Unveiling clean two-dimensional discrete time quasicrystals on a digital quantum computer

TL;DR

The study demonstrates clean two‑dimensional discrete time crystals and discrete time quasicrystals in Floquet dynamics of a kicked Ising model on a $2$D heavy‑hex lattice implemented on a large quantum processor. It combines a simple depolarising noise‑based error mitigation with hardware measurements on $L=133$ qubits, state‑vector checks at smaller sizes, and state‑of‑the‑art 2d tensor‑network simulations (2dTNS) to validate the dynamics up to $t/T=100$. The results reveal a robust period‑doubling DTC in 2D within a prethermal regime and, upon introducing a longitudinal field, the emergence of DTQC side peaks whose envelope frequency scales with the transverse‑field perturbation, signaling a DTC→DTQC crossover. This work highlights the capability of digital quantum computers to explore 2D out‑of‑equilibrium quantum dynamics beyond classical tensor‑network limits and provides a concrete benchmark for future quantum simulations of complex driven many‑body systems.

Abstract

In periodically driven (Floquet) systems, evolution typically results in an infinite-temperature thermal state due to continuous energy absorption over time. However, before reaching thermal equilibrium, such systems may transiently pass through a meta-stable state known as a prethermal state. This prethermal state can exhibit phenomena not commonly observed in equilibrium, such as discrete time crystals (DTCs), making it an intriguing platform for exploring out-of-equilibrium dynamics. Here, we investigate the relaxation dynamics of initially prepared product states under periodic driving in a kicked Ising model using the IBM Quantum Heron processor, comprising 133 superconducting qubits arranged on a heavy-hexagonal lattice, over up to $100$ time steps. We identify the presence of a prethermal regime characterised by magnetisation measurements oscillating at twice the period of the Floquet cycle and demonstrate its robustness against perturbations to the transverse field. Our results provide evidence supporting the realisation of a period-doubling DTC in a two-dimensional system. Moreover, we discover that the longitudinal field induces additional amplitude modulations in the magnetisation with a period incommensurate with the driving period, leading to the emergence of discrete time quasicrystals (DTQCs). These observations are further validated through comparison with tensor-network and state-vector simulations. Our findings not only enhance our understanding of clean DTCs in two dimensions but also highlight the utility of digital quantum computers for simulating the dynamics of quantum many-body systems, addressing challenges faced by state-of-the-art classical simulations.

Figures (24)

• Figure 1: Two-qubit gate connectivity on a heavy-hexagonal lattice and initial product states.a, The overall device geometry of ibm_torino, comprising a hevey-hexagonal lattice of $L=133$ qubits. Each circle represents a qubit and the edges indicate the qubit connectivity. Three layers of $R_{ZZ}$ gates in a single Floquet circle are highlighted in red, blue, and green (also see b). The enclosed area marked by the yellow line represents the system of $L=28$ qubits, utilised for comparison with state-vector simulations. The initial state is prepared as a product state with qubits denoted by white (black) circles initialised to be $|0\rangle$ ($|1\rangle$). b, Schematic representation of the single-cycle Floquet operator $\hat{U}_{\rm F}$. The red, blue, and green boxes correspond to the three layers of $R_{ZZ}$ gates, each applied in parallel, indicated in a, while the white boxes represent the products of $R_X$ and $R_Z$ gates. The horizontal lines represent the qubits on which the quantum gates operate.
• Figure 2: Error-mitigation protocol and comparison with classical simulations. a, Raw data of the averaged magnetisation $\langle \hat{Z}_{\rm avg}(t)\rangle_{0}$ at $(\theta_x,\theta_z)=(0.8\pi,0.5\pi)$ (cyan circles) and $f(\theta_x=\pi)=|\langle \hat{Z}_{\rm avg}(t)\rangle_{0,\theta_x=\pi}|$ at $(\theta_x,\theta_z)=(\pi,0.5\pi)$ (black crosses) for $L=28$. b, Error-mitigated data (yellow diamonds) of the averaged magnetisation $\langle \hat{Z}_{\rm avg}(t)\rangle$ at $(\theta_x,\theta_z)=(0.8\pi,0.5\pi)$ for $L=28$. Numerically exact results obtained by state-vector simulations are also shown with black squares in b. c-d, Same as a-b but for $L=133$. Results of classical tensor-network simulations with bond dimensions $\chi=100$ and 200 are also shown with green and blue triangles, respectively, in d. Error bars in b and d represent the propagated error due to sampling errors in the quantities in the numerator and the denominator of Eq. (\ref{['eq-norm']}).
• Figure 3: Dynamics of magnetisation exhibiting DTC, DTQC, and thermalisation. The raw data $\langle \hat{Z}_{\rm avg}(t) \rangle_{0}$ (cyan circles) and the error-mitigated data $\langle \hat{Z}_{\rm avg}(t) \rangle$ (yellow diamonds) obtained on the heavy-hexagonal lattice of $L=133$ qubits at various parameter sets $(\theta_x,\theta_z)$: a$(0.9\pi,0)$, b$(0.9\pi,0.5\pi)$, c$(0.9\pi,\pi)$, d$(0.85\pi,0)$, e$(0.85\pi,0.5\pi)$, f$(0.85\pi,\pi)$, g$(0.8\pi,0)$, h$(0.8\pi,0.5\pi)$, i$(0.8\pi,\pi)$, j$(0.7\pi,0.0\pi)$, k$(0.7\pi,0.5\pi)$, and l$(0.7\pi,\pi)$. Notice that the time duration in the horizontal axis in j- l is half of that in the other panels.
• Figure 4: Fourier analysis of DTQC frequencies induced by the longitudinal field. Fourier spectrum $\tilde{Z}(\omega)$ of the averaged magnetisation for a$\theta_{x}=0.9\pi$, b$\theta_{x}=0.85\pi$, and c$\theta_{x}=0.8\pi$ with various values of $\theta_z=0,0.125\pi,0.25\pi,\cdots,\pi$ (from bottom to top). The results are offset by 0.05 for adjacent values of $\theta_z$ for better visibility. d, Envelope frequency $\omega_\text{env}$ of the observed DTQCs as a function of the perturbation parameter $\varepsilon$ to the transverse field. e, Intensity of the side peaks, $A_\text{side}$, as a function of $\theta_{z}$. A schematic phase diagram showing the crossover between a DTC and a DTQC is provided at the top of the figure. Error bars in a- c and e represent the propagated error due to sampling errors in $\tilde{Z}(\omega)$. However, they are smaller than the size of symbols.
• Figure S1: Error-unmitigated raw data $\langle \hat{Z}_{\rm avg}(t) \rangle_{0}$ for the heavy hexagonal lattice of $L=28$ qubits obtained using ibm_torino (see Fig. 1a). The parameters $(\theta_x,\theta_z)$ are a$(\pi,0)$, b$(\pi,0.5\pi)$, c$(\pi,\pi)$, d$(0.9\pi,0)$, e$(0.9\pi,0.5\pi)$, f$(0.9\pi,\pi)$, g$(0.8\pi,0)$, h$(0.8\pi,0.5\pi)$, and i$(0.8\pi,\pi)$. Although smaller than the size of symbols in this scale, the statistical errors of measurements are estimated in the same manner as described in the main text.