Topological order from measurements and feed-forward on a trapped ion quantum computer

Long-range entangled quantum states are central to quantum information, condensed matter, and lattice gauge theories. While short-range entangled states can be prepared with constant-depth circuits via adiabatic encodings, long-range entangled states require extensive circuit depth under unitary dynamics due to Lieb-Robinson bounds, straining coherence on near-term devices. Introducing mid-circuit measurements with feed-forward circumvents these limitations by enabling deterministic non-unitary dynamics that can generate correlations nonlocally while pushing computational depth into classical processing. The research question is whether combining mid-circuit measurement, real-time feed-forward, and high-fidelity entangling gates can deterministically and efficiently (constant quantum depth) prepare long-range entangled topologically ordered states, specifically the toric code ground state, and further enable controlled creation and manipulation of anyonic excitations, including non-Abelian defects, on a programmable trapped-ion quantum computer.

Prior work established the challenges of preparing long-range entangled states with finite-depth unitary circuits and the necessity of deep circuits due to finite information propagation speeds (Lieb-Robinson bounds). Elements of the required toolset—high-fidelity entangling gates, mid-circuit measurements, and feed-forward—have been individually demonstrated across platforms (e.g., error correction, topological state realizations, and non-Abelian operations). However, combining all three to deterministically create long-range entangled states in a single platform had remained elusive. Theoretical proposals have shown that measurement and feed-forward can efficiently prepare complex topological orders, including non-Abelian states, sometimes with constant-depth quantum circuits and classical post-processing. This work builds on these insights to provide an experimental realization on a trapped-ion device with mobile ions and effective all-to-all connectivity to implement periodic boundary conditions and mitigate cross-talk during mid-circuit measurements.

Platform and device: Experiments were performed on Quantinuum's H1 programmable trapped-ion quantum computer with 20 171Yb+ qubits in a quantum CCD architecture enabling ion transport to suppress cross-talk. Native gates include single-qubit rotations and an arbitrary-angle RZZ entangling gate. Reported average fidelities at experiment time: two-qubit 99.7%, single-qubit 99.996%, SPAM 99.6%; memory error per depth-1 time per qubit ~0.03%. Target model and boundary conditions: A 4×4 toric code (Kitaev) with qubits on vertices and periodic boundary conditions (effective torus) is implemented using all-to-all connectivity. Stabilizers: X-type A_p = X^4 on odd plaquettes and Z-type B_p = Z^4 on even plaquettes. The targeted logical sector satisfies ⟨Z_hor⟩ = ⟨Z_vert⟩ = 1. State preparation (constant-depth, deterministic via measurement and feed-forward): - Initialize all qubits in |0⟩ so that all B_p stabilizers are +1. - Measure A_p on all odd plaquettes to project into the even/odd eigenspaces of X^4. Two measurement strategies are employed: • Ancilla-based parity checks (projector (I + X^4)/2) using one ancilla per measured plaquette via four entangling gates to map parity to the ancilla, then measure the ancilla. • Ancilla-free modified parity checks using six two-qubit gates; when the outcome is −1, an effective Z_target is applied on the measured (target) qubit, auto-correcting that plaquette and moving a potential error to a neighboring plaquette, reducing decoding burden. Plaquettes 1, 3, 9, 11 used ancilla-based measurement; plaquettes 4, 6, 12, 14 used modified ancilla-free measurement. - Real-time feed-forward: Apply conditional single-qubit Z gates to flip all plaquettes where A_p = −1 was measured, annihilating e-anyons. A simple lookup-table decoder (covering 2^12 = 4096 patterns) determines the correction locations. Due to OpenQASM 2.0 limitations (no conditioning on individual classical bits), the implemented decoder uses more conditioned gates; an alternative, more optimal decoder yields similar performance. - Heralded preparation: Because anyons should appear in pairs, runs yielding an odd number of excitations (consistent with measurement errors) are flagged and discarded (approximately 10% of repetitions). This global heralding is not exponentially costly with system size. State verification (defect-free case): - Stabilizer measurements: After preparation, destructively measure all qubits in X- or Z-basis (two settings) to estimate ⟨A_p⟩, ⟨B_p⟩, and logical strings ⟨Z_hor⟩, ⟨Z_vert⟩. 1240 total repetitions; half allocated to X-basis and half to Z-basis. - Topological entanglement entropy (TEE): Use randomized measurements to estimate second Rényi entropies for subsystems forming 2×2 and 2×3 regions and compute γ = −(S_A + S_B + S_C − S_AB − S_AC − S_BC + S_ABC). Use N_u = 72 local random unitary settings (tensor product of single-qubit unitaries) and N_M = 256 shots per setting. Bootstrapping is used for error bars. Also report values without heralding for comparison. Defect geometry and anyon dynamics: - Prepare a toric code variant with two lattice defects (15 data qubits); use five spare qubits as ancillas. Measure some X-plaquettes with ancillas (1, 3, 4, and the two defect plaquettes) and others via modified ancilla-free checks; use a lookup-table decoder that exploits that errors do not occur on modified ancilla-free plaquettes. Four measurement settings are used to obtain all stabilizers once per setting, with corner and defect plaquettes measured twice. - Anyon transmutation experiment: Create a pair of m anyons and move one across the defect line along a prescribed path (sequence X12 X13 Z6 Z5), measuring stabilizers after each step. Corner and defect stabilizers measured 1200 times per step; others 600 times. - Anyon interferometry (Hadamard test): Create a single em composite adjacent to the defect by Y10 acting on the ground state, then implement a controlled braiding operation (controlled-XXXX or equivalent CZ strings across a loop) using an ancilla initialized in |+⟩. Measure ⟨X⟩ (or equivalently ⟨Z⟩ at the end of the compiled sequence) on the ancilla to extract Re⟨ψ|U_braid|ψ⟩ for ψ ∈ {em, gs}. Mobility of ions enables implementing the controlled braiding with only four two-qubit gates. - Optional QND tracking: An alternative non-destructive protocol performs parity (stabilizer) QND measurements between anyon moves to track the anyon without collapsing the whole state, reducing shot cost at the expense of extra gates. Circuits, compilation, and error budget: - Toric code (defect-free) state-prep+decoder: 484 single-qubit gates; 40 two-qubit gates [4×4 (ancilla-based) + 6×4 (ancilla-free)]; memory accumulation over ~6 depth-1 times; SPAM on 20+4 measured qubits (ancilla reuse). - Toric code with defects: 423 single-qubit gates; 40 two-qubit gates [3×4 + 5×2 + 6×3 after compilation]. - Compilation via TKET to native gate set. Shadow density matrices (randomized measurement data) used to estimate global fidelity with target toric-code ground state. - SPAM mitigation (not used in main results) modeled via a local transition matrix A with 0.1% |0⟩→|1⟩ and 0.5% |1⟩→|0⟩ error rates; applying A^{-1}⊗n improves reported energy densities slightly; raw and mitigated results provided in Supplementary Data.

- Deterministic, constant-depth preparation of a toric code ground state on a 4×4 qubit lattice with periodic boundary conditions using mid-circuit measurements and real-time feed-forward on a trapped-ion quantum computer. - Stabilizer and energy metrics (heralded data; 1240 total repetitions with ~10% discarded): • Energy density of H = −∑_p A_p − ∑_p B_p: −0.929 ± 0.004. • Logical string operators (averaged over translations): ⟨Z_hor⟩ = 0.916 ± 0.0065, ⟨Z_vert⟩ = 0.914 ± 0.0064. • Stabilizers: ⟨A_p⟩ = 0.944 ± 0.0049; ⟨B_p⟩ = 0.914 ± 0.0063. - Topological entanglement entropy via randomized measurements: • γ/ln2 = 0.93 ± 0.055 for 2×2 regions; γ/ln2 = 1.05 ± 0.093 for 2×3 regions, consistent with Z2 topological order (γ = ln2). • Without heralding: γ/ln2 ≈ 0.87 ± 0.055 (2×2) and 1.00 ± 0.090 (2×3). - Defect geometry (two non-Abelian Ising defects): average stabilizer expectation per plaquette 0.925 ± 0.0039, comparable to defect-free case. - Anyon transmutation: After moving an m anyon across the defect line, adjacent plaquettes show stabilizer values −0.92 ± 0.017 and −0.89 ± 0.020, consistent with creation of a single em composite (fermion) permitted by the defect. - Anyon interferometry (Hadamard test): Re⟨em|U_braid|em⟩ ≈ −0.87 ± 0.018, confirming fermionic statistics of the em composite; in contrast Re⟨gs|U_braid|gs⟩ ≈ +0.87 ± 0.018. - Global fidelity with target toric-code ground state (|Z_hor⟩=|Z_vert⟩=1) estimated via shadow density matrices: 0.80 ± 0.049. - Cross-talk during mid-circuit measurement is negligible due to ion transport (ions separated by ≥180 μm), with noise dominated by two-qubit and memory errors (bias toward Z-phase flips observed in metrics).

The results demonstrate that combining mid-circuit measurements with real-time feed-forward and high-fidelity entangling gates enables deterministic, constant-depth preparation of long-range entangled topological order on a programmable ion-trap device. High stabilizer fidelities, correct logical sector selection, and near-ideal topological entanglement entropy confirm robust Z2 topological order. The ability to introduce and manipulate non-Abelian defects within the Abelian toric code and to observe anyon transmutation and fermionic braiding via interferometry shows precise control over emergent quasiparticles. The mobility of trapped-ion qubits minimizes cross-talk and reduces gate overhead for controlled braiding operations. These capabilities broaden the experimental toolkit for preparing and probing complex topological states and for implementing deterministic non-unitary dynamics, suggesting near-term applications in simulating dynamics on topologically ordered backgrounds and in algorithms that require long-range entangled inputs. The comparable fidelity cost of mid-circuit measurements to two-qubit gates signifies readiness for multi-round measurement-based protocols.

This work provides an experimental realization of deterministic, constant-depth preparation of a toric code ground state using mid-circuit measurement and feed-forward on a trapped-ion quantum computer, verified by stabilizer energies, logical string operators, and topological entanglement entropy. It further demonstrates creation and manipulation of non-Abelian defects and confirms fermionic statistics of em composites via braiding interferometry. These advances pave the way for efficient preparation of more complex, including truly non-Abelian, topological orders—potentially requiring only a single round of feed-forward—and for multi-round protocols that access exotic non-Abelian states relevant to fault-tolerant quantum information. The demonstrated approach also enables resource-efficient studies of quenches and variational dynamics in topologically ordered systems and may unlock quantum advantage in simulating lattice gauge theories and other many-body phenomena as system sizes scale.

- System size limited to 20 qubits; scaling to larger codes and more complex non-Abelian orders will require increased capacity. - Heralded preparation discards runs with odd anyon counts (~10%), reflecting sensitivity to measurement errors (though not exponentially costly with size). - Device noise and memory errors (bias toward Z-phase flips) affect stabilizer asymmetries; while mid-circuit measurement cross-talk is negligible, two-qubit and memory errors dominate. - Decoder implementation constrained by OpenQASM 2.0 (no bitwise classical conditioning), increasing the number of conditioned single-qubit operations. - Entanglement entropy estimation via randomized measurements is shot-intensive; QND alternatives trade reduced shots for added gate depth. - SPAM mitigation not applied in main results; applying simple local SPAM correction modestly improves energy density but ignores correlated SPAM errors.