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Quantum many-body systems display rich phase structure in their low-temperature equilibrium states1. However, much of nature is not in thermal equilibrium. Remarkably, it was recently predicted that out-of-equilibrium systems can exhibit novel dynamical phases2-8 that may otherwise be forbidden by equilibrium thermodynamics, a paradigmatic example being the discrete time crystal (DTC)7,9-15. Concretely, dynamical phases can be defined in periodically driven many-body-localized (MBL) systems via the concept of eigenstate order7,16,17. In eigenstate-ordered MBL 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, or from regimes in which the dynamics of afew select states can mask typical behaviour. Here we implement tunable controlled-phase (CPHASE) gates on an array of superconducting qubits to experimentally observe an MBL-DTC and demonstrate its characteristic spatiotemporal response for generic initial states7,9,10. Our work employs a time-reversal protocol to quantify the impact of external decoherence, and leverages quantum typicality to circumvent the exponential cost of densely sampling the eigenspectrum. Furthermore, we locate the phase transition out of the DTC with an experimental finite-size analysis. These results establish a scalable approach to studying non-equilibrium phases of matter on quantum processors.

Leakage of quantum information out of computational states into higher energy states represents a major challenge in the pursuit of quantum error correction (QEC). In a QEC circuit, leakage builds over time and spreads through multi-qubit interactions. This leads to correlated errors that degrade the exponential suppression of logical error with scale, challenging the feasibility of QEC as a path towards fault-tolerant quantum computation. Here, we demonstrate the execution of a distance-3 surface code and distance-21 bit-flip code on a Sycamore quantum processor where leakage is removed from all qubits in each cycle. This shortens the lifetime of leakage and curtails its ability to spread and induce correlated errors. We report a ten-fold reduction in steady-state leakage population on the data qubits encoding the logical state and an average leakage population of less than $1 \times 10^{-3}$ throughout the entire device. The leakage removal process itself efficiently returns leakage population back to the computational basis, and adding it to a code circuit prevents leakage from inducing correlated error across cycles, restoring a fundamental assumption of QEC. With this demonstration that leakage can be contained, we resolve a key challenge for practical QEC at scale., Comment: Main text: 7 pages, 5 figures

Realizing the potential of quantum computing will require achieving sufficiently low logical error rates. Many applications call for error rates in the $10^{-15}$ regime, but state-of-the-art quantum platforms typically have physical error rates near $10^{-3}$. Quantum error correction (QEC) promises to bridge this divide by distributing quantum logical information across many physical qubits so that errors can be detected and corrected. Logical errors are then exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold. QEC also requires that the errors are local and that performance is maintained over many rounds of error correction, two major outstanding experimental challenges. Here, we implement 1D repetition codes embedded in a 2D grid of superconducting qubits which demonstrate exponential suppression of bit or phase-flip errors, reducing logical error per round by more than $100\times$ when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analyzing error correlations with high precision, and characterize the locality of errors in a device performing QEC for the first time. Finally, we perform error detection using a small 2D surface code logical qubit on the same device, and show that the results from both 1D and 2D codes agree with numerical simulations using a simple depolarizing error model. These findings demonstrate that superconducting qubits are on a viable path towards fault tolerant quantum computing.

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.

As progress is made on thin-film synthesis of Heusler compounds, a more complete understanding of the surface will be required to control their properties, especially as functional heterostructures are explored. Here, the surface reconstructions of semiconducting half-Heusler NiTiSn(001), and Ni1+xTiSn(001) (x=0.0-1.0) are explored as a way to optimize growth conditions during molecular beam epitaxy. Density functional theory (DFT) calculations were carried out to guide the interpretation of the experimental results. For NiTiSn(001) a c(2x2) surface reconstruction was observed for Sn rich samples, while a (1x1) unreconstructed surface was observed for Ti-rich samples. A narrow range around 1:1:1 stoichiometry exhibited a (2x1) surface reconstruction. Electrical transport is used to relate the observed reflection high energy electron diffraction (RHEED) pattern during and after growth with carrier concentration and stoichiometry. Scanning tunneling microscopy and RHEED were used to examine surface reconstructions, the results of which are in good agreement with density functional calculations. X-ray photoelectron spectroscopy was used to determine surface termination and stoichiometry. Atomic surface models are proposed, which suggest Sn-dimers form in reconstructed Ni1+xTiSn(001) half-Heusler surfaces (x 0.25.

Epitaxial thin films of the substitutionally alloyed half-Heusler series CoTi$_{1-x}$Fe$_x$Sb were grown by molecular beam epitaxy on InAlAs/InP(001) substrates for concentrations 0.0$\leq$x$\leq$1.0. The influence of Fe on the structural, electronic, and magnetic properties was studied and compared to that expected from density functional theory. The films are epitaxial and single crystalline, as measured by reflection high-energy electron diffraction and X-ray diffraction. Using in-situ X-ray photoelectron spectroscopy, only small changes in the valence band are detected for x$\leq$0.5. For films with x$\geq$0.05, ferromagnetism is observed in SQUID magnetometry with a saturation magnetization that scales linearly with Fe content. A dramatic decrease in the magnetic moment per formula unit occurs when the Fe is substitutionally alloyed on the Co site indicating a strong dependence on the magnetic moment with site occupancy. A crossover from both in-plane and out-of-plane magnetic moments to only in-plane moment occurs for higher concentrations of Fe. Ferromagnetic resonance indicates a transition from weak to strong interaction with a reduction in inhomogeneous broadening as Fe content is increased. Temperature-dependent transport reveals a semiconductor to metal transition with thermally activated behavior for x$\leq$0.5. Anomalous Hall effect and large negative magnetoresistance (up to -18.5% at 100 kOe for x=0.3) are observed for higher Fe content films. Evidence of superparamagnetism for x=0.3 and x=0.2 suggests for moderate levels of Fe, demixing of the CoTi$_{1-x}$Fe$_x$Sb films into Fe rich and Fe deficient regions may be present. Atom probe tomography is used to examine the Fe distribution in a x=0.3 film. Statistical analysis reveals a nonhomogeneous distribution of Fe atoms throughout the film, which is used to explain the observed magnetic and electrical behavior.