PaperlayerView exact revision on arXiv

Source-linked AI summary

Non-Abelian $SU(2)$ Lattice Gauge Theories in Superconducting Circuits

A. Mezzacapo, E. Rico, C. Sabín, I. L. Egusquiza, L. Lamata, E. Solano

arXiv:1505.04720v2quant-phcond-mat.mes-hallcond-mat.supr-conhep-lat

TL;DR

The paper addresses how to digitally simulate non-Abelian pure-gauge dynamics with superconducting circuits. It constructs a minimal SU(2) quantum-link model on a triangular plaquette, proposes two transmon-based architectures, and reports convergence toward the expected gauge-invariant values while outlining scalability and error requirements.

Abstract

from arXiv · show

We propose a digital quantum simulator of non-Abelian pure-gauge models with a superconducting circuit setup. Within the framework of quantum link models, we build a minimal instance of a pure $SU(2)$ gauge theory, using triangular plaquettes involving geometric frustration. This realization is the least demanding, in terms of quantum simulation resources, of a non-Abelian gauge dynamics. We present two superconducting architectures that can host the quantum simulation, estimating the requirements needed to run possible experiments. The proposal establishes a path to the experimental simulation of non-Abelian physics with solid-state quantum platforms.

Explicit SU(2) gauge operators in terms of Schwinger modes

The paper constructs SU(2) gauge operators from Schwinger bosons, using left and right generators on each finite-dimensional link to define local gauge transformations and Gauss-law constraints.

Explicit SU(2) gauge operators in terms of Schwinger modes

  • Schwinger bosons on the two link ends implement SU(2) and generate separate left and right multiplets.The bosonic modes act on sites L and R of each link, producing generators with SU(2) commutation relations.Source: Explicit SU(2) gauge operators in… · pp. 6–7
  • The link operator U carries color between adjacent lattice points, with its two indices associated with the link’s beginning and end.Gauge transformations therefore act through right and left multiplication at the two endpoints.Source: Explicit SU(2) gauge operators in… · p. 7
  • The total color carried by a link is the difference between its right and left generators, Ra − La.This quantity enters the non-Abelian Gauss generator together with the divergence of electric fields.Source: Explicit SU(2) gauge operators in… · p. 7
  • Physical states satisfy Ga^a(x)|Ψ⟩ = 0, so each vertex transforms locally as an SU(2) singlet and obeys the corresponding non-Abelian Gauss law.The formulation is representation-independent at the level of these gauge-theory properties.Source: Explicit SU(2) gauge operators in… · p. 7

Local Hilbert space of SU(2) quantum links

A minimal SU(2) link uses a four-dimensional one-excitation Hilbert space, while two links meeting at a vertex decompose into singlet and triplet gauge sectors.

Local Hilbert space of SU(2) quantum links

  • The one-excitation sector of four bosonic modes is four-dimensional and separates into SU(2) spin and link-position degrees of freedom.The local basis is written as |n↑L, n↓L, n↑R, n↓R⟩.Source: Local Hilbert space of SU(2) quantum… · p. 7
  • Gauge-invariant interactions conserve total angular momentum at each vertex, yielding singlet and triplet sectors for two links.The singlet has total angular momentum zero, whereas the triplet has total angular momentum one.Source: Local Hilbert space of SU(2) quantum… · p. 7

Reduction to qubits - Triangular lattice

The construction restricts each link to a two-qubit representation and expands the triangular-plaquette SU(2) Hamiltonian into multiqubit interactions.

Reduction to qubits - Triangular lattice

  • Choosing one excitation per link faithfully maps the local SU(2) link state to two qubits encoding position and spin.The position and spin variables are defined from occupations of the left, right, up, and down modes.Source: Reduction to qubits - Triangular lattice · pp. 7–8
  • The minimal SU(2) link has a four-dimensional Hilbert space, and two links decompose into singlet and triplet gauge sectors.These sectors organize the local gauge-invariant subspaces used in the plaquette construction.Source: Figure 5 · p. 8
  • The triangular-plaquette Hamiltonian is a traced product of three link operators around the plaquette.The three links are labeled 12, 23, and 31 in the qubit representation.Source: Reduction to qubits - Triangular lattice · pp. 8–9
  • Expanding the trace produces 16 terms, comprising one three-body, six six-body, and nine five-body interactions.The terms arise from the nonzero traces of products of SU(2) generators.Source: Reduction to qubits - Triangular lattice · pp. 8–9

Experimental setups

The proposal considers transmon-based superconducting architectures for implementing the simulator, using either a common resonator or direct capacitive couplings.

Experimental setups

  • The first architecture uses six tunable-coupling transmon qubits connected to a single microwave resonator.Flux-controlled SQUID loops modulate qubit–resonator couplings without changing the qubits’ transition frequencies.Source: Experimental setups · pp. 9–10
  • A Cooper pair box becomes a transmon when large capacitive pads suppress sensitivity to offset-charge noise.In the appropriate regime, the device can be treated approximately as a two-level system when its anharmonicity is sufficiently large.Source: Experimental setups · pp. 9–10
  • The resonator architecture uses flux modulation to generate red and blue sidebands and thereby many-body interactions.The modulation changes the effective coupling while leaving the bare qubit energy levels unaffected.Source: Experimental setups · p. 10
  • The second architecture uses six capacitively coupled Xmon qubits arranged on a triangular geometry with a central auxiliary qubit.Because the interactions are direct rather than resonator-mediated, the many-body operations reduce to nearest-neighbor two-qubit gates.Source: Experimental setups · p. 10

Many-body interactions and gauge invariance for different sectors

The proposal decomposes the non-Abelian interaction into many-body monomials implemented with superconducting-qubit gate sequences, while digital evolution preserves the gauge-invariant sectors and supports larger lattices.

Many-body interaction implementation

  • The Hamiltonian contains 16 monomials—one three-body, six six-body, and nine five-body terms—implemented through collective or C-Phase gate decompositions.The transmon architecture requires 32 collective gates and at most 184 single-qubit gates per digital step; the ancillary-qubit architecture requires 168 C-Phase gates and at most 520 single-qubit rotations.Source: Many-body interactions and gauge… · pp. 10–11
  • The ancillary-qubit scheme realizes an arbitrary N-body interaction by coupling an X-rotated ancillary qubit to the system through sequences of ZZ and C-Phase gates.The construction uses an ancillary qubit initialized in an eigenstate of σx and reduces the many-body interaction to two-body gates plus local rotations.Source: Many-body interactions and gauge… · pp. 10–11

Gauge invariance across sectors

  • The digital gauge value converges to the expected invariants 3/4 and 11/4 as the number of digital steps increases for the two tested initial configurations.The initial state oscillates periodically in the first case but remains invariant in the second.Source: Many-body interactions and gauge… · p. 11

Gate realization and scaling

  • C-Phase gates are implemented by bringing |11⟩ near |02⟩, while optimized control suppresses leakage and enables phases up to 4 radians without significant state leakage.The cited implementation increases the gate duration from 40 ns to 55 ns; decoherence and control errors remain additional fidelity contributions.Source: Many-body interactions and gauge… · pp. 11–12

Gauge invariance across sectors

Gate realization and scaling

  • Larger lattices can use enhanced connectivity from multiple microwave resonators or capacitive couplings in superconducting pads, including a ladder architecture for singlet dynamics.The proposed Xmon layout illustrates an experiment involving two pure-gauge plaquettes.Source: Many-body interactions and gauge… · p. 12, Figure 7 · p. 12

Scaling of errors for larger lattices

The analysis bounds the Suzuki-Lie-Trotter resources needed for larger lattices and relates digital-step growth to per-step fidelity and qubit coherence constraints.

Resource scaling

  • The Suzuki-Lie-Trotter step count grows almost linearly with simulated phase and sub-polynomially with target digital error ϵ.The approximation uses k = 1 for the stated bound.Source: Scaling of errors for larger lattices · p. 13
  • The upper-bound analysis extracts a step-count scaling of Ns ∼ N^5/2 for a lattice of N plaquettes.For one plaquette, the Hamiltonian has 16 terms with one three-body, six six-body, and nine five-body contributions.Source: Scaling of errors for larger lattices · p. 13
  • The reported bound is very loose because it does not depend on the Hamiltonian’s underlying structure, although it remains non-exponential in simulator resources.The exact scaling cannot be determined without solving the larger problem numerically.Source: Scaling of errors for larger lattices · p. 13

Experimental constraints

  • Per-step experimental error must satisfy ϵs ∼ ϵ/Ns so accumulated experimental error remains comparable to the Trotter error.This condition assumes independent errors across the Ns digital steps.Source: Scaling of errors for larger lattices · p. 13
  • Accessible simulation time must remain shorter than both qubit coherence times T1 and T2, imposing an additional condition for larger lattices.The required fidelity and coherence constraints are stated alongside the step-count bound.Source: Scaling of errors for larger lattices · p. 13

Matter-field SU(2) Gauge Interactions

The paper outlines how to extend the superconducting implementation from pure-gauge plaquettes to a non-Abelian SU(2) model coupled to matter fields, while enforcing local gauge symmetries.

Model extension

  • The matter-coupled extension targets the simplest SU(2) gauge model with matter in (1+1) dimensions, generalizing ingredients previously used for Abelian gauge simulations.The extension is presented beyond the pure-gauge plaquette dynamics developed in the main text.Source: Matter-field SU(2) Gauge Interactions · p. 13

Matter-gauge interaction

  • The ideal matter-gauge interaction is a gauge-covariant hopping term b†(i)U(i,i+1)b(i+1) + H.c. involving hard-core bosonic modes.The modes carry spin labels α ∈ {↑, ↓}, link labels p ∈ {L, R}, and site index i.Source: Matter-field SU(2) Gauge Interactions · p. 13

Gauge symmetries

  • The model has local SU(2) gauge symmetry at every site and U(1) gauge symmetry on the links, generated by Ga(i) and G(i,i+1).Both symmetries are stated as local constraints of the matter-coupled Hamiltonian.Source: Matter-field SU(2) Gauge Interactions · p. 13

Gauge-symmetry enforcement

  • The construction can enforce one symmetry fundamentally and impose the other through an energy penalty, with U(1) often simpler to enforce through density-density interactions.Alternatively, interactions may satisfy the non-Abelian symmetry from the outset while the Abelian symmetry is energetically imposed.Source: Matter-field SU(2) Gauge Interactions · p. 14
  • A rotating-wave approximation separates equal-excitation sectors, while a term with energy scale Ω enforces N0 excitations per link.Number-conserving interactions remain energetically allowed in the selected subspaces.Source: Matter-field SU(2) Gauge Interactions · p. 14

Effective interaction

  • A perturbative SU(2)-invariant hopping term produces the effective Hamiltonian up to frequency renormalization and a symmetry-preserving density-density interaction.The density-density contribution can be reduced or eliminated with a counter-term in the initial Hamiltonian.Source: Matter-field SU(2) Gauge Interactions · p. 14
Loading 1505.04720v2…