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arXiv:2002.07718 · Physical Review X

Hardware-Encoding Grid States in a Nonreciprocal Superconducting Circuit

Martin Rymarz, Stefano Bosco, Alessandro Ciani, David Peter Divincenzo

Published Feb 17, 2021• 48 citations
arXivHTMLPDF

Abstract

We present a circuit design composed of two Josephson junctions coupled by a nonreciprocal element, the gyrator, whose ground space is doubly degenerate.The ground states are approximate code words of the Gottesman-Kitaev-Preskill code.We determine the low-energy dynamics of the circuit by working out the equivalence of this system to the problem of a single electron in a crystal, confined to a two-dimensional plane, and subjected to a strong, homogeneous magnetic field.We find that the circuit is naturally protected against the common noise channels in superconducting circuits, such as charge and flux noise, implying that it can be used for passive quantum error correction.We also propose realistic design parameters for an experimental realization, and we describe possible protocols to perform logical one-and two-qubit gates, state preparation, and readout.

DOI
10.1103/physrevx.11.011032

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We present a circuit design composed of two Josephson junctions coupled by a nonreciprocal element, the gyrator, whose ground space is doubly degenerate.The ground states are approximate code words of the Gottesman-Kitaev-Preskill code.We determine the low-energy dynamics of the circuit by working out the equivalence of this system to the problem of a single electron in a crystal, confined to a two-dimensional plane, and subjected to a strong, homogeneous magnetic field.We find that the circuit is naturally protected against the common noise channels in superconducting circuits, such as charge and flux noise, implying that it can be used for passive quantum error correction.We also propose realistic design parameters for an experimental realization, and we describe possible protocols to perform logical one-and two-qubit gates, state preparation, and readout.
Decoherence errors arising from noisy environments remain a central obstacle to progress in quantum computation and information processing. Quantum error correction (QEC) based on the Gottesman-Kitaev-Preskill (GKP) protocol offers a powerful strategy to overcome this challenge, with successful demonstrations in trapped ions, superconducting circuits, and photonics. Beyond active QEC, a compelling alternative is to engineer Hamiltonians that intrinsically enforce stabilizers, offering passive protection akin to topological models. Inspired by the GKP encoding scheme, we implement a superconducting qubit whose eigenstates form protected grid states - long envisioned but not previously realized - by integrating an effective Cooper-quartet junction with a quantum phase-slip element embedded in a high-impedance circuit. Spectroscopic measurements reveal pairs of degenerate states separated by large energy gaps, in excellent agreement with theoretical predictions. Remarkably, our observations indicate that the circuit tolerates small disorders and gains robustness against environmental noise as its parameters approach the ideal regime, establishing a new framework for exploring superconducting hardware. These findings also showcase the versatility of the superconducting circuit toolbox, setting the stage for future exploration of advanced solid-state devices with emergent properties.
We describe a superconducting circuit consisting of a Josephson junction in parallel with a quantum phase slip wire, which implements a Hamiltonian that is periodic in both charge and flux. This Hamiltonian is exactly diagonalisable in a double-Bloch band, and the eigenstates are shown to be code states of the Gottesman-Kitaev-Preskill quantum error correcting code. The eigenspectrum has several critical points, where the linear sensitivity to external charge and flux noise vanishes. The states at these critical points thus hold promise as qubit states that are insensitive to external noise sources.
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.
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 describe a superconducting circuit consisting of a Josephson junction in parallel with a quantum phase slip wire, which implements a Hamiltonian that is periodic in both charge and flux. This Hamiltonian is exactly diagonalizable in a double-Bloch band, and the eigenstates are shown to be code states of the Gottesman-Kitaev-Preskill quantum error correcting code. The eigenspectrum has several critical points, where the linear sensitivity to external charge and flux noise vanishes. The states at these critical points thus hold promise as qubit states that are insensitive to common external noise sources.
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.