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Quantum computation of hadron scattering in a lattice gauge theory

Zohreh Davoudi, Chung-Chun Hsieh, Saurabh V. Kadam

arXiv:2505.20408v1quant-phhep-lathep-phnucl-th

TL;DR

The paper develops a digital quantum approach to hadron scattering in a (1+1)D Z2 lattice gauge theory, focusing on efficient preparation of high-fidelity multi-wave-packet states and their collision dynamics. IonQ Forte demonstrations reach early-time agreement for local observables, while noise limits long-time evolution and precision state-sensitive measurements.

  • Problem

    Quantum simulations of hadron scattering require efficient preparation of complex, high-fidelity scattering states in non-perturbative gauge theories.

  • Method

    The paper combines a systematically improvable hybrid classical-quantum wave-packet preparation algorithm with second-order Trotter time evolution in a (1+1)D Z2 lattice gauge theory with fermionic matter.

  • Results

    The IonQ Forte experiment prepared two- and three-wave-packet states and simulated two-hadron collision dynamics, with early-time local-observable data encouraging despite decoherence.

  • Takeaways & Limitations

    High-fidelity initial scattering states are important for future precision computations of phase shifts, S-matrix elements, and decay widths.

  • Takeaways & Limitations

    Moderate coherence time, deep circuits, and small system size prevent access to interesting long-time scattering dynamics in the hardware study.

Abstract

from arXiv · show

We present a digital quantum computation of two-hadron scattering in a $Z_2$ lattice gauge theory in 1+1 dimensions. We prepare well-separated single-particle wave packets with desired momentum-space wavefunctions, and simulate their collision through digitized time evolution. Multiple hadronic wave packets can be produced using the efficient, systematically improvable algorithm of this work, achieving high fidelity with the target initial state. Specifically, employing a trapped-ion quantum computer (IonQ Forte), we prepare up to three meson wave packets using 11 and 27 system qubits, and simulate collision dynamics of two meson wave packets for the smaller system. Results for local observables are consistent with numerical simulations at early times, but decoherence effects limit evolution into long times. We demonstrate the critical role of high-fidelity initial states for precision measurements of state-sensitive observables, such as $S$-matrix elements. Our work establishes the potential of quantum computers in simulating hadron-scattering processes in strongly interacting gauge theories.

I. INTRODUCTION

The paper addresses the difficulty of simulating real-time hadron scattering by developing a digital quantum protocol for a confined Z2 lattice gauge theory in 1+1 dimensions. It combines efficient wave-packet preparation with quantum time evolution to study scattering observables on near-term hardware.

  • Real-time scattering processes remain beyond state-of-the-art lattice-QCD calculations, which are most suitable for static observables and limited to low-energy, low-inelasticity dynamics.
  • Hamiltonian lattice gauge theories retain continuous time, enabling real-time evolution of gauge-theory states on discretized spatial systems.
  • The work studies a (1+1)D Z2 lattice gauge theory with dynamical fermions, whose confinement makes it a testing ground for more complex strongly interacting theories.
  • The state-preparation algorithm creates well-separated single-particle wave packets with tunable spatial width and central momentum, while allowing systematically controlled approximations.
  • The authors evolve two wave packets through collision using a Trotter product formula and investigate which scattering observables are accessible on near-term devices.

B. Ansatz for meson creation operators in the interacting theory

The meson-creation ansatz builds momentum-resolved interacting excitations from gauge-invariant bare-meson operators and improves them order by order. Its physical-lattice translation structure and variational optimization support systematically higher-fidelity wave-packet preparation.

  • The ansatz uses gauge-invariant bare-meson operators Mm,n whose translation properties reflect the physical lattice rather than the staggered lattice alone.
  • Each operator includes the shorter forward- or backward-wrapped meson on the periodic lattice, with equal-length paths combined when necessary.
  • At order j, only bare-meson creation operators of length at most j are included, making the ansatz systematically improvable.
  • The composite ansatz assigns momentum to both meson endpoints, while a Kronecker delta enforces the desired total momentum.
  • Variational optimization begins with length-1 operators and progressively increases j while reusing previously optimized parameters.
  • F = 0.99 is achieved at j = 3 across the scanned parameter space for all momentum sectors, except where a lower-energy non-mesonic excitation is not represented by the ansatz.

C. States’ fidelities with the ansatz meson creation operators

The optimized order-by-order position-space ansatz accurately represents interacting meson momentum eigenstates across tested parameters, with fidelity improving at higher order and stronger coupling. It also performs better than the earlier two-parameter momentum-space ansatz, including at large momenta.

  • The optimized state |k^(j)⟩op is compared with exact momentum eigenstates |k⟩ex to assess hadron-state fidelity.The optimized state is prepared by applying the ansatz to the numerically obtained ground state, while the reference is the lowest-energy mesonic eigenstate from exact diagonalization.
  • 0.98 (0.99) fidelity is reached across the tested parameter range with first-(third-)order ansätze.Higher-order ansätze capture longer-range correlations more effectively, and optimization performs better in the strong-coupling regime.
  • Infidelity arises from contamination by excited states within the same momentum sector, while translation symmetry is preserved by construction.The ansatz transforms properly under lattice translations, so the contamination remains within the selected momentum sector.
  • The ansatz cannot capture a non-mesonic single-link excitation in the k = 0 sector.Where that excitation is lower in energy, fidelity is instead evaluated against the first mesonic excitation in the sector.
  • The position-space ansatz outperforms the earlier two-parameter momentum-space ansatz even at the Brillouin-zone edge.Accurate large-k eigenstates produce faster wave packets, which inject more energy and reduce the evolution time needed for collisions.

III. QUANTUM ALGORITHM AND CIRCUIT DESIGN

The scattering protocol is a hybrid classical–quantum algorithm built from ground-state preparation, well-separated wave-packet preparation, and Trotterized time evolution. It uses the MGF for circuit efficiency, while VQE parameters are classically verified in this work.

  • The three modules prepare the ground state, construct well-separated wave packets, and perform Trotterized evolution under the Hamiltonian.They begin from the strong-coupling vacuum and produce the scattering state before time evolution.
  • QGS prepares the interacting ground state using variational parameters obtained by minimizing the Hamiltonian energy.The relevant parameters are determined with a VQE procedure.
  • Additional circuit modules may be required to compute observables after the scattering evolution.The observable-measurement circuits are discussed separately for selected quantities.
  • The protocol is hybrid because ground-state and momentum-eigenstate VQE results are verified through classical evaluation in this work.Both VQE calculations could in principle be implemented on a quantum computer, but they were classically evaluated to conserve quantum-computing resources.

A. QGS: Preparing the interacting vacuum

QGS prepares the interacting vacuum from the strong-coupling vacuum using a parameterized circuit based on Hamiltonian terms. For the considered parameters and lattice sizes, a single iteration with the electric-term angle set to zero suffices for high fidelity.

  • QGS prepares the ground state through iterative evolution using Hamiltonian terms and parameters associated with hopping, mass, and electric contributions.The circuit is parameterized by the corresponding variational angles.
  • The QGS circuit acts on the strong-coupling vacuum with alternating fermion-site states and a zero bosonic-link state.The circuit layout and gate structure are shown in Fig. 3.
  • NGS = 1 and θϵ_1 = 0 suffice to prepare a high-fidelity interacting vacuum for the studied parameters and lattice sizes.The remaining optimized parameters are obtained through the circuit’s variational procedure.
  • The hopping-gate ordering in QGS preserves the global-Q symmetry.Each Trotter layer has a specified gate cost, and the optimal parameters are obtained by VQE energy minimization.

B. QInit: Preparing the initial scattering state

QInit prepares two far-separated meson wave packets by sequentially applying single-wave-packet circuits to the interacting vacuum. Using separate ancillas improves fidelity by making preparation errors more detectable, while the method assumes negligible packet overlap.

  • B. QInit: Preparing the initial scattering state: QInit prepares the initial two-wave-packet scattering state by applying QWP circuits to the interacting vacuum.Each QWP creates one wave packet using an ancilla qubit.
  • B. QInit: Preparing the initial scattering state: The circuit can reuse one ancilla between QWP(Ψ1) and QWP(Ψ2), or assign a separate ancilla to each wave packet.The one-ancilla construction inserts a σx operation between the two QWP circuits.
  • B. QInit: Preparing the initial scattering state: Sequential preparation of Ψ2 is valid when the wave packets are far separated and their overlap is approximately zero.The construction also uses approximate vacuum-annihilation and commutator relations for the interacting creation operators.
  • B. QInit: Preparing the initial scattering state: The accepted ancilla outcome identifies the intended two-wave-packet state, but ansatz imperfections and hardware noise create ancilla-violation error.The combined error includes systematic preparation error and device noise.
  • B. QInit: Preparing the initial scattering state: Two ancilla qubits produce a higher-fidelity two-wave-packet state than one ancilla in the noiseless comparison.With one ancilla, some first-packet preparation errors can propagate into the accepted outcome without detection; separate ancillas combine detectable error probabilities.

C. QTrott: Trotterized time evolution

QTrott implements real-time evolution of the two-wave-packet state with a second-order Trotter product formula. Its constituent Hamiltonian terms are compiled into circuit blocks, with a reported CNOT cost per Trotter layer.

  • C. QTrott: Trotterized time evolution: QTrott applies U(t)=e^(-itH) to evolve the prepared two-wave-packet scattering state.The same module can support return-probability measurements through a controlled evolution and Hadamard test.
  • C. QTrott: Trotterized time evolution: The QTrott circuit repeats the constituent circuit blocks for n_t Trotter steps.The displayed module organizes one Trotter step into repeated Hamiltonian-evolution blocks.
  • C. QTrott: Trotterized time evolution: The evolution uses a second-order Trotter formula with t=n_tδt and ordered H_h, H_m, and H_ε subevolutions.The H_h terms receive an additional first-order expansion to separate σ_x and σ_y contributions.
  • C. QTrott: Trotterized time evolution: Each second-order Trotter layer requires 18N_P+8 CNOT gates.The H_m and H_ε components are implemented without further approximation, while H_h is further decomposed.

IV. QUANTUM HARDWARE AND EMULATOR RESULTS

The hardware study uses IonQ Forte to implement the scattering circuit at two system sizes, corresponding to 11 and 27 qubits. The larger case has a substantially larger symmetry-restricted Hilbert space.

  • IV. QUANTUM HARDWARE AND EMULATOR RESULTS: The circuit was executed on IonQ Forte, a 32-qubit ytterbium-ion quantum computer with high-fidelity one- and two-qubit gates.Device-time availability limited the VQE optimizations required for circuit parameters.
  • IV. QUANTUM HARDWARE AND EMULATOR RESULTS: The study uses N_P=5 and N_P=13, corresponding to N=10 and N=26, or equivalently 11 and 27 qubits.These are the two system sizes used for the hardware and emulator study.
  • IV. QUANTUM HARDWARE AND EMULATOR RESULTS: The symmetry-restricted Hilbert spaces contain 504 states for N_P=5 and 20,801,200 states for N_P=13.The smaller system is accessible to exact diagonalization, while low-energy states of the larger system can be obtained with DMRG.

A. Variational-quantum-eigensolver optimization

VQE optimizes the interacting ground-state and momentum-resolved wave-packet parameters used by the scattering circuits. The reported optimizations produce target states with very high fidelity, using different Trotter counts for the two system sizes.

  • A. Variational-quantum-eigensolver optimization: VQE optimization determines θ_h* and θ_m* for preparing a ground state that closely approximates the true interacting vacuum.The resulting ground-state parameters and fidelity are summarized in Table V.
  • A. Variational-quantum-eigensolver optimization: For each target momentum k_t, VQE minimizes the energy of the state produced by QWP with Ψ(k)=δ_k,k_t.This obtains parameters for the momentum-eigenstate creation-operator ansatz.
  • A. Variational-quantum-eigensolver optimization: The momentum-state optimization uses ten second-order Trotter steps for N_P=5 and two for N_P=13.These choices were selected to keep Trotter error small while maintaining manageable emulator optimization time.
  • A. Variational-quantum-eigensolver optimization: The optimized ansatz captures the target states with very high fidelity.Table II reports the initial-state wavefunction parameters and the overlap between the two wavefunctions for two lattice sizes.

B. Preparing the initial hadronic wave packets

The paper prepares separated Gaussian meson wave packets through a systematically improvable circuit whose approximations trade preparation accuracy for quantum-resource savings. IonQ Forte results support this approach for two wave packets and demonstrate extension to three.

  • Wave-packet construction: Opposite-momentum Gaussian wave packets are placed far apart so they move toward each other during time evolution.The packets have equal-magnitude opposite momenta and position separation |μ1 − μ2| = NP.
  • Circuit construction: The QInit(Ψ1, Ψ2) circuit derives single-qubit rotation angles from optimized coefficients and retains only angles above the cutoff θc.The coefficients include zero-length and one-length meson terms, while second-order Trotterization determines the implemented rotations.
  • Resource trade-offs: Two approximations control the trade-off between systematic preparation errors and qubits, single-qubit gates, and two-qubit gates.Approximation I uses two ancillas, whereas Approximation II uses one ancilla and one second-order Trotter step for both system sizes.
  • Preparation accuracy: Approximation I agrees better with ideal results, while Approximation II remains reasonably accurate with significantly lower resource requirements.The comparisons use staggered density and electric-field observables for NP = 5 and NP = 13 systems.
  • Hardware preparation: 49.40% and 71.20% symmetry-violation errors occur for NP = 5 and NP = 13 hardware preparations, respectively.The corresponding ancilla-violation errors are 14.82% and 18.40% after symmetry-violating shots are discarded.
  • Multiple wave packets: Three spatially separated wave packets are prepared on IonQ Forte for NP = 13, with good agreement with the expected results.The same wave-packet module can be repeated using additional ancilla qubits.

C. Time-evolved observables

The experiment evolves a two-meson state with second-order Trotter circuits and finds that local observables follow ideal dynamics only at early times. Hardware noise and circuit depth prevent access to long-time scattering dynamics, while state-sensitive quantities require higher-fidelity initial states.

  • Time-evolution setup: Each Trotter-step circuit for the NP = 5 evolution uses 204 single-qubit gates and 60 CNOT gates.The circuit evolves the two-wave-packet state on IonQ Forte using Approximation II.
  • Error growth: The ancilla-violation error grows with each Trotter step because deeper circuits introduce increasing hardware error.The ancilla qubit used in initial-state preparation does not participate in time evolution, so its noiseless error is time-independent.
  • Staggered density: Noiseless staggered-density results follow the exact wave-packet profile at early times but deviate increasingly as Trotter error accumulates.Both approximations retain the qualitative shape of the exact evolution, and the hardware signal becomes noise-dominated after t = 6.
  • Electric field: The electric-field observable remains nearly at its initial value in noiseless evolution but deviates significantly on hardware.The deviation is associated with the energy cost of flipping the electric field, which is proportional to system size; emulator mitigation recovers the value.
  • Scope and limitations: Moderate coherence time, large time-evolution circuit depth, and small system size prohibit access to interesting long-time scattering dynamics.Larger systems would reduce boundary effects but substantially increase circuit depth.

D. Return probability

The return probability is used as a diagonal S-matrix element to test scattering-state fidelity. Unlike local observables, this non-local overlap is highly sensitive to approximation errors in the initial state.

  • Observable: The return probability is a diagonal entry of the scattering S-matrix for the prepared two-wave-packet state.The study restricts this computation to the initial state prepared in Sec. IV B.
  • Results: Appx II deviates significantly from the ideal return probability, whereas Appx I agrees up to Trotter errors.Appx I uses more resources and produces higher-fidelity states than Appx II.
  • Sensitivity: Non-local return probability is sensitive to contamination of the state vector through interference effects.The same approximation errors have much smaller effects on local staggered density.
  • Implications: High-fidelity initial states are crucial for precision measurements of S-matrix elements and related state-sensitive observables.The paper contrasts this requirement with the greater tractability of qualitative local-observable measurements on NISQ devices.
  • Outlook: The protocol can prepare multiple spatially separated wave packets and access exclusive scattering amplitudes using a second register and a swap test.The same framework is proposed for hadron tensors and inclusive cross sections through electroweak currents and the optical theorem.

Appendix A: Verifying the ansatz validity for a larger system using tensor networks

Tensor-network verification shows that the finite-order wave-packet ansatz remains accurate in a 26-staggered-site system. Third-order preparation reaches high fidelity across most scanned momenta, with verification caveats at the largest momentum.

  • System and verification: The larger-system test uses a Z2 LGT with 26 staggered sites, corresponding to 13 physical sites.The Brillouin zone includes momenta from -6π/13 to 6π/13.
  • Tensor-network reference: DMRG constructs approximate ground and excited states while preserving the fixed fermionic-excitation sector and gauge symmetry.The calculations use constrained orthogonality, bond dimension 600, and energy-variance checks.
  • Tensor-network reference: DMRG states achieve energy variance Var = O(10^-10) across the scanned parameter pairs.This variance is used as a check that the reference states closely approximate eigenstates.
  • Verification caveats: For nonzero momentum, DMRG outputs can mix degenerate ±k states because momentum is not specified during sweeps.The analysis therefore uses approximately orthogonal superpositions to estimate momentum-state fidelity.
  • Fidelity scan: The ansatz improves order by order and performs better in the strong-coupling regime.The parameter scan evaluates fidelity over mass and coupling values using tensor-network reference states.
  • Verification caveats: At |k| = 6π/13, nearby excited states can hinder DMRG from reliably distinguishing the target momentum sector.The affected parameter region is identified as less reliable for fidelity testing.
  • Conclusion: The authors conclude that finite-order ansätze can faithfully build momentum eigenstates in larger systems, although better MPS ansätze may be needed at large momentum.The limitation concerns verification quality rather than the stated ansatz conclusion.

Appendix B: Circuit for QWP in the minimal-gauge formalism

The appendix gives an efficient circuit construction for QWP(Ψ), the operation that prepares a wave packet from its target wavefunction. It also describes its use in multi-wave-packet preparation and return-probability estimation.

  • Appendix B: Circuit for QWP in the minimal-gauge formalism: QWP(Ψ) prepares a single-particle wave packet with wavefunction profile Ψ by exponentiating the wave-packet operator with a second-order Trotter formula.The appendix uses one Trotter step for the angle π/2.
  • Appendix B: Circuit for QWP in the minimal-gauge formalism: The QWP circuit is assembled from terms e^-iθΘ_m,n, with Θ_m,n defined by the minimal-gauge operator construction.The coefficients depend on the input wavefunction and optimized ansatz parameters.
  • Appendix B: Circuit for QWP in the minimal-gauge formalism: Each exponential is implemented using an SVD-based circuit exploiting Θ_m,n = A†⊗|1a><0a| + A⊗|0a><1a| and A^2 = A†2 = 0.The construction uses controlled unitary blocks and a diagonal SVD-derived operation.
  • Appendix B: Circuit for QWP in the minimal-gauge formalism: For the j = 1 ansatz, the implementation creates at most 1-length mesons, so nontrivial terms satisfy |m − n| ≤ 1.Terms with m = n have a trivial circuit in this restricted case.
  • Appendix B: Circuit for QWP in the minimal-gauge formalism: The same QWP circuit supports VQE optimization of a target momentum operator by choosing Ψ(k) = δ_k,k_t.The resulting circuit prepares the momentum-sector operator needed for variational optimization.
  • Appendix C: Three-wave-packet preparation: Three non-overlapping wave packets are prepared for N_P = 13 with overlap factors 0.0104, -0.0306, and 0.0059.The QInit circuit for this preparation requires 267 CNOT gates and uses Appx II.
  • Appendix D: Hadamard test for calculating the return probability: The return probability is evaluated by a Hadamard test that extracts the real and imaginary parts of the transition amplitude from ancilla probabilities.Controlled U(t) is built by controlling each Trotterized time-evolution block.

Appendix E: Time evolution of local observable for NP =13

The appendix compares noisy and ideal time evolution for a local observable in the N_P = 13 system. It finds local-observable robustness to initial-state approximations, while hardware noise strongly affects electric-field measurements and requires emulator-based mitigation.

  • Local observable: χ_n shows very small deviations from the ideal result for both Appx I and Appx II.This remains true at times selected where Appx II has large return-probability errors.
  • Local observable: Local observables are more robust to small differences in the prepared initial state than the non-local return probability.The appendix uses the contrasting behavior of χ_n and R(t) to establish this sensitivity difference.
  • Noise and mitigation: The electric-field observable E shows significant hardware error already at t = 1 and diverges further during evolution.The data were obtained on IonQ Forte using Appx II.
  • Noise and mitigation: Pauli twirling and operator decoherence renormalization are applied to recover the desired electric-field value.ODR rescales E(t) by 1 − ρ(t), with ρ(t) estimated from similarly structured U(0) circuits.
  • Limitations: The mitigation analysis uses an IonQ Forte emulator rather than the quantum processor because hardware access was limited.Its effectiveness on actual hardware remains untested in this appendix.
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