A Robust Large-Period Discrete Time Crystal and its Signature in a Digital Quantum Computer

TL;DR

This work demonstrates a robust $4T$-Discrete Time Crystal in a system of two-level particles by engineering an interacting spin-$1/2$ ladder with an emergent $Z_4$ symmetry. Using both tensor-network simulations (tMPS) and a NISQ-era IBM Q Cairo experiment, the authors show sustained subharmonic response at $Omega=pi/(2T)$ and a $pi/2$-spaced quasienergy structure that persists under disorder and finite-size effects. Finite-size scaling with $chi_{zz}$ and $s_{pi/2}$ diagnostics confirms long-range Floquet correlations consistent with a genuine DTC phase, while the IBM Q realization employs a variational circuit recompilation to maintain shallow-depth circuits and achieve agreement with theory over tens of Floquet periods. This work broadens the landscape of DTCs beyond period-doubling, offering a pathway to robust quantum memory and passive error correction on near-term quantum hardware and highlighting the potential of NISQ devices for simulating exotic non-equilibrium quantum states.

Abstract

Discrete time crystals (DTCs) are novel out-of-equilibrium quantum states of matter which break time translational symmetry. DTCs have been extensively realized in experiments, particularly their subclass that is characterized by period-doubling dynamics due to its natural occurrence in a system of periodically driven two-level, e.g., spin-1/2, particles. The realization of DTCs beyond period-doubling, including their generalizations termed discrete quasicrystals has also been made in recent years, though such experiments typically involve higher spin particles. Constructing and observing DTCs beyond period-doubling in systems of two-level particles are generally still considered an open challenge due to the latter's $\mathbb{Z}_2$ symmetry that natively only leads to period-doubling. In this work, we developed an intuitive interacting system of two-level particles (qubits) that supports the more non-trivial period-quadrupling DTCs ($4T$-DTCs). Remarkably, by utilizing a variational algorithm, we are able to observe clear signatures of such $4T$-DTCs in a quantum processor despite the presence of considerable noise and the small number of available qubits. Our findings extend the landscape of time crystalline behavior by demonstrating a distinct realization of time crystallinity beyond standard period-doubling dynamics with qubits (two-level particles) on a NISQ-era digital quantum computer, as well as the potential of existing noisy intermediate-scale quantum devices for simulating exotic non-equilibrium quantum states of matter.

Figures (13)

• Figure 1: (a) Schematics of our periodically driven spin-1/2 ladder for $N_0=4$. During the first half of the period ($0\rightarrow T/2$, solid box), the evolution is governed by externally driven Heisenberg spin exchange interactions that are continuously modulated at frequency $\omega$. In the second half of the period ($T/2 \rightarrow T$, dashed box), the interactions are switched off and instead a magnetic field $M$ is applied in the $x$ direction . (b) The 4T-periodic oscillations can be intuitively understood in the solvable limit of $JT=0$ and $hT=MT=\pi$. With all spins initialized pointing up, the system undergoes spatially uniform 4T-periodic oscillations; ironically, these oscillations become stablized if a nonzero $JT$ is introduced. We remark that an additional phase factors of $i$ or $(-1)$ is omitted in the illustration.
• Figure 2: Numerical evidence of robust $4T$-DTC for $N=16$ sites using tMPS. (a) Magnetization $\langle S_z \rangle$ as a function of time at $MT=0.98\pi$, which is slightly perturbed away from the "ideal" solvable limit value $MT=\pi$. The 4T-DTC phase (red triangles) corresponds to $hT=0.9\pi$ (which is also perturbed from $\pi$) and $JT=0.16\pi$. The thermal phase (filled orange squares) corresponds to $hT=0.52\pi$ and $JT=0.1\pi$. The $JT=0$ phase (empty blue squares) corresponds to $hT=0.9\pi$. (b) The associated stroboscopic power spectrum $\langle \tilde{S}_z \rangle$, which shows distinct frequency peaks at $\Omega = \pm \pi/2T$ only for the 4T-DTC phase.
• Figure 3: (a) Presence of $4T$-DTC behavior over a wide range of $JT$ and $hT$. The phase diagram representing the value of the subharmonic peak at $\Omega T=\pi/2$ and $MT=0.98 \pi$. (b) Illustration of how partial degeneracy in the clean system (overlapping blue and red lines) leads to the breakdown of the $\pi/2$ quasienergy spacing. The left (right)-side quasienergy spacings correspond to the ones before (after) perturbations. (c-d) The full power spectrum associated with the magnetization dynamics up to $t=100T$ under the influence of various disorders at (c) $JT=0.13\pi$, $hT=0.8\pi$, and $MT=0.98\pi$, i.e., green dot in panel a, and (d) $JT=0.16\pi$, $hT=0.9\pi$, and $MT=0.98\pi$, i.e., blue dot in panel a. The enhanced subharmonic peaks due to disorders are clearly observed near the DTC-thermal phase boundaries (panel c). All data points involving disorders are averaged over $220$ realizations.
• Figure 4: Finite size scaling analysis as a validity check of $4T$-DTC. (a) The Ising-$ZZ$ SG order $\chi_{zz}$ and the $\pi/2$-shifted spectral function $s_{\pi/2}$ as a function of the total system size for $M=1.0\pi$ and $MT=0.95\pi$. (b) The full power spectrum for different sizes from $N=4$ to $12$ for $MT=0.95\pi$. (c) Comparison of the averaged peak values of the power spectrum for both $MT=1.0\pi$ and $MT=0.95\pi$ as a function of system sizes. All results are obtained by exact diagonalization of the Floquet unitary operator for the time evolution of the total system up to $t=400T$. In all panels, other parameters used are: $JT=0.16\pi$, and $hT=0.99\pi$.
• Figure 5: Demonstration of the robustness against different initial state choices. (a,b) the full power spectrum for different impurity initial state parameters $\theta_y$ with different system sizes in (a) $N=6$ and (b) $N=12$. (c) The $\pi/2$-shifted spectral function $s_{\pi/2}$ and (d) the Ising-$ZZ$ SG order as a function of system sizes for different impurity initial state parameters $\theta_y$.