Observation of Time-Crystalline Eigenstate Order on a Quantum Processor

Abstract: Quantum many-body systems display rich phase structure in their low-temperature equilibrium states. However, much of nature is not in thermal equilibrium. Remarkably, it was recently predicted that out-of-equilibrium systems can exhibit novel dynamical phases that may otherwise be forbidden by equilibrium thermodynamics, a paradigmatic example being the discrete time crystal (DTC). Concretely, dynamical phases can be defined in periodically driven many-body localized systems via the concept of eigenstate order. In eigenstate-ordered phases, the entire many-body spectrum exhibits quantum correlations and long-range order, with characteristic signatures in late-time dynamics from all initial states. It is, however, challenging to experimentally distinguish such stable phases from transient phenomena, wherein few select states can mask typical behavior. Here we implement a continuous family of tunable CPHASE gates on an array of superconducting qubits to experimentally observe an eigenstate-ordered DTC. We demonstrate the characteristic spatiotemporal response of a DTC for generic initial states. Our work employs a time-reversal protocol that discriminates external decoherence from intrinsic thermalization, and leverages quantum typicality to circumvent the exponential cost of densely sampling the eigenspectrum. In addition, we locate the phase transition out of the DTC with an experimental finite-size analysis. These results establish a scalable approach to study non-equilibrium phases of matter on current quantum processors.

• The paper demonstrates the experimental observation of a discrete time crystal (DTC) phase in a many-body localized Floquet system on a quantum processor.
• It employs continuous CPHASE gates on 20 superconducting qubits and an echo protocol to distinguish intrinsic non-equilibrium behavior from decoherence.
• The study validates consistent eigenstate order across variable disorder and initial states, establishing a scalable method for probing exotic dynamical orders.

Observation of Time-Crystalline Eigenstate Order on a Quantum Processor

This paper presents a significant advancement in the experimental investigation of non-equilibrium phases by demonstrating the observation of a discrete time crystal (DTC) in an interacting quantum many-body system. Utilizing an array of superconducting qubits, the researchers have provided evidence for the existence of time-crystalline order in a many-body localized (MBL) Floquet system, emphasizing the potential of modern quantum processors to host and investigate such exotic phases.

The study explores systems beyond equilibrium dynamics, focusing on a DTC—a phase breaking the discrete time-translation symmetry by exhibiting period-doubled oscillations. This paper highlights the challenge of experimentally validating a stable DTC phase due to the potential for transient effects, such as decoherence and finite-size limitations, to obscure true phase behavior. The authors employ a set of continuous CPHASE gates on a linear chain of 20 superconducting qubits to implement a periodically driven Floquet circuit, revealing stable MBL-DTC behavior that persists over a wide parameter range, even when varying disorder instances.

The crux of this work involves the application of a spatiotemporal characterization of DTCs. Through the use of an echo-type protocol, the study differentiates the intrinsic thermalization from external decoherence, ensuring that observations indicate genuine non-equilibrium behavior rather than experimental artifacts. The subharmonic response, alongside the noise-corrected autocorrelator results, strongly suggests an MBL-DTC phase. The authors implement advanced techniques from quantum typicality and scaling analysis to confirm the widespread presence of eigenstate order throughout the spectrum, rather than mere prethermal phenomena restricted to specific initial states.

Strong complementary results reinforce the distinguishing features of the MBL-DTC. First, the study demonstrates equivalent responses across different initial states, verifying eigenstate order. Secondly, it probes the logarithmic growth of operator spreads characteristic of MBL phases, as opposed to thermal phases where correlation spreads algebraically. The authors additionally provide a finite-size scaling analysis to estimate the phase transition from the DTC to the thermal phase.

The implications of these findings are considerable. The demonstrated protocols illustrate a scalable approach for studying and verifying exotic, non-equilibrium phases within quantum hardware limits. As quantum processors grow, they promise to significantly extend the accessibility of this research, overcoming classical simulation barriers.

This research sets a foundational blueprint for future work in analyzing non-equilibrium quantum phases, enabling exploration of time-crystals far from equilibrium and the broad classification of novel dynamical orders in controlled quantum systems. By refining the tools to experimentally realize eigenstate-ordered phases, this study opens pathways for deeper exploration into the computational and physical properties of time crystals and related phenomena in arbitrary quantum platforms.