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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

How to simulate hadron scattering in lattice gauge theories matters because the model can serve as a testing ground for QCD and confinement physics. This work prepares meson wave packets and digitally evolves them through collision, demonstrating hardware preparation and early-time evolution while noise limits longer runs.

  • Problem

    How to simulate hadron scattering in a (1+1)D Z2 lattice gauge theory matters because the model can test QCD simulation and provide insight into confinement.

    Source: §I · p. 4

  • Method

    A hybrid classical-quantum, systematically improvable state-preparation algorithm prepares localized hadron wave packets, which are digitally evolved to simulate scattering.

    Source: §I · p. 4, §V · p. 29

  • Results

    On IonQ Forte, the authors prepared two- and three-wave-packet states and evolved a two-packet system, but noise obscured the local density signal before full collision.

    Source: §V · p. 29

  • Takeaways & Limitations

    High-fidelity initial scattering states are important for future precision calculations of state-sensitive observables such as phase shifts, S-matrix elements, and decay widths.

    Source: §V · p. 29

    Moderate hardware coherence and deep circuits prevented access to interesting long-time scattering dynamics in this hardware study.

    Source: C. Time-evolved observables · p. 28

Abstract

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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 · II. THEORETICAL FORMALISM · A. Z2 LGT in (1+1)D with dynamical matter

The paper develops digital quantum simulation of meson scattering in a (1+1)D Z2 lattice gauge theory, combining an efficient wave-packet preparation method with real-time evolution. Early-time local observables agree reasonably well with ideal results, while a state-sensitive scattering observable is highly sensitive to initial-state fidelity.

I. INTRODUCTION

  • The introduction situates scattering as a probe of hadron structure, new particles and interactions, and phases of matter, while noting that low-energy QCD is strongly interacting and confining.Classical lattice-QCD calculations support first-principles studies but are most suitable for static observables and are limited for general real-time scattering.Source: §I · pp. 2–3
  • Hamiltonian lattice gauge theories enable real-time state evolution, and digital quantum simulation maps their dynamics to gates for studying scattering beyond Euclidean-time approaches.The introduction notes that lattice-QCD methods have limitations for general real-time scattering, while tensor-network studies of high-dimensional and non-Abelian theories also remain limited.Source: §I · p. 3
  • Earlier scattering protocols used adiabatic preparation, which can require wave-packet-broadening overhead and become infeasible across phase transitions; direct interacting-theory preparation offers an alternative.The paper builds on a tensor-network approach and a hybrid classical-quantum variational method for bound-excitation wave packets.Source: §I · pp. 3–4
  • The work optimizes direct interacting-theory wave-packet preparation with a position-space parameterization, enabling tunable, systematically improvable states with polynomial parameter or correlation-length scaling.The method is demonstrated on IonQ Forte for multiple wave packets and two lattice sizes, with good agreement against high-fidelity noiseless-emulator states.Source: §I · p. 4
  • Local observables agree reasonably well with ideal results at early times, while the initial-state return probability—and thus an S-matrix-related observable—is highly sensitive to state-preparation infidelity.Symmetry-based noise mitigation has little effect on local observables; the noisy emulator suggests additional mitigation circuits can recover more-deviated signals.Source: §I · p. 4

II. THEORETICAL FORMALISM

  • The theoretical-formalism section derives an efficient Z2 LGT formulation, develops and optimizes a momentum-sector particle ansatz, and evaluates its accuracy across coupling and mass regimes.This section provides the background needed to implement the paper’s scattering protocol on quantum hardware.Source: §II · p. 5

A. Z2 LGT in (1+1)D with dynamical matter

  • The model has staggered fermions on an even-site periodic lattice, giving well-defined momentum sectors; the study restricts to Q = NP and identifies the strong-coupling vacuum as a reference state.The gauge and matter degrees of freedom are subject to local Gauss-law constraints in the explicit gauge-link formulation.Source: A. Z2 LGT in (1+1)D with dynamical… · pp. 5–6
  • With periodic boundaries, the minimal gauge-link formulation retains one gauge-link qubit instead of N, reducing qubit requirements but making the electric Hamiltonian non-local.In the fixed-Q = NP subspace, the reduced Hilbert-space dimension is 2 × (2NP)!/(NP!)^2; the simulation uses N + 1 qubits for N staggered lattice sites.Source: A. Z2 LGT in (1+1)D with dynamical… · pp. 7–8

B. Ansatz for meson creation operators in the interacting theory

The section introduces a translation-symmetric, systematically improvable ansatz for meson creation operators, built from gauge-invariant bare-meson operators with momentum-dependent kinematic factors. Its fidelity improves with ansatz order, although the mesonic ansatz does not capture low-energy non-mesonic excitations.

B. Ansatz for meson creation operators in the interacting theory

  • The ansatz incorporates physical-lattice translation symmetry through gauge-invariant Mm,n operators, which translate by two staggered sites under one physical-lattice translation.For m = n, Mm,n is the fermion number operator; for m ≠ n, it is a non-local bare-meson creation operator.Source: B. Ansatz for meson creation operators… · p. 9
  • At order j, the ansatz includes bare-meson operators of length at most j, motivated by exponential suppression of operators extending far beyond the confinement scale.The meson length is the shortest distance on the periodic lattice.Source: B. Ansatz for meson creation operators… · p. 10
  • Kinematic factors assign momentum to the two ends of each bare meson, while δk,p+q selects the desired total momentum.The kinematic factors are obtained from plane-wave solutions of the free fermion theory on a staggered lattice.Source: B. Ansatz for meson creation operators… · p. 10
  • A variational minimization starts with length-1 mesons and iteratively increases j, optimizing parameters while restricting previously optimized values to narrow ranges.The resulting operators are combined with the wave-packet profile to create the desired state from the interacting vacuum.Source: B. Ansatz for meson creation operators… · p. 11
  • F = 0.99 is achieved at j = 3 across the parameter space for all k; fidelity improves with larger mf, |ϵ|, and j, but low-energy non-mesonic excitations are not captured.The region left of the figure’s cyan contour contains a low-energy non-mesonic excitation.Source: Figure 1 · p. 12, B. Ansatz for meson creation operators… · p. 12

C. States’ fidelities with the ansatz meson creation operators

The translationally invariant position-space ansatz achieves high fidelity for meson momentum eigenstates, including at large momenta and in a larger lattice, with a specific k = 0 low-energy exception.

C. States’ fidelities with the ansatz meson creation operators

  • Fidelity compares the optimized-ansatz state prepared from the numerically obtained ground state with the lowest-energy mesonic eigenstate in the same momentum sector.The comparison is made against states obtained by exact diagonalization in small systems.Source: C. States’ fidelities with the ansatz… · p. 12
  • Fidelity reaches 0.98 with the first-order ansatz and 0.99 with the third-order ansatz across the considered parameter range; optimization works better at strong coupling.The remaining infidelity reflects contamination by excited states in the same momentum sector.Source: C. States’ fidelities with the ansatz… · p. 13
  • For sufficiently small ϵ at fixed mf, a non-mesonic electric-flux excitation can lie below the k = 0 mesonic state, so fidelity uses the first mesonic excitation instead.This exception occurs to the left of the cyan contour in Fig. 1.Source: C. States’ fidelities with the ansatz… · p. 13
  • Compared with the two-parameter momentum-space ansatz [107], this position-space ansatz performs better even at large momenta, enabling faster wave packets and shorter scattering evolutions.Faster packets inject more energy, access more interesting outgoing channels, and require fewer Trotter steps and shallower circuits.Source: C. States’ fidelities with the ansatz… · p. 13
  • In the 26-staggered-site theory, comparison with MPS shows high fidelities even for the lowest-order ansatzes.The MPS states are DMRG approximations to exact eigenstates; the parameter scan is shown in Fig. 11.Source: C. States’ fidelities with the ansatz… · p. 13

III. QUANTUM ALGORITHM AND CIRCUIT DESIGN

The algorithm starts from a strong-coupling vacuum and uses three circuit modules to prepare a multi-hadron scattering state and evolve it; the implementation uses the MGF for greater circuit efficiency.

III. QUANTUM ALGORITHM AND CIRCUIT DESIGN

  • The protocol starts from the computational-basis strong-coupling vacuum and uses three circuit modules to prepare and evolve a multi-hadron scattering state.The MGF is chosen for a more efficient circuit implementation.Source: §III · p. 14
  • QInit constructs the initial scattering state from well-separated wave packets, with the required k-momentum eigenstates prepared using VQE.The ground state and k-momentum eigenstates are verified through classical evaluation in this work to save quantum-computing resources.Source: §III · pp. 14–15
  • QTrott performs Trotterized time evolution under the Hamiltonian in Eq. (16).In theory, both VQE calculations can run on quantum hardware if Hamiltonian expectation values are measured accurately, generally requiring many measurements.Source: §III · pp. 14–15
  • The ground-state module prepares the interacting ground state with a two-parameter circuit whose parameters are optimized by VQE to minimize the Hamiltonian energy.The circuit acts on the strong-coupling vacuum and uses parameters θh and θm.Source: §III · p. 15

A. QGS: Preparing the interacting vacuum

QGS prepares the interacting vacuum through iterative evolution under Hamiltonian terms; for the parameters and lattice sizes studied, one iteration with θϵ_1 = 0 sufficed for high-fidelity preparation.

A. QGS: Preparing the interacting vacuum

  • QGS prepares the ground state by iteratively evolving an initial state with Hamiltonian terms, using a circuit similar to Ref. [107] and inspired by a VQE ground-state circuit in Ref. [131].Its parameters are related to Hh, Hm, and Hϵ.Source: A. QGS: Preparing the interacting vacuum · p. 15
  • NGS = 1 and θϵ_1 = 0 sufficed to prepare a high-fidelity |Ω⟩ for the studied mf, ε, and lattice sizes.The reported scope is the parameter choices and lattice sizes considered later in Sec. IV.Source: A. QGS: Preparing the interacting vacuum · p. 16
  • Each first-order Trotter layer preserves global-Q symmetry and requires 8NP + 4 CNOT gates.The implementation applies alternating even- and odd-link terms, with Hh_{N−1,0} identified periodically.Source: A. QGS: Preparing the interacting vacuum · p. 16
  • The QGS parameters θh and θm are optimized with VQE by minimizing the prepared state's energy, with Hamiltonian terms measured in specified qubit bases.⟨Hh⟩ requires σx and σy basis measurements for fermionic-field qubits and ˜σx for the boson qubit; ⟨Hm⟩ and ⟨Hϵ⟩ are measured in the computational basis.Source: A. QGS: Preparing the interacting vacuum · p. 16

B. QInit: Preparing the initial scattering state

QInit prepares a two-wave-packet scattering state by applying single-packet QWP circuits to the interacting vacuum. The procedure uses one or two ancilla qubits, with two ancillas enabling higher final-state fidelity by making preparation errors detectable.

B. QInit: Preparing the initial scattering state

  • QInit prepares two wave packets by applying QWP circuits to the system qubits holding the interacting vacuum, with each QWP using an ancilla initialized in |0⟩.Each wave packet is defined using non-unitary creation operators implemented in an extended Hilbert space.Source: B. QInit: Preparing the initial… · p. 17
  • The interacting-theory creation operators satisfy the vacuum-annihilation and commutator relations only approximately under the ansatz.For a normalized wavefunction profile, the resulting relations are b_Ψ|Ω⟩≈0 and [b_Ψ, b†_Ψ]≈1.Source: B. QInit: Preparing the initial… · p. 17
  • The second QWP can use the same ancilla after a σx operation or a separate ancilla, and sequential preparation assumes the wave packets have negligible overlap.This negligible-overlap condition holds for far-separated wave packets.Source: B. QInit: Preparing the initial… · pp. 17–18
  • Using two ancillas yields higher-fidelity two-packet states because preparation errors can be detected, whereas some errors in the one-ancilla circuit remain undetected.The combined systematic and hardware-noise contribution is termed the ancilla-violation error.Source: B. QInit: Preparing the initial… · p. 18

C. QTrott: Trotterized time evolution

QTrott implements U(t) with a second-order Trotter sequence and tailored circuits for its Hamiltonian terms; system-only evolution supports diagonal observables, while controlled evolution enables return-probability measurement.

C. QTrott: Trotterized time evolution

  • QTrott evolves the scattering state; controlled evolution enables a Hadamard test for return probability, while system-only evolution suffices for diagonal observables.The return probability is R(t) := |⟨Ψ1, Ψ2|U(t)|Ψ1, Ψ2⟩|2.Source: C. QTrott: Trotterized time evolution · p. 18
  • U(t) is approximated by a second-order Trotter sequence of Hh, Hm, and Hϵ terms, with t = ntδt.Each step applies Hh and Hm for half steps around a full Hϵ step.Source: C. QTrott: Trotterized time evolution · p. 19
  • Hm and Hϵ terms are implemented without further approximation, while Hh is split into σx- and σy-containing terms using a first-order Trotter formula.The Hm circuit reuses QGS subcircuits; the Hϵ circuit applies a boson-link operation conditioned on fermion occupation.Source: C. QTrott: Trotterized time evolution · p. 19
  • Each second-order Trotterized time-evolution layer requires 18NP + 8 CNOT gates.The conserved charge Q = NP lets the boson-link qubit be restored after the controlled operation on the last fermion-site qubit.Source: C. QTrott: Trotterized time evolution · p. 20

IV. QUANTUM HARDWARE AND EMULATOR RESULTS

The study tested a circuit for scattering in a (1+1)D Z2 lattice gauge theory using both a noiseless emulator and IonQ Forte hardware. The hardware prepared vacuum and wave-packet states and performed Trotterized evolution, while optimization and return-probability calculations ran on the emulator.

IV. QUANTUM HARDWARE AND EMULATOR RESULTS

  • IonQ Forte implemented vacuum and wave-packet preparation with known parameters and Trotterized evolution; emulator runs handled VQE optimizations and Hadamard tests for return probabilities.The 32-qubit Ytterbium-ion computer has high-fidelity single- and two-qubit gates [119]; noise-mitigation strategies were also employed.Source: §IV · p. 20
  • The study used 11-qubit and 27-qubit systems, corresponding respectively to NP = 5, N = 10 and NP = 13, N = 26.The symmetry-restricted Hilbert spaces contain 504 and 20,801,200 states, respectively; the smaller system permits exact diagonalization, while low-energy states of the larger system were obtained with DMRG and an MPS ansatz.Source: §IV · p. 20
  • With mf = 1.0 and ϵ = −0.3, the non-diagonal-to-diagonal vacuum-energy contribution ratio was approximately 0.221 for both system sizes.This parameter choice places the study away from the strong- and weak-coupling limits.Source: §IV · p. 20

A. Variational-quantum-eigensolver optimization

VQE optimization prepares a good approximation to the true ground state and captures target momentum states with very high fidelity. The low-lying momentum-state ansatz is optimized separately for each target momentum.

A. Variational-quantum-eigensolver optimization

  • The ground state prepared after the first module is a very good approximation to the true ground state.The VQE parameters and prepared-state fidelity are summarized in Table V in Appendix G.Source: A. Variational-quantum-eigensolver… · p. 20
  • VQE optimization captures the target momentum states with very high fidelity for each target momentum.The corresponding fidelities are summarized in Tables VI and VII in Appendix G.Source: A. Variational-quantum-eigensolver… · p. 20
  • For each target momentum, the ansatz parameters are obtained by minimizing the energy of QWP(Ψ)|Ω⟩ with Ψ(k) = δk,kt.The calculation uses a second-order Trotter formula on the angle π/2 with ten steps for NP = 5 and two steps for NP = 13, balancing small Trotter error with manageable noiseless-emulator optimization time.Source: A. Variational-quantum-eigensolver… · p. 20

B. Preparing the initial hadronic wave packets

The section prepares separated, oppositely moving hadron wave packets using resource-controllable approximations, compares their observables with ideal and simulated results, and extends the method to three packets.

B. Preparing the initial hadronic wave packets

  • The two Gaussian packets have equal-magnitude opposite central momenta and are separated by NP sites, keeping their overlap sufficiently small for repeated QWP application.Their normalization is chosen so each packet has unit norm.Source: B. Preparing the initial hadronic wave… · p. 21
  • Appx II uses one ancilla qubit, one second-order Trotter step, and discards rotations with θ ≤ 0.1; only four Cm,n values per packet exceed the cutoff.The approximations introduce systematic state-preparation errors while reducing quantum-resource requirements; for NP = 13, the cutoff can be lowered without adding contributing coefficients.Source: B. Preparing the initial hadronic wave… · pp. 21–22, Figure 6 · p. 22
  • Appx I agrees better with ideal observables, but Appx II is not far off and requires significantly fewer quantum resources.Approximation errors trade off against qubit and gate resources; the authors note this compromise is especially useful for local observables, which are robust against extensive error accumulation under time evolution.Source: B. Preparing the initial hadronic wave… · pp. 23–24
  • For two-wave-packet scattering states at NP = 5 (11 system qubits) and NP = 13 (27 system qubits), Appx II measurements on IonQ Forte agree well with noiseless-simulator counterparts for staggered density and electric field.Source: B. Preparing the initial hadronic wave… · p. 23, Figure 7 · p. 24
  • Repeating QWP with one or more ancillas extends the method to multiple spatially spaced packets; three packets were prepared on IonQ Forte in an NP = 13 system with good agreement to expected results.The work reports 11 and 27 system qubits for the NP = 5 and NP = 13 systems, respectively.Source: B. Preparing the initial hadronic wave… · p. 25

C. Time-evolved observables

Trotterized evolution captures early and qualitative two-meson scattering dynamics, but accumulated Trotter error and hardware noise limit accuracy at later times. The electric-field observable is especially noise-sensitive, and the hardware study cannot access long-time scattering.

C. Time-evolved observables

  • Using an IonQ Forte quantum computer, the study Trotter-evolved an initial two-wave-packet state with NP = 5 parameters; each step block used 204 single-qubit gates and 60 CNOT gates, with 3000 shots and δt = 1.IonQ’s noisy emulator was used to estimate the shot count and Trotter-step size.⁶Source: C. Time-evolved observables · p. 25
  • The /Q and /a errors grow with each Trotter step; the increasing /a error reflects hardware error because its noiseless systematic component is time-independent.The ancilla used for initial-state preparation does not participate in time evolution, so the noiseless /a error from approximate b†k operators does not change over time.Source: C. Time-evolved observables · p. 25
  • Staggered-density profiles track exact wave-packet evolution at early times, while hardware retains qualitative behavior only until noise dominates after t = 6.Noiseless results use 5 × 10^5 shots and deviate from exact values with increasing evolution time due to accumulated Trotter error.Source: Figure 8 · p. 26, C. Time-evolved observables · p. 27
  • For E at the boson qubit, the noiseless value nearly stays at its t = 0 value, but hardware deviates substantially, requiring further noise mitigation.Flipping the boson-qubit electric field costs energy proportional to system size; noisy-emulator recovery used Pauli twirling and operator decoherence renormalization.Source: C. Time-evolved observables · pp. 27–28, Figure 9 · p. 27
  • Moderate coherence time, deep circuits, and small system size prevent access to long-time scattering dynamics, although this was the first hadron-scattering simulation on a quantum computer.Larger systems would reduce boundary effects at long times but require considerably greater circuit depth.Source: C. Time-evolved observables · p. 28

D. Return probability · V. CONCLUSION AND OUTLOOK

The return-probability results show that state-sensitive scattering measurements require high-fidelity initial states. The conclusion reports both the promise and current noise limits of the scattering simulation, and outlines extensions to more complex gauge theories and observables.

D. Return probability

  • Appx I's return probability agrees with the ideal result up to Trotter errors, whereas Appx II deviates significantly, underscoring the need for high-fidelity states in S-matrix measurements.Return probability is a state-sensitive overlap, unlike local observables, which are comparatively insensitive to small deviations from the target state.Source: D. Return probability · p. 29

V. CONCLUSION AND OUTLOOK

  • The hybrid preparation algorithm systematically trades quantum resources for fidelity, and prepared two- and three-wave-packet states on IonQ Forte for 10- and 26-site systems.Hardware results agreed well with ideal results despite noise; symmetry-based noise mitigation yielded only small improvement.Source: §V · p. 29
  • On 10 fermionic sites, noise erased the local-density signal before full collision, while Pauli twirling and operator decoherence renormalization recovered electric-field values in noisy-emulator tests.The electric-field energy in the remnant gauge link showed significant decoherence even in early evolution steps.Source: §V · p. 29
  • Extending the ansatz to non-Abelian and higher-dimensional theories requires additional gauge-invariant meson and Wilson-loop operators, with polynomial parameter growth toward the continuum as a goal.The proposed extension aims to retain high fidelity while increasing theory complexity.Source: §V · p. 30
  • The LSH formulation [155] [156] [157] may offer a cost-efficient extension, using local gauge-invariant electric-basis degrees of freedom and only an algebraic Abelian constraint on each link.Bare-meson operators map to gauge-invariant LSH states, and the constraints may aid circuit noise mitigation; non-Abelian, higher-dimensional dynamics are resource-intensive [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153], while alternative formulations may reduce costs [145] [154] [155] [156] [157] [158] [159] [160] [161] [162] [163] [164] [165] [166] [167] [168] [169] [170] [171] [172] [173] [174] [175] [176] [177] [178] [179].Source: §V · p. 30
  • The algorithm's single-hadron momentum eigenstates could support hadron-structure studies that use real-time correlators to constrain QCD matrix elements [180], including distribution functions [181] [182] [183] [184] [185] [186] [187] [188].The paper identifies large-hadron and electron-ion colliders as contexts for those distribution functions.Source: §V · p. 30
  • For exclusive scattering amplitudes, multiple wave packets can be prepared and a swap test [189] can measure overlap with a desired final state.Adding spacetime-separated electroweak currents enables access to the hadron tensor and, via the optical theorem, inclusive cross sections; both analyses require larger lattices and deeper circuits.Source: §V · p. 30
  • The protocol accesses outgoing scattering states after collision, enabling asymptotic S-matrix measurements and other local or non-local observables, including entanglement.The authors propose studying post-collision string breaking, hadronization, fragmentation, and thermalization, phenomena of interest in quantum simulation [83] [190] [191] [192] [193] [194] [195] [196] [197] [198] [199] [200] [201] [202] [203].Source: §V · pp. 30–31

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

The appendix tests the wave-packet ansatz in a 26-staggered-site Z2 lattice gauge theory using DMRG matrix product states as approximate eigenstate references. The results support finite-order ansatz construction in larger systems, while high-momentum fidelity verification is limited by difficulty distinguishing some excited states.

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

  • The test uses a 26-staggered-site lattice, with DMRG MPS approximations to the ground state and excited states as fidelity references.The calculation includes 13 lowest-energy excited states in each momentum sector; momentum states with k ≠ 0 are inferred from degenerate DMRG outputs because momentum is not specified during the sweeps.Source: Appendix A · pp. 31–33
  • The 3rd-order ansatz achieves F > 0.99 except at the highest momentum for a small parameter range.For mf = 0.1, 0.3 and ϵ = −0.1, nearby excited states in other momentum sectors make the DMRG reference states less reliable for the |k| = 6π/13 fidelity test.Source: Figure 11 · p. 32, Appendix A · p. 34
  • The ansatz improves order by order and performs better in the strong-coupling regime, while similar fidelities are observed for negative momenta.The larger system places non-mesonic excitations well above the |k = 0⟩ eigenstate, so the cyan contour seen in the smaller-system scan is absent.Source: Figure 11 · p. 32, Appendix A · p. 34
  • DMRG outputs achieve Var = O(10^-10) for all parameter pairs, supporting their use as approximate eigenstate references.The optimizations use maximum bond dimension 600, truncate Schmidt coefficients below 10^-12, and run O(100) sweeps.Source: Appendix A · p. 32
  • The appendix concludes that finite-order ansatz states can faithfully represent momentum eigenstates in larger systems, though better MPS references may be needed to verify high-momentum fidelity.The difficulty arises because DMRG can fail to distinguish the highest-momentum state from nearby excited states in other momentum sectors.Source: Appendix A · p. 34

Appendix B: Circuit for QWP in the minimal-gauge formalism · Appendix C: Three-wave-packet preparation · Appendix D: Hadamard test for calculating the return probability

The appendices detail the wave-packet preparation circuit, demonstrate three-wave-packet preparation on hardware, and describe a Hadamard test for measuring return probability. They also report agreement of local density with ideal results despite a return-probability deviation for one preparation variant.

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

  • QWP(Ψ) implements the initial wave packet by exponentiating Eq. (33) with a second-order Trotter formula using one step at angle π/2.Its coefficients Cm,n depend on the desired input wavefunction Ψ(k) and ansatz parameters α(ℓ),k.Source: Appendix B · p. 34
  • Each QWP term e−iθΘm,n uses an efficient SVD-based circuit, with A = VSW† and A² = A†² = 0.The appendix gives U = Ha(V† ⊗ |0a⟩⟨0a| + W† ⊗ |1a⟩⟨1a|) and Dm,n = S ⊗ σz.Source: Appendix B · p. 34
  • For the j = 1 ansatz, the circuit creates at most 1-length mesons; terms with m = n are trivial, and neighboring-site terms use the illustrated construction.For terms including a boson link, the circuit includes a boson-link CNOT; circuits without that link omit it and adjust gate positions and parameters.Source: Appendix B · p. 34
  • The VQE circuit for a target momentum sector is the same QWP circuit with Ψ(k) = δk,kt, and its ansatz parameters are obtained by minimizing energy.The desired scattering-state wavefunction supplies the input coefficients, while the VQE minimizes the energy of the state prepared by b†k on |Ω⟩.Source: Appendix B · p. 35

Appendix C: Three-wave-packet preparation

  • The three-wave-packet preparation uses NP = 13, has small pairwise overlaps, and requires 267 CNOT gates.The overlaps are (Ψ2|Ψ1) = 0.0104, (Ψ3|Ψ1) = −0.0306, and (Ψ3|Ψ2) = 0.0059; the circuit was implemented with 2000 shots.Source: Appendix C · p. 35
  • The hardware-prepared three-wave-packet state exhibits target-state features, and its observable E is within two standard deviations of noiseless and ideal results.The NP = 13 preparation used 2000 hardware shots; Fig. 14 compares staggered density and the boson-qubit electric field with a 5 × 10^5-shot noiseless simulation.Source: Appendix C · p. 35, Figure 14 · p. 36

Appendix D: Hadamard test for calculating the return probability

  • Staggered-density values agree well with ideal results for both preparation variants, while return probability significantly deviates for Appx II.Fig. 15 shows staggered density at four evolution times; the return-probability comparison is to Fig. 10.Source: Figure 15 · p. 37
  • A Hadamard test extracts the real or imaginary part of A(t) from ancilla probabilities after controlled U(t), using a final Hadamard or Rx(π/4) gate.The difference p0 − p1 gives the selected part; controlled evolution is built by controlling each Trotter block, with controlled Rz gates for e−iδtHϵ.Source: Appendix D · p. 37

Appendix E: Time evolution of local observable for NP =13 · Appendix F: Pauli twirling and operator decoherence renormalization · Appendix G: Tables of variational-quantum-eigensolver results

The appendices show that local observables can remain robust despite deviations in return probability, describe emulator-based noise mitigation for the electric-field observable, and summarize VQE results for ground- and single-particle states.

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

  • At times when Appx II return probability has 15%–30% relative error, staggered density χn deviates very little from ideal for both Appxs I and II.This supports the appendix’s conclusion that local observables are more robust to small differences in initial states.Source: Appendix E · pp. 37–38

Appendix F: Pauli twirling and operator decoherence renormalization

  • The unmitigated electric-field observable from IonQ Forte hardware deviates from ideal already at t = 1 and diverges further, possibly because the boson qubit receives more entangling gates.Hardware and simulator results agreed reasonably at t = 0; the proposed gate-count explanation is conjectural.Source: Appendix F · p. 38
  • Pauli twirling improves the electric-field results and corrects the shifted value at t = 1, suggesting a coherent-error source for the uniform offset.The emulator runs used 3000 shots per Trotter step split across 10 equivalent Pauli-twirled circuits.Source: Figure 16 · p. 38, Appendix F · p. 39
  • Noise-mitigated electric-field results show much better agreement with ideal values after operator decoherence renormalization (ODR) in the IonQ Forte emulator.ODR scales E(t) by 1/(1 − ρ(t)), with ρ(t) calibrated using Pauli-twirled U(0) circuits designed to preserve the t = 0 value.Source: Figure 16 · p. 38, Appendix F · p. 39
  • The noise-mitigation analysis used the IonQ Forte emulator rather than hardware because hardware access was limited.The appendix leaves open whether the procedure also works on actual hardware.Source: Appendix F · p. 39

Appendix G: Tables of variational-quantum-eigensolver results

  • For the b†k ansatz with j = 1, the appendix tabulates optimization results for NP = 5 and NP = 13, including optimized parameters, expectation values, target energies, and fidelity measures.The VQE optimization used the noiseless Aer simulator.Source: Appendix G · p. 39
  • For NP = 5 and NP = 13, the optimized QGS energy agrees with the exact ground-state energy to sub-percent level.The appendix reports optimized QGS parameters and ground-state fidelity; optimization used the noiseless Aer simulator, with exact references from diagonalization for NP = 5 and DMRG for NP = 13.Source: Appendix G · pp. 39–40, Table V · p. 40

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