Witnessing and quantifying stellar rank with the Wigner function

AI summary

Overview

Research area: Continuous-variable (bosonic) quantum information — specifically, certifying and quantifying non-Gaussianity in experimentally prepared cavity states via the stellar rank.

Technical level: Intermediate to Advanced. The conceptual goals are accessible, but the paper relies on phase-space formalism, stellar functions, and non-convex optimization of witness thresholds.

Scope (one sentence): The paper introduces two measurement-efficient protocols — fidelity-based witnesses and few-points witnesses — for certifying the stellar rank of states using displaced parity (Wigner function) measurements, and applies them to superconducting-cavity data for Fock, binomial, cat, and GKP states.

What This Paper Is About

Bosonic quantum computers need non-Gaussian states to achieve exponential computational advantage over classical simulation, so experimenters need reliable ways to prove that the states they build actually contain non-Gaussian resources — and how much. The stellar rank is a measure with operational meaning (it tracks the cost of classical simulation), but existing certification methods were not designed for platforms like superconducting cavities, where the Wigner function can be measured directly point by point. The paper's goal is to turn those direct Wigner measurements into rigorous, quantitative certificates of stellar rank.

Key Contributions

1. A fidelity-based stellar-rank witness. The authors combine black-box fidelity estimation (which samples phase-space points according to the target state's Wigner function) with the target state's stellar fidelity profile, converting a fidelity lower bound into a certified stellar rank without full state tomography.

2. Few-points witnesses. Witness operators built as weighted sums of displaced parity measurements, requiring the Wigner function at only a handful of well-chosen points (positive weights at local maxima, negative weights at local minima of the target Wigner function), with thresholds computed via a numerically tractable optimization over Gaussian unitaries.

3. An analytical result for two-point witnesses. For two-point witnesses the authors establish a general connection between the witness's ability to detect stellar rank and the range of the Wigner function (the difference between its maximum and minimum value).

4. A phase-averaged witness for rotationally symmetric states. Averaging phase-space information yields lower thresholds than the non-averaged version, enabling certification of comparatively higher stellar rank for rotationally symmetric states.

Main Findings

• Stellar rank 8 certified for a GKP state. Applying the fidelity-based protocol to experimental GKP data (approximate zero codeword, Δ = 0.35), the estimated fidelity falls between the stellar fidelities for rank 7 and rank 8, giving a certified stellar rank of at least 8 — described by the authors as a new benchmark across all quantum platforms, exceeding the value of 6 previously detected in Ref. [44].

• Fock states certified at their theoretical maximum. Estimated fidelities were 0.9591 for |1⟩ and 0.9957 for |2⟩ (at N = 10⁶ samples, with errors 0.0087 and 0.0105). Comparing against stellar fidelities of 0.48, 1 for |1⟩ and 0.38, 0.56, 1 for |2⟩ yields stellar rank 1 and 2 respectively — the correct theoretical values.

• Binomial state certified at stellar rank 4. The estimated fidelity (0.9372 at N = 10⁶) minus half the error exceeds the rank-3 stellar fidelity but stays below rank 4, matching the theoretical rank of the state in Eq. (17).

• Cat state certified at rank at least 1. Because any cat state has infinite stellar rank and its stellar fidelities converge too slowly to 1, the fidelity approach only establishes rank ≥ 1 for the cat state.

• Few-points witnesses reproduce strong certificates with very few measurements. For the binomial state, a two-point witness gives expectation values 0.7330 (theory) and 0.6583 (experiment) against a rank-0 threshold of 0.5, already certifying rank ≥ 1; a five-point witness with a rank-3 threshold of 0.4795 yields 0.5729 (theory) and 0.5200 (experiment), certifying the maximal rank 4 for both.

• Sample complexity scaling. A single Wigner-function point needs O(1/ε²) samples by Hoeffding's inequality, so an N-point witness needs about N/ε² measurements; the fidelity-based protocol's sample cost instead scales with the Wigner negativity (1-norm) of the target state.

• Robustness analysis. Appendix D reports that few-points witnesses retain some robustness to noise and to imprecision in the location of the measurement points, with robustness decreasing as the target state's phase-space features sharpen.

Methodology in Plain English

The team works entirely from the fact that in circuit QED, a cavity mode can be displaced and then read out through a coupled qubit, giving direct access to the Wigner function at any chosen phase-space point; each such measurement is effectively a displaced parity measurement.

Protocol 1 (fidelity-based). Pick the pure state you intended to prepare. Its Wigner function, normalized by its total absolute value (the Wigner negativity), defines a probability distribution over phase space. Draw random points from that distribution, measure the actual state's Wigner function there, and combine the values with signs from the target Wigner function. The average of these samples equals the fidelity between the prepared state and the target. A concentration bound converts the number of samples and a chosen failure probability into an error bar. Subtracting that error from the fidelity and comparing with the target's stellar fidelity profile — the best fidelity achievable by any state of a given stellar rank — yields a certified rank. The comparison is conservative: because stellar fidelities come from non-convex optimization, the authors caution that the numerical values are not certified global optima, though single-mode cases keep the search space small.

Protocol 2 (few-points). Instead of random sampling, choose a few specific points: positive weights on local maxima of the target Wigner function, negative weights on local minima, with equal magnitudes. To claim a rank above k, the measured weighted sum must beat a threshold obtained by maximizing the witness expectation over all states of stellar rank ≤ k, which reduces to a largest-eigenvalue problem on a (k+1)×(k+1) matrix after optimizing over displacements, squeezing, and phases. For rotationally symmetric states, averaging the witness over phase gives tighter (lower) thresholds.

Validation. Rather than simulated noise, the authors use reconstructed density matrices from previously published superconducting 3D-cavity experiments and compute Wigner values and witness expectation values from that experimental data in post-processing.

Why This Matters

Impact on research. The stellar rank is an operationally meaningful non-Gaussianity monotone — it is non-increasing under post-selected Gaussian protocols and quantifies the added cost of classical simulation. This work makes it experimentally certifiable on the platform with the best current error-correction capabilities, and it does so without tomography, which scales unfavorably with system size and energy. It also sharpens the benchmark for how much non-Gaussianity has actually been demonstrated, moving from 6 to 8 for a GKP state.

Real-world applications (as motivated by the paper's framing):

• Benchmarking and quality control of bosonic quantum processors built on superconducting cavities.
• Certifying the resource content of bosonic error-correcting codes (binomial, cat, and GKP encodings of logical qudits).
• Validating claims of non-Gaussian resources in photonic, trapped-ion, and optomechanical platforms where the Wigner function is accessible.
• Providing resource-theoretic evidence in support of claims of exponential quantum computational advantage from bosonic architectures.

Industry relevance. Superconducting circuit QED is the leading hardware route for error-corrected quantum computing; a certification tool that runs directly on its native displaced-parity measurement, and needs only a handful of points in its cheapest form, is a practical verification primitive for hardware vendors and for third-party assessment of device performance.

Future Directions

• Removing the reliance on non-convex numerics: the thresholds from both the stellar fidelity profile and the few-points optimization come without a certificate, so rigorous global bounds (or certified outer approximations) remain open.
• Extending the framework to multi-mode states, where the optimization space grows and the authors' confidence in numerical optima is explicitly weaker than in the single-mode case.
• Generalizing beyond the specific target families studied — the few-points witnesses are described as "ad-hoc" per target, so automating the choice of number, location, and weights (the authors note numerical weight optimization may help) is a natural next step.
• Better handling of states with infinite stellar rank such as cat states, where stellar fidelities converge too slowly to 1 for the fidelity protocol to certify more than rank ≥ 1; the phase-averaged witness is offered as a route toward higher certified ranks for rotationally symmetric states.

Target Audience

Experimentalists in circuit QED and continuous-variable photonics who need to characterize non-Gaussian state preparation; quantum-information theorists working on resource theories, stellar representations, and bosonic codes; and research groups or hardware teams evaluating whether a given device has demonstrated genuine non-Gaussian quantum resources. Readers should be comfortable with Wigner functions, displacement and squeezing operators, and fidelity, but the two protocols are described concretely enough to be implemented by a reader who skips the optimization details.

Authors’ abstract

Quantifying non-Gaussianity is essential for assessing the computational power of bosonic quantum systems. We introduce a framework to witness and quantify non-Gaussianity from displaced parity measurements, and we directly apply it to experimental data obtained with superconducting cavities. Our method estimates fidelities with target pure states from the displaced parity measurements, enabling certification of non-Gaussianity via the stellar rank. We further introduce refined witnesses computed by measuring the value of the Wigner function at a few points, providing efficient certification tools outperforming existing ones. Applying these techniques to Fock, binomial, cat, and GKP experimental data, we rigorously certify non-Gaussian resources and establish a stellar rank of $8$ for a GKP state, setting a new benchmark across all quantum platforms. Our methods enable a rigorous quantification of non-Gaussian quantum resources directly applicable to superconducting devices.