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arXiv:2304.01425 · Physical Review X

Dissipative Protection of a GKP Qubit in a High-Impedance Superconducting Circuit Driven by a Microwave Frequency Comb

Lev-Arcady Sellem, Alain Sarlette, Zaki Leghtas, Mazyar Mirrahimi, Pierre Rouchon, Philippe Campagne-Ibarcq

Published Jan 22, 2025• 12 citations
arXivHTMLPDF

Abstract

We propose a novel approach to generate, protect, and control Gottesman-Kitaev-Preskill (GKP) qubits. It employs a microwave frequency comb parametrically modulating a Josephson circuit to enforce a dissipative dynamics of a high-impedance circuit mode, autonomously stabilizing the finite-energy GKP code. The encoded GKP qubit is robustly protected against all dominant decoherence channels plaguing superconducting circuits but quasiparticle poisoning. In particular, noise from ancillary modes leveraged for dissipation engineering does not propagate at the logical level. In a state-of-the-art experimental setup, we estimate that the encoded qubit lifetime could extend 2 orders of magnitude beyond the break-even point, with substantial margin for improvement through progress in fabrication and control electronics. Qubit initialization, readout, and control via Clifford gates can be performed while maintaining the code stabilization, paving the way toward the assembly of GKP qubits in a fault-tolerant quantum computing architecture.

DOI
10.1103/physrevx.15.011011

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We propose a novel approach to generate, protect, and control Gottesman-Kitaev-Preskill (GKP) qubits. It employs a microwave frequency comb parametrically modulating a Josephson circuit to enforce a dissipative dynamics of a high-impedance circuit mode, autonomously stabilizing the finite-energy GKP code. The encoded GKP qubit is robustly protected against all dominant decoherence channels plaguing superconducting circuits but quasiparticle poisoning. In particular, noise from ancillary modes leveraged for dissipation engineering does not propagate at the logical level. In a state-of-the-art experimental setup, we estimate that the encoded qubit lifetime could extend 2 orders of magnitude beyond the break-even point, with substantial margin for improvement through progress in fabrication and control electronics. Qubit initialization, readout, and control via Clifford gates can be performed while maintaining the code stabilization, paving the way toward the assembly of GKP qubits in a fault-tolerant quantum computing architecture.
We propose a simple circuit architecture for a dissipatively error corrected Gottesman-Kitaev-Preskill (GKP) qubit. The device consists of a electromagnetic resonator with impedance h/2e2≈12.91 kΩh/2e^2\approx 12.91\,{\rm k}Ωh/2e2≈12.91kΩ connected to a Josephson junction with a voltage bias oscillating at twice the resonator frequency. For large drive amplitudes, the circuit is effectively described by the GKP stabilizer Hamiltonian, whose low-energy subspace forms the code space for a qubit protected against phase-space local noise. The GKP states in the codespace can be dissipatively stabilized and error corrected by coupling the resonator to a bath through a bandpass filter; a resulting side-band cooling effect stabilizes the system in the GKP code space, dissipatively correcting it against both bit and phase flip errors. Simulations show that this dissipative error correction can enhance coherence time by factor ∼1000\sim 1000∼1000 with NbN-based junctions, for operating temperatures in the ∼100 mK\sim 100\,{\rm mK}∼100mK range. The scheme can be used to stabilize both square- and hexagonal-lattice GKP codes. Finally, a Josephson current based readout scheme, and dissipatively corrected single-qubit Clifford gates are proposed.
We show that a self-correcting Gottesman-Kitaev-Preskill (GKP) qubit can be realized with a high-impedance L C circuit coupled to a resistor and a Josephson junction via a controllable switch. When activating the switch in a particular stepwise pattern, the resonator relaxes into a subspace of GKP states that encode a protected qubit. Under continued operation, the resistor dissipatively error corrects the qubit against bit flips and decoherence by absorbing noise-induced entropy. We show that this leads to an exponential enhancement of the coherence time ( T 1 and T 2 ), even in the presence of extrinsic noise, imperfect control, and device-parameter variations. We show that the qubit supports exponentially robust single-qubit Clifford gates, implemented via appropriate control of the switch, and readout and/or initialization via supercurrent measurement. The self-correcting properties of the qubit allow it to operate at approximately 1-K temperatures and resonator Q factors down to approximately 1000 for realistic parameters, and make it amenable to parallel control through global control signals. We discuss how the effects of quasiparticle poisoning—potentially, though not necessarily, a limiting factor—might be mitigated. We finally demonstrate that a related device supports a self-correcting magic T gate.
Bosonic quantum error correction encodes a logical qubit in an oscillator, avoiding the hardware overhead of large qubit arrays. Among such encodings, Gottesman-Kitaev-Preskill (GKP) states are paticularly powerful because their phase-space grid structure protects against small displacement errors simultaneously in both conjugate quadratures. Here we provide the first protocol for preparing magnonic GKP states, which involves an ellipsoidal magnetic crystal effectively coupled to a superconducting qubit via a microwave cavity. The geometric anisotropy intrinsically squeezes the magnon mode, while the cavity-mediated qubit control realizes an effective conditional-displacement interaction. We show that two rounds of a conditional-displacement interaction and a qubit projective measurement yield three- and four-component magnonic GKP-like states. We also show how to realize single logical qubit gate operations, such as Pauli, Hadamard and phase gates, completing the logical Pauli basis of the approximate GKP code. Our results establish hybrid magnon-qubit systems as a promising platform for preparing bosonic code states, with applications in magnonic fault-tolerant quantum computation and quantum sensing.
Straightforward logical operations contrasting with complex state preparation are the hallmarks of the bosonic encoding proposed by Gottesman, Kitaev, and Preskill (GKP). The recently reported generation and error correction of GKP qubits in trapped ions and superconducting circuits thus holds great promise for the future of quantum computing architectures based on such encoded qubits. However, these experiments rely on error-syndrome detection via an auxiliary physical qubit, whose noise may propagate and corrupt the encoded GKP qubit. We propose a simple module composed of two oscillators and a physical qubit, operated with two experimentally accessible quantum gates and elementary feedback controls to implement an error-corrected GKP qubit protected from such propagating errors. In the idealized setting of periodic GKP states, we develop efficient numerical methods to optimize our protocol parameters and show that errors of the encoded qubit stemming from flips of the physical qubit and diffusion of the oscillators state in phase space may be exponentially suppressed as the noise strength over individual operations is decreased. Our approach circumvents the main roadblock towards fault-tolerant quantum computation with GKP qubits.
Straightforward logical operations contrasting with complex state preparation are the hallmarks of the bosonic encoding proposed by Gottesman, Kitaev and Preskill (GKP). The recently reported generation and error-correction of GKP qubits in trapped ions and superconducting circuits thus holds great promise for the future of quantum computing architectures based on such encoded qubits. However, these experiments rely on error-syndrome detection via an auxiliary physical qubit, whose noise may propagate and corrupt the encoded GKP qubit. We propose a simple module composed of two oscillators and a physical qubit, operated with two experimentally accessible quantum gates and elementary feedback controls to implement an error-corrected GKP qubit protected from such propagating errors. In the idealized setting of periodic GKP states, we develop efficient numerical methods to optimize our protocol parameters and show that errors of the encoded qubit stemming from flips of the physical qubit and diffusion of the oscillators state in phase-space may be exponentially suppressed as the noise strength over individual operations is decreased. Our approach circumvents the main roadblock towards fault-tolerant quantum computation with GKP qubits.
We realize a hardware-native time--frequency Gottesman--Kitaev--Preskill (GKP) logical qubit encoded in the continuous phase space of single photons, establishing a propagating photonic implementation of bosonic grid encoding. Finite-energy grid states are generated deterministically using coherently driven entangled nonlinear biphoton sources that produce single-photon frequency-comb supermodes. An optical-frequency-comb reference anchors the time--frequency phase space and enforces commuting displacement stabilizers directly at the hardware level, continuously defining the logical subspace. Timing jitter, spectral drift, and phase noise map naturally onto Gaussian displacement errors within this lattice, yielding intrinsic correctability inside a stabilizer cell. Logical operations correspond to experimentally accessible phase and delay controls, enabling deterministic state preparation and manipulation. Building on the modal time--frequency GKP framework, we identify a concrete pathway toward active syndrome extraction and deterministic displacement recovery using ancillary grid states and interferometric time--frequency measurements. These primitives establish a hardware-compatible route for integrating the time--frequency GKP logical layer into erasure-aware and fusion-based fault-tolerant photonic architectures.