Realizing the fermionic Laughlin state on a trapped-ion quantum processor

This post was contributed by Lingnan Shen, Mao Lin, Cedric Yen-Yu Lin, Di Xiao, and Ting Cao.

Quantum computing is beginning to simulate forms of matter that are difficult to study in the lab or compute classically. In this work, researchers from University of Washington and Amazon Braket demonstrate one such milestone. They realized a fundamentally new quantum state—the fermionic Laughlin state—on a programmable quantum processor.

Topological phases of matter underpin some of the most exotic phenomena in condensed matter physics. They offer deep insights into the fundamental laws of nature, such as the emergence of fractionally charged excitations, and enable next-generation computing paradigms through robust, fault-tolerant quantum computation. Yet probing these phases in real materials remains extraordinarily difficult due to the stringent conditions required for topological order to emerge. This has motivated the use of programmable quantum processors to simulate and manipulate exotic topological states in a controlled setting.

Among topological states, the Laughlin state embodies fractionalization, anyonic excitations, and incompressibility—hallmarks of the fractional quantum Hall (FQH) effect. These properties make the Laughlin state a cornerstone for understanding exotic quantum matter, while also informing the broader scientific roadmap toward topological approaches to quantum computing. While bosonic analogs of Laughlin states have been realized on photonic and cold-atom platforms, a genuine fermionic Laughlin state had not previously been demonstrated on a digital quantum processor.

In this post, researchers describe how they realized the ν = 1/3 fermionic Laughlin state on IonQ's trapped-ion quantum computers, accessed through Amazon Braket. The approach uses an efficient and scalable Hamiltonian Variational Ansatz (HVA) [1]. We demonstrate an end-to-end workflow—from Hamiltonian design and ansatz construction to error mitigation and observable extraction—that captures the defining diagnostics of the Laughlin state: density structures of the edge modes, the correlation hole of an incompressible quantum liquid, and the topological entanglement entropy. All measurements show strong agreement with exact diagonalization benchmarks. This work highlights how Amazon Braket serves as a bridge between academic discovery and real-world quantum applications—allowing researchers to prototype, test, and scale advanced quantum algorithms on the commercial cloud.

Synthetic topological orders such as the toric code can be prepared with shallow circuits thanks to their exactly solvable structure. The Laughlin state is fundamentally different: it arises from strong electron-electron interactions that lack simple mappings to short-depth quantum circuits. On a cylinder geometry, the FQH Hamiltonian reduces to an effective one-dimensional fermion chain with two-body interactions whose range and strength depend on the cylinder circumference. In the isotropic geometry regime—where the Laughlin state's defining long-range entanglement and incompressibility emerge—the full Hamiltonian contains O(N³) terms for N orbitals, making a basic variational ansatz impractical.

We address this by developing a systematic protocol to construct an effective Hamiltonian that retains only the dominant interaction terms. The selection criteria are twofold. First, quantitative fidelity: the ground state of the truncated Hamiltonian must have high wavefunction overlap with the exact Laughlin state. Second, qualitative preservation of topology and entanglement: the truncated model must remain in the same topological class, as verified by entanglement entropy scaling and symmetry classification. We find that including interactions up to the range k + m ≤ 4 satisfies both criteria, where k and m label the orbital separation of the two-body interaction terms. This truncation achieves fidelity above 0.95 between the exact Laughlin state and the ground state of the truncated Hamiltonian, computed via exact diagonalization (see Fig. 1). It also preserves the area-law entanglement scaling characteristic of a topological quantum liquid. In contrast, truncating at k + m ≤ 3 (the Tao-Thouless limit) confines the system to a Krylov subspace that is too small to capture the full Laughlin state's correlations. This leads to low fidelity and entanglement entropy that saturates rather than scaling linearly with the cylinder circumference.

Figure 1 – Cylinder geometry and interaction truncation. (a) Schematic of the cylinder in the Tao-Thouless (thin-cylinder) and isotropic geometry limits, showing the Gaussian-localized orbitals of the lowest Landau level. (b) Fidelity between the exact Laughlin state and the ground state of the truncated Hamiltonian for various interaction ranges and number of electrons Ne. Including interactions up to k + m ≤ 4 (red dashed) achieves fidelity above 0.95 in the isotropic regime, while k + m ≤ 3 (blue solid) drops significantly.

With the effective Hamiltonian identified, we construct a HVA whose circuit layers correspond to unitary evolutions generated by each interaction term. Two physical principles keep the circuit compact. First, we generalize the variational parameters across the lattice: all gates within the same unitary layer share a single parameter, yielding a constrained HVA with just five parameters per repetition, independent of system size. Second, the squeezing rule of FQH physics dictates the ordering of circuit layers, starting from the charge-density-wave initial state |100100…1001⟩.

We interpret the HVA (see Fig. 2 for an illustration) as a digitized adiabatic protocol. Lieb-Robinson bounds on the spread of correlations under local dynamics imply that the number of HVA repetitions must grow at least linearly with system size to reproduce the long-range entanglement of the topological phase faithfully. The total parameter count therefore scales as O(p), where p is the number of repetitions. For a single repetition at six electrons (16 qubits), the circuit requires 369 two-qubit (Controlled-NOT, or CNOT) gates, growing to 883 gates for 12 electrons (34 qubits). We further reduce the gate count by a factor of roughly 3× through optimized Pauli-term reordering after the Jordan-Wigner transformation.

Crucially, parameters optimized on a small 6-electron system transfer directly to larger systems as warm starts without reoptimization. Intensive observables such as local density and two-point correlations show constant deviation from exact values as the system grows. This confirms that reproducing local physics with high accuracy does not require large prefactors in the linear depth scaling.

Figure 2 – Schematic quantum circuit for preparing the ν = 1/3 Laughlin state. The initial charge-density-wave state |Ψ₀⟩ is evolved through five unitary layers (Û₂₁, Û₃₀, Û₃₁, Û₂₀, Û₁₀), each corresponding to an interaction term in the effective Hamiltonian. The circuit uses five variational parameters per repetition and scales linearly with system size.

We executed the quantum circuits on two IonQ trapped-ion quantum computers accessed through Amazon Braket. Local density and spatial correlation measurements were performed on IonQ's 25-qubit Aria-1, which reported a mean two-qubit gate fidelity of 98.5% on the day of execution. Topological entanglement entropy measurements were performed on IonQ's 36-qubit Forte-1, which reported a mean two-qubit gate fidelity of 99.7%.

With approximately 300 two-qubit gates in each qubit's light cone, a naive estimate places the raw circuit fidelity around 1% on Aria-1 — too low to extract meaningful physics without error mitigation. We employed a combined strategy: IonQ's native debiasing scheme, which is available out-of-the-box on Braket, together with a custom symmetry-verificat