The fidelity of entangling operations is a key figure of merit in quantum information processing, especially in the context of quantum error correction. High-fidelity entangling gates in neutral atoms have seen remarkable advancement recently. A full understanding of error sources and their respective contributions to gate infidelity will enable the prediction of fundamental limits on quantum gates in neutral atom platforms with realistic experimental constraints. In this work, we implement the time-optimal Rydberg controlled-Z (CZ) gate, design a circuit to benchmark its fidelity, and achieve a fidelity, averaged over symmetric input states, of $0.9971(5)$, downward corrected for leakage error, which together with our recent work [Nature 634, 321–327 (2024)] forms a new state of the art for neutral atoms. The remaining infidelity is explained by an ab initio error model, consistent with our experimental results over a range of gate speeds, with varying contributions from different error sources. Further, we develop a fidelity response theory to efficiently predict infidelity from laser noise with nontrivial power spectral densities and derive scaling laws of infidelity with gate speed. Besides its capability of predicting gate fidelity, we also utilize the fidelity response theory to compare and optimize gate protocols, to learn laser frequency noise, and to study the noise response for quantum simulation tasks. Finally, we predict that a CZ gate fidelity of $\gtrsim 0.999$ is feasible with realistic experimental upgrades.

Although the similarity between nonstabilizer states—also known as magic states—in discrete-variable systems and non-Gaussian states in continuous-variable systems has widely been recognized, the precise connections between these two notions have still been unclear. We establish a fundamental link between these two quantum resources via the Gottesman-Kitaev-Preskill (GKP) encoding. We show that the negativity of the continuous-variable Wigner function for an encoded GKP state coincides with a magic measure we introduce, which matches the negativity of the discrete Wigner function for odd dimensions. We also provide a continuous-variable representation of the stabilizer Rényi entropy—a recent proposal for a magic measure for multiqubit states. We further provide the magic measure with an operational interpretation as a runtime of a classical simulation algorithm. In addition, we employ our results to prove that implementing a multiqubit logical non-Clifford operation in the GKP code subspace requires a non-Gaussian operation even at the limit of perfect encoding, despite the fact that the ideal GKP states already come with a large amount of non-Gaussianity.

Measurements and feedback have emerged as powerful resources for creating many-body quantum states. However, a detailed understanding has been restricted to fixed-point representatives of phases of matter. Here, we go beyond this and characterize the patterns of many-body entanglement that can be deterministically created from measurement. Focusing on one spatial dimension, a framework is developed for the case where a single round of measurements is the only entangling operation. We show this creates matrix-product states and identify necessary and sufficient tensor conditions for preparability, which uniquely determine the preparation protocol. We use these conditions to both classify preparable quantum states and characterize their physical constraints. In particular, we find a trade-off between the richness of the preparable entanglement spectrum and correlation functions, which leads to a no-go theorem for preparing certain quantum states. More broadly, we connect properties of the preparation protocol to the resulting phase of matter, including trivial, symmetry-breaking, and symmetry-protected topological phases—for both uniform and modulated symmetries. This work offers a resource-theoretic perspective on preparable quantum entanglement and shows how to systematically create states of matter, away from their fixed points, in quantum devices.

The Mølmer-Sørensen gate is a widely used entangling gate for ion platforms with inherent robustness to trap heating. The gate performance is limited by coherent errors, arising from the Lamb-Dicke (LD) approximation and sideband errors. Here, we provide explicit analytical formulas for errors up to fourth order in the LD parameter, by using the Magnus expansion to match numerical precision, and overcome significant orders-of-magnitude underestimation of errors by previous theory methods. We show that fourth-order Magnus expansion terms are unavoidable, being in fact leading order in LD, and are therefore critical to include for typical experimental fidelity ranges. We show how these errors can be dramatically reduced compared to previous theory by using analytical renormalization of the drive strength, by calibration of the Lamb-Dicke parameter, and by the use of smooth pulse shaping.

Topologically ordered phases are stable to local perturbations, and topological quantum error-correcting codes enjoy thresholds to local errors. We connect the two notions of stability by constructing classical statistical mechanics models for decoding general Calderbank-Shor-Steane codes and classical linear codes. Our construction encodes correction success probabilities under uncorrelated bit-flip and phase-flip errors, and simultaneously describes a generalized ${\mathbb{Z}}_2$ lattice-gauge theory with quenched disorder. We observe that the clean limit of the latter is precisely the discretized imaginary-time path integral of the corresponding quantum code Hamiltonian when the errors are turned into a perturbative $X$ or $Z$ magnetic field. Motivated by error-correction considerations, we define general order parameters for all such generalized ${\mathbb{Z}}_2$ lattice-gauge theories, and show that they are generally lower bounded by success probabilities of error correction. For CSS codes satisfying the low-density parity-check condition and with a sufficiently large code distance, we prove the existence of a low-temperature ordered phase of the corresponding lattice-gauge theories, particularly for those lacking Euclidean spatial locality and/or when there is a nonzero code rate. We further argue that these results provide evidence for stable phases in the corresponding perturbed quantum Hamiltonians, obtained in the limit of continuous imaginary time. To do so, we distinguish space- and timelike defects in the lattice-gauge theory. A high free-energy cost of spacelike defects corresponds to a successful “memory experiment” and suppresses the energy splitting among the ground states, while a high free-energy cost of timelike defects corresponds to a successful “stability experiment” and points to a nonzero gap to local excitations.

We present a new style of fluxonium qubit, gatemonium, based on an all-superconductor-semiconductor hybrid platform. The linear inductance is achieved using $600$ planar $\mathrm{Al}$-$\mathrm{In}\mathrm{As}$ Josephson junctions (JJs) in series. By tuning the single junction with a gate voltage, we demonstrate electrostatic control of the effective Josephson energy, tuning the weight of the fictitious phase particle. One- and two-tone spectroscopy of the gatemonium transitions further reveal details of the hybrid plasmon-fluxon spectrum. Accounting for the nonsinusoidal current-phase relation of the single junction, we fit the measured spectra to extract charging and inductive energies. We conduct time-domain characterization of the plasmon modes and find that energy-relaxation times are limited by inductive loss, possibly in the thin aluminum film. We discuss future directions for this platform in gate-voltage-tunable high-plasma-frequency enhanced-impedance junction arrays, and enhanced coherence times for voltage-tunable architectures.

Isolated interacting quantum systems generally thermalize, yet there are several examples for the breakdown of ergodicity, such as many-body localization and quantum scars. Recently, ergodicity breaking has been observed in systems subjected to linear potentials, termed Stark many-body localization. This phenomenon is closely associated with Hilbert-space fragmentation, characterized by a strong dependence of dynamics on initial conditions. Here, we explore initial-state-dependent dynamics using a ladder-type superconducting processor with up to 24 qubits, which enables precise control of the qubit frequency and initial-state preparation. In systems with linear potentials, we experimentally observe distinct nonequilibrium dynamics for initial states with the same quantum numbers and energy, but with varying domain-wall numbers. Accompanied by the numerical simulation for systems with larger sizes, we reveal that this distinction becomes increasingly pronounced as the system size grows, in contrast with weakly disordered interacting systems. Our results provide convincing experimental evidence of the fragmentation in Stark systems, enriching our understanding of the weak breakdown of ergodicity.

The interplay between symmetries and entanglement in out-of-equilibrium quantum systems is currently at the center of an intense multidisciplinary research effort. Here we introduce a setting where these questions can be characterized exactly by considering dual-unitary circuits with an arbitrary number of $U(1)$ charges. After providing a complete characterization of these systems we show that one can introduce a class of solvable states, which extends that of generic dual-unitary circuits, for which the nonequilibrium dynamics can be solved exactly. In contrast to the known class of solvable states, which relax to the infinite-temperature state, these states relax to a family of nontrivial generalized Gibbs ensembles. The relaxation process of these states can be simply described by a linear growth of the entanglement entropy followed by saturation to a nonmaximal value but with maximal entanglement velocity. We then move on to consider the dynamics from nonsolvable states, combining the exact results with the entanglement membrane picture we argue that the entanglement dynamics from these states is qualitatively different from that of the solvable ones. It shows two different growth regimes characterized by two distinct slopes, both corresponding to submaximal entanglement velocities. Moreover, we show that nonsolvable initial states can give rise to the quantum Mpemba effect, where less symmetric initial states restore the symmetry faster than more symmetric ones.

Realistic multiqubit noise processes often result in error mechanisms that are not captured by the probabilistic Markovian error models commonly employed in circuit-level analyses of quantum fault tolerance. By working within an open-quantum system Hamiltonian formulation, we revisit the validity of the notion of a constant gate error in the presence of noise that is both temporally correlated and nonclassical, and the impact of which is mitigated through perfect instantaneous dynamical decoupling subject to finite timing constraints. We study a minimal exactly solvable single-qubit model under Gaussian quantum dephasing noise, showing that the fidelity of a dynamically protected idling gate can depend strongly on its location in the circuit and the history of applied control, even when the system-side error propagation is fully removed through perfect reset operations. For digital periodic control, we prove that, under mild conditions on the low-frequency behavior of the nonclassical noise spectrum, the gate fidelity saturates at a value that is strictly smaller than the one attainable in the absence of control history; the presence of high-frequency noise peaks is also found as especially harmful, due to the possible onset of control-induced resonances. We explicitly relate these features to the evolution of the bath statistics during the computation, which has not been fully accounted for in existing treatments. We find that only if decoupling can keep the qubit highly pure over a time scale larger than the correlation time of the noise, the bath approximately converges to its original statistics and a stable-in-time control performance is recovered. Our work highlights the significance of the full bath evolution between circuit locations and suggests that additional trade-offs and design constraints may arise in layered quantum fault-tolerant architectures from the need to appropriately reequilibrate the quantum bath statistics, particularly when Markovian noise is present alongside temporally correlated noise.

Neutral atoms are among the leading platforms toward realizing fault-tolerant quantum computation (FTQC). However, scaling up a single neutral-atom device beyond ${10}^4$ atoms to meet the demands of FTQC for practical applications remains a challenge. To overcome this challenge, we clarify the criteria and technological requirements for further scaling based on multiple neutral atom quantum processing units (QPUs) connected through photonic networking links. Our quantitative analysis shows that nanofiber optical cavities have the potential as an efficient atom-photon interface to enable fast entanglement generation between atoms in distinct neutral-atom modules, allowing multiple neutral-atom QPUs to operate cooperatively without sacrificing computational speed. Using state-of-the-art millimeter-scale nanofiber cavities with the finesse of thousands, over a hundred atoms can be coupled to the cavity mode with an optical tweezer array, with expected single-atom cooperativity exceeding 100 for telecom-band transition of ytterbium atoms. This enables efficient time-multiplexed entanglement generation with a predicted Bell-pair generation rate of ${10}^5{\mathrm{s}}^{-1}$ while maintaining a small footprint for channel multiplexing. These proposals and results indicate a promising pathway for building large-scale multiprocessor fault-tolerant quantum computers using neutral atoms, nanofiber optical cavities, and fiber-optic networks.

We present an experimental technique that enables the preparation of defect-free arrays of ${}^{87}$$\mathrm{Rb}$ atoms within a microscopic high-finesse optical standing-wave cavity. By employing optical tweezers, we demonstrate atom positioning with a precision well below the cavity wavelength, a crucial requirement for cavity-QED experiments in which maximum atom-cavity coupling strength is required. We leverage our control capabilities to assemble an array of up to seven atoms with an efficiency that exceeds previous probabilistic methods by 4 orders of magnitude. The atoms are subsequently transferred from the tweezer array to a two-dimensional intracavity optical lattice that offers enhanced coherence for spin qubits while maintaining strong atom confinement. Our system overcomes the efficiency limitations of previous probabilistic loading techniques of cavity-coupled atom arrays and opens the path to multiqubit quantum networks with atoms strongly coupled to optical cavities.

Scaling up quantum computing devices requires solving ever more complex quantum control tasks. Machine learning has been proposed as a promising approach to tackle the resulting challenges. However, experimental implementations are still scarce. In this work, we demonstrate experimentally a neural-network-based preparation of Schrödinger cat states in a cavity coupled dispersively to a qubit. We show that it is possible to teach a neural network to output optimized control pulses for a whole family of quantum states. After being trained in simulations, the network takes a description of the target quantum state as input and rapidly produces the pulse shape for the experiment, without any need for time-consuming additional optimization or retraining for different states. Our experimental results demonstrate more generally how deep neural networks and transfer learning can produce efficient simultaneous solutions to a range of quantum control tasks, which will benefit not only state preparation but also parametrized quantum gates.

Hybrid tensor networks (hTNs) offer a promising solution for encoding variational quantum states beyond the capabilities of efficient classical methods or noisy quantum computers alone. However, their practical usefulness and many operational aspects of hTN-based algorithms, like the optimization of hTNs, the generalization of standard contraction rules to an hybrid setting, and the design of application-oriented architectures have not been thoroughly investigated yet. In this work, we introduce a novel algorithm to perform ground-state optimizations with hybrid tree tensor networks (hTTNs), discussing its advantages and roadblocks, and identifying a set of promising applications. We benchmark our approach on two paradigmatic models, namely the Ising model at the critical point and the Toric-code Hamiltonian. In both cases, we successfully demonstrate that hTTNs can improve upon classical equivalents with equal bond dimension in the classical part.

The experimental realization of increasingly complex quantum states underscores the pressing need for new methods of state learning and verification. In one such framework, quantum state tomography, the aim is to learn the full quantum state from data obtained by measurements. Without prior assumptions on the state, this task is prohibitively hard. Here, we present an efficient algorithm for learning states on $n$ fermion modes prepared by any number of Gaussian and at most $t$ non-Gaussian gates. By Jordan-Wigner mapping, this also includes $n$-qubit states prepared by nearest-neighbor matchgate circuits with at most $t$ swap gates. Our algorithm is based exclusively on single-copy measurements and produces a classical representation of a state, guaranteed to be close in trace distance to the target state. The sample and time complexity of our algorithm is $\mathrm{poly}(n,2^t)$; thus if $t=\mathcal{O}(\log (n))$, it is efficient. We also show that, if $t$ scales slightly more than logarithmically, any learning algorithm to solve the same task must be inefficient, under common cryptographic assumptions. We also provide an efficient property-testing algorithm that, given access to copies of a state, determines whether such a state is far or close to the set of states for which our learning algorithm works. In addition to the outputs of quantum circuits, our tomography algorithm is efficient for some physical target states, such as those arising in time dynamics and low-energy physics of impurity models. Beyond tomography, our work sheds light on the structure of states prepared with few non-Gaussian gates and offers an improved upper bound on their circuit complexity, enabling an efficient circuit-compilation method.

Quantum technologies rely on the control of quantum systems at the level of individual quanta. Mathematically, this control is described by Hamiltonian or Liouvillian evolution, requiring the application of various techniques to solve the resulting dynamic equations. Here, we present a tutorial for how the quantum dynamics of systems can be solved using a Lie-algebra decoupling method. The approach involves identifying a Lie algebra that governs the dynamics of the system, enabling the derivation of differential equations to solve the Schrödinger equation. As background, we include an overview of Lie groups and Lie algebras aimed at a general-physicist audience. We then prove the Lie-algebra decoupling theorem and apply it to both closed and open dynamics. The results represent a broad methodology to find the dynamics of quantum systems with applications across many fields of modern quantum research.

We show that quantum-number-preserving ansatzes for variational optimization in quantum chemistry find an elegant mapping to ultracold fermions in optical superlattices. Using native Hubbard dynamics, trial ground states of molecular Hamiltonians can be prepared and their molecular energies measured in the lattice. The scheme requires local control over interactions and chemical potentials and global control over tunneling dynamics, but foregoes the need for optical tweezers, shuttling operations, or long-range interactions. We describe a complete compilation pipeline from the molecular Hamiltonian to the sequence of lattice operations, thus providing a concrete link between quantum simulation and chemistry. Our work enables the application of recent quantum algorithmic techniques, such as double factorization and quantum tailored coupled cluster, to present-day fermionic optical lattice systems with significant improvements in the required number of experimental repetitions. We provide detailed quantum resource estimates for small nontrivial hardware experiments.

The barren plateau phenomenon, characterized by loss gradients that vanish exponentially with system size, poses a challenge to scaling variational quantum algorithms. Here we explore the potential of warm starts, whereby one initializes closer to a solution in the hope of enjoying larger loss variances. Focusing on an iterative variational method for learning shorter-depth circuits for quantum real-time evolution we conduct a case study to elucidate the potential and limitations of warm starts. We start by proving that the iterative variational algorithm will exhibit substantial (at worst vanishing polynomially in system size) gradients in a small region around the initializations at each time step. Convexity guarantees for these regions are then established, suggesting trainability for polynomial-size time steps. However, our study highlights scenarios where a good minimum shifts outside the region with trainability guarantees. Our analysis leaves open the question whether such minima jumps necessitate optimization across barren plateau landscapes or whether there exist gradient flows, i.e., fertile valleys away from the plateau with substantial gradients, that allow for training. While our main focus is on this case study of variational quantum simulation, we end by discussing how our results work in other iterative settings.

We study methods to replace entangling operations with random local operations in a quantum computation, at the cost of increasing the number of required executions. First, we consider “spacelike cuts” where an entangling unitary is replaced with random local unitaries. We propose an entanglement measure for quantum dynamics, the product extent, which bounds the cost in a procedure for this replacement based on two copies of the Hadamard test. In the terminology of prior work, this procedure yields a quasiprobability decomposition with minimal 1-norm in a number of cases, which addresses an open question of Piveteau and Sutter. As an application, we give an improved algorithm for clustered Hamiltonian simulation. Specifically we show that interactions can be removed at a cost, which is exponential in the sum of their strengths times the evolution time, and vanishing in the limit of weak interactions. We also give an improved upper bound on the cost of replacing wires with measure-and-prepare channels using “timelike cuts.” We prove a matching information-theoretic lower bound when estimating output probabilities.

The classification and characterization of topological phases of matter is well understood for ground states of gapped Hamiltonians that are well isolated from the environment. However, decoherence due to interactions with the environment is inevitable—thus motivating the investigation of topological orders in the context of mixed states. Here, we take a step toward classifying mixed-state topological orders in two spatial dimensions by considering their (emergent) generalized symmetries. We argue that their 1-form symmetries and the associated anyon theories lead to a partial classification under two-way connectivity by quasilocal quantum channels. This allows us to establish mixed-state topological orders that are intrinsically mixed, i.e., that have no ground-state counterpart. We provide a wide range of examples based on topological subsystem codes, decohering $G$-graded string-net models, and “classically gauging” symmetry-enriched topological orders. One of our main examples is an Ising string-net model under the influence of dephasing noise. We study the resulting space of locally indistinguishable states and compute the modular transformations within a particular coherent space. Based on our examples, we identify two possible effects of quasilocal quantum channels on anyon theories: (1) anyons can be incoherently proliferated—thus reducing to a commutant of the proliferated anyons, or (2) the system can be “classically gauged,” resulting in the symmetrization of anyons and an extension by transparent bosons. Given these two mechanisms, we conjecture that mixed-state topological orders are classified by premodular anyon theories, i.e., those for which the braiding relations may be degenerate.

Decoherence is a major obstacle to the preparation of topological order in noisy intermediate-scale quantum devices. Here, we show that decoherence can also give rise to new types of topological order. Specifically, we construct concrete examples by proliferating fermionic anyons in the toric code via local quantum channels. The resulting mixed states retain long-range entanglement, which manifests in the nonzero topological entanglement negativity, though the topological quantum memory is destroyed by decoherence. By comparison with the gapless spin liquid in pure states, we show that the identified states represent a novel intrinsic mixed-state topological order, which has no counterpart in pure states. Through the lens of quantum anomalies of 1-form symmetries, we then provide general constructions of intrinsic mixed-state topological order and reveal the existence of nonbosonic deconfined anyons as another key feature of these novel phases. The extended meaning and characterization of deconfined excitations and their statistics in mixed states are clarified. Moreover, when these deconfined anyons have nontrivial braiding statistics, we prove that the mixed states cannot be prepared via finite-depth local quantum channels from any bipartite separable states. We further demonstrate our construction using the decohered Kitaev honeycomb model and the decohered double-semion model. In the latter case, a surprising scenario arises where decoherence gives rise to additional types of deconfined anyons.