PaperlayerView exact revision on arXiv

Source-linked AI summary

Digital quantum simulations of scattering in quantum field theories using W states

Roland C. Farrell, Nikita A. Zemlevskiy, Marc Illa, John Preskill

arXiv:2505.03111v2quant-phhep-lathep-phnucl-th

TL;DR

Predicting particle production and collective dynamics after quantum-field-theory scattering remains a fundamental challenge. This work introduces a wavepacket-preparation algorithm and uses it in a 104-qubit Ising-field-theory scattering simulation, observing evidence of inelastic particle production and an outgoing heavy particle.

  • Problem

    Predicting particle abundances and collective dynamics generated by high-energy quantum-field-theory scattering remains a forefront theoretical problem.

    Source: §I · p. 1

  • Method

    The paper prepares scattering wavepackets with a W-state-based algorithm using mid-circuit measurement and feedforward, then simulates Ising-field-theory collisions on a quantum processor.

    Source: A. Inelastic scattering on IBM’s… · p. 5, §IV · p. 8

  • Results

    The 104-qubit simulation observes evidence of inelastic particle production and identifies an outgoing heavy particle from the skewness of the measured energy density.

    Source: §IV · p. 7

  • Takeaways & Limitations

    Wavepackets can be prepared with circuit depth independent of wavepacket size in finite spatial dimensions, supporting scattering simulations across lattice models.

    Source: §IV · p. 8

    Device noise was asymmetric, with ⟨Z_n⟩ measurements effectively about 3× noisier than ⟨X_n⟩ and ⟨Z_n Z_{n+1}⟩ measurements.

    Source: A. Inelastic scattering on IBM’s… · p. 7

Abstract

from arXiv · show

High-energy particle collisions can convert energy into matter through the inelastic production of new particles. Quantum computers are an ideal platform for simulating the out-of-equilibrium dynamics of collisions and the formation of subsequent many-particle states. In this work, evidence for inelastic particle production is observed in one-dimensional Ising field theory using IBM's quantum computers. The scattering experiment is performed on 104 qubits of ibm_marrakesh and uses up to 5,589 two-qubit gates to access the post-collision dynamics. An outgoing heavy particle produced in the collision is identified from the skewness of the measured energy density. Integral to this computation is a new quantum algorithm for preparing the initial state (wavepackets) of a quantum field theory scattering simulation. This method efficiently prepares wavepackets by extending recent protocols for creating W states with mid-circuit measurement and feedforward. The required circuit depth is independent of wavepacket size and spatial dimension, representing a superexponential improvement over previous methods. Our wavepacket preparation algorithm can be applied to a wide range of lattice models and is demonstrated in one-dimensional Ising field theory, scalar field theory, the Schwinger model and two-dimensional Ising field theory.

I. INTRODUCTION

The work demonstrates particle production in an inelastic quantum-field-theory scattering simulation and introduces an efficient wavepacket-preparation algorithm based on ancilla-free, constant-depth W-state preparation.

Introduction

  • The wavepacket algorithm prepares |W(k0)⟩ for the target spatial profile, momentum, and quantum numbers, then projects it onto single-particle eigenstates by symmetry-preserving energy minimization.The energy minimization is performed with ADAPT-VQE on classical computers, with future quantum implementation proposed.Source: §I · p. 1
  • The method extends mid-circuit measurement and feedforward W-state preparation to an ancilla-free protocol with constant depth, and generalizes it to initialize wavepackets.This is presented as the first ancilla-free constant-depth W-state preparation protocol.Source: §I · p. 1
  • The required wavepacket-preparation depth is independent of wavepacket size and scales only with correlation length, improving over prior methods with exponential classical overhead or polynomial size-dependent depth.The algorithm applies to a wide range of lattice models.Source: §I · p. 2
  • Evidence for particle production is observed for the first time in a quantum simulation of inelastic scattering, using post-collision energy-density skewness as the signature.The collision creates a slow-moving heavy particle, skewing the post-collision energy density.Source: §I · pp. 1–2
  • The approach is demonstrated in one- and two-dimensional Ising field theory, scalar field theory, and the Schwinger model, including a 104-qubit ibm marrakesh scattering simulation using 5,589 two-qubit gates.Up to 45 Trotterized time-evolution steps evolve the one-dimensional Ising simulation beyond the collision.Source: §I · p. 2

II. OVERVIEW OF THE WAVEPACKET PREPARATION ALGORITHM

The algorithm prepares quantum-field-theory wavepackets by combining constant-depth, mid-circuit-measurement W-state preparation with symmetry-preserving variational energy minimization. Its circuit depth is independent of wavepacket volume and extends across lattice models and finite spatial dimensions.

Wavepacket construction

  • A wavepacket is a localized superposition of single-particle momentum eigenstates centered at position x0 and momentum k0, with position spread proportional to σ−1.Small momentum spread produces a large position spread and allows the packet to propagate for a long time as a localized particle.Source: §II · p. 2

Algorithm overview

  • The method first prepares |W(k0)⟩ with the target wavepacket’s momentum amplitudes and phases, then minimizes energy while preserving momentum and other quantum numbers.Translational invariance preserves amplitudes in each momentum block, while real, symmetry-preserving circuits preserve phases and remove unwanted components.Source: Figure 1 · p. 3, §II · p. 4

W-state preparation

  • Depth 7 prepares |W(k0)⟩ using mid-circuit measurement and feedforward, independent of lattice geometry, system size, and target wavepacket size.The circuit improves earlier W-state preparation methods and generalizes them to momentum-shifted wavepackets.Source: §II · p. 3
  • I = O(δ2) and psuccess ≥0.43δ follow from odd-parity post-selection, with both success probability and infidelity independent of wavepacket size for large wavepackets.The initial state also contains multi-particle and heavier single-particle components, which the subsequent energy minimization removes.Source: §II · pp. 3–4

Scope and applications

  • The method applies to lattice models in any finite spatial dimension, with depth independent of wavepacket volume when qubit connectivity matches the target lattice.It is demonstrated for one-dimensional Ising and scalar field theories, the Schwinger model, and two-dimensional Ising field theory.Source: §II · p. 4

III. QUANTUM SIMULATIONS OF SCATTERING IN ONE-DIMENSIONAL ISING FIELD THEORY

The simulations study one-dimensional Ising field theory with two stable particles, targeting the threshold-opening inelastic channel 11 →12 that produces a heavy particle. Energy-density tracks distinguish elastic scattering from this inelastic process, while higher-energy multiparticle channels remain outside the simulated regime.

Scattering processes

  • The energy-density heatmap shows elastic 11 →11 scattering for low-energy collisions and inelastic 11 →12 scattering with heavy-particle production for high-energy collisions.Propagating particles appear as energy beams with constant velocity across lattice position and time.Source: Figure 2 · p. 4

Simulation setup

  • The simulations use gx = 1.25 and gz = 0.15, giving a light |1⟩ particle with mass m1 = 1.59 and a heavy |2⟩ particle with mass m2 = 2.98.The initial state comprises two spatially separated |1⟩ wavepackets that travel toward one another, collide, and separate.Source: §III · p. 4

Scattering processes

  • At low center-of-mass energies, only elastic 11 →11 scattering is kinematically allowed, whereas the 11 →12 channel opens above the particle-production threshold.This threshold separates the low-energy elastic regime from the targeted inelastic process.Source: §III · pp. 4–5
  • Above the threshold Etot > Ethr = m1 +m2 = 4.57, collisions can produce a heavy |2⟩ particle through 11 →12, identifiable from additional outgoing energy-density tracks.The displayed inelastic regime contains elastic scattering and heavy-particle motion in either direction in superposition, yielding four outgoing tracks despite two particles per outcome.Source: Figure 2 · p. 4, §III · p. 5

Scope and limitations

  • At energies Etot ≫Ethr, additional multiparticle channels generate significant entanglement, but current hardware cannot access these ultrahigh-energy collisions and other channels are neglected here.The simulated energies are restricted so that channels besides 11 →12 can be neglected.Source: §III · p. 5

A. Inelastic scattering on IBM’s quantum computers

IBM’s 104-qubit simulation of one-dimensional Ising-field-theory scattering provides evidence for inelastic production of a heavy |2⟩ particle, identified through post-collision energy-density tracks and skewness. The experiment uses variationally prepared wavepackets, Trotterized evolution, vacuum subtraction, and noise mitigation despite increasing systematic errors at later times.

Experimental setup

  • The experiment uses L = 104 ibm marrakesh qubits with open boundaries, variationally optimized symmetry-preserving wavepacket circuits, and second-order Trotterized evolution with δt = 0.55.Wavepackets have support on d ≳21 sites and total energy Etot/Ethr = 1.2, selected for a strong inelastic signal at limited circuit depth.Source: A. Inelastic scattering on IBM’s… · p. 5, Figure 3 · p. 5

Inelastic-scattering signal

  • Two lower-velocity inner tracks in MPS energy-density dynamics represent heavy |2⟩ particles produced through 11 →12, while elastic and inelastic processes occur in superposition.Outgoing |1⟩ trajectories overlap elastic tracks, and no four-particle wavefunction components are present.Source: A. Inelastic scattering on IBM’s… · p. 5

Experimental dynamics

  • At t2 = 16.5, IBM energy densities agree well with MPS; by t3 = 24.75, systematic errors grow, and noise prevents simulations beyond t = 24.75.The wavepackets collide around t1 = 8.25, begin separating at t2, and form asymptotic particles by t3.Source: A. Inelastic scattering on IBM’s… · p. 6, Figure 4 · p. 6

Noise and resources

  • 5,589 two-qubit gates and depth 130 are required at the latest time, motivating Pauli twirling and Operator Decoherence Renormalization for noise mitigation.The larger simulation light cone increases sensitivity to faulty gates and explains stronger systematic errors at t3.Source: A. Inelastic scattering on IBM’s… · pp. 6–7

Inelastic-scattering signal

  • 0.6σ and 1σ larger skewness than single-particle propagation at t2 and t3, respectively, provides evidence that post-collision asymmetry partly arises from inelastic |2⟩ production.The post-collision state is right-skewed, with skewness increasing at t3 as the heavy particle emerges; systematic errors and boundary effects also contribute.Source: A. Inelastic scattering on IBM’s… · p. 6, Figure 5 · p. 7

IV. DISCUSSION

The work extends quantum scattering simulations into non-equilibrium QFT dynamics while demonstrating that far-from-continuum simulations can reveal qualitative processes. Its wavepacket algorithm offers dimension-independent depth, supporting future two-dimensional and higher-energy studies.

Simulation regime

  • The simulations are the first to probe post-collision non-equilibrium QFT dynamics, reaching particle propagation distances of approximately 40 on 104 qubits and revealing inelastic production.Observing this effect required state-of-the-art hardware and balancing device errors, algorithmic errors, and quantum resources.Source: §IV · p. 7

Limitations

  • The simulations are far from the continuum limit and target qualitative non-equilibrium QFT features rather than precise quantitative predictions.Compared with MPS simulations, their smaller systematic scales do not isolate individual inelastic channels or enable robust underlying-QFT predictions.Source: §IV · p. 7

Future directions

  • Future error-corrected quantum simulations may reach ultra-high-energy scattering with highly entangled many-particle states beyond the reliable range of MPS methods.Such states are expected to render classical tensor-network approaches unreliable.Source: §IV · p. 8

Wavepacket preparation

  • MCM-FF prepares wavepackets in any finite spatial dimension with circuit depth independent of wavepacket size when qubit connectivity matches the target lattice.The speedup builds long-range entanglement through W-state structure and local modifications over approximately ξ sites, followed by local operations and classical communication.Source: §IV · p. 8

Two-dimensional applications

  • Efficient wavepacket preparation could enable quantum-advantaged two-dimensional QFT scattering, opening studies of spin, anyonic statistics, and orbital angular momentum.Native two-dimensional connectivity on emerging quantum computers supports this prospective research frontier.Source: §IV · p. 8

METHODS · A. A constant-depth circuit for initializing |W(k0)⟩ · B. Quantum circuits for preparing wavepackets in lattice quantum field theories

The methods develop a constant-depth, ancilla-free protocol for preparing generalized W states and apply it to wavepacket construction across several lattice quantum field theories. Wavepackets are benchmarked against exact states using statevector or MPS circuit simulation and symmetry-preserving ADAPT-VQE optimization.

A. A constant-depth circuit for initializing |W(k0)⟩

  • The protocol prepares generalized |W(k0)⟩ states by extending constant-depth parity-measurement methods to arbitrary amplitudes cn and phases ϕn without ancilla qubits.Odd-parity postselection prepares the state with success probability bounded by psuccess ≥ 0.43δ and infidelity I = O(δ^2) in the cited special case.Source: A. A constant-depth circuit for… · p. 10
  • The circuit uses mid-circuit measurement and feedforward to kick parity back to q0, postselect odd parity, reset even qubits, spread excitation probability, and add phases with RZ rotations.The final state |ψ2⟩ approximates |W(k0)⟩ after the controlled-RY/CNOT spreading step and single-qubit phase rotations.Source: A. A constant-depth circuit for… · pp. 10–11
  • Both psuccess and infidelity I become independent of wavepacket size d for large d because the coefficients satisfy Pn = O(d^-1).The resulting preparation therefore avoids a wavepacket-size dependence in these quantities under the stated scaling.Source: A. A constant-depth circuit for… · p. 11

B. Quantum circuits for preparing wavepackets in lattice quantum field theories

  • The wavepacket-construction framework targets one- and two-dimensional Ising field theories, λϕ^4 scalar field theory, and the Schwinger model.Circuits with up to 28 qubits are obtained using a qiskit statevector simulator and benchmarked against exact wavepackets from exact diagonalization; L = 256 one-dimensional circuits use Quimb MPS simulation.Source: B. Quantum circuits for preparing… · p. 11
  • Classical simulations use the unitary W-state circuit because MCM-FF branches exponentially with the number of measurements in circuit simulation.Additional lower-depth circuits using beyond-linear connectivity and/or MCM-FF are provided separately.Source: B. Quantum circuits for preparing… · p. 11
  • Approximate wavepackets are prepared by variationally minimizing energy with translationally invariant, real, symmetry-preserving ansätze.For statevector-simulator studies, the wavepacket size d equals the system size L.Source: B. Quantum circuits for preparing… · p. 11
  • ADAPT-VQE greedily builds symmetry-preserving ansätze by appending the circuit-layer candidate that most reduces energy from a Hamiltonian-algebra-inspired operator pool.At each iteration, the selected layer and variational parameters are optimized before further layers are added.Source: B. Quantum circuits for preparing… · p. 12

1. One-dimensional Ising field theory … D. MPS simulations of scattering in one-dimensional Ising field theory

The wavepacket-preparation method is applied across several lattice field theories and validated through MPS simulations of one- and two-dimensional Ising scattering. These studies assess wavepacket quality and convergence while examining elastic and inelastic scattering behavior.

1. One-dimensional Ising field theory

  • In one-dimensional Ising theory, a W-state circuit prepares wavepackets with CNOT depth 2⌊d/2⌋ + 2, while symmetry-preserving ADAPT-VQE reaches infidelity < 0.12 by depth 16.The variational pool uses translationally invariant, real, and anti-symmetric operators generated from Hamiltonian commutators; convergence plateaus can require fifth- and seventh-order commutators, especially as the mass gap decreases.Source: §1 · pp. 12–13

2. One-dimensional scalar field theory

  • In one-dimensional scalar field theory, digitized harmonic-oscillator and field bases enable W-state initialization followed by symmetry-constrained ADAPT-VQE, reaching infidelity < 0.05 by the second step.For nq = 3 and L = 9, the harmonic-oscillator-to-field-basis circuit has CNOT depth 11, while the symmetric Fourier transform has depth 6.Source: §2 · pp. 15–16

3. The Schwinger model

  • In the Schwinger model, charge-neutral strong-coupling hadron wavepackets are prepared with charge-conjugation- and parity-symmetric ADAPT-VQE, whose convergence mostly plateaus after two steps.The method truncates the bosonic gauge-field zero mode to one state; fidelities are considered sufficient for near-term scattering demonstrations but may require improvement toward the continuum limit.Source: §3 · pp. 16–18

4. Two-dimensional Ising field theory

  • The preparation method generalizes to two-dimensional tilted-field Ising theory on a periodic square lattice with translational symmetry in both directions and π/2 rotational symmetry.Recovery of continuum SO(2) rotational symmetry may require lattice and operator smearing.Source: §4 · p. 18
  • On a 5 × 5 lattice, ADAPT-VQE converges toward the desired two-dimensional Ising wavepacket, with slower convergence for smaller mass gaps.The operator ordering begins with one-site, nearest-neighbor two-site, and three-site L- and line-configured operators, reflecting stronger short-distance correlations in gapped systems.Source: Figure 11 · p. 19, §4 · p. 20

C. Quantum circuits for simulating scattering in one-dimensional Ising field theory

  • The one-dimensional scattering circuit prepares opposite-momentum wavepackets with a shared real ADAPT-VQE circuit before applying second-order Trotterized time evolution.The vacuum-subtracted energy density is used throughout, including subtraction of the time-evolved approximate vacuum in scattering simulations.Source: C. Quantum circuits for simulating… · p. 21, Figure 12 · p. 21

D. MPS simulations of scattering in one-dimensional Ising field theory

  • ADAPT-VQE wavepackets on L = 256 and L = 28 lattices have energy-density differences ΔE_n < 10^-3, approximate-vacuum agreement of 0.01%, and variational parameters matching to four decimal places.The MPS scattering parameters are L = 256, σ = 0.13, d = 22, gx = 1.25, gz = 0.15, with mass gap m = 1.59; optimization minimizes energy density over 28 sites to avoid vacuum-dominated updates.Source: D. MPS simulations of scattering in… · p. 21, Figure 13 · p. 22
  • Only elastic scattering appears at k0 = 0.18π, while increasing momentum above the inelastic threshold reveals slower outgoing |2⟩ tracks from 11 → 12 production.The predicted threshold is |kthr| ≈ 0.24π, but variational wavepacket energies shift the effective threshold closer to |kthr| ≈ 0.22π; simulations use maximum bond dimension 350 and achieve 0.01% energy-density convergence.Source: D. MPS simulations of scattering in… · p. 22, Figure 14 · p. 23

E. Details on the simulations performed using IBM’s quantum computers

The IBM simulations use parameter choices optimized to reveal inelastic scattering while controlling hardware, Trotter, truncation, and boundary errors. Open-boundary wavepackets remain valid because they are localized far from boundaries, although boundary-induced ripples emerge at later times.

Simulation parameters

  • The simulations use L = 104, δt = 0.55, tmax = 24.75 with nT = 45, k0 = 0.32π, σ = 0.13, and d = 21 to balance signal strength and numerical errors.These choices maximize the inelastic signal while limiting device, Trotter, wavepacket-spreading, and truncation errors; two-site separation keeps initial-wavepacket interactions negligible.Source: E. Details on the simulations… · p. 23

Open-boundary implementation

  • Wavepackets localized at least 30 sites from open boundaries use periodic-boundary preparation circuits with boundary-coupling terms removed, while time evolution uses the open-boundary Hamiltonian.Approximate bulk translational symmetry preserves the wavepacket momentum content, but the vacuum is slightly lower quality near boundaries.Source: E. Details on the simulations… · p. 24

Boundary effects

  • At t = 20, open- and periodic-boundary energy densities are nearly identical, whereas boundary fluctuations appear between outgoing particles by t = 30.The fluctuations arise when perturbations from the lower-quality boundary vacuum propagate inward and interact with the post-collision state.Source: E. Details on the simulations… · p. 24, Figure 15 · p. 24

Circuit construction

  • The scattering circuit combines a d = 21 unitary W-state preparation, 7 ADAPT-VQE steps, and nT second-order Trotter steps, with two-qubit depth 22 + 18 + 2nT.The unitary preparation was less noisy than the mid-circuit-measurement-and-feedforward alternative on ibm_marrakesh.Source: E. Details on the simulations… · pp. 23–24

Hardware implementation

  • Using native RZZ gates and time-reversal symmetry reduces each Trotter step’s two-qubit gate depth from 4 to 2 while avoiding extra single-qubit gates.RZZ calibration circuits are run before each experiment to tune rotation angles and reduce coherent errors.Source: E. Details on the simulations… · pp. 24–25

Classical validation

  • The measured energy density is compared with noise-free Quimb MPS simulations, which require max bond = 2250 at δt = 0.55 and t = 40 for convergence.For comparison, max bond = 350 suffices for simulations using δt = 1/16.Source: E. Details on the simulations… · p. 25

1. Error mitigation

A combined error-mitigation pipeline substantially improves noisy quantum-simulation observables, although residual state-dependent noise remains near outgoing particles. The approach combines dynamical decoupling, Pauli twirling, TREX, ODR, filtering, and symmetry-based averaging.

Mitigation methods

  • The pipeline combines dynamical decoupling, Pauli twirling, TREX readout mitigation, and ODR-based observable rescaling to suppress idle, coherent, gate, and measurement errors.ODR estimates each observable’s signal strength from a classically tractable mitigation circuit with a similar noise profile.Source: §1 · p. 25

Systematic errors

  • Trotter errors cause energy-density fluctuations of up to 15%, which are mitigated by enforcing energy conservation during post-processing.Approximate vacuum-preparation errors are negligible in the simulations, while the mitigation circuit uses Trotterized vacuum evolution.Source: §1 · p. 25

Filtering and symmetry

  • Observables with signal strength p_O < 0.01 or readout error ϵ_readout > 0.035 are discarded, while reflection symmetry effectively doubles the available shots.The symmetry combines measurements satisfying E_n = E_{L−1−n}.Source: §1 · p. 26

Limitations

  • The Z_n signal strength is consistently approximately 3× lower than those of X_n and Z_n Z_{n+1}, producing larger ODR fluctuations and uncertainty.Possible causes include incomplete RZZ twirling and the coupling imbalance g_x = 1.25, g_z = 0.15.Source: §1 · p. 26

Error-mitigation performance

  • After error mitigation, device energy-density results lie close to MPS expectations, whereas raw data approaches the inaccurate pure-depolarizing-noise limit, with residual disagreements near outgoing particles.The discrepancy indicates that ODR’s vacuum reference circuit does not capture state-dependent noise affecting wavepackets.Source: §1 · p. 26, Figure 17 · p. 27

F. Calculating skewness · APPENDICES · A. A lower bound on the success probability of initializing |W(k0)⟩

The paper uses a noise-aware, cutoff-averaged skewness of the energy density to distinguish elastic from inelastic scattering and identify inelastic particle production. An appendix derives a lower bound on the MCM-FF protocol’s success probability by bounding its odd-parity failure contribution with Jensen’s inequality.

F. Calculating skewness

  • The heavy |2⟩ particle travels more slowly than |1⟩ and skews the post-collision energy profile toward the collision point, producing evidence of inelastic particle production.At late times, the heavy particle appears as a distinct, slower-moving energy-density bump.Source: F. Calculating skewness · p. 27
  • Skewness is computed over a contiguous positive-energy window selected by En + σn ≥ ϵ, with bootstrap-resampled uncertainties and cutoff variation used to control noise and systematic errors.The window captures excitations while omitting noise-driven outliers, and the third moment is evaluated for each cutoff.Source: F. Calculating skewness · p. 27, Figure 18 · p. 28
  • The reported γ combines the cutoff-dependent third moments, while its error bars include both bootstrap statistical errors and variation across energy cutoffs.Positive γ denotes right skewness and negative γ denotes left skewness; substantial γϵ variation motivates including cutoff dependence in the uncertainty.Source: F. Calculating skewness · p. 28
  • After the collision, skewness increases for inelastic scattering but remains near zero for elastic scattering, providing a quantitative indicator of inelastic effects.Both channels are consistent with zero before collision; the distinction emerges around t ≈ 12−15. The comparison is shown for large PBC systems and IBM-marrakesh-like simulations.Source: F. Calculating skewness · p. 28, Figure 19 · p. 29
  • The skewness analysis compares elastic k0 = 0.18π and inelastic k0 = 0.32π scattering, with differences between system variants attributed primarily to boundary effects and Trotter errors.Figure 19 includes both a larger PBC system and simulations replicating the IBM-marrakesh parameters.Source: F. Calculating skewness · p. 28, Figure 19 · p. 29

A. A lower bound on the success probability of initializing |W(k0)⟩

  • The MCM-FF initialization succeeds when odd parity is measured in the state specified by Eq. (10), so its success probability is expressed through that parity probability.The derivation uses the tensor-product structure of the state and assumes δ ≤ 1.Source: A. A lower bound on the success… · p. 30
  • A lower bound on initialization success is obtained by upper-bounding the relevant product term using concavity of log(1 − 2p) and Jensen’s inequality.The inequality is exponentiated and substituted into the success-probability expression to produce the stated bound.Source: A. A lower bound on the success… · p. 30

B. Single-particle spectra, inelastic thresholds and wavepacket spreading

The Ising-field-theory spectrum is well converged at accessible system sizes, while dispersion-controlled kinematics identify suitable inelastic-scattering momenta and explain wavepacket spreading. For the parameters used, 11 →12 is the dominant inelastic channel.

Single-particle spectra

Inelastic thresholds

  • At kthr, outgoing |1⟩ and |2⟩ particles have strongly different group velocities, enabling trajectories to separate and making inelastic effects distinguishable.Energy and total momentum conservation determine the outgoing velocities, whose predictions agree well with MPS simulations.Source: B. Single-particle spectra, inelastic… · p. 31, Figure 21 · p. 32

Wavepacket spreading

  • At k0 = 0.18π, wavepacket energy density delocalizes, whereas at k0 = 0.32π it remains focused during propagation.The corresponding group-velocity ranges are 0.9 ≲v ≲1.45 and 1.5 ≲v ≲1.6, with estimated velocities v(0.18π) ≈1.25 and v(0.32π) ≈1.6.Source: B. Single-particle spectra, inelastic… · p. 32, Figure 22 · p. 32

Inelastic thresholds

  • 11 →12 is the dominant inelastic process because higher-energy channels are kinematically suppressed and 11 →111 has a much smaller branching ratio.At k0 = 0.32π and σ = 0.13, multiple channels with kthr ≤0.42π are accessed in superposition, but higher-energy processes are suppressed.Source: B. Single-particle spectra, inelastic… · pp. 32–33, Table III · p. 33

C. Additional circuits for preparing |W(k0)⟩

The appendix presents three additional |W(k0)⟩ preparation circuits that reduce CNOT depth using beyond-linear connectivity and/or mid-circuit measurement with feedforward. They include heavy-hex and all-to-all constructions, plus a probabilistic fusion protocol for even d.

Common construction

Heavy-hex connectivity

  • CNOT depth 7 + 2 is asymptotically half that of the baseline circuit on heavy-hex connectivity.The method uses heavy-hex connections as ancillas to split the state into two segments, which are built simultaneously; it is most efficient for d = 2+4n.Source: C. Additional circuits for preparing… · pp. 33–34

All-to-all connectivity

Mid-circuit measurement and feedforward

  • For any even d, the protocol fuses two halves with a Bell measurement followed by feedforward and succeeds with probability psuccess = 1/2.In the d = 6 instance, the protocol succeeds for measurement outcomes q3q2 = {10, 11}.Source: C. Additional circuits for preparing… · p. 35

D. More details on using ADAPT-VQE to prepare wavepackets and vacua

This section details an ADAPT-VQE wavepacket-preparation procedure using symmetry-preserving operator pools, sequential energy minimization, and lowest-energy operator selection. It finds that the third-order commutator pool best balances convergence and circuit depth, while wavepacket-derived circuits can prepare higher-quality vacua than vacuum-specific circuits.

ADAPT-VQE procedure

  • The procedure uses translationally invariant imaginary operators respecting Hamiltonian symmetries, initializes the desired momentum-block amplitudes and phases, then iteratively optimizes and appends the lowest-energy operator until tolerance is reached.The initial state also carries relevant quantum numbers such as Z2 parity or electric charge.Source: D. More details on using ADAPT-VQE to… · pp. 36–37

Operator-selection strategy

  • Wavepacket ADAPT-VQE selects the operator producing the lowest energy rather than the largest energy gradient, improving convergence at computational cost proportional to pool size.The overhead is not limiting because symmetry constraints make the operator pools small.Source: D. More details on using ADAPT-VQE to… · p. 37

Operator-pool comparison

  • The third-order commutator operator pool {O}3 provides the best balance between ADAPT-VQE convergence and circuit depth and is used in the quantum simulations.At later steps, the largest pool {O}7 performs best, especially for the smaller mass gap, but its longer Pauli strings generally increase circuit depth.Source: Figure 24 · p. 37, D. More details on using ADAPT-VQE to… · p. 38

Vacuum preparation

  • Wavepacket-derived circuits can yield higher-quality vacua than vacuum-specific ADAPT-VQE circuits, which initially converge faster but plateau after a few steps.The wavepacket-derived vacuum converges more slowly, avoids as prominent a plateau, and can outperform the vacuum-specific construction; convergence is slower for the smaller mass gap.Source: D. More details on using ADAPT-VQE to… · pp. 38–39, Figure 25 · p. 39

E. State preparation and Trotter errors … H. Operators and variational parameters for preparing wavepackets in one-dimensional Ising field theory

The paper calibrates wavepacket truncation and Trotterization for reliable Ising-field-theory scattering, while detailing exact-diagonalization, circuit-identity, and ADAPT-VQE procedures for state preparation. These choices preserve inelastic signatures and support scalable preparation across system sizes.

E. State preparation and Trotter errors

  • Wavepackets are truncated to an interval d when L ≫σ−1, reducing spatial support before state preparation.The truncation is defined for a wavepacket centered at x0 and underlies the support-size studies used to choose d.Source: E. State preparation and Trotter errors · p. 39
  • d = 21 balances momentum resolution against collision time, preserving distinguishable elastic and inelastic scattering while avoiding the truncation effects seen for smaller supports.For the tested parameters, infidelity shows essentially no truncation effects at d ≥20; reducing d to 9 completely washes out the inelastic signal and introduces elastic-channel interference.Source: E. State preparation and Trotter errors · p. 40, Figure 27 · p. 40
  • δt = 1/16 best approximates exact evolution, while δt = 0.55 retains inelastic signatures and minimizes circuit depth and Trotter errors for quantum simulations.Larger steps slow propagation, heat the system, wash out inelastic bumps, and create persistent energy-density offsets; the quantum simulations therefore use δt = 0.55.Source: E. State preparation and Trotter errors · pp. 40–41, Figure 28 · p. 41

F. Determining wavepackets with exact diagonalization

  • Exact wavepackets are constructed by projecting the Hamiltonian into translation-symmetry momentum blocks, diagonalizing each block, and superposing the resulting single-particle eigenstates.A consistent overall phase convention is required so the relative e−ikx0 phases correctly define the wavepacket.Source: F. Determining wavepackets with exact… · pp. 41–42

G. Useful circuit identities

  • Circuit identities combine basis changes, brickwall CZ sequences, and separately implemented Pauli-string exponentials to realize the wavepacket-preparation operator circuits.The constructions include the transformed Hadamard-like gate ˜H ≡ S.H.S† and ordered evolution of the operator sets {Ô1} and {Ô2}.Source: G. Useful circuit identities · pp. 42–43, Figure 29 · p. 43

H. Operators and variational parameters for preparing wavepackets in one-dimensional Ising field theory

  • ADAPT-VQE operators and variational parameters are tabulated for quantum, statevector, MPS, and vacuum-preparation workflows across the one-dimensional Ising simulations.Table VI covers the quantum simulations, while Tables VII–X provide parameters for statevector, MPS, and vacuum preparations.Source: H. Operators and variational… · p. 43, Table VI · p. 44, Table VII · p. 44
  • Variational parameters converge exponentially with system size, but small-wavepacket operators are not optimal for larger wavepackets because SC-ADAPT-VQE fixes d/L.For gx = 1.25, gz = 0.15, and k0 = 0.36π, parameters for L = 28 and L = 256 agree to 4 decimal places.Source: H. Operators and variational… · p. 43
  • Direct MPS simulation of large-L preparation avoids parameter extrapolation here, whereas SC-ADAPT-VQE remains promising for higher dimensions beyond faithful classical simulation.The work reports no parameter extrapolations for these one-dimensional circuits, while identifying extrapolation workflows as potentially useful in higher-dimensional settings.Source: H. Operators and variational… · pp. 43–44

I. Error mitigation with energy rescaling

A post-processing energy-rescaling step restores energy conservation and reduces systematic errors from Trotterization and device noise. It also enables static-vacuum calculations, while its effectiveness decreases at later times because of a positive-energy bias.

Method

  • The method multiplicatively rescales the ODR-predicted vacuum-subtracted energy density using the initial-state total energy.This corrects energy-conservation violations caused by Trotter and device errors and reduces ODR’s classical computing overhead.Source: §I · p. 45
  • Enforcing energy conservation permits the unevolved vacuum to replace the time-evolved vacuum when computing predicted observables and signal strength.This avoids large MPS bond dimensions associated with the large Trotter step δt = 0.55.Source: §I · p. 45

Limitations

  • The ∼15% error from using the static rather than time-evolved vacuum is corrected when energy conservation is enforced.At later simulation times, increasing positive-energy bias makes the rescaling ineffective; light-cone-based zeroing could improve it.Source: §I · p. 45

Results

  • Energy rescaling reduces the systematic error near the collision point, with rescaled quantum energy density aligning with MPS after enforcing energy conservation.The improvement is largest in the lattice center, although residual bias remains away from the collision point.Source: §I · pp. 44–45

J. Table of results

Tables XI and XII report vacuum-subtracted energy densities from MPS and error-mitigated IBM Marrakesh simulations for scattering and single-wavepacket dynamics. The results use parity symmetry to omit equivalent sites, while noisy-readout filtering causes missing entries in the single-wavepacket data.

J. Table of results

  • The simulations use L = 104 with open boundary conditions, gx = 1.25, gz = 1.5, δt = 0.55, k0 = 0.32π, and σ = 0.13.The setup applies to the quantum simulations of inelastic scattering summarized in Table XI.Source: J. Table of results · p. 45
  • Parity symmetry relates En and EL−1−n, so energy densities for symmetry-equivalent sites are omitted from the tables.MPS values outside the simulation light cone are not exactly zero because of the finite Trotter step size.Source: J. Table of results · p. 45, Table XI · p. 47
  • At t1 = 8.25, Fig. 30 compares MPS predictions with IBM Marrakesh data before and after energy rescaling with energy conservation enforced.The figure distinguishes unrescaled results from energy-conserving rescaled results.Source: Figure 30 · p. 46
  • Tables XI and XII compare MPS and error-mitigated IBM Marrakesh energy densities for scattering and single-wavepacket simulations at the reported evolution times.Table XI corresponds to Fig. 4 at t = 0, 8.25, 16.5, 24.75; Table XII corresponds to Fig. 5 at t = 16.5, 24.75.Source: J. Table of results · pp. 45, 47, Table XI · p. 47, Table XII · p. 48
  • Several single-wavepacket entries are missing because qubits with noisy readout were filtered out.This limitation is specified for the quantum-simulation columns in Table XII.Source: Table XII · p. 48

Related papers

Loading 2505.03111v2…