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

Nature Physics (2026) Cite this article High-energy particle collisions can convert energy into matter through the inelastic production of new particles. Quantum computers offer a route to simulating these out-of-equilibrium processes, but accessing post-collision dynamics and determining the abundance of produced particles remain challenging. Here we report evidence for inelastic particle production in one-dimensional Ising field theory using a digital quantum processor. We collide two wavepackets, each containing the lightest particle in the theory, and use the skewness of the measured energy density to identify an inelastic component containing one outgoing light particle and one heavier particle. The experiment uses 104 qubits and up to 5,589 two-qubit gates to access the post-collision dynamics. The computation relies on a quantum algorithm that extends protocols for efficiently creating W states—multipartite entangled states with a single excitation distributed across many qubits—to prepare the initial wavepackets. With mid-circuit measurements followed by operations conditioned on their outcomes, the circuit depth is independent of the wavepacket’s spatial volume, rather than scaling polynomially as in previous methods. We construct wavepacket-preparation circuits for one-dimensional Ising and scalar field theories, the Schwinger model and two-dimensional Ising field theory. 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Kandala for sharing methods for improvements in RZZ gate calibrations in our simulations.R.C.F. and J.P. acknowledge support from the US Department of Energy QuantISED programme through the theory consortium ‘Intersections of QIS and Theoretical Particle Physics’ at Fermilab, from the US Department of Energy, Office of Science, Accelerated Research in Quantum Computing, Quantum Utility through Advanced Computational Quantum Algorithms (QUACQ), and from the Institute for Quantum Information and Matter, an NSF Physics Frontiers Center (grant no. PHY-2317110). R.C.F. additionally acknowledges support from a Burke Institute prize fellowship. N.A.Z. acknowledges support provided by the DOE, Office of Science, Office of Nuclear Physics, InQubator for Quantum Simulation (IQuS) under Award Number DOE (NP) Award DE-SC0020970 via the programme on Quantum Horizons: QIS Research and Innovation for Nuclear Science. N.A.Z. is also supported by the Department of Physics and the College of Arts and Sciences at the University of Washington. M.I. acknowledges support provided by the Quantum Science Center (QSC), which is a National Quantum Information Science Research Center of the US Department of Energy. J.P. acknowledges funding provided by the US Department of Energy Office of High Energy Physics (grant no. DE-SC0018407), the US Department of Energy, Office of Science, Accelerated Research in Quantum Computing, Fundamental Algorithmic Research towards Quantum Utility (FAR-Qu), and the US Department of Energy, Office of Science, National Quantum Information Science Research Centers, Quantum Systems Accelerator. The computations presented in this work were conducted in the Resnick High Performance Computing Center, a facility supported by the Resnick Sustainability Institute at Caltech and also enabled by the use of advanced computational, storage and networking infrastructure provided by the Hyak supercomputer system at the University of Washington. R.C.F. and N.A.Z. acknowledge the use of IBM Quantum Credits for this work. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Quantum team.

This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the US Department of Energy under contract no. DE-AC02-05CH11231 using NERSC award no. NERSC DDR-ERCAP0034353.

This research used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under contract no. DE-AC05-00OR22725.Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA, USARoland C. Farrell & John PreskillDepartment of Physics, California Institute of Technology, Pasadena, CA, USARoland C. Farrell & John PreskillInQubator for Quantum Simulation, Department of Physics, University of Washington, Seattle, WA, USANikita A. Zemlevskiy & Marc IllaAWS Center for Quantum Computing, Pasadena, CA, USAJohn PreskillSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarSearch author on:PubMed Google ScholarR.C.F. and J.P. conceived the research direction of simulating one-dimensional Ising field theory on a quantum computer. R.C.F. developed the wavepacket preparation algorithm, and R.C.F. and N.A.Z. performed classical simulations of the state preparation and scattering circuits to validate the approach. R.C.F., N.A.Z. and M.I. worked on the design of the quantum circuits, and R.C.F. and N.A.Z. carried out the experiments on IBM’s quantum computers. All authors contributed to interpreting the experimental data and to writing the manuscript.Correspondence to Roland C. Farrell, Nikita A. Zemlevskiy, Marc Illa or John Preskill.J.P. is a shareholder and part-time employee of Oratomic, Inc., which is developing fault-tolerant quantum computers. The other authors declare no competing interests.Nature Physics thanks the anonymous reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.Upper: Circuits used to prepare wavepackets in one-dimensional Ising field theory. Left: a unitary circuit that prepares \(\left\vert W({k}_{0})\right\rangle\) in Eq. 9 across 9 sites. The rotation angles are given in Eq. 15. Right: circuit elements used in ADAPT-VQE to prepare wavepackets in Ising field theory. They implement the unitary evolution of the operators in Eq. 17. a) implements \(\exp [-{\rm{i}}(\uptheta /2)({\rm{YZ}}+{\rm{ZY}})]\) and b) implements \(\exp [-{\rm{i}}(\uptheta /2)({\rm{YX}}+{\rm{XY}})]\). Examples of circuits that implement \(\exp [-{\rm{i}}(\uptheta /2)\sum {\rm{ZYZ}}]\) and \(\exp [-{\rm{i}}(\uptheta /2)\sum ({\rm{ZXY}}+{\rm{YXZ}})]\) across 8 qubits with PBCs are shown in c) and d), respectively. Lower: Quality of prepared wavepackets in one-dimensional Ising field theory. The deviation in energy (left) and infidelity (right) of the prepared wavepacket as a function of CNOT depth. The CNOT depth is monotonically related to the step of ADAPT-VQE, and corresponds to the additional depth after \(\left\vert W({k}_{0})\right\rangle\) preparation. Results are shown for wavepacket parameters σ = 0.13, k0 = 0.36Π and L = 28 for various couplings gx, gz.Left: a circuit that prepares \(\left\vert W({k}_{0})\right\rangle\) in Eq. 9. Right: the circuit from ref. 64 that implements the global controlled unitary in constant depth.a) Circuits for simulating scattering in one-dimensional Ising field theory. A circuit that simulates the scattering of two d=5 wavepackets on a L=12 lattice. The pieces of the circuit highlighted in light red prepare \(\left\vert W({k}_{0})\right\rangle\) and \(\left\vert W(-{k}_{0})\right\rangle\), the layer in gray represents the symmetry-preserving energy minimization circuit U(θ*), and the layer in blue is two steps of second-order Trotterized time evolution U2(t). b) Comparison of wavepackets prepared on large and small lattices in one-dimensional Ising field theory. The vacuum-subtracted energy density En of a wavepacket with gx = 1.25, gz = 0.15, k0 = 0.36π, σ = 0.13 and d=22. The approximate wavepackets \(\left\vert {\psi }_{{\rm{ansatz}}}\right\rangle\) are prepared from 8 steps of ADAPT-VQE on a L=256 lattice (dark blue) and a L=28 lattice (light blue). The energy density of the exact L=28 wavepacket is also shown (tan). The inset shows the difference in the energy density ΔEn between the approximate L=28 and L=256 wavepackets. Only the energy density of the 56 sites in the center of the lattice is shown, and the L=28 wavepackets are translated to align with the L=256 wavepacket.The vacuum-subtracted energy density En throughout the scattering process for a selection of center-of-mass energies Etot. The initial wavepackets are separated by 10 sites and evolved with a Trotter time step of δt = 1/16. The wavepackets are constructed from 8 steps of ADAPT-VQE except for k0 = 0.2π which is constructed from 10 steps. Results for L = 256 and σ = 0.13, d = 22, gx = 1.25, gz = 0.15 and max bond dimension 350 are shown.Left: the vacuum-subtracted energy density En throughout a MPS simulation of scattering on a L=100 lattice with OBCs. Wavepacket parameters σ = 0.13, k0 = 0.36π and a time step of δt = 1/16 are used. Right: the energy density at t = 20 and t=30 for OBCs (blue) and periodic boundary conditions (PBCs) (tan).The Pauli gates Pi are appended before and after RZZ(θ) as shown on the left. The table on the right shows the sets of Pi that leave RZZ(θ) invariant for all θ.The data corresponds to the quantum simulation results in Fig. 4 for L=104 and t3 = 24.75 (130 two-qubit gate depth). Top left: the median signal strengths pO defined in Eq. 25 as a function of lattice position n for O = Zn, Xn, ZnZn+1. The signal strength is computed for each n by averaging over the TREX twirls, and then computing the median over the set of PTs. Top right: the effect of ODR on the expectation values of the same O for one lattice site, n=40. The data points are determined by averaging over TREX twirls for each PT. The gray dotted lines represent the expectation value of a completely decohered state, while the gray dashed lines are the MPS predictions. The colored bands represent ± 1 standard deviation of the mitigated results. Bottom: the vacuum-subtracted energy density En of the raw device data (diamonds) and after all error mitigation (circles). The raw device data represents the median over PTs (averaged over TREX twirls). The error bars for both the raw and the mitigated data are obtained via bootstrap resampling.Upper: Calculation of skewness for t2 = 16.5. Left: the range of energy cutoffs ϵ (dotted lines) are overlaid on the energy density from Fig. 4. The sample size is 1.28 × 106 and the statistical error bars are the bootstrap standard error centered around the mean. Middle: the third moment of the energy density distribution γϵ is calculated for a contiguous interval of points where En + σn ≥ ϵ. The error bars represent the standard deviation computed via bootstrap resampling centered around the mean. Right: the reported value of γ is computed via Eq. 28, with error bars centered at the mean that cover the range of γϵ coming from the variation of the energy cutoff and, for the quantum results, statistical errors from the bootstrap standard error. Lower: Skewness as an indicator of inelasticity. a) The skewness γMPS calculated for MPS simulations of elastic (k0 = 0.18π) and inelastic (k0 = 0.32π) scattering with L=256, δt = 1/16, d = 27, σ = 0.1, PBCs, and max bond dimension 350. The energy cutoffs ϵ are chosen to be [EL/2 + 0.01, EL/2 + 0.03] in increments of 0.005, and γMPS is set to 0 if the maximum value of En occurs at L/2 or if there are no points with En > ϵ. b) The same but for L = 104, δt = 0.55, d = 21, σ = 0.13, OBCs, and a max bond dimension 1,500. The vertical lines show the times t2 = 16.5 and t3 = 24.75 at which the skewness in Fig. 5 is calculated.Supplementary Sections A–L, Figs. 1–14 and Tables 1–10.Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.Reprints and permissionsFarrell, R.C., Zemlevskiy, N.A., Illa, M. et al. Digital quantum simulations of scattering in quantum field theories using W states. Nat. Phys. 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