Time Crystals as Passively Protected Oscillating Qubits

Abstract

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 time-crystalline dynamics as a mechanism for passive protection of dynamical quantum information, enabling autonomously stabilized oscillating qubits.

Figures (5)

• Figure 1: Eigenvalue of $r_{eo}$ for the eigenoperator with the eigenvalue whose real part is closest to zero as a function of the drive $\tilde{F}$: real (a) and imaginary (b) parts plotted for different $N$. A phase transition around $\Tilde F=0.93$ manifests itself as a level crossing of the dissipative gap (a) and frequency discontinuity (b). Upward/downward triangles represent the eigenoperator above/below the transition point. The full eigenspectra for $N=3$ were found with exact solvers, whereas sparse solvers were used for $N=10,20$ (data for $\Tilde F=1.0, 1.1$ not shown as sparse solvers failed near the transition point supplements). In (b), the semiclassical frequency \ref{['eq:sc_frequency']} is also shown.
• Figure 2: Fock representations (absolute values) and Wigner functions (insets) for the partial traces $r_{ee}^B=\Tr_A(r_{ee})$ (a), $r_{ee}^A=\Tr_B(r_{ee})$ (b), $r_{oo}^A=\Tr_B(r_{oo})$ (c), $r_{eo}^A=\Tr_B(r_{eo})$ (d), of the eigenoperators in the bonding (a) and antibonding (b--d) modes at $\Tilde F = 1.8$ and $N = 20$. For the bonding mode, the operators $r^B_{ee}$, $r^B_{oo}$ and $r^B_{eo}$ are very similar to each other supplements, hence only $r^B_{ee}$ is displayed in (a). For the Wigner function in (d), the real part is plotted.
• Figure 3: (a) Hilbert–Schmidt (HS) distances between pairs of right eigenoperators as functions of the normalized system drive strength ${\Tilde F}$ for $N=4,10,20$. At driving strengths larger than $\Tilde F = 0.93$ the eigenoperators approach one another towards the thermodynamic limit, indicating the emergence of a NS that encodes a qubit. (b) Scaling of HS distances at representative driving strength values, demonstrating that the subsystem becomes asymptotically noiseless in the large $N$ limit above the transition point. The exponential scaling of the distances plotted implies that the pairwise distances of all eigenoperators $\{r_{oo},r_{ee},r_{eo},r_{oe}\}$ scale similarly because $r_{eo}$ and $r_{oe}$ are Hermitian conjugates of each other.
• Figure 4: (a) Real and imaginary parts of the leading order of the dephasing perturbation with the rate $\gamma/10$. (b) Scaling of HS distances at $\tilde{F}=1.8$. Both the real part in (a) and the distances in (b) exhibit exponential decay, indicating existing of an NS in spite of dephasing.
• Figure 5: Evolution of the state \ref{['eq:four_ss']} initialized with $b=c(0)=1/2$ with $\tilde{F}=1.8$ at $N=20$ after an instantaneous phase kick at $t=2.5$ of magnitude $\delta\phi=1$ in the rotating frame of reference for partial traces over bonding and antibonding modes. Bonding mode recovers due to dissipative dynamics. The complex amplitude of the bonding mode is coupled to the phase of the antibonding mode, pulling the phase back during its evolution. (a) Snapshots of Wigner functions. (b) Distances of the evolving kicked and unkicked states with respect to the unkicked state $\rho(t)$ (top) and $\rho(0)$ (bottom).