Secure multiparty computation enables the joint evaluation of multivariate functions across distributed users while ensuring the privacy of their local inputs. This field has become urgent due to the demand for computationally intensive deep learning inference. These computations are typically offloaded to cloud servers, leading to vulnerabilities. To solve this problem, we introduce a linear algebra engine that leverages the quantum nature of light for information-theoretically secure multiparty inference using telecommunication components. We apply this linear algebra engine to deep learning and derive rigorous upper bounds on the information leakage of both the deep neural network weights and the client’s data, enabling double-blind operations. Applied to the modified National Institute of Standards and Technology classification task, we obtain test accuracies exceeding 95% while guaranteeing leakage of less than 0.1 bits per weight and data element. This leakage is an order of magnitude below the minimum bit precision required for accurate deep learning using state-of-the-art quantization techniques. Our work lays the foundation for practical quantum-secure computation and unlocks secure cloud deep learning as a field.

Efficient, low-noise, and high-bandwidth transduction between optical and microwave photons is key to long-range quantum communication between distant superconducting quantum processors. Recent demonstrations of microwave-optical transduction using the broadband direct electro-optic (Pockels) effect in optical thin films made of AlN or ${\mathrm{LiNbO}}_3$ have shown promise. To improve efficiency and added noise, materials with larger Pockels coefficients, such as the soft ferroelectrics ${\mathrm{BaTiO}}_3$ or ${\mathrm{SrTiO}}_3$, are needed. However, these materials require adapted designs and fabrication approaches due to their nonlinear and, in some cases, hysteretic electro-optic response. Here, we engineer an on-chip, triply resonant transducer comprising low-loss ${\mathrm{BaTiO}}_3$-on-${\mathrm{SiO}}_2$ waveguides monolithically integrated with a superconducting microwave resonator made of Nb. We demonstrate bidirectional microwave-optical transduction and reach total off-chip efficiencies of $1\times {10}^{-6}$ using pulsed pumping. The device design permits in situ poling of the ferroelectric material without introducing excess microwave loss, exploiting superconducting air bridges fabricated using only subtractive patterning steps. In addition, we investigate optically induced heating, revealing fast thermalization and quasiparticle resilience of the microwave resonator. Our transducer concept and fabrication process are applicable to other materials with a large bias-induced Pockels effect like ${\mathrm{SrTiO}}_3$ and pave the way for efficient, low-power quantum interconnects.

Elucidating fundamental limitations inherent in physical systems is a central subject in physics. For important thermodynamic operations such as information erasure, cooling, and copying, resources like time and energetic cost must be expended to achieve the desired outcome within a predetermined error margin. In the context of cooling, the unattainability principle of the third law of thermodynamics asserts that infinite “resources” are needed to reach absolute zero. However, the precise identification of relevant resources and how they jointly constrain achievable error remains unclear within the frameworks of stochastic and quantum thermodynamics. In this work, we introduce the concept of separated states, which consist of fully unoccupied and occupied states, and formulate the corresponding thermokinetic cost and error, thereby establishing a unifying framework for a broad class of thermodynamic operations. We then uncover a three-way trade-off relation between time, cost, and error for thermodynamic operations aimed at creating separated states, simply expressed as $\tau \mathcal{C}{ϵ}_{\tau }\ge 1-\eta$. This fundamental relation is applicable to diverse thermodynamic operations, including information erasure, cooling, and copying. It provides a profound quantification of the unattainability principle in the third law of thermodynamics in a general form. Building upon this relation, we explore the quantitative limitations governing cooling operations, the preparation of separated states, and a no-go theorem for exact classical copying. Furthermore, we extend these findings to the quantum regime, encompassing both Markovian and non-Markovian dynamics. Specifically, within Lindblad dynamics, we derive a similar three-way trade-off relation that quantifies the cost of achieving a pure state with a given error. The generalization to general quantum dynamics involving a system coupled to a finite bath implies that the dissipative cost becomes infinite as the quantum system is exactly cooled down to the ground state or perfectly reset to a pure state, thereby resolving an open question regarding the thermodynamic cost of information erasure.

Continuous-variable cat codes are encodings into a single photonic or phononic mode that offer a promising avenue for hardware-efficient fault-tolerant quantum computation. Protecting information in a cat code requires measuring the mode’s occupation number modulo two, but this can be relaxed to a linear occupation-number constraint using the alternative two-mode pair-cat encoding. We construct multimode codes with similar linear constraints using any two integer matrices satisfying a Calderbank-Shor-Steane-like homological condition of a quantum rotor code. Just like the pair-cat code, syndrome extraction can be performed in tandem with stabilizing dissipation using current superconducting-circuit designs. The framework includes codes with various finite- or infinite-dimensional code spaces and codes with finite or infinite Fock-state support. It encompasses two-component cat, pair-cat, dual-rail, two-mode binomial, various bosonic repetition codes, and aspects of $\chi$-squared encodings, while also yielding codes from homological products, lattices, generalized coherent states, and algebraic varieties. Among our examples are analogs of repetition codes, the Shor code, and a surface-code-like construction that is not a concatenation of a known cat code with the qubit surface code. Code words are coherent states projected into a Fock-state subspace defined by an integer matrix, and their overlaps are governed by Gelfand-Kapranov-Zelevinsky hypergeometric functions.

Entanglement can improve the measurement precision of quantum sensors beyond the shot noise limit. Neutral atoms, the basis of some of the most precise and accurate optical clocks and interferometers, do not naturally exhibit the all-to-all interactions traditionally used to generate such entangled states. On the other hand, these systems exhibit exceedingly high degrees of experimental control over parameters such as temperature, spatial entropy, and itinerancy. In this work, we investigate spin squeezing in a highly coherent itinerant system of neutral atoms with magnetic dipole-dipole interactions. We achieve 7.1 dB of metrologically useful squeezing using finite-range spin-exchange interactions in an erbium quantum gas microscope, and we demonstrate that introducing atomic motion, realizing a dipolar $t\text{−}J$ model, protects the spin sector coherence at low fillings, significantly improving the achievable spin squeezing in a 2D dipolar system. This work’s protocol can be implemented with most neutral atoms, opening the door to quantum-enhanced metrology in other itinerant dipolar systems, such as molecules or optical lattice clocks, and serves as a novel method for studying itinerant quantum magnetism with long-range interactions.

We provide a polynomial-time classical algorithm for noisy quantum circuits. The algorithm computes the expectation value of any observable for any circuit, with a small average error over input states drawn from an ensemble (e.g., the computational basis). Our approach is based upon the intuition that noise exponentially damps nonlocal correlations relative to local correlations. This enables one to classically simulate a noisy quantum circuit by keeping track of only the dynamics of local quantum information. Our algorithm also enables sampling from the output distribution of a circuit in quasipolynomial time, so long as the distribution anticoncentrates. A number of implications are discussed, including a fundamental limit on the efficacy of noise mitigation strategies: For constant noise rates, any quantum circuit for which error mitigation succeeds in polynomial-time on most input states can also be classically simulated in polynomial-time on most input states. Our algorithms scale exponentially in the inverse noise rate, which is fundamental and makes them impractical for current quantum devices.

The most advanced techniques using fault-tolerant quantum computers to estimate the ground-state energy of a chemical Hamiltonian involve compression of the Coulomb operator through tensor factorizations, enabling efficient block encodings of the Hamiltonian. A natural challenge of these methods is the degree to which block-encoding costs can be reduced. We address this challenge through the technique of spectral amplification, which magnifies the spectrum of the low-energy states of Hamiltonians that can be expressed as sums of squares. Spectral amplification enables estimating ground-state energies with significantly improved cost scaling in the block encoding normalization factor $\mathrm{\Lambda }$ to just $\sqrt{2\mathrm{\Lambda }{E}_{\mathrm{gap}}}$, where $E_{\mathrm{gap}}\ll \mathrm{\Lambda }$ is the lowest energy of the sum-of-squares Hamiltonian. To achieve this, we show that sum-of-squares representations of the electronic structure Hamiltonian are efficiently computable by a family of classical simulation techniques that approximate the ground-state energy from below. In order to further optimize, we also develop a novel factorization that provides a trade-off between the two leading Coulomb integral factorization schemes—namely, double factorization and tensor hypercontraction—that when combined with spectral amplification yields a factor of 4 to 195 speedup over the state of the art in ground-state energy estimation for models of iron-sulfur complexes and a ${\mathrm{CO}}_2$-fixation catalyst.

Control landscape phase transitions (CLPTs) occur as abrupt changes in the cost function landscape upon varying a control parameter and can be revealed by nonanalytic points in statistical order parameters. A prime example are quantum speed limits, which mark the onset of controllability as the protocol duration is increased. Here, we lay the foundations of an analytical theory for CLPTs by developing Dyson, Magnus, and cumulant expansions for the cost function that capture the behavior of CLPTs with a controlled precision. Using linear and quadratic stability analysis, we reveal that CLPTs can be associated with different types of instabilities of the optimal protocol. This allows us to explicitly relate CLPTs to critical structural rearrangements in the extrema of the control landscape: Utilizing path-integral methods from statistical field theory, we trace back the critical scaling of the order parameter at the quantum speed limit to the topological and geometric properties of the set of optimal protocols, such as the number of connected components and its dimensionality. We verify our predictions by introducing a numerical sampling algorithm designed to explore this optimal set via a homotopic stochastic update rule. We apply this new toolbox explicitly to analyze CLPTs in the single- and two-qubit control problems whose landscapes are analytically tractable and compare the landscapes for bang-bang and piecewise continuous protocols. Our work provides the first steps toward a systematic theory of CLPTs and paves the way for utilizing statistical field theory techniques for generic complex control landscapes.

The entropy accumulation theorem—and its subsequent generalized version—is a powerful tool in the security analysis of many device-dependent and device-independent cryptography protocols. However, it has the drawback that the finite-size bounds it yields are not necessarily optimal; furthermore, it relies on the construction of an affine min-tradeoff function, which can often be challenging to construct optimally in practice. In this work, we address both of these challenges simultaneously by deriving a new entropy accumulation bound. Our bound yields significantly better finite-size performance and can be computed as an intuitively interpretable convex optimization, without any specification of affine min-tradeoff functions. Furthermore, it can be applied directly at the level of Rényi entropies if desired, yielding fully-Rényi security proofs. Our proof techniques are based on elaborating on a connection between entropy accumulation and the frameworks of quantum probability estimation or $f$-weighted Rényi entropies; in the process, we obtain some new results with respect to those frameworks as well. In particular, those findings imply that our bounds apply to prepare-and-measure protocols without the virtual tomography procedures or repetition-rate restrictions previously required for entropy accumulation.

The realization of large-scale quantum computers requires not only quantum error correction but also fault-tolerant (FT) operations to handle errors that propagate into harmful errors. Recently, flag-based protocols have been introduced that use ancillary qubits to flag harmful errors. However, there is no clear recipe for finding a FT quantum circuit with flag-based protocols, especially when we consider hardware constraints, such as the qubit connectivity and available gate set. This work presents a novel approach to automatically discover compact and hardware-adapted FT quantum circuits to make significant progress towards scalable FT quantum computing. We employ reinforcement learning (RL) as an enabling tool, leveraging a fast, parallelized stabilizer quantum circuit simulator and a nontrivial reward function specifically adapted to the problem. We show that, in the task of FT logical state preparation, RL discovers not only circuits with fewer gates and ancillary qubits than published results but also novel circuits without and with hardware constraints of up to distance-5 codes with 25 physical qubits, and they can be implemented directly in experiments. Furthermore, RL allows for straightforward exploration of different qubit connectivities and the use of transfer learning to accelerate the discovery. More generally, our work sets the framework towards the use of RL or other machine learning techniques for FT quantum circuit discovery with hardware constraints to make real progress towards the realization of large-scale quantum computers, addressing tasks beyond state preparation, including magic state preparation, logical gate synthesis, and syndrome measurement.

Recent investigations have unveiled exotic quantum phases that elude characterization by simple bipartite correlation functions. In these phases, long-range entanglement arising from tripartite correlations plays a central role. Consequently, the study of multipartite correlations has become a focal point in modern physics. Here, conditional mutual information (CMI) is one of the most well-established information-theoretic measures, adept at encapsulating the essence of various exotic phases, including topologically ordered ones. Within the realm of quantum many-body physics, it has been a long-sought goal to establish a quantum analog to the Hammersley-Clifford theorem that bridges the two concepts of the Gibbs state and the Markov network. This theorem posits that the correlation length of CMI remains short-range across all thermal equilibrium quantum phases. In this work, we demonstrate that CMI exhibits exponential decay with respect to distance, with its correlation length increasing polynomially with respect to the inverse temperature. While this clustering theorem has previously been believed to hold for high temperatures devoid of thermal phase transitions, it has remained elusive at low temperatures, where genuine long-range entanglement is corroborated to exist by the quantum topological order. Our findings unveil that, even at low temperatures, a broad class of tripartite entanglement cannot manifest in the long-range regime. To achieve the proof, we establish a comprehensive formalism for analyzing the locality of effective Hamiltonians on subsystems, commonly known as the “entanglement Hamiltonian” or “Hamiltonian of mean force.” As one outcome of our analyses, we enhance the prior clustering theorem concerning bipartite entanglement. In essence, we investigate genuine bipartite entanglement that extends beyond the limitations of the positive-partial-transpose class.

Preparation of high-fidelity logical magic states has remained as a necessary but daunting step towards building a large-scale fault-tolerant quantum computer. One approach is to fault-tolerantly prepare a magic state in one code and then switch to another, a method known as code switching. We experimentally demonstrate this protocol on an ion-trap quantum processor, yielding a logical magic state encoded in an error-correcting code with state-of-the-art logical fidelity. Our experiment is based on the first demonstration of code switching between color codes, from the fifteen-qubit quantum Reed-Muller code to the seven-qubit Steane code. We prepare an encoded magic state in the Steane code with 82.58% probability, with an infidelity of at most $5.1(2.7)\times {10}^{-4}$. The reported infidelity is lower than the leading infidelity of the physical operations utilized in the protocol by a factor of at least 2.7, indicating the quantum processor is below the pseudothreshold. Furthermore, we create two copies of the magic state in the same quantum processor and perform a logical Bell basis measurement for a sample-efficient certification of the encoded magic state. The high-fidelity magic state can be combined with the already-demonstrated fault-tolerant Clifford gates, state preparation, and measurement of the 2D color code, completing a universal set of fault-tolerant computational primitives with logical error rates equal or better than the physical two-qubit error rate.

The ability to distribute heralded entanglement between distant matter nodes is a primitive for the implementation of large-scale quantum networks. Some of the most crucial requirements for future applications include high heralding rates at telecom wavelengths, multiplexed operation, and on-demand retrieval of stored excitations for synchronization of separate quantum links. Despite tremendous progress in various physical systems, the demonstration of telecom-heralded entanglement between quantum nodes featuring both multiplexed operation and on-demand retrieval remains elusive. In this work, we combine narrow band parametric photon-pair sources and solid-state quantum memories based on rare-earth doped crystals to demonstrate telecom-heralded entanglement between spatially separated spin-wave quantum memories with fully adjustable recall time and temporal multiplexing of 15 modes. In a first experiment, the storage in the spin state is conditioned on the entanglement heralding. We take advantage of the control over readout pulse phase to achieve feedforward conditional phase shifts on the stored photons depending on which heralding detector clicked. We exploit this effect to double the entanglement heralding rate for a given quantum state up to $510\text{counts}/\mathrm{s}$, with an associated detection rate of $0.32\text{counts}/\mathrm{s}$ and measured positive concurrence by up to 6 standard deviations. In a second experiment, we simulate the communication time of a long-distance link by implementing an unconditional storage scheme with a dead time of $100\mathrm{\mu }\mathrm{s}$. We take advantage of temporal multiplexing to increase the entanglement rates by a factor of 15 with respect to single mode storage, reaching a value of $22\text{counts}/\mathrm{s}$ per heralding detector. These results establish our architecture as a prime candidate for the implementation of scalable high-rate quantum network links.

Quantum communication is needed to build powerful quantum computers and establish reliable quantum networks. At its basis lies the ability to generate and distribute entanglement to separate quantum systems, which can be used to run remote quantum operations on them or teleport quantum states from one system to another with the help of classical channels. To this end, it is useful to harness the resource of continuous-variable (CV) entanglement, since it can be efficiently and unconditionally produced by squeezing light in a nonlinear medium and can be easily manipulated, distributed, and measured using standard components. While various aspects of CV-based quantum communication have been successfully demonstrated in the optical domain, some key capabilities, such as entanglement swapping, have been lacking in the microwave domain. Here, we demonstrate three key elements of CV-based microwave quantum communication: (i) a Josephson mixer operating as a nondegenerate two-mode entangler with maximum measured logarithmic negativity $E_N=1.5$, (ii) a quantum teleportation apparatus, capable of teleporting vacuum and coherent states with a maximum fidelity of 73%, which exceeds the 50% classical limit and is mainly limited by intermediate losses in the setup, and (iii) an entanglement-swapping system which generates entanglement between two remote noninteracting modes via entanglement-swapping operations applied to input vacuum and coherent states with maximum measured logarithmic negativity $E_N=0.53$. Such hardware-efficient CV entanglement building blocks that are based on nondegenerate Josephson mixers could enable wide-ranging applications in modular quantum computation, quantum cryptography, and quantum communication.

Understanding the physics of the two-dimensional Hubbard model is widely believed to be a key step in achieving a full understanding of high-$T_{\mathrm{c}}$ cuprate superconductors. In recent years, progress has been made by large-scale numerical simulations at finite doping and, on the other hand, by microscopic theories able to capture the physics of individual charge carriers. In this work, we study single pairs of dopants in a cylindrical system using the density-matrix renormalization group algorithm. We identify two coexisting charge configurations that couple to the spin environment in different ways: a tightly bound configuration featuring (next-)nearest-neighbor pairs and a stripelike configuration of dopants on opposite sides of the cylinder, accompanied by a spin domain wall. Thus, we establish that the interplay between stripe order and uniform pairing, central to the models’ phases at finite doping, has its origin at the single-pair level. By interpolating between the Hubbard and the related $t\text{−}J$ model, we are able to quantitatively understand discrepancies in the pairing properties of the two models through the three-site hopping term usually omitted from the $t\text{−}J$ Hamiltonian. This term is closely related to a next-nearest-neighbor tunneling $t^{\prime }$, which we observe to upset the balance between the competing stripe and pair states on the two-dopant level.

We initiate the study of state complexity for continuous-variable quantum systems. Concretely, we consider a setup with bosonic modes and auxiliary qubits, where available operations include Gaussian one- and two-mode operations and single- and two-qubit operations as well as qubit-controlled phase-space displacements. We define the (approximate) complexity of a bosonic state by the minimum size of a circuit that prepares an approximation to the state in trace distance. We propose a new circuit which prepares an approximate Gottesman-Kitaev-Preskill (GKP) state $|{\mathsf{G}\mathsf{K}\mathsf{P}}_{\kappa ,\mathrm{\Delta }}⟩$. Here, ${\kappa }^{-2}$ is the variance of the envelope, and ${\mathrm{\Delta }}^2$ is the variance of the individual peaks. We show that the circuit accepts with constant probability and—conditioned on acceptance—the output state is polynomially close in $(\kappa ,\mathrm{\Delta })$ to the state $|{\mathsf{G}\mathsf{K}\mathsf{P}}_{\kappa ,\mathrm{\Delta }}⟩$. The size of our circuit is linear in $(\log 1/\kappa ,\log 1/\mathrm{\Delta })$. To our knowledge, this is the first protocol for GKP-state preparation with fidelity guarantees for the prepared state. We also show converse bounds, establishing that the linear circuit-size dependence of our construction is optimal. This fully characterizes the complexity of GKP states.

The Středa formula establishes a fundamental physical connection between the topological invariants characterizing the bulk of topological matter and the presence of gapless edge modes. In this work, we extend the Středa formula to periodically driven systems, providing a rigorous framework to elucidate the unconventional bulk-boundary correspondence of Floquet systems while offering a physically grounded link between Floquet winding numbers and tractable response functions. Using the Sambe representation of periodically driven systems, we analyze the response of the unbounded Floquet density of states to a magnetic perturbation. This Floquet-Středa response is regularized through Cesàro summation, yielding a well-defined, quantized result within spectral gaps. The response features two physically distinct contributions: a quantized charge flow between edge and bulk and an anomalous energy flow between the system and the drive, offering new insight into the nature of anomalous edge states. This fundamental result rigorously connects Floquet winding numbers to the orbital magnetization density of Floquet states and holds broadly, from clean to disordered and inhomogeneous systems. This is further supported by providing a real-space formulation of the Floquet-Středa response, which introduces a local topological marker suited for periodically driven settings. In translationally invariant systems, the framework yields a remarkably simple expression for Floquet winding numbers involving geometric properties of Floquet-Bloch bands. A concrete experimental protocol is proposed to extract the Floquet-Středa response via particle-density measurements in systems coupled to engineered baths. Finally, by expressing the topological invariants through the magnetic response of the Floquet density of states, this approach opens a promising route toward the topological characterization of interacting driven phases.

Non-Abelian gauge theories provide the most accurate description of fundamental interactions, showing remarkable agreement with experimental data in cosmology and particle physics. Highly precise predictions can be made using standard techniques, both in the continuum and in the lattice frameworks. However, classical methods have limitations, particularly when attempting to extrapolate the continuum limit from the study of lattice gauge theories. Complementing classical computations or combining them with quantum computational methods, to improve the predictions toward the continuum limit with current quantum resources, is a formidable open challenge. In this paper, we propose a resource-efficient method to compute the running of the coupling in non-Abelian gauge theories beyond one spatial dimension. We first represent the Hamiltonian on periodic lattices in terms of loop variables and conjugate loop electric fields, exploiting the Gauss law to retain the gauge-independent ones. Then, we identify a local basis for small and large loops variationally to minimize the truncation error while computing the running of the coupling on small tori. Our method enables computations at arbitrary values of the bare coupling and lattice spacing with current quantum computers, simulators, and tensor-network calculations, in regimes otherwise inaccessible.

Recent progress in tabletop experiments offers the opportunity to show, for the first time, that gravity is not compatible with a classical description. In all current experimental proposals, such as the generation of gravitationally induced entanglement between two quantum sources of gravity, gravitational effects can be explained solely with the Newton potential, namely, in a regime that is consistent with the weak-field limit of general relativity and does not probe the field nature of gravity. Hence, the Newtonian origin of the effects is a limitation to the conclusions on the nature of gravity that can be drawn from these experiments. Here, we identify two effects that overcome this limitation: They cannot be reproduced using the Newton potential, and they are independent of graviton emission. First, we show that the interaction between two generic quantum sources of gravity, e.g., in wide Gaussian states, cannot be reproduced with the Newton potential nor with a known classical theory of gravity. Second, we show that the quantum commutator between the gravitational field and its canonically conjugate momentum appears as an additional term in the relative phase of a generic quantum source interacting with a test particle. Observing these two effects would give further quantitative information and stronger evidence, compared to experiments only involving the Newton potential, to show that gravity is nonclassical. More broadly, identifying stronger quantum aspects of gravity than those reproducible with the Newton potential is crucial to prove the nonclassicality of gravity and to plan a new generation of experiments testing quantum aspects of gravity in a broader sense than what has been proposed so far.

The classification of topological phases of matter is a fundamental challenge in quantum many-body physics, with applications to quantum technology. Recently, this classification has been extended to the setting of adaptive finite-depth local unitary (AFDLU) circuits, which allow global classical communication. In this setting, the trivial phase is the collection of all topological states that can be prepared via AFDLU. Here, we propose a complete classification of the trivial phase by showing how to prepare all solvable anyon theories that admit a gapped boundary via AFDLU, extending recent results on solvable groups. Our construction includes non-Abelian anyons with irrational quantum dimensions, such as Ising anyons, and more general acyclic anyons. Specifically, we introduce a sequential gauging procedure, with an AFDLU implementation, to produce a string-net ground state in any topological phase described by a solvable anyon theory with gapped boundary. In addition, we introduce a sequential ungauging and regauging procedure, with an AFDLU implementation, to apply string operators of arbitrary length for anyons and symmetry twist defects in solvable anyon theories. We apply our procedure to the quantum double of the group $S_3$ and to several examples that are beyond solvable groups, including the doubled Ising theory, the ${\mathbb{Z}}_3$ Tambara-Yamagami string net, and doubled $\mathrm{SU}(2{)}_4$ anyons.