We introduce a qudit-native framework for engineering robust discrete time crystals (DTCs) by leveraging their internal multilevel structure. Our approach confines the periodic drive to specified on-site subspaces, creating an embedded kick that suppresses heating by preventing population leakage to inactive levels. We underpin DTC stability with a normal-form analysis that decomposes the effective dynamics into distinct components: the carrier locks the subharmonic frequency, neutral terms govern the slow decay and dephasing of the subharmonic response, and charged terms scatter spectral weight away from the locked modes. This framework's predictive power is demonstrated across various qudit platforms: in spin-1 chains, we enhance the stability of DTC by confining the drive to a subspace; in spin-3/2 systems, we show that robustness is dictated by the symmetry of the subspace partition; and in spin-2 platforms, we realize concurrent 2T and 3T DTCs under a unified drive. These findings establish a systematic, hardware-efficient methodology for designing stable and multifunctional Floquet phases of matter on modern qudit-based quantum processors.
We introduce a qudit-native framework for engineering robust discrete time crystals (DTCs) by leveraging their internal multilevel structure. Our approach confines the periodic drive to specified on-site subspaces, creating an embedded kick that suppresses heating by preventing population leakage to inactive levels. We underpin DTC stability with a normal-form analysis that decomposes the effective dynamics into distinct components: the carrier locks the subharmonic frequency, neutral terms govern the slow decay and dephasing of the subharmonic response, and charged terms scatter spectral weight away from the locked modes. This framework's predictive power is demonstrated across various qudit platforms: in spin-1 chains, we enhance the stability of DTC by confining the drive to a subspace; in spin-3/2 systems, we show that robustness is dictated by the symmetry of the subspace partition; and in spin-2 platforms, we realize concurrent 2T and 3T DTCs under a unified drive. These findings establish a systematic, hardware-efficient methodology for designing stable and multifunctional Floquet phases of matter on modern qudit-based quantum processors.
We propose a Floquet time crystal model that responds in arbitrary multiples of the driving period. Such an n-tuple discrete time crystal is theoretically constructed by permutations in a disordered spin chain and is well suited for experimental implementations. Transitions between these time crystals with different periods give rise to a novel phase of matter that we call subspace-thermal discrete time crystals, where states within subspaces of definite charges are fully thermalized at an early time. However, the whole system still robustly responds to the periodic driving subharmonically, with a period being the greatest common divisor of the original two periods. Existing theoretical analysis from many-body localization cannot be used to understand the rigidity of such subspace-thermal time crystal phases. To resolve this, we develop a theoretical framework for the robustness of DTCs from the perspective of the robust $2\ensuremath{\pi}/n$ quasi-energy gap. Its robustness is rigorously proved if the system satisfies a certain condition where the mixing length, defined by the Hamming distance of the symmetry charges, does not exceed a global threshold. Although whether the condition is satisfied in generic disordered systems is unclear, the rigorous proof for DTC properties applies beyond the models considered here and extends to other existing DTCs realized by kicking disordered MBL systems, where the condition is automatically satisfied and conventional MBL-DTCs can be regarded as a special case of the subspace-thermal DTC with the subspace dimension being one, thus offering a systematic way to construct discrete time crystal models. We also introduce the notion of DTC charges that allow us to probe observables that spontaneously break the time-translation symmetry in both regular discrete time crystals and subspace-thermal discrete time crystals. Moreover, our discrete time crystal models can be generalized to systems with higher spin magnitudes or qudits, as well as to higher spatial dimensions.
The simulation of non-equilibrium quantum many-body systems on noisy intermediate-scale quantum (NISQ) processors represents a central challenge and opportunity in modern condensed matter physics. Periodically driven (Floquet) systems can host exotic phases of matter with no equilibrium counterpart, most notably discrete time crystals (DTCs) characterized by spontaneous breaking of discrete time-translation symmetry and robust subharmonic responses. However, the realization of DTCs faces fundamental obstacles: driven many-body systems generically absorb energy and thermalize to an infinite-temperature state, destroying time-crystalline order. Two primary mechanisms have been identified to stabilize DTCs—many-body localization (MBL) through strong disorder, and Floquet prethermalization wherein high-frequency driving suppresses heating over exponentially long timescales. This work investigates the digital quantum simulation of prethermal DTCs on current superconducting quantum processors, with a focus on understanding and mitigating the impact of hardware noise. Recent experiments on 133-qubit heavy-hexagonal devices have demonstrated the realization of clean two-dimensional DTCs and incommensurately modulated DTCs (IM-DTCs) using kicked Ising models, observing period-doubling oscillations persisting for up to 100 Floquet cycles . Crucially, these simulations reveal that quantum noise, often perceived as detrimental, can in certain regimes stabilize novel prethermal DTC phases—a phenomenon identified as noise-induced DTCs supported by symmetry charge pumping or random gate fluctuations . The heavy-hexagonal lattice geometry, compared to square lattices, exhibits significantly slower quantum scrambling, thereby extending the prethermal plateau and enhancing DTC stability. We address two key challenges limiting the scalability of DTC simulations: limited qubit connectivity and environmental noise. By leveraging the universality of native gate sets, we demonstrate how to implement effective couplings among physically disconnected qubits at the cost of increased circuit depth, enabling the study of interaction-range effects on DTC stabilization—a phenomenon previously unobserved experimentally . Furthermore, we benchmark the performance of error mitigation protocols based on depolarising noise models, achieving excellent agreement between error-mitigated hardware data and tensor-network simulations in regimes where classical computations remain tractable . Our results highlight the complementary roles of quantum hardware and classical simulation: quantum processors provide access to regimes where entanglement growth outpaces tensor-network methods, while careful validation against classical benchmarks ensures the reliability of quantum outcomes. This study establishes a concrete pathway for using NISQ devices as experimental platforms for synthesizing and probing novel non-equilibrium quantum phases, advancing both our fundamental understanding of Floquet many-body dynamics and the practical capabilities of quantum simulation. The identification of noise as an active ingredient in phase stabilization suggests new directions for quantum engineering of spatiotemporal orders on near-term hardware.
Discrete time crystals (DTCs) are non-equilibrium phases of matter that break the discrete time-translation symmetry and is characterized by a robust subharmonic response in periodically driven quantum systems. Here, we explore the DTC in a disorder-free, periodically kicked XXZ spin chain, which is stabilized by the Floquet strong Hilbert space fragmentation. We numerically show the period-doubling response of the conventional DTC order, and uncover a multiple-period response with beating dynamics due to the coherent interplay of multiple π-pairs in the Floquet spectrum of small-size systems. The lifetime of the DTC order exhibits independence of the driving frequency and a power-law dependence on the ZZ interaction strength. It also grows exponentially with the system size, as a hallmark of the strong fragmentation inherent to the Floquet model. We analytically reveal the approximate conservation of the magnetization and domain-wall number in the Floquet operator for the emergent strong fragmentation, which is consistent with numerical results of the dimensionality ratio of symmetry subspaces. The rigidity and phase regime of the DTC order are identified through finite-size scaling of the Floquet-spectrum-averaged mutual information, as well as via dynamical probes. Our work establishes the Floquet Hilbert space fragmentation as a disorder-free mechanism for sustaining nontrivial temporal orders in out-of-equilibrium quantum many-body systems.
Discrete Time Crystals (DTCs) are a non-equilibrium phase of matter that breaks the time-translation symmetry. This manifests as robust oscillations observed in driven Hamiltonians. Realizing DTCs on current Noisy Intermediate-Scale Quantum (NISQ) computers can be challenging due to the environmental noise, decoherence of qubits and computational errors. This thesis investigates the resilience and robustness of DTC and examines recent developments, and possible improvements of DTC stabilization using current quantum processors. Generally, an externally driven many-body system is expected to heat up to an infinite temperature state. This process, which is called thermalization, prevents the time-crystalline order. The two primary mechanisms that protect time-crystalline order are many-body localization (MBL) which has a strong disorder that prevents the thermalization, and Floquet prethermalization where its high frequency driving prevents the thermalization. These two processes allow us to realize and investigate the time crystals before a quantum system thermalizes.
Periodically driven quantum many-body systems can spontaneously break discrete time-translation symmetry, realizing discrete time crystals. To date, both experimental and theoretical efforts have largely focused on the simplest case of spontaneous period-doubling in Z2 discrete time crystals realized with qubits. This owes, in part, to the challenge of stabilizing eigenstate order in higher discrete symmetry (Zn) time crystals, due to the presence of richer domain wall physics. Here, we demonstrate the realization of a Z3 discrete time crystal by implementing a Floquet chiral clock model in a chain of 15 superconducting qutrits. Unlike the conventional Ising setting, our system features a tunable chiral angle that governs domain-wall dynamics, spectral degeneracies, and crucially, the stability of time-crystalline order. Using disordered nearest-neighbor chiral interactions, we observe robust subharmonic period tripling that persists across a wide range of drive strengths and is independent of initial state. Finally, we highlight the special role that chirality plays in our Z3 discrete time crystal -- in its absence, the system's Floquet dynamics exhibit a marked initial state dependence governed by domain wall degeneracies. Our results establish native qudit hardware as a powerful platform to access a broader landscape of non-equilibrium phases.
We propose a novel protocol to realize discrete time-crystal (DTC) order in clean, periodically driven spin-1/2 chains. In each drive cycle, a global spin flip is followed by a two-tone flat-band segment. This flat-band segment engineers a fully degenerate Floquet quasienergy spectrum, suppresses thermalization, and stabilizes a robust period-doubled subharmonic response. Using exact time evolution, we identify a pronounced subharmonic peak at half the drive frequency in the Fourier spectrum of the order parameter, thereby providing clear evidence for the emergence of stable DTC. The resulting phase is insensitive to system size, interaction strength, and interaction range; however, it remains sensitive to spin-rotation errors (εr), which can destabilize the subharmonic response. Compared with disorder-induced many-body localized (MBL) and disorder-free dynamically many-body localized (DMBL) DTCs, we find that the exact flat-band protocol offers a broader tunability of drive parameters, whereas MBL and DMBL based DTCs are more resistant to εr. In particular, the εr sensitivity can be suppressed by incorporating additional spin-spin interactions that have modest deviations from the ideal flat-band protocol. This manifests itself in a robust DTC response over a finite window of spin-coupling strengths and drive frequencies. Our results establish flat-band driving as a versatile and experimentally relevant route to DTC order in disorder-free spin systems and motivate further exploration of nonequilibrium phases.