Universal gates for protected superconducting qubits using optimal control

Andras Gyenis; Andrew A. Houck; Brian Baker; David Schuster; Jens Koch; Mohamed Abdelhafez; Pranav Mundada

REVIEW 4 major objections 5 minor 62 references

Universal gates for protected superconducting qubits using optimal control

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Optimal control can produce high-fidelity gates for protected superconducting qubits by using higher-lying states and, for the 0-pi qubit, randomizing offset charge during optimization to make pulses charge-insensitive.

desk verdict The offset-charge randomization trick is a genuine new idea, and the heavy-fluxonium gates are solid; the 0-pi numbers are conditional and the abstract overreaches. read the letter →

arxiv 1908.07637 v1 pith:2WIKASBN submitted 2019-08-20 quant-ph cond-mat.mes-hall

Mohamed Abdelhafez , Brian Baker , Andras Gyenis , Pranav Mundada , Andrew A. Houck , David Schuster , Jens Koch This is my paper · ORCID

classification quant-phcond-mat.mes-hall

keywords superconductingqubitsprotectedheavyfluxonium0-piqubitoptimalcontrolautomaticdifferentiationoffsetchargegatefidelity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Protected superconducting qubits such as heavy fluxonium and the 0-pi circuit store quantum information in computational states that live in separate potential wells, so ordinary microwave pulses cannot directly move population between them. This paper shows that quantum optimal control can nonetheless produce fast, high-fidelity universal gates by routing population through higher-lying delocalized states, temporarily lifting the protection for part of the pulse. The enabling tool is an automatic-differentiation optimizer that can simultaneously minimize gate infidelity, leakage into forbidden states, pulse roughness, and pulse power. For the 0-pi qubit, the key step is to randomize the offset charge $n_g$ at every optimizer iteration, steering the search toward pulses insensitive to charge fluctuations. If these pulses work on real devices, protected qubits—chosen for their long coherence—gain a practical route to the high-fidelity operations quantum error correction requires.

What carries the argument

The motor of the argument is a multi-objective cost functional minimized with automatic differentiation: the gate infidelity $C_1=1-\frac{1}{n^2}|\mathrm{Tr}(U_t^\dagger U_f)|^2$, a penalty $C_2$ for occupation of forbidden states, a penalty $C_3$ on pulse derivatives, and a penalty $C_4$ on pulse power. Because gradients are computed by automatic differentiation rather than hand-derived formulas, these targets can be added or modified flexibly. The drive enters dispersively through a resonator, producing an effective qubit Hamiltonian $V(t)=2g\omega_r v(t)\sum_{l,l'}\frac{\langle l|n_\varphi|l'\rangle}{(\epsilon_l-\epsilon_{l'})^2-\omega_r^2}|l\rangle\langle l'|$ for fluxonium, with $n_\theta$ replacing $n_\varphi$ for 0-pi. The decisive addition for 0-pi is to draw a fresh random offset charge $n_g$ uniformly from $[0,1)$ at every optimizer iteration, turning the gradient descent into stochastic gradient descent and converging to a pulse that works on average across all offset charges. Physically, the common enabler in both circuits is the set of higher-lying states that delocalize across the potential wells and carry population between computational states while the protection is momentarily lifted.

What would settle it

Build a 0-pi device with the paper's simulated parameters, run the optimized X, H, and T pulses while sweeping the offset charge $n_g$, and compare the measured average process fidelities to the closed-system values of 98.6%, 99.4%, and 99.95% and to the open-system lower bounds of 95.8%, 97.9%, and 99.7% (for X, H, and T respectively). If the measured fidelities fall clearly below those bounds, or if they vary strongly with $n_g$, the charge-randomization claim fails.

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Extended reading notes

Core claim

The central claim is that optimal control with automatic differentiation can synthesize realistic, smooth drive pulses that realize high-fidelity gates for protected superconducting qubits even though the computational states have exponentially suppressed transition matrix elements. For heavy fluxonium at $\Phi_{\text{ext}}=0.45\Phi_0$ with $E_C/h=0.5$ GHz, $E_L/h=0.25$ GHz, and $E_J/h=4.0$ GHz, optimized 60-ns pulses realize X, Hadamard, and T gates with closed-system fidelities of 99.94%, 99.93%, and 99.93%, and open-system fidelities of 99.66%, 99.60%, and 99.59%, together with a resonator-mediated controlled-Z gate at 99.4% closed and 99.0% open. For the 0-pi qubit, using the optimistic parameter set from [33], the same approach gives offset-charge-averaged closed-system fidelities of 98.6% for X, 99.4% for H, and 99.95% for T, with conservative worst-case open-system lower bounds of 95.8%, 97.9%, and 99.7% once zeta-mode shot noise, charge noise, and dielectric loss are included. The underlying mechanism is temporary occupation of delocalized higher levels: X and H gates move population out of the computational subspace, through states such as $|2\rangle,|3\rangle$ for fluxonium and $|13\rangle,|14\rangle$ for 0-pi, and back, while the T gate only makes a brief excursion to $|3\rangle$ to accumulate its phase.

Load-bearing premise

The 0-pi results hold only if the optimistic, not-yet-demonstrated circuit parameters ($E_L/h=E_C/h=40$ MHz, $E_J/h=10$ GHz, $E_{CJ}/h=20$ GHz) are physically realizable and if the noise model with 5% disorder in the zeta-mode coupling captures the dominant error sources.

Editorial extensions

If this is right

Heavy-fluxonium devices can be controlled by a complete gate set—X, H, T, and CZ—at 60 ns per gate with open-system fidelities above 99%, without changing the protected circuit design.
The 0-pi qubit, whose protection is stronger but whose ideal parameters are harder to reach, can be driven by microwave pulses that realize a universal single-qubit gate set with worst-case open fidelities of 95.8% or higher.
Penalizing forbidden-state occupation and pulse roughness keeps the optimized pulses smooth and bounded, so the predicted fidelities are not tied to idealized unbounded drives.
For 0-pi, zeta-mode shot-noise dephasing, offset-charge dephasing, and dielectric loss are the dominant open-system error channels, and the paper identifies active cooling or open-system optimization as the natural next step to push these fidelities higher.
The offset-charge randomization scheme directly removes the main obstacle that earlier 0-pi gate proposals faced—the strong $n_g$ dependence of matrix elements among high-lying states—without requiring charge control in the experiment.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same randomize-a-parameter-per-iteration trick could be applied to any qubit whose gate performance drifts with a slowly varying parameter such as flux bias, qubit frequency, or coupling strength; the cost would be a trade-off between peak fidelity and robustness.
Because the optimized gates deliberately occupy unprotected high-energy states, gate operation and qubit protection are in tension: the same mechanism that opens the gate window also exposes the system to relaxation and charge noise during the pulse, so even shorter gate times are unlikely to make fidelity improve monotonically.
If the optimistic 0-pi parameters cannot be realized in the lab, the 0-pi fidelities are best read as upper bounds for this method; the heavy-fluxonium results, which use demonstrated parameter values, are the more immediately actionable predictions.
An experiment on existing heavy-fluxonium hardware running the optimized 60-ns X, H, and T pulses described in the paper and measuring process fidelities would directly test whether the predicted roughly 0.3% gap between closed- and open-system fidelity is correct, or whether additional noise channels are missing.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the short version. The genuinely new thing in this paper is the offset-charge randomization during optimization: drawing a fresh ng each iteration steers the pulses toward solutions that work across the full offset-charge range. That is a clever, transferable technique and the main reason to read the paper. The heavy-fluxonium work is also a clean demonstration of optimal control on a protected qubit: >99.9% closed-system fidelities for X, H, T, a CZ gate at 99.4%, and a careful check that the resonator stays essentially unoccupied. The open-system numbers there, around 99.5% or better, are believable. The 0-pi section is more fragile than the abstract suggests. First, the abstract says closed-system fidelities are 99% or higher, but the X gate averages 98.6%. That's a straightforward overstatement. Second, the 0-pi simulations use the dispersively filtered drive of Eq. (8) without the resonator-occupation check the authors perform for heavy fluxonium. Given that the maximal drive amplitude reaches about 1.5 GHz, that omission is not minor. If the drive populates the resonator or the second-order Schrieffer-Wolff reduction breaks down, the optimized unitaries don't represent the physical gate. Third, the 0-pi parameters are explicitly called "optimistic." That's acceptable for a design study, but the noise-model lower bounds of 95.8% and 97.9% are not device-level predictions. Finally, no code, pulse data, or optimizer hyperparameters are given, so reproducing the 0-pi results would require guesswork. I checked the stress-test worry about the missing 0-pi dispersive validation, and it holds up. The paper's own conclusion lists zeta-mode coupling and offset charge as barriers, but it does not address this more basic model-validity question. Citations are fine: the authors cite the relevant experimental and theoretical work on fluxonium, 0-pi, and optimal control, and the main self-citation is to their own automatic-differentiation optimizer, which is appropriate. Who is this for? Researchers working on gates for protected superconducting qubits, and to a lesser extent optimal-control practitioners. It deserves a serious referee. I would send it to review with a recommendation for major revision: fix the abstract, add the missing 0-pi dispersive validation and parameter values, and release the pulse data or code.

Referee Report

4 major / 5 minor

Summary. The paper uses quantum optimal control with automatic differentiation to design microwave pulses for two protected superconducting qubits, heavy fluxonium and 0-pi, whose computational states have disjoint support. For heavy fluxonium, optimized 60 ns pulses realize X, H, and T gates with closed-system fidelities above 99.9% and open-system fidelities around 99.6%, and a two-qubit controlled-Z gate with 99.4% closed and 99.0% open fidelity. For the 0-pi qubit, the authors introduce a stochastic-gradient scheme in which the offset charge is randomized during optimization, yielding X, H, and T gates with average closed-system fidelities of 98.6%, 99.4%, and 99.95%, and conservative open-system lower bounds of 95.8%, 97.9%, and 99.7% when coupling to the zeta-mode is included. The paper argues that optimal control naturally exploits higher-lying delocalized states to overcome protection, and that including multiple cost targets yields realistic pulse shapes.

Significance. If the reported results hold, the paper provides a practical route toward gate sets for protected superconducting qubits, a problem that is important because disjoint-support protection suppresses exactly the matrix elements needed for direct gates. The heavy-fluxonium results appear internally consistent: the optimized pulses are characterized in both time and frequency domain, the resonator occupation is checked, and the open-system model includes flux noise and dielectric loss with clearly stated parameters. The idea of randomizing the offset charge during optimization to obtain charge-insensitive pulses is a useful and transferable methodological contribution. The 0-pi results, however, are less secure: they rely on an explicitly 'optimistic' parameter set and on a dispersive drive reduction whose validity is not verified for the large drive amplitudes used. The paper does not ship code or pulse data, and omits several optimization hyperparameters, which limits reproducibility. Overall, the central approach is sound and the heavy-fluxonium part is convincing; the 0-pi part needs additional validation before the reported fidelities can be considered predictive for experiments.

major comments (4)

[Section IV, Eq. (8)] The 0-pi optimization relies on the dispersively filtered drive Hamiltonian of Eq. (8) without the resonator-occupation check that is explicitly performed for heavy fluxonium in Section III.A. Section IV gives no resonator frequency, no coupling strength g, no verification that the drive leaves the resonator near vacuum, and no statement of the Hilbert-space truncation or leakage penalty C2. This is load-bearing because the optimized X and H pulses reach |v(t)| ~ 1.5 GHz, about five times the fluxonium amplitude, and because the protocol intentionally populates high-lying delocalized states whose transition frequencies may approach the resonator frequency. The second-order Schrieffer-Wolff reduction in Appendix B can fail if such transitions are near resonant. The authors should provide the 0-pi resonator parameters and a numerical check of <a†a> << 1 for the optimized pulses, ideally by comparing the effective model against the full generalized Jaynes-Cummings Hamiltonian for representative offset-charge values.
[Section IV, parameter set] The 0-pi gate fidelities are computed with the parameter set EL/h = EC/h = 40 MHz, EJ/h = 10 GHz, and ECJ/h = 20 GHz, which the paper itself calls 'optimistic' and which has not been demonstrated experimentally. Consequently, the reported lower bounds of 95.8% (X) and 97.9% (H) do not transfer to currently realizable devices. The manuscript should either present these results explicitly as a design study for future devices or include a sensitivity analysis showing how the fidelities degrade when realistic parameter variations and disorder beyond the 5% level are included.
[Sections II-IV, reproducibility] The central results are numerical, but the paper does not provide the optimized pulse data, the cost-functional weights (C1, C2, C3, C4), the optimizer hyperparameters (learning rate, iteration count, batch size for the offset-charge randomization), or the Hilbert-space cutoffs used for the fluxonium and 0-pi simulations. Without these, the reported fidelities cannot be reproduced or checked by independent groups. At minimum, the authors should list all hyperparameters and supply the optimized pulse arrays as supplementary material or a public repository.
[Abstract and Fig. 6] The abstract states that 'Closed-system fidelities obtained are 99% or higher,' but the 0-pi X gate has an average closed-system fidelity of 98.6% (Fig. 6 and Section IV). This is a direct quantitative inconsistency in the abstract. The sentence should be qualified, for example by noting that most gates exceed 99% while the charge-randomized X gate reaches only 98.6%.

minor comments (5)

[Eq. (9) and text around it] The symbol for the low-frequency cutoff appears as 'ωir' in the text and 'ω_ir' in the equation; please unify the notation.
[Fig. 2 caption] The caption reports a fidelity of '99.933%' for the Hadamard gate while the text gives '99.93%'; these values should be made consistent.
[Section IV, Eq. (8) usage] When the authors state that nφ is replaced by nθ for the 0-pi qubit, they should explicitly rewrite Eq. (8) with the 0-pi eigenbasis and charge operator so that the notation is self-contained.
[Section IV, paragraph after Fig. 6] The statement that states |4⟩ and |5⟩ do not contribute 'due to the lack of connecting matrix elements' would benefit from a brief symmetry explanation, since the reader might otherwise wonder whether this is a numerical artifact.
[Section III.A, cost function] Table I introduces the leakage penalty C2, but the text describing the fluxonium optimizations mentions only C1, C3, and C4. The authors should clarify whether C2 was used and, if not, why leakage into the truncated Hilbert space is not a concern.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the reported fidelities are computed from explicitly optimized pulses under stated Hamiltonians and noise models, not from fitted parameters or self-citation chains.

full rationale

The paper's central claims are numerical gate fidelities obtained from optimal-control simulations. The optimizer minimizes the infidelity C1 subject to additional penalties, and the reported closed-system fidelities are evaluations of that same optimized objective. This is the standard use of optimal control, not a hidden circularity: no parameter is fitted to a subset of data and then 'predicted' for a closely related quantity. For the 0-pi qubit, the offset-charge randomization during optimization directly targets the average fidelity over ng, and the paper additionally reports fidelity-versus-ng plots and open-system lower bounds that include the zeta-mode and noise channels; these are independent checks of the found pulses rather than re-statements of the training objective. The self-citations to Ref. [22] (the automatic-differentiation optimizer) and Refs. [33,34] (device parameters and zeta-mode coupling model) are inputs and tools with stated assumptions, not invoked as an external uniqueness theorem or as a substitute for the derivation. The 'optimistic' parameter set is explicitly acknowledged as such, and the dispersive drive Hamiltonian of Eq. (8) is derived in Appendix B via a Schrieffer-Wolff transformation; its validity is a modeling assumption, separately verified for heavy fluxonium, not a circular reduction. No equation in the paper is defined in terms of the quantity it is claimed to predict, and no fitted parameter is renamed as a prediction. The derivation chain is therefore self-contained for the purpose of the claims made.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The central numerical results rest on standard open-quantum-system modeling, perturbation theory for the dispersive drive, finite Hilbert-space truncation, and chosen device and noise parameters. No new physical entities are introduced, but several hand-chosen parameters (cost weights, noise constants, and the optimistic 0-pi parameters) are load-bearing for the reported fidelities.

free parameters (6)

Cost functional weights for C1, C2, C3, C4 = not stated
The optimized pulse shapes and fidelities depend on these weights; Section IV says weights were carefully tuned but gives no values, so the exact pulses cannot be reproduced.
Dielectric loss rate constant Gamma = set by 1/gamma_02 = 50 microseconds
Used to set depolarization rates for both qubits; the value comes from a personal communication and dielectric loss theory (Ref. 53), not from a measurement in this paper.
Flux-noise amplitude A_Phi = 1 micro-Phi_0
Input from Ref. 52 for the dephasing rate in Eq. (9); directly affects the heavy-fluxonium open-system fidelities.
0-pi circuit parameters EL, EC, EJ, ECJ = EL/h = EC/h = 40 MHz, EJ/h = 10 GHz, ECJ/h = 20 GHz
Chosen as an "optimistic" parameter set from Groszkowski et al. [33]; the 0-pi gate fidelities and protection properties depend entirely on this choice.
Optimizer hyperparameters and Hilbert-space cutoff = not stated
Learning rate, iteration count, minimization tolerance, and level truncation are not reported; these affect convergence and the reported fidelities.
Zeta-mode disorder strengths dL, dC = 5% each
Used in the 0-pi open-system lower bounds (Eqs. 16-17); the lower-bound fidelities scale with this assumption.

assumptions (4)

domain assumption Lindblad master equation with Markovian rates is adequate for open-system fidelity estimates
Used in Eq. (3) and Eq. (11); the authors explicitly note that 1/f flux noise is non-Markovian and that the Lindblad treatment is a compromise accepted for a fidelity-loss bound (Section III A).
domain assumption Dispersive Schrieffer-Wolff transformation for the filtered drive and qubit-qubit coupling is valid
Appendix B assumes lambda_ll' = g_ll' / Delta_ll' is much less than 1 and small photon number to derive Eqs. (8) and (B5); violations would change the control Hamiltonian.
domain assumption Finite-dimensional truncation plus the leakage penalty captures the relevant dynamics
Cost term C2 suppresses occupation of forbidden states, but the exact truncation level is not given; the results assume negligible population above the cutoff.
domain assumption Weak coupling to the zeta-mode with 5% disorder represents dominant circuit disorder for 0-pi
Section IV says the authors verify that weak coupling to the zeta-mode does not significantly reduce gate fidelities; the lower bounds depend on this model and on the stated disorder level.

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Pith. "Pith review of Universal gates for protected superconducting qubits using optimal control." pith.science (2026). https://pith.science/paper/2WIKASBN

@misc{pith2026190807637,
  author       = {Pith},
  title        = {Pith review of: Universal gates for protected superconducting qubits using optimal control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2WIKASBN}},
  note         = {Machine review of arXiv:1908.07637}
}

abstract

We employ quantum optimal control theory to realize quantum gates for two protected superconducting circuits: the heavy-fluxonium qubit and the 0-$\pi$ qubit. Utilizing automatic differentiation facilitates the simultaneous inclusion of multiple optimization targets, allowing one to obtain high-fidelity gates with realistic pulse shapes. For both qubits, disjoint support of low-lying wave functions prevents direct population transfer between the computational-basis states. Instead, optimal control favors dynamics involving higher-lying levels, effectively lifting the protection for a fraction of the gate duration. For the 0-$\pi$ qubit, offset-charge dependence of matrix elements among higher levels poses an additional challenge for gate protocols. To mitigate this issue, we randomize the offset charge during the optimization process, steering the system towards pulse shapes insensitive to charge variations. Closed-system fidelities obtained are 99% or higher, and show slight reductions in open-system simulations.

Figures

Figures reproduced from arXiv: 1908.07637 by the authors.

**Figure 2.** FIG. 2. High-fidelity single-qubit gates for heavy fluxonium. (a) Optimized pulse shape [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Time-evolution of state populations for 60 ns high-fidelity single-qubit gates. (a) The time-evolution of states involved in the X gate [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Controlled-Z gate for two heavy-fluxonium qubits with a gate time of 60 ns [ [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. Spectrum and matrix elements of the [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: FIG. 6. Optimized pulses for [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

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Reviewed August 14, 2026 · model on record in the stance chip above.

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