REVIEW 3 major objections 3 minor 1 cited by
Exponentially robust non-Clifford gate in a driven-dissipative circuit
T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read This paper proposes a protected non-Clifford $\sqrt{T}$ gate on a self-correcting GKP qubit in a driven-dissipative superconducting circuit, using a quartic flux potential whose phase-revival dynamics rotate the logical state by $\pi/8$…
desk verdict A well-argued protocol for a protected physical sqrt(T) gate, undermined only by an overbroad exponential-robustness claim for static parameter errors. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing object is the phase-revival identity $N^4 \pmod{16} = N \pmod{2}$, applied to the low-energy flux-well eigenstates $|N,\nu\rangle$ of the GKP circuit. It converts the quartic part of the dynamical phase into a logical rotation by $\pi/8$ at the revival time $t_4$, just as the quadratic part $N^2 \pmod{4} = N \pmod{2}$ produces the protected $S^\dagger$ gate. Supporting machinery includes the dissipative stabilization segments and cleanup steps of the underlying GKP architecture, adiabatic elimination of higher circuit modes to obtain the effective potential $V_{\rm eff}(\varphi)$, and dynamic decoupling with Uhrig pulse sequences during the $\varphi^4$ segment to suppress $1/f$ flux noise.
What would settle it
A decisive test would be to build the Fig. 1(a) circuit with parameter set 3, initialize a GKP logical $|X\rangle$ state, run the $\varphi^4$/ $\varphi^2$/cleanup protocol, and measure the logical Bloch vector as a function of gate time and of deliberately injected timing noise; the central claim fails if infidelity is not exponentially suppressed as $t_{\rm gate}$ increases or if it exceeds $10^{-3}$ for roughly 10 ps timing uncertainty and 1% parameter mistargeting. A purely theoretical falsifier is to compute the exact multi-mode spectrum including ancillary junction capacitances and show that higher-mode hybridization or thermal activation breaks the $\varphi^4$ revival condition.
Extended reading notes
Core claim
Adding an effective quartic flux potential $(\epsilon_4/\varphi_0^4)\varphi^4$ to the dissipatively stabilized GKP qubit lets one perform a protected $\sqrt{T}$ gate. Low-lying eigenstates localized in flux wells labeled by integer $N$ acquire dynamical phases $e^{-iN^4\epsilon_4 t/\hbar}$, and because $N^4 \pmod{16} = N \pmod{2}$, all like-parity well amplitudes re-align at the quartic revival time $t_4 = \pi\hbar/(8|\epsilon_4|)$, while even- and odd-parity wells differ in phase by $\pi/8$. This relative phase is the $\sqrt{T}$ rotation on the logical qubit, and it survives weak perturbations because any smooth deformation of the protocol keeps the state confined in the GKP code space. In the full protocol the quartic segment is followed by a quadratic $\varphi^2$ segment and cleanup stabilization steps, and a family of concrete circuits with three ancillary Josephson junctions is shown by numerical search to realize the required nearly pure $\varphi^4$ potential.
Load-bearing premise
The load-bearing premise is that a fast switch can suppress Josephson couplings on 10-50 ps timescales and that the ancillary junctions can be made with capacitances below roughly 0.1 fF, so the higher circuit modes stay frozen out and the potential is nearly pure quartic; the paper itself flags the switch as challenging and says it is not immediately clear that capacitance-per-area scaling holds at that size.
Editorial extensions
If this is right
- Together with the protected Clifford gates of the underlying GKP architecture, the $\sqrt{T}$ gate gives a universal set of logic gates on a single physical qubit, all with exponential suppression of control and device-error infidelities.
- The need for magic-state distillation could be reduced or eliminated for single-qubit non-Clifford operations, since the device directly produces $\sqrt{T}$ rotations at the physical level.
- Because timing tolerance scales with the LC inductance, moving to larger hyperinductances proportionally relaxes the picosecond timing requirement.
- Flux noise of realistic $1/f$ strength can be pushed below $10^{-3}$ infidelity with just a few Uhrig dynamic-decoupling steps during the quartic segment.
- The large separation between $t_4$ and the revival time suggests the ancillary circuit can be left permanently connected, treating its high-order potential as correctable phase-space-local noise.
Reading between the lines
- If the phase-revival mechanism generalizes to other polynomial potentials, the same architecture could implement other non-Clifford angles, and possibly two-qubit entangling gates, giving universal self-correcting logic without distillation.
- A decisive experimental extension would be to fabricate a Josephson-junction circuit with an effective quartic potential, prepare a GKP state, run the pulse sequence, and measure the logical Bloch vector; an infidelity curve decreasing exponentially in gate time would directly confirm the mechanism.
- The paper's reliance on sub-$0.1$ fF junction capacitances could be bypassed by alternative high-impedance or kinetic-inductance implementations preserving the quartic revival idea, if conventional junction scaling fails at that size.
- A useful theoretical check is to compute exact multi-mode eigenstates without the adiabatic elimination, to see whether the 10-50 ps switch and $0.1$ fF capacitances are necessary or merely sufficient for the exponential protection.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proposes a protocol for a protected non-Clifford sqrt(T) gate on a dissipatively stabilized GKP qubit. The key idea is to supplement the LC/Josephson Hamiltonian with a quartic flux potential, so that the N^4 well-index phases revive at t4 = pi hbar/(8|V4|) and differ by pi/8 between even and odd wells. A phi^2 segment and cleanup steps are then used to restore the code space. The authors derive conditions on the effective potential, give a circuit realization with ancillary Josephson junctions, and present stochastic Schrodinger equation simulations of infidelity versus gate time, timing errors, parameter mistargeting, and flux noise.
Significance. The protocol addresses a real bottleneck: a native protected non-Clifford gate for self-correcting GKP qubits. Its central revival identity (Eq. 4) is parameter-free, and the Supplemental Material contains explicit bounds for eigenstate modification and residual potential coefficients. The numerics are extensive, with trajectory counts and convergence checks. However, the headline claim of exponential robustness under device imperfections is not yet established for the most basic imperfection, a relative error in the quartic coefficient or segment timing, and the flux-noise simulation uses an acknowledged approximation. With revision, the protocol could still be valuable; as written, the theoretical support for the strongest claim is incomplete.
major comments (3)
- [Protected sqrt(T) gates via phi^4 potential; Fig. 2(c)] The exponential robustness claim is not established for static mistargeting of the quartic coefficient, which is equivalent to mistiming of the phi^4 segment. The revival time is set to t4 = pi hbar/(8|V4|). If V4 is replaced by (1+u)V4, the accumulated phase in well N is -i(pi/8)(1+u)N^4, so the deviation from the revival condition is an N-dependent phase error of order (pi u/8)N^4 modulo 2pi. Since the GKP envelope has support on wells with N up to O(1/lambda0), this is not a logical Z rotation but a direction out of the code space. The SM bounds (S30) and (S40) control eigenstate distortion and residual V_k with k != 4, but they do not bound this gate-angle error; the cleanup steps are asserted to refocus it but no scaling argument is provided. Fig. 2(c) is consistent with a non-exponential sensitivity: 1-F reaches about 10^-3 at u ~ 1%, and the text says stability improves for smaller J (larger lambda0), the opposite trend from the claimed exponential protection as lambda0 -> 0. Please provide an analytic bound on the residual logical error after cleanup in terms of u and lambda0, or revise the statement that gate infidelity is exponentially suppressed in control and device imperfections.
- [SM III.B.1; Fig. 2(d)] The flux-noise simulation is not exact for the stabilizer and phi^4 segments. The method evolves the noise-free system and then applies exp(-i alpha(t1,t2) phi), which in the co-rotating frame discards the term -q/(C hbar) alpha(t1,t) in the Hamiltonian and the corresponding shift of the jump operator in Eqs. (S54)-(S55). The SM states that this method is not exact and heuristically expects the resistor to correct the omitted charge diffusion. Because the suppression of 1/f flux noise is one of the central resilience claims, this approximation needs a quantitative error estimate or a benchmark against exact integration for representative parameters.
- [SM Fig. S1] The circuit realization requires ancillary junction capacitances below roughly 0.1 fF, and SM Fig. S1 shows that the higher-mode gap closes as C_junc approaches this value. The SM itself says it is not immediately clear that the capacitance-per-area scaling holds for such small junctions. This is an acknowledged feasibility limitation, but since the protocol's promise rests on the physical circuit, the manuscript should either present additional evidence for achievable sub-0.1 fF junctions or explicitly frame the device as a long-term target rather than a near-term one.
minor comments (3)
- [Introduction] The phrase 'analogous the the ones described' in the paragraph introducing the phi^4 protocol should read 'analogous to those described'.
- [Throughout] The notation for the quartic coefficient is inconsistent: both epsilon_4 and eps_4 appear. Please choose one and use it consistently.
- [Numerical Results] The protocol steps are described only in prose; a compact table or timing diagram listing the durations of the phi^4, phi^2, stabilizer, and free segments would substantially improve reproducibility.
Circularity Check
No circularity found: the sqrt(T) gate follows from the modular revival identity and the circuit parameters are design inputs rather than fitted predictions.
full rationale
The paper's central derivation is self-contained. The sqrt(T) gate arises from the modular arithmetic identity N^4 (mod 16) = N (mod 2), combined with the definition t4 = pi hbar / (8|epsilon4|) for the quartic potential. This is a parameter-free phase-revival mechanism, not a fit to the target gate fidelity. The circuit parameters in Table I are obtained by numerical search for an approximately pure quartic effective potential; they are design inputs, not parameters fitted to reproduce the predicted logical rotation. The self-citations to Ref. [35] supply the pre-existing GKP stabilization architecture and the phase-space-locality argument for code-space confinement, which are inputs to the protocol rather than the protocol's predicted output. The paper's own simulations (Fig. 2) independently probe sensitivity to timing uncertainty, parameter mistargeting, and flux noise. The SM's caveat about junction capacitance scaling is a feasibility limitation, not a circular reduction. The static parameter-error concern raised by the skeptic is a substantive correctness/quantitative issue, but it does not make the derivation equivalent to its inputs by construction. Therefore no significant circularity is present.
Assumptions & free parameters
free parameters (2)
- quartic coefficient epsilon_4 =
negative, magnitude sets the gate time t4 = pi*hbar/(8|epsilon_4|)
- ancillary junction parameters (L_i, J'_i, node positions) =
five parameter sets in Table I, e.g. Set 3: L1=2.40 uH, L2=.0556 uH, L3=.0436 uH, J'_1/h=.843 GHz, J'_2/h=-.289 GHz…
assumptions (6)
- domain assumption The GKP codespace, logical operators, dissipative stabilization, and exponential robustness of the qubit from Ref. [35] are correct and transfer to this protocol.
- ad hoc to paper For small epsilon_4 and lambda_0, the low-lying eigenstates of H4 coincide with the well states |N,nu> of HLCJ, with energies EN,nu approximately nu epsilon_0 + N^4 epsilon_4.
- domain assumption Dissipation from the resistor preserves inter-well coherence up to exponentially suppressed corrections.
- domain assumption The higher circuit modes of the ancillary junctions can be adiabatically eliminated because their frequencies are much larger than the qubit and temperature scales.
- domain assumption Any phase-space local perturbation that keeps the state confined in the codespace leaves the logical state unchanged.
- standard math The universal Lindblad equation and stochastic Schrodinger equation provide an accurate open-system model for these circuits.
Cite this review
Pith. "Pith review of Exponentially robust non-Clifford gate in a driven-dissipative circuit." pith.science (2026). https://pith.science/paper/JTPBF2AS
@misc{pith2026250719713,
author = {Pith},
title = {Pith review of: Exponentially robust non-Clifford gate in a driven-dissipative circuit},
year = {2026},
howpublished = {\url{https://pith.science/paper/JTPBF2AS}},
note = {Machine review of arXiv:2507.19713}
}
abstract
Recent work (Nathan et al, arXiv:2405.05671) proposed an architecture for a dissipatively stabilized GKP qubit, and protocols for protected Clifford gates. Here we propose a protocol for a protected non-Clifford $\sqrt{T}$ gate at the physical qubit level, based on the inclusion of a quartic flux potential generated by ancillary Josephson junctions. We show that such a gate is topologically robust with exponentially suppressed infidelity from control or device imperfections, and operates on microsecond timescales for GHz resonators. We analyze the resilience of the protocol to noise, imperfect control, and imperfect targeting of circuit parameters.
Figures
Forward citations
Cited by 1 Pith paper
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Protected measurements for protected superconducting qubits
Protected Z and X measurements of the 0-pi qubit are proposed, with exponentially suppressed errors via GKP-state encoding and charge-parity mapping.
Reference graph
Works this paper leans on
-
[1]
D. Gottesman, The heisenberg representation of quan- tum computers (1998), arXiv:quant-ph/9807006 [quant- ph]
arXiv 1998
-
[2]
Aaronson and D
S. Aaronson and D. Gottesman, Improved simulation of stabilizer circuits, Phys. Rev. A 70, 052328 (2004)
2004
-
[3]
Kliuchnikov, D
V. Kliuchnikov, D. Maslov, and M. M., Fast and efficient exact synthesis of single qubit unitaries generated by clif- ford and t gates, Quantum Inf. Comput. 13, 607–630 (2013)
2013
-
[4]
Illustrating this, in Fig
In the Supple- mentary Material (SM) [58], we estimate this minimal gate time to be of order ∼ 10/fLC. Illustrating this, in Fig. 2(a) we plot gate infidelity (defined below) ver- sus gate time with fLC = 0.82 GHz (corresponding to experimentally achievable L = 2.5µH [39, 43]) and for several values of J, in the absence of any noise. Our sim- ulation demons...
-
[5]
S. Forest, D. Gosset, V. Kliuchnikov, and D. McKinnon, Exact synthesis of single-qubit unitaries over Clifford-cyclotomic gate sets, Journal of Mathematical Physics 56, 082201 (2015), https://pubs.aip.org/aip/jmp/article- pdf/doi/10.1063/1.4927100/15804902/082201 1 online.pdf
-
[6]
Kitaev, Quantum computations: algorithms and error correction, RUSSIAN MATHEMATICAL SUR VEYS52, 1191 (1997)
A. Kitaev, Quantum computations: algorithms and error correction, RUSSIAN MATHEMATICAL SUR VEYS52, 1191 (1997)
1997
-
[7]
M. A. Dawson, Christopher M. and Nielsen, The solovay- kitaev algorithm, Quantum Information and Computa- tion 6, 81 (2006)
2006
-
[8]
Eastin and E
B. Eastin and E. Knill, Restrictions on transversal en- coded quantum gate sets, Phys. Rev. Lett. 102, 110502 (2009)
2009
Show all 100 references
-
[9]
Bravyi and A
S. Bravyi and A. Kitaev, Universal quantum computa- tion with ideal clifford gates and noisy ancillas, Phys. Rev. A 71, 022316 (2005)
2005
-
[10]
A. G. Fowler and C. Gidney, Low overhead quantum com- putation using lattice surgery (2019), arXiv:1808.06709 [quant-ph]
2019 arXiv
-
[12]
Bravyi and J
S. Bravyi and J. Haah, Magic-state distillation with low overhead, Phys. Rev. A 86, 052329 (2012)
2012
-
[13]
A. G. Fowler, S. J. Devitt, and C. Jones, Surface code implementation of block code state distillation, Scientifi c Reports 3, 1939 (2013)
2013
-
[14]
A. M. Meier, B. Eastin, and E. Knill, Magic-state distil- lation with the four-qubit code (2012), arXiv:1204.4221 [quant-ph]
2012 arXiv
-
[15]
Jones, Multilevel distillation of magic states for qu an- tum computing, Phys
C. Jones, Multilevel distillation of magic states for qu an- tum computing, Phys. Rev. A 87, 042305 (2013)
2013
-
[16]
Duclos-Cianci and K
G. Duclos-Cianci and K. M. Svore, Distillation of nonsta- bilizer states for universal quantum computation, Phys. Rev. A 88, 042325 (2013)
2013
-
[17]
Duclos-Cianci and D
G. Duclos-Cianci and D. Poulin, Reducing the quantum- computing overhead with complex gate distillation, Phys. Rev. A 91, 042315 (2015)
2015
-
[18]
E. T. Campbell and M. Howard, Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost, Phys. Rev. A 95, 022316 (2017)
2017
-
[19]
O’Gorman and E
J. O’Gorman and E. T. Campbell, Quantum computa- tion with realistic magic-state factories, Phys. Rev. A 95, 032338 (2017)
2017
-
[20]
Haah and M
J. Haah and M. B. Hastings, Codes and Protocols for Distilling T , controlled- S, and Toffoli Gates, Quantum 2, 71 (2018)
2018
-
[21]
E. T. Campbell and M. Howard, Magic state parity- checker with pre-distilled components, Quantum 2, 56 (2018)
2018
-
[22]
Gidney and A
C. Gidney and A. G. Fowler, Efficient magic state fac- tories with a catalyzed |CCZ ⟩ to 2 |T ⟩ transformation, Quantum 3, 135 (2019)
2019
-
[23]
Litinski, Magic State Distillation: Not as Costly as You Think, Quantum 3, 205 (2019)
D. Litinski, Magic State Distillation: Not as Costly as You Think, Quantum 3, 205 (2019)
2019
-
[24]
Babbush, C
R. Babbush, C. Gidney, D. W. Berry, N. Wiebe, J. Mc- Clean, A. Paler, A. Fowler, and H. Neven, Encoding elec- tronic spectra in quantum circuits with linear t complex- 6 ity, Phys. Rev. X 8, 041015 (2018)
2018
-
[25]
Calderbank, E
A. Calderbank, E. Rains, P. Shor, and N. Sloane, Quan- tum error correction via codes over gf(4), IEEE Transac- tions on Information Theory 44, 1369 (1998)
1998
-
[26]
Gottesman, Theory of fault-tolerant quantum compu- tation, Phys
D. Gottesman, Theory of fault-tolerant quantum compu- tation, Phys. Rev. A 57, 127 (1998)
1998
-
[27]
Jones, P
C. Jones, P. Brooks, and J. Harrington, Gauge color codes in two dimensions, Phys. Rev. A 93, 052332 (2016)
2016
-
[28]
Bravyi and A
S. Bravyi and A. Cross, Doubled color codes (2015), arXiv:1509.03239 [quant-ph]
2015 arXiv
-
[29]
Jochym-O’Connor and S
T. Jochym-O’Connor and S. D. Bartlett, Stacked codes: Universal fault-tolerant quantum computation in a two- dimensional layout, Phys. Rev. A 93, 022323 (2016)
2016
-
[30]
Bombin, 2d quantum computation with 3d topological codes (2018), arXiv:1810.09571 [quant-ph]
H. Bombin, 2d quantum computation with 3d topological codes (2018), arXiv:1810.09571 [quant-ph]
2018 arXiv
-
[31]
Chamberland and A
C. Chamberland and A. W. Cross, Fault-tolerant magic state preparation with flag qubits, Quantum 3, 143 (2019)
2019
-
[32]
B. J. Brown, A fault-tolerant non-clifford gate for the surface code in two dimen- sions, Science Advances 6, eaay4929 (2020), https://www.science.org/doi/pdf/10.1126/sciadv.aay4929
2020 doi
-
[33]
G. H. Low, V. Kliuchnikov, and L. Schaeffer, Trading T gates for dirty qubits in state preparation and unitary synthesis, Quantum 8, 1375 (2024)
2024
-
[34]
Lachance-Quirion, M.-A
D. Lachance-Quirion, M.-A. Lemonde, J. O. Simoneau, L. St-Jean, P. Lemieux, S. Turcotte, W. Wright, A. Lacroix, J. Fr´ echette-Viens, R. Shillito, F. Hopf- mueller, M. Tremblay, N. E. Frattini, J. Camirand Le- myre, and P. St-Jean, Autonomous quantum error correc- tion of gott...
2024
-
[35]
Sellem, A
L.-A. Sellem, A. Sarlette, Z. Leghtas, M. Mirrahimi, P. Rouchon, and P. Campagne-Ibarcq, Dissipative pro- tection of a gkp qubit in a high-impedance superconduct- ing circuit driven by a microwave frequency comb, Phys. Rev. X 15, 011011 (2025)
2025
-
[36]
Nathan, L
F. Nathan, L. O’Brien, K. Noh, M. H. Matheny, A. L. Grimsmo, L. Jiang, and G. Refael, Self-correcting gkp qubit and gates in a driven-dissipative circuit (2024), arXiv:2405.05671 [cond-mat.mes-hall]
2024 arXiv
-
[37]
Geier and F
M. Geier and F. Nathan, Self-correcting gkp qubit in a superconducting circuit with an oscillating voltage bias (2024), arXiv:2412.03650 [quant-ph]
2024 arXiv
-
[38]
Gottesman, A
D. Gottesman, A. Kitaev, and J. Preskill, Encoding a qubit in an oscillator, Phys. Rev. A 64, 012310 (2001)
2001
-
[39]
Campagne-Ibarcq, A
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, and M. H. Devoret, Quantum error correction of a qubit encoded in grid states of an oscillator, Nature 584...
2020
-
[40]
I. V. Pechenezhskiy, R. A. Mencia, L. B. Nguyen, Y.- H. Lin, and V. E. Manucharyan, The superconducting quasicharge qubit, Nature 585, 368 (2020)
2020
-
[41]
T. B. Brennan de Neeve, Thanh-Long Nguyen and J. P. Home, Error correction of a logical grid state qubit by dissipative pumping, Nature Physics 18, 296–300 (2022)
2022
-
[42]
Eickbusch, V
A. Eickbusch, V. Sivak, A. Z. Ding, S. S. Elder, S. R. Jha, J. Venkatraman, B. Royer, S. M. Girvin, R. J. Schoelkopf, and M. H. Devoret, Fast universal control of an oscillator with weak dispersive coupling to a qubit, Nature Physics 18, 1464–1469 (2022)
2022
-
[43]
V. V. Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsiout- sios, S. Ganjam, A. Miano, B. L. Brock, A. Z. Ding, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, and M. H. De- voret, Real-time quantum error correction beyond break- even, Nature 616, 50–55 (2023)
2023
-
[44]
Manset, J
P. Manset, J. Palomo, A. Schmitt, K. Gerashchenko, R. Rousseau, H. Patange, P. Abgrall, E. Flurin, S. Del´ eglise, T. Jacqmin, and L. Balembois, Hyperin- ductance based on stacked josephson junctions (2025), arXiv:2505.02764 [quant-ph]
2025 arXiv
-
[45]
Nathan and M
F. Nathan and M. S. Rudner, Universal Lindblad equa- tion for open quantum systems, Phys. Rev. B 102, 115109 (2020)
2020
-
[46]
Breuer, F
H.-P. Breuer, F. Petruccione, H.-P. Breuer, and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, Oxford, New York, 2007)
2007
-
[47]
C. W. Gardiner and P. Zoller, Quantum Noise (Springer Berlin, Heidelberg, 2004)
2004
-
[48]
Note that we are able to describe the state of our system with state vectors—even in the presence of dissipation— thanks to the SSE formalism
-
[49]
This is the principal mode of decoherence when λ0 ≪ 1, since—crucially—the resistor-induced dissipation pre- serves inter-well coherence up to exponentially sup- pressed corrections [35]
-
[50]
J. P. Barnes and W. S. Warren, Automatic Quantum Er- ror Correction, Physical Review Letters 85, 856 (2000), publisher: American Physical Society
2000
-
[51]
Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303, 2 (2003)
A. Kitaev, Fault-tolerant quantum computation by anyons, Annals of Physics 303, 2 (2003)
2003
-
[52]
B. M. Terhal, Quantum error correction for quantum memories, Rev. Mod. Phys. 87, 307 (2015)
2015
-
[53]
B. J. Brown, D. Loss, J. K. Pachos, C. N. Self, and J. R. Wootton, Quantum memories at finite temperature, Rev. Mod. Phys. 88, 045005 (2016)
2016
-
[54]
Liu and S
Y.-J. Liu and S. Lieu, Dissipative phase transitions and passive error correction, Phys. Rev. A 109, 022422 (2024)
2024
-
[55]
The latter equality follows using the condition √ L/C = h/2e2
-
[56]
Recall that σz|N, ν⟩ = ( −1)N |N, ν⟩, implying (−i)N mod 2 |N, ν⟩ = e−i π 2 (σz −1)|N, ν⟩
-
[57]
As with the S† gate, corrections from the finite modifi- cations do not take the system out of the code subspace when ϵ4 and λ0 are small enough
-
[58]
Technically speaking, we realize a √ T gate for ϵ4 < 0; ϵ4 > 0 realizes a √ T † gate
-
[59]
See Supplemental Material at [URL TBD] for analysis of the circuit Hamiltonian, discussion of adiabtic elimi- nation of additional resonator modes, and details about the inclusion of noise in the simulations
-
[60]
C. W. J. Beenakker, Universal limit of critical-current fluctuations in mesoscopic josephson junctions, Phys. Rev. Lett. 67, 3836 (1991)
1991
-
[61]
Furusaki, H
A. Furusaki, H. Takayanagi, and M. Tsukada, Joseph- son effect of the superconducting quantum point contact, Phys. Rev. B 45, 10563 (1992)
1992
-
[62]
J. P. Cleuziou, W. Wernsdorfer, V. Bouchiat, T. On- dar¸ cuhu, and M. Monthioux, Carbon nanotube supercon- ducting quantum interference device, Nature Nanotech- nology 1, 53 (2006)
2006
-
[63]
M. L. Della Rocca, M. Chauvin, B. Huard, H. Pothier, D. Esteve, and C. Urbina, Measurement of the current- 7 phase relation of superconducting atomic contacts, Phys. Rev. Lett. 99, 127005 (2007)
2007
-
[64]
Sochnikov, L
I. Sochnikov, L. Maier, C. A. Watson, J. R. Kirtley, C. Gould, G. Tkachov, E. M. Hankiewicz, C. Br¨ une, H. Buhmann, L. W. Molenkamp, and K. A. Moler, Non- sinusoidal current-phase relationship in josephson junc- tions from the 3d topological insulator hgte, Phys. Rev. Lett. 1...
2015
-
[65]
C. D. English, D. R. Hamilton, C. Chialvo, I. C. Moraru, N. Mason, and D. J. Van Harlingen, Observation of non- sinusoidal current-phase relation in graphene josephson junctions, Phys. Rev. B 94, 115435 (2016)
2016
-
[66]
D. J. van Woerkom, A. Proutski, B. van Heck, D. Bouman, J. I. V¨ ayrynen, L. I. Glazman, P. Krogstrup, J. Nyg ˚ ard, L. P. Kouwenhoven, and A. Geresdi, Mi- crowave spectroscopy of spinful andreev bound states in ballistic semiconductor josephson junctions, Nature Physics 13, 8...
2017
-
[67]
E. M. Spanton, M. Deng, S. Vaitiek˙ enas, P. Krogstrup, J. Nyg ˚ ard, C. M. Marcus, and K. A. Moler, Cur- rent–phase relations of few-mode inas nanowire josephson junctions, Nature Physics 13, 1177 (2017)
2017
-
[68]
Kringhøj, L
A. Kringhøj, L. Casparis, M. Hell, T. W. Larsen, F. Kuemmeth, M. Leijnse, K. Flensberg, P. Krogstrup, J. Nyg ˚ ard, K. D. Petersson, and C. M. Marcus, Anhar- monicity of a superconducting qubit with a few-mode josephson junction, Phys. Rev. B 97, 060508 (2018)
2018
-
[69]
Leblanc, C
A. Leblanc, C. Tangchingchai, Z. S. Momtaz, E. Kiyooka, J.-M. Hartmann, G. T. Fernandez-Bada, Z. Scher¨ ubl, B. Brun, V. Schmitt, S. Zihlmann, R. Maurand, E. Du- mur, S. De Franceschi, and F. m. c. Lefloch, From nonre- ciprocal to charge-4e supercurrent in ge-based josephson de...
2024
-
[70]
Banszerus, C
L. Banszerus, C. W. Andersson, W. Marshall, T. Linde- mann, M. J. Manfra, C. M. Marcus, and S. Vaitiek˙ enas, Hybrid josephson rhombus: A superconducting element with tailored current-phase relation, Phys. Rev. X 15, 011021 (2025)
2025
-
[71]
M. Sanz, E. Solano, and I. L. Egusquiza, Applications + practical conceptualization + mathematics = fruitful innovation: Proceedings of the forum of mathematics for industry 2014 (Springer Japan, 2016) Chap. Beyond adi- abatic elimination: Effective Hamiltonians and singular pe...
2014
-
[72]
This is achievable due to the nonlinear dependence of the ancillary junction phases on φ, which can lead to cancellation of the unwanted powers of φ when Taylor expanding the ancillary junction potentials
-
[73]
This can be achieved in practice by threading π fluxes through the loops of the blue subcircuit of Fig
Note that, in our computational search, we allow the Josephson energies J ′ i to be either positive or negative. This can be achieved in practice by threading π fluxes through the loops of the blue subcircuit of Fig. 1(a)
-
[74]
We show data at or below these thresholds in any case, to illustrate the general exponential trend
Note that, while all of our data appear to follow the gen- eral trend of exponentially suppressed fidelities, we can technically only resolve infidelities of O(1/Ntraj), where Ntraj is the number of SSE trajectories sampled. We show data at or below these thresholds in any case,...
-
[75]
Kirˇ sanskas, M
G. Kirˇ sanskas, M. Francki´ e, and A. Wacker, Phenomeno- logical position and energy resolving Lindblad approach to quantum kinetics, Phys. Rev. B 97, 035432 (2018)
2018
-
[76]
Nathan, Topological Phenomena in Periodically Driven Systems , Ph.D
F. Nathan, Topological Phenomena in Periodically Driven Systems , Ph.D. thesis, University of Copenhagen (2018)
2018
-
[77]
Nathan, I
F. Nathan, I. Martin, and G. Refael, Topological fre- quency conversion in a driven dissipative quantum cavity, Physical Review B 99, 094311 (2019)
2019
-
[78]
Davidovi´ c, Geometric-arithmetic master equation in large and fast open quantum systems, Journal of Physics A: Mathematical and Theoretical 55, 455301 (2022)
D. Davidovi´ c, Geometric-arithmetic master equation in large and fast open quantum systems, Journal of Physics A: Mathematical and Theoretical 55, 455301 (2022)
2022
-
[79]
Nathan and M
F. Nathan and M. S. Rudner, Quantifying the accuracy of steady states obtained from the universal lindblad equa- tion, Phys. Rev. B 109, 205140 (2024)
2024
-
[80]
Dalibard, Y
J. Dalibard, Y. Castin, and K. Mølmer, Wave-function approach to dissipative processes in quantum optics, Phys. Rev. Lett. 68, 580 (1992)
1992
-
[81]
Carmichael, An open systems approach to quantum optics (Springer, 1993)
H. Carmichael, An open systems approach to quantum optics (Springer, 1993)
1993
-
[82]
Note that this definition is equivalent to the squared ove r- lap |⟨Ψtarget|Ψf⟩|2, where |Ψtarget⟩ and |Ψf⟩ are the logical wavefunctions for the target and final states with Bloch vectors σtarget and σf, respectively
-
[83]
This way, the frequency of the driving (which can be very precisely realized in real devices) is preserved, even if the details of each driving period are not
-
[84]
Flux noise, conversely, is not cor- rected until after a free segment
The principal effects of charge noise (stochastic displac e- ment along φ in phase space) are immediately corrected by the resistor [35]. Flux noise, conversely, is not cor- rected until after a free segment
-
[85]
G. S. Uhrig, Keeping a quantum bit alive by optimized π-pulse sequences, Phys. Rev. Lett. 98, 100504 (2007)
2007
-
[86]
We can then relate δφ to the effective mistargeting u via δφ = φ0 arccos(1 − u)/2π
This estimate can be obtained by treating flux noise as a quasistatic correction φ ↦→φ + δφ, and using the angle addition formula cos ( 2π φ0 (φ + δφ) ) = cos ( 2π φ0 δφ ) cos ( 2π φ0 φ ) − sin ( 2π φ0 δφ ) sin ( 2π φ0 φ ) . We can then relate δφ to the effective mistargeting u ...
-
[87]
Hastrup, M
J. Hastrup, M. V. Larsen, J. S. Neergaard-Nielsen, N. C. Menicucci, and U. L. Andersen, Unsuitability of cubic phase gates for non-clifford operations on gottesman- kitaev-preskill states, Phys. Rev. A 103, 032409 (2021)
2021
-
[88]
Kitaev, Protected qubit based on a superconducting current mirror (2006), arXiv:cond-mat/0609441 [cond- mat.mes-hall]
A. Kitaev, Protected qubit based on a superconducting current mirror (2006), arXiv:cond-mat/0609441 [cond- mat.mes-hall]
2006 arXiv
-
[89]
Brooks, A
P. Brooks, A. Kitaev, and J. Preskill, Protected gates for superconducting qubits, Phys. Rev. A 87, 052306 (2013)
2013
-
[90]
Exponentially robust non-Clifford gate i n a driven-dissipative circuit
X. C. Kolesnikow, T. B. Smith, F. Thomsen, A. Alase, and A. C. Doherty, Protected phase gate for the 0- π qubit using its internal modes (2025), arXiv:2503.14634 [quant- ph]. Supplementary Information for “Exponentially robust non-Clifford gate i n a driven-dissipative circuit”...
2025
-
[91]
For states confined to 3 FIG. S1. Normal mode (angular) frequencies ωi as a function of ancillary junction capacitance Cjunc. Left and right vertical axes show the frequencies in THz and equivalent t emperature in Kelvin, respectively. Black dashed line (where visible) indicate...
-
[92]
+C ′ 2 ¨φ′ 1 = − 4π2 φ2 0 [J ′ 1(φ′ 1 −φ′
-
[93]
+J ′ 2φ′ 1] + φ −φ′ 1 L1 − φ1 −φ2 L2 , (S11b) −C ′ 1( ¨φ′ 1 − ¨φ′
-
[94]
+C ′ 3 ¨φ′ 2 = − 4π2 φ2 0 [−J ′ 1(φ′ 1 −φ′
-
[95]
(S11c) These equations define a linear system C ¨⃗ φ= V⃗ φ+⃗b of differential equations, where ⃗ φ= ( φ φ′ 1 φ′ 2 ) T and ⃗b = ( 4π2JN/φ 0 0 0 ) T
+J3φ′ 2] + φ1 −φ2 L2 − φ2 L3 . (S11c) These equations define a linear system C ¨⃗ φ= V⃗ φ+⃗b of differential equations, where ⃗ φ= ( φ φ′ 1 φ′ 2 ) T and ⃗b = ( 4π2JN/φ 0 0 0 ) T . Note that the presence of ⃗b only adds a constant (in time) shift to the solution vector ⃗ φ, so we...
-
[96]
The eigenstates of HLCJ are effectively unchanged relative to the Vk ≡ 0 case, to maintain phase coherence between wells
-
[97]
We address each of these conditions in subsections II A and II B, respe ctively
The phases from the k ̸= 4 terms neither take the system out of the codespace nor affect the logi cal information. We address each of these conditions in subsections II A and II B, respe ctively. The conditions above lead us to upper limits on the gate speed and on the magnitud...
-
[98]
This Hamiltonian, for a given realization of ξφ along with the associated jump operators {Li(t)}i, defines a master equation with a time-dependent Liouvillian
Stabilizer and φ4 Segments During the stabilizer and φ4 segments, we have the following system Hamiltonian: H =HLCJ +V (φ) + ξφ(t) L φ (S52) where V (φ) is the (effective) potential in φ arising from the ancillary Josephson junctions (if any). This Hamiltonian, for a given real...
-
[99]
with χφ(t) ≡ 0
Simulate SSE evolution from t1 to t2 in the absence of noise, i.e. with χφ(t) ≡ 0
-
[100]
Act on the state with a unitary exp( −iα(t1,t 2)φ), where α(t1,t 2) = 1 ℏL ∫ t2 t1 ξφ(t) dt. (S53) Step 1 can be done efficiently, by pre-computing and storing logarithmi cally spaced evolution operators, and step 2 involves only a single integral of a scalar function (rather th...
-
[101]
Self-correcting gkp qubit and gates in a driven-dissipative circuit,
Free Segment During the free segment, we model the system as evolving ( unitarily) under the Hamiltonian H = φ2 2L + q2 2C + φ Lξφ(t). (S56) In the co-rotating frame of the LC Hamiltonian HLC, we have: ˜H(t) = 1 L [ cos(2πfLCt)φ + φ0ν 2e sin(2πfLCt)q ] ξφ(t) (S57) where ν = 4e...
2014
Reviewed August 6, 2026 · model on record in the stance chip above.
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