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Quantum error correction of a qubit encoded in grid states of an oscillator

P. Campagne-Ibarcq, A. Eickbusch, S. Touzard, E. Zalys-Geller, N. E. Frattini, V. V. Sivak, P. Reinhold, S. Puri, S. Shankar, R. J. Schoelkopf, L. Frunzio, M. Mirrahimi, M. H. Devoret

arXiv:1907.12487v3quant-ph

TL;DR

GKP error correction needs nondestructive oscillator syndrome measurements, which have been experimentally challenging. This experiment implements them with a transmon–microwave-cavity system and feedback, preparing square and hexagonal grid states while demonstrating suppression of logical errors.

  • Problem

    GKP error correction requires measuring oscillator code stabilisers without destroying the encoded state.

  • Method

    The experiment uses conditional oscillator displacements within transmon Ramsey sequences to measure GKP stabilisers and implement feedback-based correction.

  • Results

    The generated states achieved F3 = 0.976, with 1−F3 = 0.2%, while peak broadening limited state-preparation fidelity.

  • Takeaways & Limitations

    The protocol supports both square and hexagonal grid-state codes and provides a framework for feedback-based oscillator error correction.

  • Takeaways & Limitations

    State-preparation fidelity remains limited by broadening of the grid-state peaks.

Abstract

from arXiv · show

Quantum bits are more robust to noise when they are encoded non-locally. In such an encoding, errors affecting the underlying physical system can then be detected and corrected before they corrupt the encoded information. In 2001, Gottesman, Kitaev and Preskill (GKP) proposed a hardware-efficient instance of such a qubit, which is delocalised in the phase-space of a single oscillator. However, implementing measurements that reveal error syndromes of the oscillator while preserving the encoded information has proved experimentally challenging: the only realisation so far relied on post-selection, which is incompatible with quantum error correction (QEC). The novelty of our experiment is precisely that it implements these non-destructive error-syndrome measurements for a superconducting microwave cavity. We design and implement an original feedback protocol that incorporates such measurements to prepare square and hexagonal GKP code states. We then demonstrate QEC of an encoded qubit with unprecedented suppression of all logical errors, in quantitative agreement with a theoretical estimate based on the measured imperfections of the experiment. Our protocol is applicable to other continuous variable systems and, in contrast with previous implementations of QEC, can mitigate all logical errors generated by a wide variety of noise processes, and open a way towards fault-tolerant quantum computation.

I. MEASUREMENT OF DISPLACEMENT OPERATORS · II. CONVERGENCE TO THE GKP CODE MANIFOLD · III. LOGICAL QUBIT INITIALISATION

The experiment measures GKP displacement operators through conditional oscillator–transmon couplings, uses feedback to converge toward the code manifold, and initializes logical states through heralded Pauli measurements followed by QEC rounds.

  • I. MEASUREMENT OF DISPLACEMENT OPERATORS: Conditional displacement gates couple oscillator quadratures to transmon σz rotations, enabling direct measurement of displacement operators and GKP stabilisers.For a transmon prepared on the equator, the measured coherence satisfies ⟨σx −iσy⟩= ⟨D(β)⟩.
  • I. MEASUREMENT OF DISPLACEMENT OPERATORS: The same conditional-displacement Ramsey sequence measures arbitrary ⟨D(β)⟩ values and therefore reconstructs the oscillator’s characteristic function, the Fourier transform of its Wigner function.Conditional displacements embedded in the sequence provide the stabiliser measurements required for GKP error correction.
  • II. CONVERGENCE TO THE GKP CODE MANIFOLD: Feedback maps binary stabiliser outcomes to corrective displacements answering whether the grid moved up or down, or left or right.The protocol balances grid-peak resolution against increased sensitivity to dissipation through the envelope width.
  • II. CONVERGENCE TO THE GKP CODE MANIFOLD: 20 rounds are sufficient for stabiliser values to converge rapidly toward a steady state, with four-round oscillations between 0.62 and 0.5.Beyond this periodic behavior, the stabilisers remain stable over hundreds of rounds.
  • II. CONVERGENCE TO THE GKP CODE MANIFOLD: After 200 rounds, the steady state is a maximally mixed logical qubit, identified by null characteristic-function values at the three logical Pauli operators.The characteristic-function representation is Fourier-conjugate to the theoretical Wigner representation and remains grid-like for grid states.
  • III. LOGICAL QUBIT INITIALISATION: Logical X, Y, or Z measurements replace one QEC round and herald preparation of the corresponding logical state through transmon σx readout.The conditional-displacement amplitudes are β = a/2, (a+b)/2, and b/2 for X, Y, and Z, respectively.
  • III. LOGICAL QUBIT INITIALISATION: A few subsequent QEC rounds project the imperfectly initialized state back onto the code manifold, making the readout non-demolition and repeatable for higher fidelity.The finite squeezing of the code causes the initial logical-Pauli readout to have non-unit fidelity.

IV. COHERENCE OF THE ERROR-CORRECTED LOGICAL QUBIT … DAC n

The protocol extends logical-qubit coherence, with TX = TZ = 275 µs and TY = 160 µs, while also implementing a symmetric hexagonal code and identifying hardware limits and improvement paths.

  • IV. COHERENCE OF THE ERROR-CORRECTED LOGICAL QUBIT: 275 µs is the measured coherence time for the error-corrected X and Z logical components, while TY reaches 160 µs.The protocol extends coherence in all three tested Pauli components compared with no QEC.
  • IV. COHERENCE OF THE ERROR-CORRECTED LOGICAL QUBIT: Photon-dissipation-induced diffusive phase-space shifts cause more logical Y flips, explaining its shorter coherence time.Master-equation simulations reproduce the measured results quantitatively.
  • V. HEXAGONAL CODE: The hexagonal code has equal decay times for all three Pauli operators by symmetry.It is implemented as a variant of the square code.
  • V. HEXAGONAL CODE: After 200 rounds, the oscillator reaches a fully mixed logical state that reveals the hexagonal code structure.The protocol sharpens grid-state peaks and trims the envelope along three directions while stabiliser expectations converge to steady state.
  • V. HEXAGONAL CODE: The hexagonal GKP code uses commuting stabilisers whose phase-space vectors enclose an area of 4π.The stabilisers satisfy Im(a∗b) = 4π.
  • 3. The Pauli operators correspond to dis-: QEC alternates measurements of three hexagonal stabilisers and three short displacement operators with corrective feedback.The stabiliser measurements sharpen peaks along three directions, while short-displacement measurements trim the envelope.
  • 3. The Pauli operators correspond to dis-: The logical-qubit coherence is characterised by preparing Pauli eigenstates and measuring the time decay of their mean values.The |−XL⟩, |−YL⟩, and |−ZL⟩ states are prepared using single-round measurements of Re(X), Re(Y), or Re(Z).
  • VI. LOGICAL ERRORS AND OUTLOOK: Coherence is limited by finite QEC-round duration and could improve with a noise-biased ancilla and a higher-quality-factor cavity.Transmon readout and processing account for about half the round duration, with the conditional displacement gate accounting for the other half.

SUPPLEMENTARY INFORMATION · EXPERIMENTAL DESIGN · THE CONDITIONAL DISPLACEMENT GATE

The experiment uses two coaxial microwave cavities bridged by a superconducting transmon and implements a conditional displacement gate through echoed storage displacements and transmon rotations. Residual unconditional displacement is small and compensated by feedback.

  • EXPERIMENTAL DESIGN: Two rectangular coaxial microwave cavities machined from a single 6061 aluminum block provide the storage and readout oscillator modes.The cavities are extended by rectangular waveguides and closed with copper-powder-coated lids containing a copper braid.
  • EXPERIMENTAL DESIGN: A single transmon superconducting circuit bridges the cavities, using a double-angle-evaporated Al/AlOx/Al Josephson junction with 7.3 nH inductance.The junction bridges two 0.7 mm-by-0.4 mm rectangular aluminum pads.
  • THE CONDITIONAL DISPLACEMENT GATE: The conditional displacement sequence combines storage-mode pulses with a π transmon rotation to produce a conditional phase-space displacement.The first two resonant pulses create a large storage-state excursion parametrized by α(t), followed by the transmon flip and two additional storage pulses.
  • EXPERIMENTAL DESIGN: The transmon is fabricated on a 5 mm-by-37.5 mm, 0.43 mm-thick c-plane sapphire chip clamped between copper blocks with 200 µm indium foil.The fabrication uses a bridge-free technique and double-side-polished sapphire.
  • THE CONDITIONAL DISPLACEMENT GATE: The gate uses fast 30 ns transmon rotations and storage displacements induced by resonant microwave drives in the dispersive cQED regime.The rotation pulses are shaped to minimize leakage to the second excited state.
  • THE CONDITIONAL DISPLACEMENT GATE: A double-echo sequence synchronously reverses α(t) and flips the transmon, preserving the desired coupling while eliminating spurious terms.The storage is first displaced by a large amplitude, reaching |α|^2 ≃320 photons for stabilizer measurements and up to |α|^2 ≃2500 photons in the cited sequence.
  • THE CONDITIONAL DISPLACEMENT GATE: For square-grid stabilization, the residual unconditional displacement is |γ| = 0.04 ∼|β|/60 and is compensated by the subsequent feedback displacement.The resulting asymmetric feedback shifts are δq = +0.04 ± 0.2 for q sharpening and δp = −0.04 ∓0.2 for p sharpening.

EXPERIMENTAL PARAMETERS CHARACTERIZATION · Transmon readout · Displacement length characterization

The experiment uses a fast, minimally dephasing transmon readout and separately calibrates conditional and unconditional storage-oscillator displacements. Conditional-displacement calibration is further optimized through logical-qubit coherence under QEC, while unconditional lengths are extracted from geometric-phase oscillations.

  • Transmon readout: The transmon is read out through an overcoupled dedicated resonator with photon exit rate κr = 2π × 2.5 MHz and dispersive shift χr = 2π × 1 MHz.Readout uses a near-quantum-limited amplification chain and the largest photon number that does not degrade transmon T1.
  • Transmon readout: The readout measurement induces only 2Kt ∼2π × 70 Hz, small enough that the storage mode is not dephased.This condition supports nondestructive transmon measurement while preserving the storage oscillator.
  • Transmon readout: A fast unloading protocol alternates positive and negative resonator-drive amplitudes to separate and recombine readout-oscillator trajectories conditioned on the transmon state.The modified dispersive-readout sequence is designed to shorten resonator ringdown and reduce dead time before further transmon manipulation.
  • Transmon readout: Optimizing the amplitudes and phases of ϵ1,2,3 minimizes residual photons after 600 ns, allowing transmon rotation 100 ns later with no detectable spurious dephasing.The optimization accounts for the photon lifetime being comparable to the sequence duration.
  • Displacement length characterization: Because input-line transmission is uncertain, Gaussian 5 ns feedback pulses require direct calibration of the storage-oscillator displacement D(δ) and conditional displacement CD(β).The two scaling factors are not reliably inferred from one another because conditional displacements arise from large unconditional drives on different timescales.
  • Displacement length characterization: Conditional-displacement scaling is set by measuring the vacuum characteristic function C(β) and fitting it to a Gaussian with standard deviation 2.The calibration assumes vacuum equilibrium while neglecting transmon bit flips during the displacement and spurious oscillator thermal excitations.
  • Displacement length characterization: The conditional-displacement calibration is finely tuned by about 1% by empirically maximizing logical-qubit coherence under QEC.This tuning is applied alongside calibration of other parameters relevant to QEC performance.
  • Displacement length characterization: Unconditional displacement length is calibrated with a conditional/unconditional phase-space sequence whose geometric-phase oscillations are recorded after the trajectories recombine.The sequence disentangles oscillator and transmon, while opposed enclosed areas produce a transmon geometric phase used to determine the displacement length.

Single-photon lifetime in the storage mode · Storage mode resonance frequency and dispersive coupling to the transmon

The storage mode’s photon lifetime is measured from coherent-state amplitude decay, yielding 2Ts = 490 µs. Resonance and dispersive parameters are calibrated from transmon-state-dependent frequencies and logical-qubit dynamics, including χ = 2π × 28 kHz and Ks = 2π×1 Hz.

  • Single-photon lifetime in the storage mode: The photon lifetime is estimated by recording coherent-state amplitude decay while neglecting pure dephasing.The oscillator is initially displaced by δ0 with the transmon in its ground state.
  • Single-photon lifetime in the storage mode: The coherent-state characteristic function is measured along β ∈R and β ∈iR to fit the time-dependent amplitude δ(t).The detuning ∆s is defined between the displacement radiation and oscillator resonance frequency.
  • Single-photon lifetime in the storage mode: 2Ts = 490 µs: the coherent-state amplitude decays exponentially with this characteristic time while rotating in phase space.The rotation occurs at −χ/2 for the storage oscillator dressed by the transmon in |g⟩.
  • Storage mode resonance frequency and dispersive coupling to the transmon: With the transmon in |g⟩, the storage mode resonates at ωs −χ/2, while in |e⟩ it resonates at ωs + χ/2.These measurements provide initial estimates of the oscillator frequency ωs and dispersive shift χ.
  • Storage mode resonance frequency and dispersive coupling to the transmon: ωs is calibrated within ±100 Hz by varying oscillator-control-pulse frequencies and maximizing logical-qubit coherence time.Transmon relaxation during decay and characteristic-function measurements limits the initial calibration precision.
  • Storage mode resonance frequency and dispersive coupling to the transmon: Ks = 2π×1 Hz: the storage oscillator Kerr anharmonicity is estimated from its hybridization with the transmon mode.The estimate uses Kt = 2π×193 MHz for the transmon anharmonicity.
  • Storage mode resonance frequency and dispersive coupling to the transmon: The decays of ⟨Re(X)⟩OFF, ⟨Re(Y)⟩OFF and ⟨Re(Z)⟩OFF establish a baseline for evaluating quantum error-correction performance.These components are reconstructed from circular characteristic-function cuts after preparing | + XL⟩, | + YL⟩ and | + ZL⟩.

Excitation to higher levels of the transmon · Master equation simulations

Transmon leakage to |f⟩ depolarizes the logical qubit, while master-equation simulations model the QEC dynamics and reproduce most experimental observables. The simulations quantify the leakage contribution to logical decay and identify limitations in agreement and error-budget separation.

  • Excitation to higher levels of the transmon: Excitation to |f⟩ disables |g⟩↔|e⟩ control, causing oscillator rotation at ∆s = −3 2χ until random relaxation returns the transmon to the computational manifold.Because T1/2 is much longer than 1/∆s, the oscillator is equally likely to decode as |+ZL⟩ or |−ZL⟩.
  • Excitation to higher levels of the transmon: The transmon’s |f⟩ excitations arise thermally or from fast |g⟩↔|e⟩ pulses lasting 30 ns, and DRAG shaping is used to limit them.The equilibrium occupation of |e⟩ is approximately 1%, while the pulse spectral width is comparable to the transmon anharmonicity.
  • Excitation to higher levels of the transmon: Γ→f = (3 ms)−1 is extracted from |f⟩ occupation dynamics using a hidden Markov model and the fitted excitation rate Γef.The protocol drives P(|e⟩) to 0.5, after which P(|f⟩) approaches equilibrium with characteristic time 1/(Γef/2+Γfe).
  • Master equation simulations: Master-equation simulations use a 300×300 density matrix for the joint storage-transmon state and solve Lindblad dynamics with independently calibrated parameters.The Hamiltonian is expressed in the displaced frame, includes oscillator Kerr anharmonicity, and incorporates transmon pure dephasing through Tφ.
  • Master equation simulations: Each QEC round models control pulses and feedback as instantaneous unitaries, readout as a projection, and outcome-dependent feedback through separate conditional evolutions.The two measurement outcomes are propagated separately before applying the corresponding oscillator displacement and transmon rotation.
  • Master equation simulations: The simulations quantitatively reproduce measured stabilizer expectations and logical-state preparation fidelities, while incorporating |f⟩-induced depolarization when predicting Bloch-vector lifetimes.The simulated coherence decay is fitted exponentially, and the supplementary dephasing rate Γ→f is added to obtain the predicted decay.
  • Master equation simulations: Agreement is within ∼2 % overall and reaches 5 % for hexagonal-code stabilizers, likely because those data were recorded later without retuning drifting setup parameters.The simulations also estimate separate error-channel impacts, but compounding errors prevent a quantitative independent error budget.

Non-linearity and pure dephasing of the storage mode … FINITELY SQUEEZED GKP CODE

The supplement characterizes storage-mode imperfections and the structure of ideal, square, and hexagonal GKP codes, while cautioning that finitely squeezed code properties are inferred from simulations reproducing the measured data. It identifies approximately 1 Hz-scale Kerr non-linearity and pure dephasing as relevant coherence limits and describes symmetric three-stabilizer measurements for isotropic hexagonal-code correction.

  • Non-linearity and pure dephasing of the storage mode: The storage oscillator’s residual Kerr non-linearity is estimated as Ks ≈ 2π × 1 Hz.Larger Kerr non-linearity would distort grid states and limit logical-qubit coherence.
  • Non-linearity and pure dephasing of the storage mode: The storage mode’s pure dephasing rate is bounded by κφ ≲ 2π × 1 Hz.Simulations indicate that larger pure dephasing would shorten the logical-qubit coherence time.
  • IDEAL GKP CODES: Infinitely extended GKP codes are defined by at least two commuting displacement stabilizers, Sa = D(a) and Sb = D(b).Different choices of complex a and b define infinitely many possible GKP codes.
  • Square code: The square GKP code uses stabilizers Sa = D(a = 2√π) and Sb = D(b = ia), with logical Pauli operators X = D(a/2), Y = D((a + b)/2), and Z = D(b/2).The shorter lifetime of the logical Y component is attributed to its states being closer to |−YI⟩ than to the corresponding X or Z eigenstates.
  • Square code: Square-code stabilizers make code-word Wigner functions a-periodic along the corresponding phase-space axes, while Pauli eigenstates acquire a/2-periodicity along their respective axes.The supplied description also relates reciprocal-lattice marginals to the stabilizer structure.
  • Hexagonal code: The hexagonal GKP code uses commuting displacement stabilizers and symmetrically measures three stabilizers so corrected grid-state peaks have rotational symmetry.This symmetry is required for isotropic decay of the logical-qubit Bloch vector.
  • FINITELY SQUEEZED GKP CODE: For finitely squeezed GKP codes, code properties are extrapolated from master-equation simulations reproducing the measured data with independently measured parameters.The resulting figures of merit should be treated with caution because they characterize realistic grid states through simulations rather than direct maximum-likelihood Wigner reconstruction.

Optimal envelope size

Photon loss distorts GKP grid states through competing drift and diffusion, so the optimal envelope balances these effects to minimize logical flips. For the experimental protocol, this gives an estimated envelope size of Δ∼4, with the caveat that the estimate is only order-of-magnitude.

  • Optimal envelope size: Photon loss causes both deterministic contraction and diffusion of the oscillator’s phase-space distribution, which distort grid states and can induce logical flips.The drift contracts the distribution at rate κ_s, while diffusion has constant κ_s/2; in steady state they balance in the vacuum.
  • Optimal envelope size: The optimal envelope is found by requiring drift and diffusion to produce the same average traveled distance before error correction.Logical flips occur when a fraction of the distribution travels more than a/4 in phase space.
  • Optimal envelope size: 25 µs is the characteristic convergence time used for the experimental discrete-time protocol, yielding an estimated optimal envelope size of Δ∼4.The estimate is only order-of-magnitude because a rigorous treatment would include QEC details, higher moments, and decoherence mechanisms beyond photon loss.
  • Optimal envelope size: The simulated experimentally error-corrected steady state uses an envelope width Δ=3.2, with dissipation broadening peaks to ˜σ = 0.29 > σ = 1/2∆.The envelope width was chosen to maximize the logical qubit coherence time.

Characteristics of the error-corrected code · State preparation fidelity

The error-corrected square GKP code reaches a steady-state envelope optimized for logical coherence, with finite peak squeezing and negligible overlap between orthogonal logical states. Preparation fidelity depends on the metric, ranging from experimentally relevant readout fidelity F1 = 0.86 to an ideal code-manifold estimate F3 = 0.976.

  • Characteristics of the error-corrected code: ∆= 3.2 is the steady-state grid-state envelope width chosen to maximize the logical qubit coherence time.The distribution is widest immediately before a q-peak sharpening round, with an analogous p distribution before p sharpening.
  • Characteristics of the error-corrected code: σ = 0.15 (13.4 dB squeezing) characterizes the broader-than-ideal probability-distribution peaks caused by oscillator dissipation and other decoherence.The oscillator state purity is correspondingly below that of a fully mixed logical qubit state.
  • Characteristics of the error-corrected code: 9.5 dB squeezing describes the thinner peaks after a q-peak sharpening round.The code-state wavefunction peak width is 2σ, larger than the probability-distribution peak width.
  • Characteristics of the error-corrected code: The orthogonal-state overlap is |⟨+XL| −XL⟩|2 = |⟨+ZL| −ZL⟩|2 ≪ |⟨+YL| −YL⟩|2 = 4.10−7, enabling a logical qubit definition.The overlap is negligible for the finitely squeezed code.
  • State preparation fidelity: F1 = 0.86 defines preparation fidelity from the expectation value of the real part of the ideal code Pauli operator, matching the subsequent experimental X readout.This definition neglects dissipation during conditional displacement and transmon errors.
  • State preparation fidelity: F2 = 0.90 is obtained by integrating the generated q-probability distribution over regions assigned to | + XL⟩.It represents the probability that ideal q-homodyne detection assigns the state to the target state.
  • State preparation fidelity: F3 = 0.976 is the best-assignment fidelity from the full simulated density matrix after ideal projection onto the code manifold.Although 1 −F3 = 0.2 %, this metric is considered irrelevant for assessing quantum computational resources; its near-unity value indicates peak broadening limits preparation.
  • State preparation fidelity: A second Re(X) measurement with post-selection on outcome +1 boosts preparation fidelity, while logical Bloch-vector component decay is independent of the fidelity definition.Post-selection removes the most ambiguous q-detection results and cannot improve q-detection-based logical readout fidelity.

DETAILS OF THE QEC PROTOCOL · Discrete-time Markovian feedback QEC in the square code

The square-code protocol uses transmon-assisted conditional displacements and σy readout to sharpen oscillator peaks, while feedback displacements preserve stabilizers. Alternating sharpening and envelope-trimming rounds counteract measurement-induced envelope expansion and tune the steady-state error distribution.

  • Discrete-time Markovian feedback QEC in the square code: Each QEC round prepares the transmon in | + x⟩, applies a conditional displacement, and measures the transmon along σy.The oscillator dynamics are described by Kraus operators associated with the | ± y⟩ detection outcomes.
  • Discrete-time Markovian feedback QEC in the square code: Feedback displacements shift the collapsed distribution back toward the origin without changing the code stabilizers.For the envelope-trimming step, displacements by a/2 compensate peak-sharpening-induced envelope expansion and commute with the stabilizers.
  • Discrete-time Markovian feedback QEC in the square code: Measurement backaction partly collapses and sharpens the q-probability peaks, while shifting their centers and temporarily skewing their shapes.Dissipation rapidly restores a Gaussian peak shape.
  • Discrete-time Markovian feedback QEC in the square code: Along p, q-peak sharpening expands the probability distribution by generating a new outward peak and shifts it by a/2 modulo a.The shift deterministically flips the logical qubit, implementing a Z-gate for q-peak sharpening and an X-gate for p-peak sharpening.
  • Discrete-time Markovian feedback QEC in the square code: The protocol alternates peak-sharpening rounds with envelope-trimming rounds using a shorter conditional displacement CD(ϵ), with |ϵ| ≈a/20.This mitigates undesired expansion of the envelope caused by peak sharpening.
  • Discrete-time Markovian feedback QEC in the square code: The envelope-trimming backaction also partly collapses the q distribution, after which feedback shifts the whole distribution back toward the origin.These feedback shifts are large enough to compensate the envelope expansion induced by peak-sharpening rounds.
  • Discrete-time Markovian feedback QEC in the square code: The parameter ϵ controls measurement strength: ϵ →0 produces marginal backaction and a larger steady-state envelope, whereas larger ϵ produces stronger collapse and a smaller envelope.The optimal feedback shift is tuned through stabilizer measurements conditioned on the transmon outcome.

Optimization of the feedback displacements · Measurement of the logical Pauli operators in the square code

Feedback displacements are optimized by simulation and steady-state stabilizer measurements, with separate tuning for different QEC rounds. Logical Pauli measurements in the square code use conditional displacements and transmon readout, followed by correction of measurement-induced stabilizer sign changes.

  • Optimization of the feedback displacements: Feedback displacement lengths are first estimated in simulations, then finely adjusted by maximizing the real part of steady-state stabilizer expectations.The protocol reaches steady state before the final-round shifts are varied independently according to the transmon measurement outcome.
  • Optimization of the feedback displacements: For p-envelope trimming, the optimal feedback shifts are symmetric and have lengths close to a/2.Enforcing exactly a/2 provides a more precise displacement-length calibration than the steady-state optimization method.
  • Optimization of the feedback displacements: Shifts following q-peak sharpening and q-envelope trimming rounds are also tuned by nullifying ⟨Sb⟩|±y⟩.These additional optimizations account for feedback displacements associated with other stabilization steps.
  • Optimization of the feedback displacements: The hexagonal-code feedback sequence uses position- and momentum-dependent shifts from 6 directions to prevent peak spreading and envelope expansion.Displacements along a⊥, b⊥ and c⊥ sharpen the grid peaks, while those along a, b and c constrain the envelope.
  • Measurement of the logical Pauli operators in the square code: Square-code Re(X), Re(Y), and Re(Z) measurements replace q-peak sharpening with conditional displacements β = a/2, (a + b)/2, and b/2, respectively, followed by σx readout.Each conditional displacement implements the corresponding logical-Pauli measurement through the transmon readout.
  • Measurement of the logical Pauli operators in the square code: Measurement backaction multiplies the β⊥-probability distribution by 1±cos β⊥.The conjugate-variable backaction can damp one peak of the distribution out of two.
  • Measurement of the logical Pauli operators in the square code: On the conjugate variable, the measurement displaces the state by ±β/2, reversing the sign of one or both stabilizers.This shift moves the grid state by half the lattice period; an unconditional displacement D(−β/2) flips the stabilizer sign back.

QEC and measurement of the Pauli operators in the hexagonal code · TRANSMON ERRORS AND FEEDBACK PERFORMANCES · Readout errors and phase-flips of the transmon

The hexagonal-code QEC protocol measures all three stabilizers through conditional displacements, transmon readout, and feedback, preserving stabilizers while applying deterministic logical Pauli gates. Transmon readout errors and phase-flips reduce convergence according to the readout contrast, while the experiment’s phase-flips contribute only ∼1.5 % of logical errors.

  • QEC and measurement of the Pauli operators in the hexagonal code: Measuring only two of the three redundant hexagonal-code stabilizers would produce grid-state peaks without rotational symmetry, making the Pauli operators X, Y, and Z unequal.The three stabilizers are Sa, Sb, and Sc; the corresponding Pauli operators are X = D(a/2), Y = D(b/2), and Z = D(c/2).
  • QEC and measurement of the Pauli operators in the hexagonal code: The protocol sharpens peaks sequentially along the three orthogonal directions a⊥, b⊥, and c⊥ using conditional displacements, σy readout, and feedback shifts of ±δ = ±0.2.The feedback shifts recenter peaks at β⊥= 0 mod 2π and generate effective dissipation that prevents peak spreading.
  • QEC and measurement of the Pauli operators in the hexagonal code: Sharpening along all three directions avoids the ellipse-shaped peaks produced by a two-stabilizer protocol, which would leave peaks more elongated along an unsharpened direction.The protocol’s trimming directions preserve stabilizer values even though the corresponding feedback displacements deterministically apply X, Y, and Z gates.
  • QEC and measurement of the Pauli operators in the hexagonal code: Real parts of X, Y, or Z are measured by shortening the corresponding conditional displacement to CD(a/2), CD(b/2), or CD(c/2), reading out σx, and recentering the grid state.An additional unconditional displacement D(−a/2), D(−b/2), or D(−c/2) is applied after readout.
  • Readout errors and phase-flips of the transmon: Transmon phase-flips commute with the interaction Hamiltonian and are therefore equivalent to readout errors that cause an incorrect feedback displacement.This equivalence holds because the conditional-displacement interaction acts on the transmon through σz only.
  • Readout errors and phase-flips of the transmon: The convergence rate is Γ = δ(1 −2ϵ)a/τ, and the steady-state variance is σ2Γ = δ (1−2ϵ)a, so decreasing δ improves squeezing but slows stabilization.Here τ = 4Tround is the time between q-peak sharpening rounds, and Γ is proportional to the transmon readout contrast 1−2ϵ.
  • Readout errors and phase-flips of the transmon: For photon-loss rate κs, the optimal feedback displacement balances dissipation-induced diffusion against readout inaccuracy, favoring longer displacements for diffusion and shorter ones for inaccurate readout.Envelope-trimming rounds also contribute to peak broadening by acting as conditional displacements by ϵ.
  • Readout errors and phase-flips of the transmon: ∼1.5 % of the logical errors are attributed to transmon phase-flips in simulations, with a pure dephasing rate Γφ = (140 µs)−1 and readout fidelity above 99.5 %.Under the experimental parameters, the impact of transmon phase-flips on error-correction performance is negligible.

Bit-flips of the transmon

Transmon bit-flips do not commute with the interaction and can perturb the oscillator, producing timing-dependent logical errors during peak-sharpening rounds. In the real interaction, they also cause small oscillator rotations that slightly increase logical error rates.

  • Mechanism: Transmon bit-flips perturb the oscillator because random σx-gates do not commute with the interaction Hamiltonian.The experiment uses an interaction duration Tint = 1.1 µs.
  • Mechanism: During peak sharpening, conditional displacements deterministically flip the logical qubit, while a bit-flip reverses the interaction sign and changes the resulting operation.The displacement is ±a/2 for p sharpening and ±b = ±ia for q sharpening.
  • Logical errors: Transmon bit-flips induce X, Z, or Y logical errors only during the corresponding p-, q-, or either peak-sharpening round and within a specific interaction-time window.The cited condition depends on the bit-flip time t relative to Tint.
  • Logical errors: Amplitude damping and regular echoes yield a transmon bit-flip rate of 1/2T1 because the excited-level occupation averages to 0.5.The echoes tend to symmetrize the damping channel.
  • Limitations: In the real interaction, bit-flips additionally produce rotations of angle φ ≤χTe ≈10°; more frequent echoes could mitigate these rotation-induced errors.The rotations arise from dispersive coupling and slightly increase the logical error rate.

TELEPORTED GATES AND ARBITRARY STATE PREPARATION

A teleported-gate protocol enables arbitrary manipulation and preparation of logical qubit states by combining conditional and unconditional oscillator displacements with transmon measurement and feedback. The experiment uses this protocol to prepare a logical magic state in both square and hexagonal codes, with deterministic preparation possible through conditional feedback.

  • Protocol overview: The protocol enables arbitrary logical-qubit manipulation and arbitrary logical-state preparation using teleported gates.The analysis considers logical Pauli operators as oscillator displacements for an infinitely squeezed code.
  • Protocol overview: For a target rotation around V = X, Y or Z, any logical state is expressed as |ΨL⟩ = α| + VL⟩ + β| −VL⟩.The protocol operates in the corresponding | ± VL⟩ basis.
  • Teleported gate implementation: The gate prepares the transmon in | + x⟩, applies conditional displacement CD(γ), recenters with D(−γ/2), and measures the transmon along a rotated axis.A V-gate feedback correction is applied for one measurement outcome, yielding an unconditional rotation RV(−φ).
  • Arbitrary state preparation: The experiment prepares |ML⟩ = cos(π/8)|+XL⟩ − sin(π/8)|−XL⟩ in both square and hexagonal codes.Preparation is heralded in the experiment and can become deterministic by applying a Y-gate after the orthogonal transmon outcome; this state is a first step toward magic-state distillation.
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