Dissipative Time Crystals as Passively Protected Oscillating Qubits

The gist The driven-dissipative Bose-Hubbard dimer (BHD) can encode a persistently oscillating qubit within a noiseless subsystem, realized explicitly in the Bose–Hubbard dimer (BHD)<ref:2602.20269#pg2>.

Passive Error Correction Mechanism

The study demonstrates that dissipative timecrystalline dynamics serve as a mechanism for passive protection of dynamical quantum information, enabling autonomously stabilized oscillating qubits The results establish dissipative timecrystalline dynamics as a mechanism for passive protection of dynamical quantum information, enabling autonomously stabilized oscillating qubits. This is achieved by utilizing the strong parity symmetry of the system to encode dynamical information that can be maintained and passively error-corrected We propose the driven-dissipative Bose-Hubbard dimer (BHD), known to host continuous time crystal phases, as a platform for quantum information processing For a certain phase of the BHD we encode dynamical information that can be maintained and passively error-corrected, utilizing the strong parity symmetry of the system.

Time Crystal Dynamics and Phase Transition

The model exhibits persistent oscillations in non-stationary states above a dissipative phase transition point As the driving force increases, the Liouvillian spectrum of these states features a phase transition Above the transition point, the model hosts a dynamical qubit encoded in a noiseless subsystem that oscillates persistently within a time crystal phase. This non-stationary state is characterized by its persistent oscillations in the thermodynamic limit The transition manifests as an abrupt change in the frequency of the non-stationary state in the imaginary spectrum.

Protection Against Noise and Dephasing

The noiseless subsystem feature persists even in the presence of global dephasing errors, yielding a qubit with a slightly different frequency but similar behavior The stability of the time crystal phase is manifested in its stability to global dephasing errors, suggesting the BHD as a possible medium for autonomously correcting quantum gates. In perturbation analysis, the real part of the leading order correction to the eigenvalue diminishes with N, demonstrating that the state does not decay in the thermodynamic limit under dephasing The system recovers after instantaneous phase kicks due to dissipative dynamics and nonlinear interaction terms This recovery is understood semiclassically as a guardian where the bonding mode's dissipative recovery pulls the antibonding phase back through nonlinear coupling.

Encoding and Control of Qubits

The oscillating qubit is preserved in the presence of global dephasing, which is compatible with experiments where both modes naturally couple to a common reservoir With such protection mechanisms, strong symmetric open systems can provide a platform for controlled phase shifts As the frequency of the oscillations can be controlled through system parameters, single-qubit gates can be implemented by a judiciously constructed detuned pulse sequence. The emergence of the noiseless oscillating qubit is understood as a separation of scales where the antibonding mode is weakly populated and quantum, while the bonding mode is macroscopic and semiclassical, relaxing to a single stable point under dissipation

Experimental Relevance

The restriction to global dephasing and loss is compatible with experiments where both modes naturally couple to a common reservoir These could be probed in contemporary photonic platforms, including driven Kerr resonators and cat-qubit architectures, where parity protection, engineered dissipation, and high-coherence bosonic modes are readily available. Strong symmetry constraints similar to those in the BHD also arise in superconducting circuits and spin ensembles with global dissipation This suggests several experimentally accessible routes for realizing noiseless dynamical subsystems. The kick dynamics provide a direct visualization of the protection mechanism, showing how the bonding mode’s dissipative recovery pulls the antibonding phase back through nonlinear coupling, restoring coherence after a macroscopic phase perturbation.

Conclusion and Outlook

The driven-dissipative BHD goes through a dynamical transition relevant for quantum information storage at a critical driving strength magnitude The high frequency branch supports a qubit encoded in the noiseless subsystem of the BHD By choosing appropriate system parameters, one can therefore encode a persistently oscillating noiseless qubit, passively protecting against global dephasing errors. A systematic gate-level construction will be addressed in future research as time crystals have been suggested to be utilized as not gates in a recent colloquium. The bonding mode’s dissipative recovery pulls the antibonding phase back through nonlinear coupling, restoring the encoded coherence after a macroscopic phase perturbation. This mechanism enables the encoding of dynamically stabilized oscillating qubits. The study confirms that dissipative timecrystalline dynamics enable passively protected quantum information storage.

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Dissipative Time Crystals as Passively Protected Oscillating Qubits

Mert Esencan,1 A. I. Lvovsky,1 and Berislav Buˇca2, 3, 1

1Clarendon Laboratory, University of Oxford, Parks Road, OX1 3PU, Oxford, United Kingdom

2Universit´e Paris-Saclay, CNRS, LPTMS, 91405, Orsay France

3Niels Bohr Institute University of Copenhagen Blegdamsvej 17 2100 Copenhagen Denmark

Protecting information against decoherence in open quantum systems remains a central challenge for quantum computing. In particular, passive error correction schemes have so far been limited to static memories rather than dynamical qubits. We demonstrate that a driven-dissipative bosonic system can encode a persistently oscillating qubit within a noiseless subsystem, realized explicitly in the Bose–Hubbard dimer (BHD). The strong parity symmetry of the model leads to degenerate stationary states. This symmetry is further broken into non-stationary states in the thermodynamic limit, which exhibit persistent oscillations. As the driving force increases, the Liouvillian spectrum of these states features a phase transition. Above the transition point, the non-stationary state encodes quantum information, preserving it in a noiseless subsystem. In addition to global loss that affects both bosonic modes identically, we further add global dephasing and show that the oscillating qubit is preserved. Finally, in order to gain additional physical insight, we study the effect of phase perturbation to both modes and observe that likewise they are passively protected, returning approximately to their initial configurations. These results establish dissipative timecrystalline dynamics as a mechanism for passive protection of dynamical quantum information, enabling autonomously stabilized oscillating qubits.

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Lˆρ(t) = ∂ρ/∂t = −i[H, ρ] + γDˆ[L]ρ. (1)

The dissipator with strength γ is defined by Dˆ[L]ρ = LρL† − 1/2

G L†L, ρ】, where L is the jump operator describing open interaction with the environment. The driven-dissipative BHD is an open many-body system with two coupled bosonic modes represented by operators a1 and a2. Due to the strong symmetry (defined below), we move to a beam-splitter basis aB,A = (a1 ± a2)/√2, which we refer to as the bonding and antibonding modes, respectively. Our Hamiltonian is H = √2F(a†B + aB) + (−∆ − J)a†BaB + (−∆ + J)a†AaA

• U squared (a†Ba†BaBaB + a†AaA

• a†Ba † BaAaA

• aBaBA † AaA).

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F is the driving amplitude, ∆ is the detuning between the drive frequency and resonant frequencies of the two modes, J is the coupling between the modes, and U is the nonlinear interaction strength. The jump operator L = aB dissipates only the bonding mode, highlighting the strong symmetry of the model. The dissipation is symmetric such that in the lab basis, a1 and a2 have identical couplings to a common reservoir.

To investigate the system in the thermodynamic limit, we take F → ∞ while keeping F√U fixed, by introducing the scaling parameter N. Thus we redefine the drive and the nonlinearity as F = F˜√N and U = U /N ˜[120, 121], and increase N to predict the behavior in the thermodynamic limit.

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In generic systems in the absence of special symmetries, the system decays into a unique steady state r0 ≡ρ(∞), as λ0 = 0 whereas all other exponents λj have negative real parts and lead to the decay of their corresponding states [1]. The BHD, however, hosts a Z2 strong parity symmetry P = P nB,nA≥0 (−1)nA nB, nA⟩ ⟨nB, nA where nB, A are the number operators nk = a†k ak in each mode B or A. P commutes with the Hamiltonian and the jump operators, separately, i.e. [H, P] = [L, P] = 0. The strong symmetry of the BHD block-diagonalizes Lˆ into four distinct even or odd blocks Lˆ = Lˆee⊕Lˆeo⊕Lˆoe⊕Lˆoo according to the action of P from the left and right on eigenoperators of L

[114].

1. Passive Error Correction for Dynamical Qubits: The improved AI system can encode a persistently oscillating qubit within a noiseless subsystem by utilizing dissipative timecrystalline dynamics, enabling autonomously stabilized oscillating qubits that are protected against global dephasing errors.

2. Controlled Phase Shifts via Time Crystals: The system can implement single-qubit gates by controlling the frequency of oscillations, as As frequency of the oscillations can be controlled through system parameters, single-qubit gates can be implemented by a judiciously constructed detuned pulse sequence.

3. Robust Quantum Memory: The AI system will feature a quantum memory that is protected against dephasing errors because the noiseless subsystem is not prevented by dephasing in the large N limit, and the state does not decay in the thermodynamic limit under global dephasing.