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

The paper addresses how to simulate out-of-equilibrium QFT scattering and inelastic particle production on quantum hardware. It introduces a W-state-based wavepacket-preparation algorithm with symmetry-preserving energy minimization, then applies it to Ising-field-theory scattering on IBM hardware. The study observes evidence for inelastic production of a heavy particle and demonstrates wavepacket preparation across several lattice models.

  • Problem

    QFT scattering simulations require efficient preparation of localized single-particle wavepackets to study particle production and subsequent many-particle dynamics.

  • Method

    The method prepares W-state-based wavepackets with mid-circuit measurement and feedforward, then applies symmetry-preserving energy minimization to reach the target wavepacket.

  • Results

    The study observes evidence for inelastic particle production in one-dimensional Ising field theory using a 104-qubit ibm marrakesh simulation.

  • Takeaways & Limitations

    The algorithm prepares wavepackets with depth independent of wavepacket size in finite dimensions and supports scattering simulations across multiple lattice models.

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 paper addresses quantum simulation of QFT scattering and particle production by introducing an efficient wavepacket-preparation algorithm. It demonstrates the approach across lattice models and uses it to prepare a 104-qubit Ising-field-theory scattering simulation.

  • Quantum simulations of QFT scattering could study particle production and out-of-equilibrium dynamics relevant to high-energy collisions and extreme environments.The paper frames predicting produced-particle abundances and collective dynamics as a forefront theoretical problem.
  • The wavepacket algorithm first prepares |W(k0)⟩ with the target spatial profile, momentum content, and quantum numbers, despite multi-particle contributions.
  • Symmetry-preserving energy minimization then projects |W(k0)⟩ onto the desired single-particle wavepacket.
  • The initial W-state preparation extends constant-depth mid-circuit-measurement and feedforward protocols and removes the need for ancillas.
  • The required circuit depth is independent of wavepacket size and scales only with correlation length, improving over methods with exponential classical overhead or polynomial wavepacket-size depth.
  • The method is demonstrated in Ising, scalar, Schwinger, and two-dimensional Ising field theories, including a 104-qubit ibm marrakesh scattering simulation.

II. OVERVIEW OF THE WAVEPACKET PREPARATION ALGORITHM

The algorithm prepares a localized wavepacket by establishing its momentum-sector amplitudes and phases, then minimizing energy within symmetry-preserving circuits. W-state preparation provides a constant-depth first step, while the second step removes unwanted higher-energy components.

  • A wavepacket is a localized superposition of single-particle eigenstates, with small momentum spread σ producing position spread proportional to σ^-1.
  • Translational invariance block-diagonalizes the Hamiltonian by momentum, whose lowest-energy state in each block is the corresponding single-particle eigenstate.
  • The initial state |W(k0)⟩ is constructed with the target momentum-sector amplitudes and phases, using Gaussian spatial weighting and a circuit of two-qubit gate depth 7.
  • Mid-circuit measurement and feedforward prepare |W(k0)⟩ with success probability psuccess ≥0.43δ and infidelity I = O(δ2), independent of wavepacket size for large wavepackets.
  • The W-state initialization retains unwanted multi-particle and heavier single-particle components that must be removed by the second step.
  • Energy minimization uses translationally invariant, real circuits that preserve momentum amplitudes and phases while projecting onto the target single-particle wavepacket.
  • The method applies in any finite spatial dimension with depth independent of wavepacket volume when qubit connectivity matches the target lattice.

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

The Ising-field-theory simulation distinguishes elastic scattering from the lowest inelastic channel, where two light particles produce one light and one heavy particle. Above threshold, the heavy-particle channel appears through additional energy-density tracks.

  • The Ising field theory emerges by tuning the Ising model toward criticality, with energy density serving as an important observable.
  • Heatmaps encode energy density over lattice position and time, with constant-velocity beams distinguishing elastic 11 →11 from inelastic 11 →12 scattering.
  • The simulation uses two stable particles, a light |1⟩ particle with mass m1 = 1.59 and a heavy |2⟩ particle with mass m2 = 2.98.
  • At low center-of-mass energy, only elastic scattering 11 →11 is allowed, while the lowest inelastic channel is 11 →12.
  • The inelastic channel opens above Etot > Ethr = m1 +m2 = 4.57 and can be detected from additional outgoing energy-density tracks.
  • At energies Etot ≫Ethr, many-particle inelastic processes become relevant and are challenging for MPS methods and current quantum hardware.

A. Inelastic scattering on IBM’s quantum computers

The experiment uses 104 qubits to simulate inelastic Ising-field-theory scattering and identifies heavy-particle production through post-collision energy-density asymmetry. Quantum data agree with MPS through early post-collision times, while later times show skewness associated with the slower heavy particle amid substantial device-noise challenges.

  • Simulation setup: L = 104 qubits on ibm marrakesh simulate scattering with open boundaries, variationally prepared wavepackets, and second-order Trotterized time evolution.The wavepackets are initialized from |W(k0)⟩ and optimized with symmetry-preserving circuits determined by ADAPT-VQE.
  • Scattering dynamics: Two outgoing track pairs appear in MPS energy-density calculations: fast outer |1⟩ particles and slower inner |2⟩ particles from 1_1 1_1 → 1_1 2_1.The elastic and inelastic outcomes occur in superposition, without four-particle wavefunction components.
  • Scattering dynamics: At t2 = 16.5, IBM energy densities agree well with MPS, while by t3 = 24.75 asymptotic particles begin forming and systematic quantum-data errors grow.The wavepackets collide around t1 = 8.25, separate at t2, and develop asymmetry at later times.
  • Evidence for inelastic production: Post-collision energy density becomes center-skewed because the heavy |2⟩ particle travels more slowly than the light particles.The skewness γ is computed from the third moment and increases at t3, although systematic errors also contribute.
  • Resources and errors: 45 Trotter steps require 5,589 two-qubit gates and a two-qubit gate depth of 130, making error mitigation essential.The simulations encounter asymmetric measurement noise and sensitivity to individual faulty gates across a light cone of approximately 40 qubits.

IV. DISCUSSION

The work demonstrates non-equilibrium collision dynamics on quantum hardware while remaining far from the continuum limit and current ultra-high-energy regime. Its wavepacket algorithm offers size-independent depth under connectivity conditions, supporting future higher-dimensional scattering simulations.

  • Scope and achievement: The simulations are the first to probe non-equilibrium QFT dynamics generated after particle collisions on quantum hardware.They use L = 104, propagation distance vtmax ∼40, wavepacket size d ∼21, and correlation length ξ ∼0.6.
  • Scope and limitations: The simulations are far from the continuum limit and are intended to demonstrate informative non-equilibrium features rather than make precise underlying-QFT predictions.The authors distinguish this scope from future simulations aimed at more precise channel-level predictions.
  • Future prospects: Ultra-high-energy scattering remains inaccessible to current hardware, while many-particle production can make MPS methods unreliable.The authors identify error correction as a likely requirement for reaching that regime.
  • Algorithmic significance: The wavepacket algorithm has depth independent of wavepacket size in finite dimensions when qubit connectivity matches the target lattice.Its advantage comes from combining local operations with mid-circuit measurement and feedforward to distribute long-range entanglement.
  • Future prospects: Efficient wavepacket preparation is a prerequisite for pursuing quantum advantage in two-dimensional scattering, where classical dynamics simulations are challenging.The paper identifies two dimensions as a prospective regime rather than a demonstrated quantum advantage.

METHODS

The methods construct momentum-resolved W-like states, project them toward single-particle wavepackets, and benchmark applications across several lattice field theories. Mid-circuit measurement and feedforward provide constant-depth preparation, while classical simulation uses alternative unitary circuits because branching is costly.

  • Conventions: The notation defines L as system size, d as wavepacket size, m as the lightest-particle mass, and |ψwp⟩, |ψansatz⟩, and |ψvac⟩ as exact, approximate, and vacuum states.The default convention uses little-endian qubit ordering and periodic boundaries unless stated otherwise.
  • W-state preparation: The W-state protocol prepares a spatially and momentum-structured state by rotations, parity measurement, post-selection, probability spreading, and phase addition.The final state approximates |W(k0)⟩, with success probability and infidelity characterized analytically.
  • W-state preparation: The generalized ancilla-free protocol prepares |W(k0)⟩ with two-qubit gate depth 7 independent of lattice geometry, system size, and target wavepacket size.For large d, both success probability and infidelity are independent of d under the stated scaling.
  • W-state preparation: The controlled-parity circuit uses mid-circuit measurement and feedforward to correlate a control qubit with the parity of odd-numbered sites.This parity correlation supports the subsequent projection and redistribution steps.
  • Applications and benchmarking: The algorithm is applied to one-dimensional Ising and scalar theories, the Schwinger model, and two-dimensional Ising theory, with small systems benchmarked against exact diagonalization.Classical circuit simulations instead use unitary preparation because measurement branching can scale exponentially with the number of measurements.
  • Energy minimization: ADAPT-VQE builds symmetry-preserving ansätze layer by layer by selecting the circuit layer that most effectively minimizes energy.This supplies the energy-minimization stage used to obtain approximately prepared wavepackets.

1. One-dimensional Ising field theory

In one-dimensional Ising field theory, wavepackets are initialized with a W-state-like circuit and refined by symmetry-preserving ADAPT-VQE energy minimization. The method is extended across scalar, Schwinger, and two-dimensional Ising models, while scattering simulations identify elastic and inelastic particle-production signatures.

  • Wavepacket preparation: The initial wavepacket circuit prepares |W(k0)⟩, which establishes spatial profile and momentum content before energy minimization projects it toward single-particle eigenstates.ADAPT-VQE uses translationally invariant, real, symmetry-preserving circuits to minimize the energy.
  • Wavepacket preparation: CNOT depth: 2⌊d/2⌋+ 2 for the unitary |W(k0)⟩ preparation circuit, scaling linearly with wavepacket size without MCM-FF.This scaling is described as likely optimal for linear connectivity when mid-circuit measurement and feedforward are unavailable.
  • Wavepacket quality: A convergence plateau appears around the 6th ADAPT-VQE step at depth 16 for most tested couplings, and expanding the operator pool with higher-order commutators can overcome it.Convergence generally worsens as the mass gap decreases and the correlation length increases.
  • Generalization: Minimizing the energy rapidly decreases infidelity in scalar field theory and converges toward the desired wavepacket in two-dimensional tilted-field Ising theory.The method is also described as overcoming wavepacket scalability and circuit-optimization limitations in the Schwinger model.
  • Finite-size validation: For L = 256 and L = 28 lattices, the prepared wavepackets have energy-density differences ∆En < 10^-3 after 8 ADAPT-VQE steps, supporting similar wavepacket quality.The approximate vacuum energies agree to 0.01% between the two lattice sizes.
  • Scattering: MPS scattering simulations distinguish elastic 11 →11 scattering below threshold from inelastic 11 →12 scattering, whose heavier-particle tracks become identifiable at k0 ≥0.28π.The inelastic process converts kinetic energy into mass, producing slower outgoing particles.

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

The IBM scattering simulations use open boundaries, hardware-compatible gate choices, and parameter settings selected to expose inelastic effects while controlling truncation, Trotter, and device errors. Boundary-induced perturbations remain small early in the evolution but become visible later.

  • Time evolution: δt = 0.55 is the largest Trotter step size without significant Trotter errors, while tmax = 24.75 is the maximum time before device errors overwhelm results.The simulation reaches tmax with nT = 45 Trotter steps.
  • Parameter choices: k0 = 0.32π is chosen for the strongest inelastic-scattering signal at tmax = 24.75, while σ = 0.13 limits wavepacket spreading enough to resolve elastic and inelastic processes.The wavepacket extent and spatial size are selected jointly to control spreading during evolution.
  • State preparation: The unitary |W(k0)⟩ circuit is used instead of the constant-depth MCM-FF circuit because it is significantly less noisy on ibm_marrakesh.MCM-FF also creates exponentially branching classical simulation costs in the worst case.
  • Boundary effects: At t = 20, OBC and PBC energy densities are nearly identical, but boundary-generated fluctuations appear between outgoing particles at t = 30.These perturbations arise because the boundary vacuum is lower quality and can be mitigated with boundary operators in the ADAPT-VQE pool.
  • Gate implementation: By using RZZ instead of CZ gates, the two-qubit gate depth per Trotter step is reduced from 4 to 2 through time-reversal-based evolution.The circuit avoids the extra single-qubit gates otherwise needed to reverse the RZZ angle.

1. Error mitigation

Error mitigation combines dynamical decoupling, Pauli twirling, TREX, and ODR to estimate observables from noisy device data. Mitigated results generally approach MPS expectations, though state-dependent noise remains problematic near outgoing particles.

  • Mitigation methods: Error mitigation applies dynamical decoupling, Pauli twirling, TREX, and ODR to reduce idle, coherent, measurement, and observable-dependent noise.ODR rescales measured observables using signal strengths estimated from a matched vacuum-evolution circuit.
  • Mitigation methods: The signal strength |p_O| ≤ 1 quantifies how strongly the Pauli noise channel impacts an observable.Measured expectation values satisfy ⟨O⟩meas = p_O⟨O⟩pred.
  • Limitations and corrections: Trotter errors cause vacuum-reference energy-density fluctuations of up to 15%, which are mitigated by enforcing energy conservation during post-processing.Approximate vacuum-preparation errors are reported as negligible in this calculation.
  • Results: At t3 = 24.75, error-mitigated device results lie close to MPS expectations, unlike raw data near the pure-depolarizing-noise limit.Some mitigated points still disagree by several standard deviations, especially around outgoing particles.
  • Limitations and corrections: The Z-observable signal is consistently ∼3× lower than those of the other observables, indicating stronger impact from anticommuting Pauli errors.This weaker signal contributes to disagreements that remain after mitigation.

F. Calculating skewness

The paper defines a cutoff-robust skewness metric for the positive-energy half of the lattice and uses it to distinguish elastic from inelastic scattering. In simulations, skewness rises after inelastic collisions but remains near zero for elastic ones.

  • Metric construction: The skewness window includes contiguous sites on one lattice half where E_n + σ_n ≥ ϵ, capturing the positive-energy region.Energy values and uncertainties are obtained through bootstrap resampling.
  • Metric construction: The energy density is symmetrized about the collision center as part of error mitigation before calculating skewness.This preprocessing is stated explicitly for the reported metric.
  • Metric construction: The reported γ aggregates skewness values γ_ϵ across energy cutoffs, with error bars incorporating bootstrap uncertainty and cutoff variation.Positive γ denotes right skew, while negative γ denotes left skew.
  • Elastic versus inelastic scattering: After collision at t ≈ 12–15, skewness increases for inelastic k0 = 0.32π scattering but stays near zero for elastic k0 = 0.18π scattering.Both channels have skewness consistent with zero before collision.

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

The analysis establishes converged single-particle spectra, uses dispersion relations and conservation laws to select inelastic-scattering kinematics, and characterizes wavepacket propagation. The chosen momentum accesses additional channels while making outgoing particles kinematically distinguishable.

  • Single-particle spectra: For the gapped Ising field theory, single-particle spectra converge exponentially with system size, with E(k) well converged by L = 28.Finite-size effects are most pronounced for particle |2⟩ at low momentum.
  • Wavepacket spreading: At larger momentum, the group velocity reaches a relativistic plateau near c ≈ 1.6 before decreasing because of lattice artifacts.At low momentum, propagation is nonrelativistic and v(k) increases approximately linearly with momentum.
  • Inelastic thresholds: Energy and momentum conservation determine outgoing velocities in 11 → 12 scattering, and near threshold the |1⟩ and |2⟩ trajectories separate strongly.This separation guides momentum selection for a clearer inelastic signal.
  • Wavepacket spreading: For k0 = 0.18π and 0.32π, the simulated wavepacket group-velocity supports are approximately 0.9 ≲ v ≲ 1.45 and 1.5 ≲ v ≲ 1.6, respectively.These ranges describe the 68% probability regions of the momentum-space wavepackets.
  • Inelastic thresholds: The k0 = 0.32π, σ = 0.13 wavepacket accesses additional inelastic processes beyond 11 → 12, although 11 → 12 remains dominant because 11 → 111 has a much smaller branching ratio.The relevant thresholds are summarized for processes with k_thr ≤ 0.42π.

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

The paper presents alternative circuits for preparing |W(k0)⟩ using connectivity, all-to-all constructions, and mid-circuit measurement with feedforward. These methods trade connectivity or probabilistic success for reduced CNOT depth.

  • State construction: The circuits focus first on preparing amplitudes c_n, after which single-qubit RZ rotations add the phases needed for |W(k0)⟩.This separates amplitude preparation from phase encoding.
  • Heavy-hex connectivity: A heavy-hex circuit uses available connectivity and ancillas to build two wavepacket segments simultaneously, reducing depth relative to a linear construction.The heavy-hex layout is viewed as a chain with additional inter-segment connections.
  • All-to-all connectivity: An all-to-all-connected circuit prepares |W(k0)⟩ with CNOT depth 2⌈log2(d)⌉.The rotation angles are determined from a d × d matrix relating amplitudes to controlled-RY operations.
  • MCM-FF fusion: The fusion protocol resets the measured qubits and applies a conditional Z feedforward before completing the controlled-RY and CNOT sequence.The feedforward is applied for the {11} outcome, while the {10} outcome requires no correction.
  • Resource comparison: Table V compares qubits, connectivity, CNOT gates, measurements, success probability, and infidelity across W-state preparation methods.The constant-depth resource values are adjusted for linear connectivity.

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

ADAPT-VQE prepares wavepackets and vacua by optimizing symmetry-preserving circuits from suitable operator pools. Larger pools improve later convergence but increase circuit depth, while vacuum preparation depends on the chosen initialization and coupling regime.

  • Algorithm: ADAPT-VQE defines symmetry-preserving operator pools, initializes the desired momentum and quantum-number sectors, and iteratively adds the lowest-energy operator until tolerance is reached.The procedure optimizes every operator in the pool at each step rather than selecting only the largest energy gradient.
  • Operator pools: For later optimization steps, the largest pool { Ô}7 gives the best wavepacket infidelity, especially at the smaller mass gap.Longer Pauli strings likely build long-range correlations more effectively, but their circuits are generally deeper.
  • Operator pools: The pool { Ô}3 is selected as the best balance between convergence and circuit depth for the quantum simulations.The larger pools converge better at later steps, but the comparison does not directly report their circuit depths.
  • Vacuum preparation: Vacuum preparation from the wavepacket circuit converges more slowly but avoids the prominent plateau seen when starting from the vacuum-specific circuit.In some cases, the wavepacket circuit also produces a higher-quality vacuum because greedy operator ordering is not always optimal.
  • Vacuum preparation: Vacuum convergence is slower for the coupling set with the smaller mass gap.The comparison uses (gx, gz, m) = (1.25, 0.15, 1.6) and (1.08, 0.03, 0.7).

E. State preparation and Trotter errors

Wavepacket truncation and Trotterization create distinct accuracy–cost tradeoffs in scattering simulations. The chosen wavepacket size balances momentum resolution against collision time, while large Trotter steps can drive dynamics toward heating and chaos.

  • Wavepacket truncation: For L ≫ σ^-1, truncating the initial wavepacket to d sites reduces spatial support, with d = 21 selected for the quantum simulations.At σ = 0.13 and k0 = 0.36π, truncation effects are essentially absent for d ≥ 20.
  • Wavepacket truncation: Decreasing d from 27 to 9 washes out the inelastic signal, while small d also produces interference that obscures elastic and inelastic channels.Larger wavepackets spread less but take longer to overlap and scatter.
  • Trotter errors: The Trotter step δt balances approximation error against circuit depth: smaller steps are more accurate but require deeper circuits to reach a fixed final time.The choice also depends on device noise and the target simulation time.
  • Trotter errors: At δt = 0.66, Trotter evolution effectively quenches the state into chaotic dynamics, while δt > 1/16 reduces wavepacket peaks and suggests thermalization.Larger steps also modify group velocity, causing slower wavepacket propagation.
  • Exact wavepackets: The exact single-particle wavepacket is constructed by diagonalizing the Hamiltonian in a momentum block and coherently superposing the resulting eigenstates with consistent phases.Translation-related basis states span each momentum block before projection and diagonalization.
  • Scaling: Wavepacket variational parameters converge exponentially with system size, but circuits optimized for small wavepackets are not generally optimal for larger ones.This limits extrapolation approaches that keep d/L fixed.

I. Error mitigation with energy rescaling

Energy conservation is used as a post-processing constraint to reduce systematic errors in quantum-simulated energy densities. The rescaling improves agreement with MPS near the collision, though its effectiveness decreases at later times.

  • Motivation and method: Quantum noise and Trotter errors violate total-energy conservation, motivating multiplicative rescaling of the vacuum-subtracted energy density.The rescaling uses the initial-state energy and the ODR-predicted energy density.
  • Results: Using the static vacuum instead of the time-evolved vacuum introduces an approximately 15% error that is corrected when energy conservation is enforced.This also avoids the large bond dimensions needed when the large Trotter step effectively quenches the vacuum.
  • Results: At t1 = 8.25, energy-conservation enforcement aligns the rescaled quantum energy density with MPS and reduces the systematic error near the collision.The improvement is largest in the middle of the lattice, although a residual positive-energy bias remains elsewhere.
  • Limitations: At later simulation times, increasing positive-energy bias makes energy rescaling ineffective.The rescaling value is reduced because the bias accumulates when summed over many sites.

J. Table of results

The reported tables compare MPS and error-mitigated IBM quantum results for vacuum-subtracted energy densities in scattering and single-wavepacket simulations. The dataset uses an L = 104 open-boundary system and includes missing entries from noisy-readout filtering.

  • Simulation settings: The corresponding simulations use L = 104, open boundary conditions, k0 = 0.32π, σ = 0.13, and Trotter step δt = 0.55.The quantum and MPS columns provide the comparison for each reported time.
  • Inelastic scattering: The inelastic-scattering results report vacuum-subtracted energy density at t = 0, 8.25, 16.5, and 24.75 using MPS and error-mitigated ibm marrakesh data.Unshown sites are recovered using the parity relation E_n = E_{L−1−n}.
  • Single wavepacket: The single-wavepacket results report vacuum-subtracted energy density at t = 16.5 and 24.75 from MPS and error-mitigated ibm marrakesh simulations.Several entries are missing because qubits with noisy readout were filtered out.
Loading 2505.03111v2…