Dissipative protection of a GKP qubit in a high-impedance superconducting circuit driven by a microwave frequency comb

TL;DR

The paper introduces a dissipative, autonomous protection scheme for GKP qubits embedded in a high-impedance superconducting circuit, driven by a broadband microwave frequency comb to implement four modular Lindblad dissipators that stabilize a finite-energy GKP code. It shows that this modular-dissipation approach exponentially suppresses dominant low-weight noise channels and analyzes realistic imperfections, including ancilla noise, finite control bandwidth, fabrication disorder, flux noise, and quasi-particles, with quantitative device parameters. The authors also outline protected Clifford gates and Pauli measurements within this framework and demonstrate how to realize the required modular interactions in a rotating-frame Josephson circuit, paving a path toward fault-tolerant, scalable GKP-based quantum computing with modest hardware overhead. Overall, the work shifts complexity from hardware to microwave control, offering a practical route to long-lived GKP qubits and highlighting remaining challenges, particularly quasi-particle poisoning, that future refinements could address.

Abstract

We propose a novel approach to generate, protect and control 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 quasi-particle 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 two 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.

Figures (20)

• Figure 1: a) Low-weight interactions.${\bf H}=-g{\bf p} {\bf B}$ is an example of low-weight Hamiltonian employed in recent experiments stabilizing the GKP code. It entails a continuous displacement of a GKP state along the $q$ quadrature of an oscillator (plain black lines, initial state represented by dashed black lines), conditioned on an ancillary mode observable ${\bf B}$. The interaction duration $\delta t$ is chosen such that the state is displaced by one period of the square GKP lattice after the evolution. However, if noise modifies the value of ${\bf B}$ during the interaction (red lightning), the final target state is shifted (red lines) and the GKP qubit may be flipped (see Sec. \ref{['sec:gkp']}). b) High-weight interactions.${\bf H}=-g \mathrm{cos}(2\sqrt{\pi}{\bf p}) {\bf B}$ is a high-weight (modular) Hamiltonian that entails a hopping dynamics along the GKP lattice. If noise modifies the value of ${\bf B}$ during the interaction, the relative weights of the final state peaks may be affected but not their positions, so that no logical flip may occur.
• Figure 2: Schematic representation of modular dissipation engineering. A switch controls the coherent tunneling of Cooper pairs (charge $2e$) across a Josephson junction placed in parallel with a two-mode circuit. The target mode (top) has a high impedance $Z$ such that, in normalized phase-space coordinates, tunneling events translate its state by $\pm 2\sqrt{\pi}$ along the charge axis. The switch is controlled with a train of sharp pulses (duration $\delta t$) activating tunneling every quarter of a period $T$ of the target oscillator. In between pulses, the oscillator state rotates freely in phase-space at $\omega=2\pi/T$. Overall, the target mode dynamics is generated by discrete shifts along a square grid matching the GKP lattice (gray grid with period $2\sqrt{\pi}$ overlaid with Wigner diagrams of the oscillator state). A lower impedance ancillary mode (bottom), also driven by Cooper pair tunneling, dissipates excitations into a cold load (purple wriggled arrow) to ensure that the target mode dynamics is irreversible, autonomously stabilizing the GKP code.
• Figure 3: Modular dissipation phase portraits. For a finite-energy code state ($\mathrm{sinh}(\Delta)=0.2/\eta$) displaced by $\alpha+i\beta$ in phase-space, arrows encode the evolution of the state center of mass (top panel) and modular coordinates (bottom panel) entailed by the Lindblad operators \ref{['eq:lindissip']} over a short time step $\mathrm{d}t\ll 1/\Gamma$. Arrows length are rescaled to arbitrary units.
• Figure 4: GKP qubit protection by modular dissipation. The decay rate $\Gamma_L$ of the Pauli operators ${\bf Z}$ and ${\bf X}$ is extracted from numerical simulations (dots) when varying the strength of some intrinsic noise channel relative to the modular dissipation rate $\Gamma$. For all low-weight noise channels considered, errors appear to be exponentially suppressed in the weak noise limit. a) Quadrature noise modeled by two Lindblad operators $\sqrt{\kappa}{\bf q}$ and $\sqrt{\kappa}{\bf p}$. Dashed lines are predictions by spectral analysis of the Lindblad superoperator (see \ref{['sm__ssec__eigenvalues']}). b) Single-photon dissipation modeled by a Lindblad operator $\sqrt{\kappa_{1\mathrm{ph}}}{\bf a}$. c) Pure dephasing modeled by a Lindblad operator $\sqrt{\kappa_{\phi}}{\bf a}^{\dagger}{\bf a}$. d) Kerr Hamiltonian perturbation of the form $\frac{K}{2}({\bf a}^{\dagger}{\bf a})^2$. For (c-d), note the rescaling of the x-axis by $\eta/\epsilon=2\overline{n}$. For (b-d), dashed gray lines reproduce the dashed colored lines in (a), un-rescaled, for comparison.
• Figure 5: Engineering of the GKP Hamiltonian.a) A Josephson ring is placed in parallel with an $LC$ resonator of large impedance $Z=\sqrt{\frac{L}{C}}=2R_Q$. Magnetic fluxes threading the circuit loops $\Phi_{J}^{\mathrm{ext}}(\xi)$ and $\Phi_{L}^{\mathrm{ext}}(\xi)$ are both functions of a control signal $\xi$. They are used to tune the circuit effective Josephson energy $2E_J\xi$ and the phase of Josephson tunneling. b) The laboratory frame flux coordinate $\Phi$ periodically aligns with the coordinates $q$ and $p$ of the frame rotating at $\omega=1/\sqrt{LC}$. c) The signal $\xi(t)$ controlling the circuit effective Josephson energy consists in a train of short bias pulses, so that the Josephson energy takes non-zero values at these instants only. In the RWA, the effective Hamiltonian contains modular functions of ${\bf q}$and${\bf p}$, with spatial frequencies $2\sqrt{\pi}$ for the chosen circuit impedance, both stemming from the Josephson modular flux operator.