of chiral symmetry breaking, while the energy density is defined as ε = ⟨H⟩/L.

III. SC-ADAPT-VQE FOR THE LATTICE SCHWINGER MODEL

SC-ADAPT-VQE adapts ADAPT-VQE for Schwinger-model vacuum preparation by selecting energy-improving operators and optimizing their parameters, while determining circuits classically to avoid quantum-device measurement overhead.

  • ADAPT-VQE procedure: ADAPT-VQE builds the ansatz iteratively by appending the pool operator whose commutator with the Hamiltonian has the largest magnitude.The algorithm terminates when the selected commutator or newest optimized parameter falls below a predetermined threshold.
  • ADAPT-VQE procedure: The operator pool is constrained by the system’s symmetries, and the ansatz begins from a strategically selected state with the target quantum numbers.
  • Circuit implementation: First-order Trotterization approximates exponentials of sums of non-commuting pool terms, introducing systematic deviations from the target unitaries.
  • Circuit implementation: Trotterized implementations trade off gate depth, coherence time, gate-operation errors, and Trotter errors on NISQ devices.
  • SC-ADAPT-VQE procedure: SC-ADAPT-VQE determines state-preparation circuits with a classical simulator, avoiding the large number of quantum-computer measurements normally required by hybrid ADAPT-VQE.The resulting circuits can be scaled to arbitrarily large lattices.

A. A Scalable Operator Pool for the Lattice Schwinger Model

The Schwinger-model operator pool combines translationally invariant volume operators with boundary-localized surface operators while enforcing charge neutrality and discrete symmetries.

  • Symmetry constraints: The pool is constrained to be charge neutral, CP symmetric, and invariant under time reversal.
  • Volume and surface operators: Volume operators respect translational symmetry, whereas surface operators account for boundary effects localized over penetration depths set by the mass gap.
  • Operator construction: Only odd-distance hopping operators are retained because even-distance operators break CP.
  • Symmetry constraints: Time-reversal symmetry motivates an imaginary, anti-symmetric pool, so the real operators are replaced by their commutators.
  • Volume and surface operators: The volume operators contribute O(L), while surface operators contribute O(1) because their support is confined near the boundaries.

IV. SCALABLE QUANTUM CIRCUITS FROM CLASSICAL COMPUTING

SC-ADAPT-VQE performs ADAPT-VQE across sufficiently large systems using a noiseless classical simulator, producing parameterized circuits that can be scaled beyond the systems used to determine them.

  • ADAPT-VQE is applied to a series of systems large enough to support robust scaling of the parameterized circuits.The scalable circuits can be determined with classical computing or a smaller partition of a larger quantum computer.

A. Trotterized Quantum Circuits for the Scalable Operator Pool

The scalable operator pool is implemented with Trotterized gate circuits, using compact R± constructions and circuit rearrangements that enable cancellations and reduce CNOT costs.

  • Trotterized implementation: Individual pool terms are implemented with Trotterization, and the gate decomposition is crucial for successful quantum-computer simulations.
  • R± gate construction: A 2-CNOT R±(θ) circuit implements the d = 1 unitary, and extended X-pattern circuits implement longer-range terms.
  • R± gate construction: The R± gate implements exp[−iθ/2(Y X ± X Y)] and is extended to exponentials involving X Z2 Y − Y Z2 X and X Z4 Y − Y Z4 X, with a sign change in the latter.
  • Circuit simplification: The optimized arrangement reduces CNOT count by two and CNOT depth by a factor of ×2 relative to staircase-based circuits.
  • Circuit simplification: Different Trotter orderings can change convergence while remaining equivalent up to higher-order Trotter errors.
  • Circuit simplification: Arranging neighboring terms with offsets of d − 1 qubits allows outermost gates to cancel, while interleaving X-pattern legs can reduce circuit depth for d ≥ 5.

B. Building Scalable State Preparation Quantum Circuits using SC-ADAPT-VQE with Classical Computing

SC-ADAPT-VQE constructs Schwinger-model vacuum circuits classically on modest lattices, using exponentially decaying correlations to extrapolate their parameters and structure to larger systems. Errors in intensive observables decrease exponentially with algorithm step and become independent of lattice size.

  • Classical circuit construction: SC-ADAPT-VQE approximates the Schwinger-model vacuum on L ≤14 spatial sites (28 qubits) using classically simulated circuits.The circuits are generated through SC-ADAPT-VQE simulations with qiskit’s classical simulator.
  • Systematic improvement: The error in energy density, chiral condensate, and infidelity density decreases exponentially with algorithm step.The exponential trend is observed through 10 steps and reaches convergence comparable to systematic errors from later L-extrapolations.
  • Systematic improvement: For a fixed algorithm step, observable errors become independent of system size, supporting circuit extrapolation to arbitrarily large lattices.The figure compares deviations across L = 6 to L = 14 for three parameter sets.
  • Ansatz structure: The ansatz adds localized operators first, then longer-range and surface operators; longer correlation lengths require larger-range operators.This ordering follows the correlation-length dependence of the state-preparation problem.
  • Volume extrapolation: The order of operators and variational parameters converge with increasing L, enabling exponential extrapolation to L = ∞.Convergence sets in for L > 7,13, and the extrapolated parameters are used to initialize larger-lattice vacua.

V. PREPARING THE VACUUM OF THE SCHWINGER MODEL ON LARGE LATTICES

Circuits determined on lattices with L ≤14 are scaled without further optimization to prepare Schwinger-model vacua on much larger systems. The scaled circuits reach L = 500 in classical MPS simulation and L = 50, or 100 qubits, on IBM Eagle processors.

  • Large-lattice preparation: Circuits determined for L ≤14 are scaled without further optimization to prepare vacua on lattices up to L = 500.The large-lattice preparations use a qiskit matrix product state circuit simulator.
  • Large-lattice preparation: The scaled circuits prepare the vacuum on L = 50, corresponding to 100 qubits, using IBM’s Eagle-processor quantum computers.The circuits were originally determined with an exact statevector classical simulator.
  • Validation: The large-lattice section evaluates the scaled-circuit results against DMRG results for m = 0.5 and g = 0.3.Table III reports the comparison for seven SC-ADAPT-VQE steps.

A. Classical Simulation

Classical simulations establish exact and large-volume reference results, then show that SC-ADAPT-VQE circuit structure and intensive-observable errors remain stable as the lattice grows. These findings support using classically determined circuits for large-scale vacuum preparation.

  • Reference calculations: Exact diagonalization determines vacuum energy density and chiral condensate for L ≤14, while DMRG extends the reference calculations to L ≤103.Results for L ≥9 are extrapolated consistently to L →∞ with 1/L scaling.
  • Scalable circuits: SC-ADAPT-VQE intensive quantities converge exponentially with circuit depth, while circuit structure and errors become independent of L.The variational parameters can therefore be extrapolated to arbitrarily large lattice sizes.
  • Large-volume validation: For seven-step scaled circuits, energy-density and chiral-condensate deviations remain independent of L on large lattices.The deviations agree with those found on L ≤14 systems.
  • Conclusion: Classically determined SC-ADAPT-VQE circuits can prepare the Schwinger-model vacuum at scale with precision independent of system size.This is the stated consequence of the classical and large-lattice simulations.

B. Quantum Simulations on 100 Qubits using IBM’s Quantum Computers

Scaled SC-ADAPT-VQE circuits prepared Schwinger-model vacua on up to 100 qubits, with error mitigation producing observables consistent with classical references. Charge correlations and condensate measurements remained compatible with simulator results despite device noise.

  • Error mitigation: Operator Decoherence Renormalization estimates decoherence separately for each operator after Pauli twirling, enabling operator-specific error correction.The mitigation stack also included readout-error mitigation and dynamical decoupling.
  • Chiral condensate: After mitigation, each local chiral-condensate value was within 3σ of the MPS simulator result, with 1σ uncertainties of approximately 15% of the mean.The reported uncertainties were expected to decrease with more statistics and twirlings.
  • Volume scaling: The chiral-condensate results stayed within 3σ of MPS values across L = 14, 20, 30, 40, and 50, while the CNOT count grew linearly with L to 788 gates over 100 qubits.Boundary deviations affected only a few qubits near the lattice edges.
  • Charge correlations: For L = 50, measured connected charge-charge correlations were within 3σ of MPS results, and volume-averaged correlations were consistent with zero for distances d > 2 within 2σ.The correlation sum omitted the first and last two spatial sites to reduce boundary effects.

VI. SUMMARY AND OUTLOOK

SC-ADAPT-VQE prepares the Schwinger-model vacuum on up to 100 qubits by scaling circuits identified on smaller lattices, exploiting translational invariance and finite correlation length. The framework supports ground-state preparation and future dynamical simulations, while its asymptotic parameter extrapolations retain uncertainties requiring larger-volume validation.

  • Summary: 100 qubits: The Schwinger-model vacuum was prepared on IBM Eagle processors using SC-ADAPT-VQE circuits determined on lattices with L ≤14.The circuits were built with a classical simulator and scaled to L = 50, corresponding to 100 qubits.
  • Summary: Finite correlation length and translational symmetry allow local circuit structures to be replicated across arbitrarily large lattices.The prepared-state quality becomes independent of lattice length up to O(e^-ξ/L) corrections, under the stated gapped-system assumptions.
  • Outlook: SC-ADAPT-VQE provides low-depth ground-state circuits with only classical computing overhead, supporting later simulations of ground-state properties and dynamics.The paper also discusses extensions to single- and multi-hadron states and localized excitations.
  • Model simplifications: The Schwinger-model electric-sector restriction reduces Hamiltonian term count from 1−3L+2L^2 to 1+L^2−2L and changes required connectivity from all-to-all to half-to-half.Further correlation-based truncation could reduce the Hamiltonian to O(Lξ) terms with connectivity limited to separations d ≲ξ.
  • Circuit optimization: Deeper Trotterizations improve SC-ADAPT-VQE convergence, while exact unitaries generally outperform Trotterized ones except for a single step.For a single step, optimized Trotter errors can reduce the vacuum-overlap error below that of the exact unitary.
  • Limitations: Larger volumes are needed to resolve the plateau region and fully quantify uncertainties in exponentially extrapolated variational parameters.The selected exponential form reproduces DMRG results for the analyzed systems, suggesting small systematic errors there, but polynomial corrections remain insufficiently explored.

Appendix F: Additional Results From Classical Simulations

Additional simulations examine parameter regimes, circuit extrapolation, and hardware implementation of SC-ADAPT-VQE. The results also quantify why classical optimization is used and describe the mitigation procedures applied to IBM-device measurements.

  • Classical simulations: The operator structure through L = 14 converges for m = 0.1, g = 0.8, enabling consistent extrapolation of circuits to larger L.The sixth algorithmic step is selected because its operator structure agrees through L = 14 with the comparison at seven steps.
  • Classical simulations: Longer correlation lengths at m = 0.1 prevent the available MPS simulations from reaching L = 500.The scalable circuits for m = 0.1, g = 0.3 and m = 0.1, g = 0.8 were nevertheless evaluated with Qiskit’s MPS simulator.
  • Classical versus quantum optimization: Approximately 10^10 shots would be required for L = 14 SC-ADAPT-VQE on a noiseless device, rising to nearly a trillion after accounting for device noise.The estimate follows from roughly 6000 optimizer calls, about 500 commutator evaluations, and approximately 10^-3 precision per measured observable.
  • Quantum-device implementation: Three-step circuits for L = 30 and L = 50 were run on ibm_brisbane and ibm_cusco, respectively, using twirled instances and 8 × 10^3 shots per instance.The resulting local chiral condensate and charge-charge correlators are shown in the corresponding figures.
  • Error mitigation: Operator Decoherence Renormalization is used instead of zero-noise extrapolation with probabilistic error correction, with lower sampling overhead than those methods.The circuits use Pauli twirling to transform coherent errors into incoherent errors before mitigation.
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