Preparation of a Laughlin-like fractional Chern insulator on a superconducting quantum processor

AI summary

Overview

Research area: Quantum simulation of topological quantum matter — specifically fractional Chern insulators (FCIs) — on gate-based superconducting quantum hardware.

Technical level: Advanced. The paper assumes familiarity with fractional quantum Hall physics, Chern bands, parent Hamiltonians, and variational circuit optimization, though the core results are described in accessible terms.

Scope: A digital quantum-simulation experiment preparing a bosonic Laughlin-like FCI ground state and a two-quasihole state on a 4 × 4 lattice of an IBM Nighthawk superconducting processor, characterizing them through densities, configuration probabilities, and chiral bond currents.

What This Paper Is About

Fractional Chern insulators are lattice versions of fractional quantum Hall states, hosting topological order and fractionalized anyonic excitations, but preparing them in a controllable quantum system is hard. The authors use a superconducting qubit processor to digitally prepare a Laughlin-like FCI state that is an exact zero-energy ground state of a long-range "Kapit–Mueller" parent Hamiltonian on a torus, so that there are no physical edges. They also prepare a state with two localized quasiholes and measure the circulating chiral currents that accompany them.

Key Contributions

1. Digital preparation of a bosonic Laughlin-like FCI ground state. On a 4 × 4 qubit lattice at flux density 1/2 per plaquette, the authors prepare the four-boson ground state of the Kapit–Mueller Hamiltonian using a classically optimized, particle-number-conserving variational circuit with p = 4 steps and 264 CZ gates (2-qubit gate depth 44), reaching fidelity F = 0.95 in noiseless simulation.

2. Preparation of a two-quasihole state. By reducing the particle number to N = 3 and adding strong local repulsive potentials (μ = 100) at two pinning sites at opposite corners, (0,0) and (3,3), they prepare a state hosting two quasiholes with a shallower p = 3 circuit of 168 CZ gates (2-qubit gate depth 28), also at F = 0.95 noiseless fidelity.

3. Experimental characterization on hardware. Using a 4 × 4 patch of the 120-qubit ibm_miami Nighthawk R1 processor with particle-number post-selection and XpXm dynamical decoupling, they measure densities, conditional correlations, and symmetry-reduced Fock-state configuration probabilities that quantitatively track the target states.

4. Reconstruction of chiral bond currents around quasiholes. From two-qubit X and Y basis measurements, they reconstruct Kapit–Mueller bond-current expectation values along a 12-bond closed loop and observe a finite net circulation of I_total^exp = −0.19 ± 0.06, consistent in sign and spatial structure with theory.

Main Findings

• Target states are exact and degenerate. At filling ν = N/N_φ = 1/2 the Laughlin-like FCI is an exact zero-energy ground state of the Kapit–Mueller Hamiltonian; on the torus the ground-state manifold is twofold degenerate, and the authors optimize fidelity to the full two-dimensional manifold rather than a single vector.

• High noiseless fidelities with shallow circuits. Both the p = 3 (N = 3) and p = 4 (N = 4) protocols reach F = 0.95 in noiseless simulation. The N = 3 circuit uses 168 CZ gates at 2-qubit depth 28; the N = 4 circuit uses 264 CZ gates at 2-qubit depth 44.

• Post-selection retention. For N = 3, 80,000 measured shots yielded a retention rate of 34.6%; for N = 4, 60,000 shots yielded 33.3%. This corresponds to 27,661 and 19,965 post-selected outcomes, respectively.

• Density profiles match targets. For N = 3 the density is suppressed at the two quasihole pinning sites (0,0) and (3,3); for N = 4 the density is approximately uniform with ⟨n_i⟩ ≃ 0.25 across the 16 sites.

• Conditional correlations reproduce target structure. Conditioning on a particle at (1,1) suppresses density on neighboring sites; conditioning additionally on a particle at (2,3) confines remaining density to sites maximizing particle separation. For N = 3, the quasiholes additionally suppress density at (0,3), adjacent to the pinning sites across the magneto-periodic boundary.

• Configuration distributions are attenuated, not destroyed. Comparing experimental class probabilities to the maximally mixed fixed-N baseline gives linear-fit slopes λ = 0.813 for N = 3 and λ = 0.742 for N = 4. A separate noise-scaling analysis estimates error-free fractions after post-selection of 60(2)% for N = 3 and 46(5)% for N = 4.

• Chirality is robust across the ground-state manifold. The net quasihole circulation has the same sign throughout the entire target manifold, with −0.342 ≤ I_total ≤ −0.234.

• Measured currents are attenuated but linear. The experimental bond currents remain approximately linear in the noiseless values with fitted slope λ = 0.624 and intercept b = 10⁻⁴. Net circulation: I_total^exp = −0.19 ± 0.06, versus I_total^sim = −0.300 and I_total^target = −0.319.

• Dominant limitation is hardware noise. Remaining errors primarily attenuate configuration probabilities and coherences toward a mixed-state background and cannot be removed by particle-number post-selection alone.

Methodology in Plain English

The team encodes a lattice of hard-core bosons directly onto qubits: each site is one qubit, with |0⟩ empty and |1⟩ occupied, and the hard-core limit (U → ∞) forbids double occupancy. They choose a 4 × 4 lattice with periodic boundary conditions at magnetic flux φ = 1/2 per plaquette, which gives N_φ = L²/2 = 8 flux quanta and, at filling 1/2, N = L²/4 = 4 bosons. Removing one boson creates two quasiholes, giving the N = 3 target.

The target states are computed by exact diagonalization of the Kapit–Mueller Hamiltonian, whose fine-tuned long-range complex hopping produces an exactly flat Chern band mimicking a continuum Landau level. Exact diagonalization gives two orthonormal states |ψ_target,α⟩ spanning the degenerate manifold, and the figure of merit is the manifold fidelity F = Σ|⟨ψ_target,α|ψ_sim⟩|².

Rather than implementing the long-range Hamiltonian dynamics, the authors classically optimize a shallow variational circuit built from local operations. Each variational step is a first-order Trotterized evolution with complex nearest-neighbor hopping J, nearest-neighbor interaction V, and on-site potential μ, with independent parameters per bond, site, and step. The Trotterized gate structure is built into the classical optimization, so Trotter errors are absorbed into the optimized parameters. Steps decompose into four layers of non-overlapping bonds for parallelism, plus R_z rotations for on-site potentials. A hopping operation costs two CZ gates via Qiskit's XXPlusYYGate (θ = −2|J|, β = arg(J)); adding an interaction costs a third CZ via a controlled-phase gate with φ = −V_ij. To limit depth for N = 3, interactions appear in only one of the three steps.

Parameter counts are K = 216 for N = 3 and K = 328 for N = 4 (a step without interactions has 48 real hopping parameters plus 16 on-site potentials = 64; adding 24 interaction strengths gives 88). Optimization uses JAX with reverse-mode automatic differentiation, restricted to the fixed-N subspace (dimension 560 for N = 3, 1820 for N = 4). Gradient descent typically converges within 2000 iterations; the selected protocols converged after 1426 (N = 3) and 1913 (N = 4) iterations. Circuits were compiled at optimization_level=3 into the native CZ, SX, X, R_z gate set and run with the SamplerV2 primitive, active initialization, a 4 ms repetition delay, and eight logical-to-physical mappings related by rotations and reflections, averaged over with standard deviation as error bars.

Why This Matters

This work shows that gate-based superconducting processors can prepare and probe strongly correlated topological lattice states with fractional quantum Hall character using circuits of experimentally accessible depth — a regime previously limited to fermionic Laughlin states on seven- to sixteen-qubit processors. It provides a concrete digital alternative to analog platforms such as rotating atomic traps, ultracold atoms with synthetic gauge fields, and optical cavities, which struggle with the large angular momentum and fragile adiabatic preparation required for the FQH regime.

Potential applications (as framed by the paper):

• Fault-tolerant quantum computing built on anyonic quasiparticles with fractional statistics.
• Interferometric extraction of fractional braiding phases by manipulating quasihole positions.
• Randomized-measurement protocols that infer the many-body Chern number directly from statistical correlations of a prepared wave function.
• Topological entanglement entropy as a complementary nonlocal diagnostic of the FCI phase.

Industry relevance: The experiment runs on commercial IBM hardware (ibm_miami, Nighthawk R1, 120 qubits) with standard Qiskit tooling, indicating that near-term cloud-accessible processors can host meaningful topological-matter benchmarks. The particle-number-conserving ansatz is doubly useful industrially: it reduces circuit depth through gate parallelism and enables efficient mitigation of particle-loss and bit-flip errors via global post-selection.

Future Directions

1. Measuring topological invariants. The authors propose accessing the many-body Chern number through randomized-measurement protocols and measuring topological entanglement entropy as a nonlocal diagnostic of the FCI phase.

2. Braiding and interferometry of quasiholes. Having prepared localized quasiholes, the natural next step is to manipulate their positions and extract fractional braiding phases interferometrically.

3. Scaling to larger lattices and particle numbers. The authors note this is constrained by hardware fidelity, achievable circuit depth, and the difficulty of finding a sufficiently large, well-connected patch of physical qubits with consistently high gate fidelity; long-range entanglement in topologically ordered states makes preparation harder as the lattice grows.

4. Removing dependence on classically known targets. The present optimization relies on classically accessible target states, which limits scalability; beyond that regime the authors suggest analytically constructed target states or alternative variational approaches, and they emphasize that recovering unbiased observables will matter increasingly once classical benchmarks are unavailable. Improved gate fidelity, coherence, and sampling rate should make more demanding error-mitigation strategies practical.

Target Audience

Quantum-simulation experimentalists working on superconducting and other programmable platforms; condensed-matter theorists interested in fractional Chern insulators, parent Hamiltonians, and topological order; quantum-algorithm researchers focused on variational state preparation and error mitigation; and graduate students or advanced readers seeking a concrete case study of how far current hardware can push strongly correlated topological state preparation. Readers without a background in fractional quantum Hall physics will find the conceptual framing accessible, but the fidelity metrics, current reconstruction, and Hamiltonian details require intermediate-to-advanced familiarity.

Authors’ abstract

Fractional Chern insulators (FCIs) are lattice analogues of fractional quantum Hall states, hosting topological order and fractionalized anyonic excitations. Their realization poses a demanding target for quantum simulation. Here, we realize a bosonic FCI on an IBM Nighthawk superconducting quantum processor using classically optimized variational circuits. We target a Laughlin-like state that is an exact zero-energy ground state of a long-range parent Hamiltonian on a torus, eliminating physical edges and providing a favorable geometry for accessing fractional quantum Hall physics in a small system. On a $4\times4$ qubit lattice, we prepare both the four-boson FCI ground state and a three-boson state hosting two quasiholes. Measurements of particle densities and correlations reproduce the characteristic structure of the target states, while reconstructed chiral magnetization currents resolve the circulating current pattern associated with the quasiholes. The preparation of these strongly correlated topological states with shallow circuits establishes a route toward the digital simulation and manipulation of fractionalized excitations, including measurements of topological invariants and anyonic braiding.