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Grid-state qubits in a superconducting cavity reach combined preparation and measurement error below one in a thousand.

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2026-07-10 22:47 UTC pith:U2ICNT5M

load-bearing objection Solid experimental result: two-order SPAM reduction for single-mode GKP via post-selected sBs + repeated finite-energy measurement, with QEC left intact and magic states from vacuum included.

arxiv 2607.06718 v1 pith:U2ICNT5M submitted 2026-07-07 quant-ph

Quantum error correction of a grid-state qubit with state preparation and measurement errors below 10⁻³

Sara Turcotte (1 , 2) , Lucas St-Jean (1) , Am\'elie L. Pessonneaux (1) , Ross Shillito (1) , Bohdan Kulchytskyy (1) , Eliott Ouellet (1) , Jean Olivier Simoneau (1)
show 8 more authors
Florian Hopfmueller (1) Matthew Hamer (1) Pascal Lemieux (1) Dany Lachance-Quirion (1) Baptiste Royer (2) Nicholas E. Frattini (1) ((1) Nord Quantique (2) D\'epartement de Physique et Institut quantique Universit\'e de Sherbrooke)
This is my paper
classification quant-ph
keywords grid-state qubitGKP codebosonic quantum error correctionstate preparation and measurementfinite-energy measurementsBs stabilizationmagic statescircuit QED
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Grid-state (GKP) qubits encode logical information in a microwave oscillator so that photon loss can be corrected without many physical qubits. Until now their preparation and readout errors were far higher than those of ordinary superconducting qubits, limiting real algorithms. This experiment uses the same stabilization cycle that corrects errors to prepare the six cardinal states and H-type magic states by post-selecting on clean auxiliary-qubit outcomes, and measures the logical Paulis by repeating finite-energy measurements until they all agree. Together the two protocols drive the total SPAM error below 10^{-3} for cardinal states (and roughly 8 imes10^{-3} for magic states) while remaining compatible with autonomous error correction that still beats break-even. The result removes a long-standing practical barrier and places grid-state SPAM on the same footing as ordinary transmon SPAM.

Core claim

When post-selected sBs stabilization is used for state preparation and repeated finite-energy measurements are used for readout, the combined SPAM error of a single-mode grid-state qubit, averaged over the six cardinal states, falls below 7(7) imes10^{-4} (and to 8(5) imes10^{-3} for H-type magic states). This is two orders of magnitude better than previous grid-state experiments and matches typical transmon SPAM levels, without degrading the logical error rate of subsequent autonomous quantum error correction.

What carries the argument

Post-selected small-big-small (sBs) stabilization interleaved with mid-circuit auxiliary measurements, combined with multi-round finite-energy Pauli measurements that retain only all-agree outcomes; together they suppress auxiliary readout errors, finite-energy envelope errors, and photon-loss errors while remaining fully compatible with autonomous QEC.

Load-bearing premise

That discarding every shot in which successive mid-circuit auxiliary outcomes disagree fully removes residual logical and measurement errors without biasing the reconstructed Pauli expectation values used to compute logical fidelity.

What would settle it

Repeat the same post-selected preparation and multi-round measurement protocol on independently prepared cardinal states and check whether the extracted logical fidelity still matches the claimed SPAM figure when an independent tomography method (for example full characteristic-function reconstruction without post-selection) is used as ground truth.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Grid-state logical qubits can now be prepared and read out at error rates comparable to ordinary superconducting qubits, removing a major bottleneck for circuit-level algorithms.
  • H-type magic states prepared from vacuum by the same stabilization cycle become practical resource states for non-Clifford gates inside a GKP architecture.
  • The same SPAM protocols can be layered under existing autonomous QEC without increasing the logical error per round, enabling longer error-corrected computations.
  • Because the protocols are hardware-efficient and use only a single oscillator mode, they lower the resource overhead for multi-mode or multi-qubit GKP processors.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The large reduction in SPAM error implies that previous GKP experiments were limited more by auxiliary readout and finite-energy effects than by the intrinsic quality of the oscillator itself.
  • If the all-agree policy can be relaxed to a controlled number of allowed flips without reintroducing bias, the exponential survival-probability cost could be mitigated for deeper circuits.
  • Extending the same post-selected stabilization to two-mode grid codes would test whether the SPAM improvement scales with the higher protection those codes provide.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The manuscript reports experimental SPAM improvement for a single-mode GKP (grid-state) qubit in a superconducting cavity-transmon architecture. Using post-selected sBs stabilization (from vacuum or 9 dB squeezed states) for preparation of the six cardinal states and H-type magic states, combined with repeated finite-energy measurements under an all-agree post-selection policy, the authors achieve a total SPAM error averaged over cardinal states of 7(7)×10^{-4} (survival ~0.24 prep / ~0.39 meas) and 8(5)×10^{-3} for magic states. These protocols are shown to be compatible with autonomous QEC, yielding a logical error per sBs round of 8.1(2)×10^{-3} that is statistically unchanged from the unimproved baseline. Device parameters, reset error, and readout visibility are independently calibrated; logical fidelity is extracted from Pauli expectations via the standard average (Eq. 3).

Significance. If the reported numbers hold under the stated post-selection, this closes a long-standing practical gap for bosonic GKP qubits: SPAM errors that previously limited computational performance are brought two orders of magnitude below prior GKP experiments and into the range of typical transmon SPAM. The work is concrete and hardware-relevant: it re-uses high-performance autonomous sBs QEC for repeat-until-success preparation (including magic states from vacuum), demonstrates that the same protocols do not degrade QEC gain, and supplies transparent calibrations plus a noise model that reproduces the main trends. The result strengthens the case for grid-state qubits as a hardware-efficient route to fault-tolerant computation.

minor comments (5)
  1. Abstract and Sec. I claim “below 10^{-3}” while the body (Fig. 4a, text) quotes 7(7)×10^{-4} for cardinal states and 8(5)×10^{-3} for magic states; a single clarifying sentence that the headline figure is the cardinal average under all-agree post-selection would avoid any ambiguity.
  2. Eq. (3) and the surrounding text define FL via Pauli expectations of the six cardinal states; it would help the reader if the precise mapping from the post-selected mid-circuit bit-strings {mk} to ⟨μ0⟩± were written out once (even if standard).
  3. Appendix D.2 and Figs. D.2–D.3 already quantify the survival–infidelity trade-off under milder policies; a short pointer in the main text (Sec. II D or II E) would make clear that the all-agree choice is conservative rather than the only viable option.
  4. Fig. 2b bottom panel and Fig. 3b use single-round vs multi-round finite-energy measurements; labeling the measurement protocol explicitly on each panel would reduce the need to cross-reference the caption.
  5. A few typographical items: “presqueezing” / “Presqueezing” capitalization is inconsistent; “primarly” (Sec. II E) should be “primarily”; arXiv date stamp appears as July 9, 2026.

Circularity Check

0 steps flagged

No significant circularity: experimental SPAM measurement under explicit post-selection protocols, not a derivation that reduces to its inputs.

full rationale

The paper's central claim is an empirical measurement of combined SPAM error (ϵ_SPAM = 1 − F_L from the standard average of Pauli expectations over the six cardinal states, Eq. 3) under two explicitly defined protocols: post-selected sBs stabilization (repeat-until-success compatible, all-agree on mid-circuit auxiliary outcomes) and repeated finite-energy measurements (likewise all-agree). These protocols are described in Sec. II C–D and Appendices C–D with concrete pulse sequences, gauge updates, and survival probabilities; the reported numbers (7(7)×10^{-4} for cardinal states, 8(5)×10^{-3} for H-type magic states) are direct experimental outcomes, not predictions obtained by fitting a parameter to a subset of the same data and re-using it. Self-citations (e.g., [24] for the device and sBs implementation, [21, 29–31] for finite-energy measurement and GKP background) supply prior experimental techniques and theory; they are not invoked as uniqueness theorems that force the present result, nor do they smuggle an ansatz that is then re-labeled a first-principles derivation. Simulations in the appendices use an independent noise model (storage decay + transmon T1/Tϕ + classical bit-flip) and recover the same qualitative saturation of ϵ_SPAM, providing consistency checks rather than circular support. The logical-error-per-round comparison (0.0081(2) vs 0.0085(2)) further shows the SPAM protocols do not alter the autonomous QEC performance they are claimed to be compatible with. No step reduces by construction to its own inputs; the work is self-contained experimental reporting.

Axiom & Free-Parameter Ledger

3 free parameters · 3 axioms · 0 invented entities

The central experimental claim rests on a small set of calibrated device parameters, the standard dispersive Hamiltonian of circuit QED, and the finite-energy GKP encoding with a chosen envelope Δ. No new physical entities are postulated; free parameters are the usual experimental knobs (envelope size, number of rounds, post-selection thresholds).

free parameters (3)
  • finite-energy envelope Δ = 0.38
    Chosen by hand as Δ = 0.38 (average photon number ≈ 2.96); controls both peak width and error-correction rate and directly enters the reported fidelities.
  • number of sBs rounds N_sBs and RFE rounds N_RFE = typically 8–11 (prep), 5–8 (meas)
    Selected to saturate Pauli expectation values while keeping survival probability acceptable; different values used for cardinal vs. magic states.
  • auxiliary readout visibility V = 0.95(1)
    Measured mid-circuit visibility V = 0.95(1) sets the single-shot limit that post-selection is designed to overcome.
axioms (3)
  • domain assumption The system is accurately described by the dispersive Hamiltonian of a cavity mode dispersively coupled to a transmon (including first- and second-order cross-Kerr terms).
    Invoked throughout Sec. II A and Appendix A; all control and measurement sequences are derived from this model.
  • domain assumption sBs stabilization with the listed displacement amplitudes and gauge updates drives the oscillator into the finite-energy GKP code space.
    Taken from Royer et al. (2020) and used as the primitive for both state preparation and QEC (Sec. II B).
  • ad hoc to paper All-agree post-selection on auxiliary mid-circuit outcomes yields an unbiased sample of the logical state whose Pauli expectations equal the true logical fidelity.
    Central to the SPAM metric (Eq. 3 and Sec. II D); Appendix D explores alternatives but the main claim uses the strictest policy.

pith-pipeline@v1.1.0-grok45 · 53168 in / 2716 out tokens · 34874 ms · 2026-07-10T22:47:19.341643+00:00 · methodology

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read the original abstract

Grid state qubits offer a hardware-efficient approach to large-scale fault-tolerant quantum computing. They access the information redundancy required for quantum error correction by exploiting the large Hilbert space naturally available in harmonic oscillators. Superconducting architectures are particularly suitable to implement grid state qubits due to their fast and high-fidelity operations. Grid states in superconducting circuits enable quantum error correction (QEC) with performance beyond break-even. However, the state preparation and measurements (SPAM) errors of grid states has been a significant limitation to computational performances. In this work, we leverage high-performance QEC to enable repeat-until-success state preparation of both cardinal and magic states of the single-mode grid-state qubit. We combine this with an improved measurement protocol that corrects for both finite-energy envelope and auxiliary qubit readout errors, and increases robustness to photon loss. Our experiments, using both techniques, achieve a combined state-preparation and measurement error below $10^{-3}$. This represents two orders-of-magnitude improvement over the state of the art, bringing this platform on par with standard SPAM error levels measured in transmon qubits.

Figures

Figures reproduced from arXiv: 2607.06718 by 2), (2) D\'epartement de Physique et Institut quantique, Am\'elie L. Pessonneaux (1), Baptiste Royer (2), Bohdan Kulchytskyy (1), Dany Lachance-Quirion (1), Eliott Ouellet (1), Florian Hopfmueller (1), Jean Olivier Simoneau (1), Lucas St-Jean (1), Matthew Hamer (1), Nicholas E. Frattini (1) ((1) Nord Quantique, Pascal Lemieux (1), Ross Shillito (1), Sara Turcotte (1, Universit\'e de Sherbrooke).

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: (a). The measurement outcomes are used to imple￾ment a procedure compatible with a repeat-until-success approach, in which postselection keeps only instances where the auxiliary qubit is measured in its ground state after each round. Starting from a squeezed state, which is an eigenstate of one of the code stabilizers, can improve the state-preparation survival probability. The mid-circuit measurement sequ… view at source ↗
Figure 3
Figure 3. Figure 3: (b). For a single round, SPAM fidelity is primarily limited by readout errors of the auxiliary qubit, which are signif￾icantly suppressed through repetition and postselection. As shown by the dashed grey line in [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

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Reference graph

Works this paper leans on

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