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SU(2) hadrons on a quantum computer

Yasar Atas, Jinglei Zhang, Randy Lewis, Amin Jahanpour, Jan F. Haase, Christine A. Muschik

arXiv:2102.08920v3quant-phhep-lat

TL;DR

The paper addresses the challenge of simulating non-Abelian gauge theories with both gauge and matter fields on quantum hardware. Using a resource-efficient variational quantum eigensolver for SU(2), it realizes the theory on an IBM platform and studies baryon and meson states. The experiments access their masses and observe the expected mass degeneracy in the physical limit, while motivating extensions toward larger groups and dimensions.

  • Problem

    Quantum simulation of non-Abelian gauge theories with dynamical matter remains limited, although such simulations could reach lattice-gauge-theory regimes unattainable by classical methods.

  • Method

    The study uses a resource-efficient hybrid variational quantum eigensolver for the full SU(2) theory, reducing resources through gauge-field elimination, efficient circuits, and classical preprocessing.

  • Results

    The IBM experiment realizes a non-Abelian gauge theory with gauge and matter fields, accesses baryon and meson energies, calculates their masses, and observes mass degeneracy in the physical limit.

  • Takeaways & Limitations

    This proof-of-concept establishes a route toward quantum simulations of non-Abelian lattice gauge theories and future studies of QCD-related models.

  • Takeaways & Limitations

    Extensions to smaller fermion masses require circuits beyond current quantum-hardware capabilities and further experimental development.

Abstract

from arXiv · show

We realize, for the first time, a non-Abelian gauge theory with both gauge and matter fields on a quantum computer. This enables the observation of hadrons and the calculation of their associated masses. The SU(2) gauge group considered here represents an important first step towards ultimately studying quantum chromodynamics, the theory that describes the properties of protons, neutrons and other hadrons. Quantum computers are able to create important new opportunities for ongoing essential research on gauge theories by providing simulations that are unattainable on classical computers. Our calculations on an IBM superconducting platform utilize a variational quantum eigensolver to study both meson and baryon states, hadrons which have never been seen in a non-Abelian simulation on a quantum computer. We develop a resource-efficient approach that not only allows the implementation of a full SU(2) gauge theory on present-day quantum hardware, but further lays out the premises for future quantum simulations that will address currently unanswered questions in particle and nuclear physics.

INTRODUCTION

Quantum simulation of non-Abelian gauge theories remains an outstanding challenge despite their central role in particle physics and the limitations of classical lattice methods. This work addresses that challenge by using a resource-efficient hybrid approach to realize SU(2) gauge theory with matter and study hadron masses.

  • Non-Abelian gauge theories are central to the Standard Model, with QCD describing strong interactions between quarks and gluons.
  • Classical lattice methods face sign-problem limitations for real-time dynamics and highly entangled matter, motivating quantum simulations.
  • Before this work, experiments had realized one-dimensional Abelian theories and pure-gauge non-Abelian theories, but not a non-Abelian theory with dynamical matter.
  • SU(2) is the smallest non-Abelian Lie group and supports a gauge-singlet baryon made solely from valence fermions, alongside a meson.
  • A hybrid variational quantum eigensolver reduces present-day hardware demands through gauge-field elimination, efficient ansätze, and classical preprocessing.
  • The IBM implementation accesses baryon and meson energies and shows their masses become equal as lattice artifacts vanish.

RESULTS

The study implements an SU(2) gauge theory with matter on a quantum computer and uses it to access non-Abelian baryon and meson states. The measured results agree with exact calculations and show the expected approach to baryon–meson mass degeneracy toward the continuum limit.

  • Effective qubit model: 2N qubits represent N spatial sites after integrating out gauge degrees of freedom while retaining their interactions through the effective Hamiltonian.The resulting model contains long-range and non-diagonal interactions originating from color-electric-field energy and non-Abelian structure.
  • Baryon and meson calculations: The experimental VQE baryon mass agrees well with exact diagonalisation on the IBM Casablanca processor.The calculation uses a resource-efficient circuit adapted to the processor topology.
  • Baryon and meson calculations: The meson mass is obtained as M_m = E_m − E_v, where E_m is the first excited-state energy and E_v is the vacuum energy.The meson is accessed through an overlap-based excited-state VQE procedure in the B = 0 sector.
  • Continuum-limit behavior: The meson-to-baryon mass ratio approaches 1 at larger x, restoring the expected mass degeneracy toward the continuum limit.This degeneracy follows from a global SU(2) symmetry, while staggered fermions restore it only in the continuum limit.
  • Continuum-limit behavior: Finite-size effects are already quite limited for the small systems compared in exact diagonalisation, but larger-regime circuits exceed current hardware capabilities.The reported extensions to smaller fermion masses and broader parameter regimes require further experimental development.

DISCUSSION

The work demonstrates a complete non-Abelian SU(2) gauge theory with gauge and matter fields on a quantum computer. This resource-efficient proof of concept measures hadron properties and establishes a route toward broader non-Abelian lattice-gauge simulations.

  • The study realizes a non-Abelian gauge theory containing both gauge and matter fields on a quantum computer.
  • A resource-efficient VQE approach makes the full SU(2) implementation feasible on present-day quantum hardware.
  • The simulation includes gauge-invariant baryons and mesons, enabling quantum-computer calculations of their masses.
  • Within the one-dimensional SU(2) theory, the framework can be extended to other hadrons, including tetraquarks, while future work targets SU(3) and higher spatial dimensions.
  • The complete non-Abelian benchmarking model provides an important first step toward quantum computation of non-Abelian lattice gauge theories.

METHODS

The methods section explains how the experiment measures the VQE Hamiltonian expectation value and optimizes the variational parameters while reducing quantum-resource requirements.

  • The methods measure Hamiltonian expectation values with minimal state preparations and describe the optimization algorithm used by the VQE.

A. Hamiltonian decomposition and classical optimisation

The Hamiltonian is decomposed into Pauli operators whose measured expectation values are combined classically during optimization. Commuting-group measurements and reusable data reduce the quantum-processor workload.

  • The Hamiltonian is expressed as a weighted sum of 2N-qubit Pauli operators, with coefficients depending on the Hamiltonian parameters.
  • The number of Pauli strings is n = 6N^2 − 11N + 9, which grows quadratically with the number of qubits.
  • Commuting Pauli operators are grouped so that measurements of one representative support classical evaluation of the others, using local measurements.
  • Measured Pauli expectations are stored across Hamiltonian-parameter changes because changing x only changes the coefficients.
  • Jointly measuring operators needed for vacuum and baryon energies enables error reduction and reduces quantum-processor calls.
  • Optimization combines grid-based exploration with Bayesian exploitation to search the parameter space while limiting optimization overhead.

B. Adaptations for NISQ hardware

The hardware adaptations transform and simplify VQE circuits to fit current devices. They reduce circuit depth, connectivity demands, and active-qubit counts while using shared sampling to suppress certain errors.

  • The circuit construction is tailored to the targeted baryon-number sector and illustrated for N = 4.
  • An effective Hamiltonian transformation allows a shorter ansatz circuit while preserving the relevant cost-function evaluation.
  • The circuit transformation reduces both gate depth and the qubit connectivity required by the superconducting experiment.
  • Inactive encoded qubits are removed after their entanglement is traded for additional measurements, leaving separable active and inactive states.
  • For B = 0, inactive qubits are removed to obtain a six-qubit circuit; a similar reduction removes qubits seven and eight for B = 1.
  • Using the same circuit sample for vacuum and baryon energies makes their difference independent of the modeled error-state contribution under the stated error assumption.

C. VQE computation of the meson mass

The meson mass is obtained by estimating the ground and first excited energies with VQE. Because overlap measurements deepen the circuit and amplify errors, the experimental protocol uses an alternative excited-state cost function.

  • The meson mass is calculated as the energy difference between the ground and first excited states in the B = 0 subsector.The energies E_v and E_m are identified as the ground and first excited-state energies, respectively.
  • The overlap method obtains the required overlap by applying the inverse variational circuit and measuring the initial computational-basis state.The inverse circuit reverses the gate sequence and changes each parameter's sign.
  • Approximately doubled circuit depth makes overlap estimation more susceptible to gate errors and environmental noise.These effects can render the overlap measurement experimentally infeasible.
  • A Gram–Schmidt procedure searches for the first excited state in the space orthogonal to the variational ground state.The associated cost function penalizes components overlapping with the ground state.
  • The orthogonalized cost function can recover excited-state energy from small excited-state components, but its denominator makes errors in the ground-state energy strongly distort results near unit overlap.The protocol therefore uses the main-text cost function for the experimental meson calculation.

D. Implementation on the IBM processors

The reduced circuits are mapped onto IBM hardware with limited connectivity and calibrated error-mitigation procedures. Circuit modifications accommodate the processor topology while readout and CNOT errors are mitigated experimentally.

  • Reduced circuits require only nearest-neighbor coupling because further connectivity requirements are incorporated into effective-Hamiltonian measurements.This allows implementation on hardware arranged as a simple chain.
  • The six-qubit baryon circuit runs on the seven-qubit ibmq_casablanca processor and requires at least one SWAP operation.The circuit and effective-Hamiltonian operator labels are modified so the SWAP need not be reversed.
  • Readout errors are mitigated by calibrating a map from true measurement probabilities to observed probabilities and inverting it.The calibration estimates the mixing map Λ before probability correction.
  • CNOT errors are mitigated by replacing each CNOT with either three or five CNOT gates and linearly interpolating the results.The replacement artificially amplifies CNOT errors for extrapolation.

E. Extension for future quantum computers

The paper presents the current resource-reduction measures as tools for investigating the model on more powerful future quantum hardware. These measures address restrictions imposed by present-day NISQ devices.

  • Mass cutoffs and circuit-splitting techniques are necessary because of present-day NISQ hardware restrictions.The authors frame these measures as preparation for studies on more powerful future quantum computers.

1. The N “ 6 baryon mass

A classical proof-of-principle VQE simulation estimates the baryon mass on a six-site lattice using a symmetry-preserving ansatz. The construction starts from a strong-coupling state and explores the relevant parameter space while preserving baryon-number sectors.

  • 1. The N “ 6 baryon mass: The proof-of-principle VQE estimates the baryon mass for a six-site lattice represented by 12 qubits.Statistical quantum measurement noise is not included in these numerical simulations.
  • 1. The N “ 6 baryon mass: The initial state is the strong-coupling ground state, chosen as a red-green particle pair for B = 1 and the bare vacuum for B = 0.For small finite x, second-order perturbation theory motivates the red-green pair at the N-th site as the lowest-energy choice.
  • 1. The N “ 6 baryon mass: The variational circuit uses layers of pairwise excitation-preserving gates between neighboring qubits.The gates are parameterized SWAP gates.
  • 1. The N “ 6 baryon mass: Each circuit layer preserves total spin magnetisation, keeping the final state within the selected baryon-number subspace.In the qubit formulation, fixed baryon number corresponds to fixed total magnetisation.
  • 1. The N “ 6 baryon mass: Using 10 layers in the B = 0 sector and 15 layers in the B = 1 sector gives baryon masses across the parameter space shown in Fig. 5.The procedure is applied for different values of the rescaled mass parameter.

2. General circuit for the N “ 4 baryon

The N = 4 baryon circuit reduces the variational description by combining symmetry-compatible basis states into color singlets. Its nine-parameter ansatz has high fidelity with the exact ground state in noiseless classical VQE simulations.

  • Circuit design: The proposed N = 4 circuit reduces both circuit depth and the number of variational parameters for the lightest baryon ansatz.It is designed for arbitrary Hamiltonian parameters.
  • Symmetry sector: The B = 1 sector contains 16 basis states, with 12 combined pairwise into color-singlet states satisfying the non-Abelian charge constraints.States sharing a row in Table I are combined to form singlets.
  • Circuit design: The circuit starts from |0000⟩, generates the 16 B = 1 basis states, and combines them into color singlets using controlled Y-axis rotations and static gates.The static part can be incorporated into the Hamiltonian to reduce computational effort.
  • Parameterization: Only nine variational parameters are required because normalization fixes the tenth coefficient in the hyperspherical parametrization.The parameter vector is θ = (θ1, θ2, …, θ9).
  • Validation: Classical noiseless VQE simulations show high fidelity with the exact ground state throughout the B = 1 sector and for arbitrary Hamiltonian parameters.The circuit’s static blocks combine the relevant basis states into singlets.

H. Qubit encoding

The qubit encoding removes explicit gauge fields by absorbing color into a doubled lattice and then mapping single-component fermions to qubits. This produces long-range spin interactions, including off-diagonal terms characteristic of the non-Abelian theory.

  • Decolorization: Color is absorbed by doubling the lattice and assigning red and green fermions to alternating sites of a single-component fermionic field.Odd and even sites carry the two color components.
  • Qubit mapping: The single-component fermionic field is mapped to spin operators through a Jordan-Wigner transformation.The transformation preserves the fermionic anticommutation relations through its string factor.
  • Staggered interpretation: The staggered encoding distinguishes matter, antimatter, and vacuum through occupation patterns on even and odd cells.The mapping uses the cell index modulo 4 to identify matter and antimatter types.
  • Hamiltonian structure: The transformed Hamiltonian contains no explicit gauge fields but introduces long-range spin-spin interactions in the chromoelectric term.The gauge-field elimination trades explicit links for nonlocal spin interactions.
  • Non-Abelian structure: Some resulting interactions are off-diagonal and directly reflect the non-Abelian character absent from the U(1) Abelian Schwinger model.This formulation enables implementation of an SU(2) model with matter and gauge fields on a quantum computer.

I. Symmetries and eigenstates

The eigenstate construction uses total non-Abelian charge and baryon-number symmetries to identify physical basis states. For larger lattices, enforcing color symmetry in the circuit becomes non-trivial and may require variational restoration.

  • Charge constraints: Physical basis states in the neutral Qz sector must contain equal numbers of the two antiparallel cell types.This condition identifies the zero-eigenvalue states of the diagonal charge component.
  • Baryon number: For a lattice with N sites, the allowed baryon quantum numbers are the integers from −N/2 through N/2.The baryon number is determined by the counts of the two cell types.
  • Color singlets: States satisfying Qz = 0 may still fail the Qx and Qy constraints, so they must be combined with companion basis states to form complete color singlets.A representative pair is combined with a relative minus sign and normalization factor.
  • Scalability: For arbitrary N, constructing simultaneous zero-eigenvalue states of all non-Abelian charges is a non-trivial task.The construction becomes more involved as the lattice grows.
  • Scalability: The circuit can impose color symmetry directly for small lattices, whereas the N = 6 results rely on the variational algorithm to restore the correct symmetry.This distinguishes explicit symmetry encoding from variational enforcement.
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