Stability and decoherence rates of a GKP qubit protected by dissipation

We analyze an experimentally accessible Lindblad master equation for a quantum harmonic oscillator. It approximately stabilizes finite-energy periodic grid states called Gottesman-Kitaev-Preskill (GKP) states, that can be used to encode and protect a logical qubit. We give explicit upper bounds for the energy of the solutions of the Lindblad master equation. Using three periodic observables to define the Bloch sphere coordinates of a logical qubit, we show that their dynamics is governed by a diffusion partial differential equation on a 2D-torus with a Witten Laplacian. We show that the evolution of these logical coordinates is exponentially slow even in presence of small diffusive noise processes along the two quadratures of the phase space. Numerical simulations indicate similar results for other physically relevant noise processes.Comment: 16 pages, 1 figure. This work has been accepted to IFAC for publication under a Creative Commons Licence CC-BY-NC-N

Exponential convergence of a dissipative quantum system towards finite-energy grid states of an oscillator

Based on the stabilizer formalism underlying Quantum Error Correction (QEC), the design of an original Lindblad master equation for the density operator of a quantum harmonic oscillator is proposed. This Lindblad dynamics stabilizes exactly the finite-energy grid states introduced in 2001 by Gottesman, Kitaev and Preskill for quantum computation. Stabilization results from an exponential Lyapunov function with an explicit lower-bound on the convergence rate. Numerical simulations indicate the potential interest of such autonomous QEC in presence of non-negligible photon-losses.Comment: Submitted, 15 pages, 1 figure

Quantum control of a cat-qubit with bit-flip times exceeding ten seconds

Binary classical information is routinely encoded in the two metastable states of a dynamical system. Since these states may exhibit macroscopic lifetimes, the encoded information inherits a strong protection against bit-flips. A recent qubit - the cat-qubit - is encoded in the manifold of metastable states of a quantum dynamical system, thereby acquiring bit-flip protection. An outstanding challenge is to gain quantum control over such a system without breaking its protection. If this challenge is met, significant shortcuts in hardware overhead are forecast for quantum computing. In this experiment, we implement a cat-qubit with bit-flip times exceeding ten seconds. This is a four order of magnitude improvement over previous cat-qubit implementations, and six orders of magnitude enhancement over the single photon lifetime that compose this dynamical qubit. This was achieved by introducing a quantum tomography protocol that does not break bit-flip protection. We prepare and image quantum superposition states, and measure phase-flip times above 490 nanoseconds. Most importantly, we control the phase of these superpositions while maintaining the bit-flip time above ten seconds. This work demonstrates quantum operations that preserve macroscopic bit-flip times, a necessary step to scale these dynamical qubits into fully protected hardware-efficient architectures

Convergence of bipartite open quantum systems stabilized by reservoir engineering

33 pages. Comments welcomeWe study a generic family of Lindblad master equations modeling bipartite open quantum systems, where one tries to stabilize a quantum system by carefully designing its interaction with another, dissipative, quantum system - a strategy known as quantum reservoir engineering. We provide sufficient conditions for convergence of the considered Lindblad equations; our setting accommodates the case where steady states are not unique but rather supported on a given subspace of the underlying Hilbert space. We apply our result to a Lindblad master equation proposed for the stabilization of so-called cat qubits, a system that received considerable attention in recent years due to its potential applications in quantum information processing

Convergence of bipartite open quantum systems stabilized by reservoir engineering

33 pages. Comments welcomeWe study a generic family of Lindblad master equations modeling bipartite open quantum systems, where one tries to stabilize a quantum system by carefully designing its interaction with another, dissipative, quantum system - a strategy known as quantum reservoir engineering. We provide sufficient conditions for convergence of the considered Lindblad equations; our setting accommodates the case where steady states are not unique but rather supported on a given subspace of the underlying Hilbert space. We apply our result to a Lindblad master equation proposed for the stabilization of so-called cat qubits, a system that received considerable attention in recent years due to its potential applications in quantum information processing

Stability and decoherence rates of a GKP qubit protected by dissipation

16 pages, 1 figure. This work has been accepted to IFAC for publication under a Creative Commons Licence CC-BY-NC-NDInternational audienceWe analyze an experimentally accessible Lindblad master equation for a quantum harmonic oscillator. It approximately stabilizes finite-energy periodic grid states called Gottesman-Kitaev-Preskill (GKP) states, that can be used to encode and protect a logical qubit. We give explicit upper bounds for the energy of the solutions of the Lindblad master equation. Using three periodic observables to define the Bloch sphere coordinates of a logical qubit, we show that their dynamics is governed by a diffusion partial differential equation on a 2D-torus with a Witten Laplacian. We show that the evolution of these logical coordinates is exponentially slow even in presence of small diffusive noise processes along the two quadratures of the phase space. Numerical simulations indicate similar results for other physically relevant noise processes

Exponential convergence of a dissipative quantum system towards finite-energy grid states of an oscillator

Submitted, 15 pages, 1 figuresBased on the stabilizer formalism underlying Quantum Error Correction (QEC), the design of an original Lindblad master equation for the density operator of a quantum harmonic oscillator is proposed. This Lindblad dynamics stabilizes exactly the finite-energy grid states introduced in 2001 by Gottesman, Kitaev and Preskill for quantum computation. Stabilization results from an exponential Lyapunov function with an explicit lower-bound on the convergence rate. Numerical simulations indicate the potential interest of such autonomous QEC in presence of non-negligible photon-losses

A GKP qubit protected by dissipation in a high-impedance superconducting circuit driven by a microwave frequency comb

61 pages, 20 figures, 1 tableWe 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

Quantum control of a cat-qubit with bit-flip times exceeding ten seconds

Binary classical information is routinely encoded in the two metastable states of a dynamical system. Since these states may exhibit macroscopic lifetimes, the encoded information inherits a strong protection against bit-flips. A recent qubit - the cat-qubit - is encoded in the manifold of metastable states of a quantum dynamical system, thereby acquiring bit-flip protection. An outstanding challenge is to gain quantum control over such a system without breaking its protection. If this challenge is met, significant shortcuts in hardware overhead are forecast for quantum computing. In this experiment, we implement a cat-qubit with bit-flip times exceeding ten seconds. This is a four order of magnitude improvement over previous cat-qubit implementations, and six orders of magnitude enhancement over the single photon lifetime that compose this dynamical qubit. This was achieved by introducing a quantum tomography protocol that does not break bit-flip protection. We prepare and image quantum superposition states, and measure phase-flip times above 490 nanoseconds. Most importantly, we control the phase of these superpositions while maintaining the bit-flip time above ten seconds. This work demonstrates quantum operations that preserve macroscopic bit-flip times, a necessary step to scale these dynamical qubits into fully protected hardware-efficient architectures

Quantum control of a cat-qubit with bit-flip times exceeding ten seconds

Binary classical information is routinely encoded in the two metastable states of a dynamical system. Since these states may exhibit macroscopic lifetimes, the encoded information inherits a strong protection against bit-flips. A recent qubit - the cat-qubit - is encoded in the manifold of metastable states of a quantum dynamical system, thereby acquiring bit-flip protection. An outstanding challenge is to gain quantum control over such a system without breaking its protection. If this challenge is met, significant shortcuts in hardware overhead are forecast for quantum computing. In this experiment, we implement a cat-qubit with bit-flip times exceeding ten seconds. This is a four order of magnitude improvement over previous cat-qubit implementations, and six orders of magnitude enhancement over the single photon lifetime that compose this dynamical qubit. This was achieved by introducing a quantum tomography protocol that does not break bit-flip protection. We prepare and image quantum superposition states, and measure phase-flip times above 490 nanoseconds. Most importantly, we control the phase of these superpositions while maintaining the bit-flip time above ten seconds. This work demonstrates quantum operations that preserve macroscopic bit-flip times, a necessary step to scale these dynamical qubits into fully protected hardware-efficient architectures