Topological Order from Measurements and Feed-Forward on a Trapped Ion Quantum Computer

I. TORIC CODE PREPARATION WITH FEED-FORWARD

• The target is the toric code ground state on a square lattice with periodic boundaries, where four ground states are distinguished by logical strings wrapping around the torus.The chosen state has logical expectations ⟨Zhori⟩=⟨Zvert⟩=1.Source: §I · p. 2
• The three-step constant-depth procedure initializes qubits in |0⟩, measures Ap on odd plaquettes, then applies conditional single-qubit Z gates where Ap = −1.A lookup-table decoder identifies where to apply the corrections; measurements use both ancilla-free and one-ancilla strategies.Source: §I · p. 3
• Odd defect counts are heralded and discarded to account for syndrome-measurement errors, with roughly 10% of 1240 preparation runs discarded.Even for a completely depolarized state, the authors report that this procedure would discard only half the data.Source: §I · p. 3
• An energy density of −0.929±0.004 and logical-string expectations Zhori = 0.916 ± 0.0065 and Zvert = 0.914 ± 0.0064 indicate high-fidelity preparation of the target ground state.The measured X-type plaquette expectation is 0.944 ± 0.0049, versus 0.914 ± 0.0063 for Z-type plaquettes.Source: §I · p. 3
• γ/ ln 2 is 0.93 ± 0.055 for 2×2 regions and 1.05 ± 0.093 for 2×3 regions, consistent with Z2 topological order.The randomized-measurement scheme computes second Rényi entropies for connected regions of up to six qubits.Source: §I · p. 3

II. ANYON TRANSMUTATION AND INTERFEROMETRY

• The defect state has an average expectation value per plaquette of 0.925 ± 0.0039, comparable to the toric code ground state.The geometry contains two defects formed by removing the central qubit and redefining stabilizers.Source: §II · p. 3, Figure 3 · p. 4
• Moving one magnetic anyon across the line connecting the defects transmutes it into an electric anyon, producing an electric-magnetic pair that cannot be created singly in a defect-free toric code.The final adjacent-plaquette stabilizer expectations are −0.92 ± 0.017 and −0.89 ± 0.020.Source: §II · pp. 3–4, Figure 3 · p. 4
• The interferometry creates an electric-magnetic pair next to the defect and applies a controlled braid, measuring its phase through an ancilla Hadamard test.The composite is fermionic because its wavefunction acquires a minus sign under braiding one constituent around the other.Source: §II · p. 4
• The interferometric braid signal is ℜ⟨em|Ubraid|em⟩= −0.87 ± 0.018, versus ℜ⟨gs|Ubraid|gs⟩= +0.87 ± 0.018 without the fermionic excitation, verifying fermionic exchange statistics.The controlled braiding operation uses only four two-qubit gates, enabled by effective all-to-all connectivity.Source: §II · p. 4

III. DISCUSSION AND OUTLOOK

• The demonstrated preparation combines mid-circuit measurement, feed-forward, and low-error gates to prepare topologically ordered states deterministically, in constant depth and with high fidelity.Effective all-to-all connectivity was vital for implementing periodic two-dimensional geometry and conditional dynamics.Source: §III · p. 5
• The work studies Abelian topological orders and defects with non-Abelian character within Abelian phases, laying groundwork for preparing true non-Abelian topological orders.The authors note that some such states may require the same overhead as Abelian counterparts, while multiple rounds of measurements and feed-forward can access more exotic states.Source: §III · p. 5

A. Entropy measurements from randomised measurements

• The protocol estimates subsystem purity from randomized measurements and uses it to evaluate second-order Rényi entropies.It follows the purity-estimation protocol in; local unitaries are drawn from the circular unitary ensemble.Source: A. Entropy measurements from… · p. 8
• The experiment used 72 random local-unitary settings and 256 shots per setting, with parameters selected by grid search on the H1-1 emulator.An unbiased estimator was used to evaluate the squared outcome probabilities.Source: A. Entropy measurements from… · p. 8
• Without discarding shots with an odd number of heralded anyons, γ/ln 2 was 0.87 ± 0.055 for 2 × 2 regions and 1.00 ± 0.090 for 2 × 3 regions.These estimates are reported separately from the main-text procedure that discards such shots.Source: A. Entropy measurements from… · p. 8
• The topological entanglement entropy was estimated for 2 × 2 and 2 × 3 regions by averaging subsystem Rényi entropies across regions.The resulting uncertainty was evaluated by bootstrapping new samples from the dataset.Source: A. Entropy measurements from… · p. 8

B. Effects of Measurement Error Mitigation

• The local transition matrix is characterized by readout-confusion probabilities: 0.1% for |0⟩ read as |1⟩ and 0.5% for |1⟩ read as |0⟩.Applying the inverse local transition matrix recovers the modeled ideal distribution efficiently.Source: B. Effects of Measurement Error… · p. 8
• SPAM correction improves energy density from −0.93 to −0.95 when the main-text heralded-error procedure is used.The model ignores correlated SPAM errors across multiple sites.Source: B. Effects of Measurement Error… · p. 8
• Without discarding heralded errors, SPAM mitigation improves energy density from −0.89 to −0.91.The correction models the noisy distribution as a local transition-matrix transform of the ideal distribution.Source: B. Effects of Measurement Error… · p. 8

C. Measurement and Decoding of the Ap-operator with and without ancillae

• Parity-check measurements project the state into stabilizer eigenspaces; the modified ancilla-free check autocorrects a −1 outcome while moving a potential error to an adjacent plaquette.This limits error accumulation to half of the X-type plaquettes and reduces subsequent decoding cost; the modification is convenient, not essential for high fidelity.Source: C. Measurement and Decoding of the… · pp. 8–9
• The 20-qubit device used ancilla-based checks on four X-type plaquettes and modified ancilla-free checks on the remaining four.The −1 outcome on a modified check moves the error or anyon to the diagonally adjacent target plaquette.Source: C. Measurement and Decoding of the… · p. 9
• The lookup-table decoder handles eight error possibilities, and the reported simple and later optimized decoders achieve almost the same energy density.Reusing ancillae saves eight two-qubit gates but generally increases depth, execution time, and potential memory errors; measuring all X-type plaquettes with ancillae gives a slight energy-density improvement, particularly for X-type plaquettes.Source: C. Measurement and Decoding of the… · p. 9

D. State Preparation and dynamics of the model with a defect

• For the defective model, five available qubits serve as ancillas for three X-type plaquettes and two defect plaquettes, while the other three X-type plaquettes use modified ancilla-free checks.Its stabilizers are measured in four settings, with corner and defect plaquettes measured twice.Source: D. State Preparation and dynamics of… · p. 9
• The anyon-transmutation sequence is followed by stabilizer measurements, while interferometry reads the real part of the braiding phase from the ancilla’s final ⟨Z⟩ expectation.Quantum nondemolition stabilizer measurements also track a magnetic anyon as it is moved into and then out of a defect plaquette.Source: D. State Preparation and dynamics of… · p. 9

E. Circuit Construction, Gate Count and Error Budget

• At the time of the experiment, the device specifications reported average two-qubit gate fidelity of 99.7%, one-qubit fidelity of 99.996%, and state-preparation-and-measurement fidelity of 99.6%.The reported memory error was 1 − 99.97% per depth-1 circuit time per qubit.Source: E. Circuit Construction, Gate Count… · p. 9
• The defect-free circuit uses 484 one-qubit gates and 40 two-qubit gates; the defective circuit uses 423 one-qubit gates and 40 two-qubit gates.Both circuits were compiled to the native gate set and sent to the device using TKET.Source: E. Circuit Construction, Gate Count… · p. 9
• The measured global fidelity with the toric-code ground state was 0.80 ± 0.049.It was estimated from shadow density matrices using the randomized-measurement dataset also used for topological entanglement entropy.Source: E. Circuit Construction, Gate Count… · pp. 9–10
• The estimated global damping factor is approximately 0.762, compatible with the measured global fidelity.The estimate multiplies contributions from two-qubit gates, one-qubit gates, memory over six depth-1 circuit times, and SPAM on 20 data plus four reused qubits.Source: E. Circuit Construction, Gate Count… · p. 10

• A single Z-flip during the (I + X^⊗4)/2 projection flips two adjacent Z-plaquettes while leaving X-plaquettes invariant.Source: Figure 4 · p. 11
• Panel (a) shows X-plaquette covariances on the left and Z-plaquette covariances on the right, marking neighboring and non-neighboring plaquettes differently.Source: Figure 4 · p. 11
• Panel (b) compares average Z-plaquette correlations for adjacent and distant plaquette pairs.Source: Figure 4 · p. 11

B. Topological Entanglement Entropy

• Figure 5 plots entanglement-entropy correction γ for labeled 2 × 2 and 2 × 3 regions on the 4 × 4 torus.The 2 × 2 values average over rotations; maximum error bars are ±0.35 for 2 × 2 regions and ±0.54 for 2 × 3 regions.Source: Figure 5 · p. 11

C. Parity check circuits and stabilizer measurement layout

• Figure 6 contrasts ancilla and modified ancilla-free X^⊗4 parity-check circuits and marks the measured plaquettes on a 4 × 4 torus.Twelve colored plaquettes are stabilized after projection; the four empty plaquettes require feed-forward correction.Source: Figure 6 · p. 12

D. Non-demolition anyon tracing

• Figure 7 depicts magnetic-to-electric anyon transmutation using non-demolition measurements, contrasting destructive measurements after each step with a single-shot trajectory that costs extra gates.Source: Figure 7 · p. 12

• Panels (a–d) examine discarding heralded shots and applying state preparation and measurement error mitigation.Source: Figure 8 · p. 13
• Panel (e) shows energy density for the optimized decoder, while panel (f) uses ancilla qubits for every stabilizer measurement.Source: Figure 8 · p. 13
• Panel (g) leaves one plaquette unprojected and infers it from the constraint that the number of anyons is even.Source: Figure 8 · p. 13