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.

We propose a general framework for studying two-dimensional (2D) topologically ordered states subject to local correlated errors and show that the resulting mixed state can display intrinsically mixed-state topological order (imTO)—topological order that is not expected to occur in the ground state of 2D local gapped Hamiltonians. Specifically, we show that decoherence, previously interpreted as anyon condensation in a doubled Hilbert space, is more naturally phrased as, and provides a physical mechanism for, “gauging out” anyons in the original Hilbert space. We find that gauging out anyons generically results in imTO, with the decohered mixed state strongly symmetric under certain anomalous 1-form symmetries. This framework lays bare a striking connection between the decohered density matrix and topological subsystem codes, which can appear as anomalous surface states of three-dimensional topological orders. Through a series of examples, we show that the decohered state can display a classical memory, encode logical qubits (i.e., exhibit a quantum memory), and even host chiral or nonmodular topological order. We argue that a partial classification of imTO is given in terms of nonmodular braided-fusion categories.

We investigate how the computational difficulty of contracting tensor networks depends on the sign structure of the tensor entries. Using results from computational complexity, we observe that the approximate contraction of tensor networks with only positive entries has lower computational complexity as compared to tensor networks with general real or complex entries. This raises the question of how this transition in computational complexity manifests itself in the hardness of different tensor-network-contraction schemes. We pursue this question by studying random tensor networks with varying bias toward positive entries. First, we consider contraction via Monte Carlo sampling and find that the transition from hard to easy occurs when the tensor entries become predominantly positive; this can be understood as a tensor-network manifestation of the well-known negative-sign problem in quantum Monte Carlo. Second, we analyze the commonly used contraction based on boundary tensor networks. The performance of this scheme is governed by the number of correlations in contiguous parts of the tensor network (which by analogy can be thought of as entanglement). Remarkably, we find that the transition from hard to easy—i.e., from a volume-law to a boundary-law scaling of entanglement—already occurs for a slight bias of the tensor entries toward a positive mean, scaling inversely with the bond dimension $D$, and thus the problem becomes easy the earlier the larger $D$ occurs. This is in contrast both to expectations and to the behavior found in Monte Carlo contraction, where the hardness at fixed bias increases with the bond dimension. To provide insight into this early breakdown of computational hardness and the accompanying entanglement transition, we construct an effective classical statistical-mechanical model that predicts a transition at a bias of the tensor entries of $1/D$, confirming our observations. We conclude by investigating the computational difficulty of computing expectation values of tensor-network wave functions (projected entangled-pair states, PEPSs) and find that in this setting, the complexity of entanglement-based contraction always remains low. We explain this by providing a local transformation that maps PEPS expectation values to a positive-valued tensor network. This not only provides insight into the origin of the observed boundary-law entanglement scaling but also suggests new approaches toward PEPS contraction based on positive decompositions.

Quantum computing has been pursued with various hardware platforms and an optical system is one of the most reasonable choices for large-scale computation. In the optical continuous-variable computation scheme, the incorporation of Gaussian gates and a highly nonclassical non-Gaussian state enables universal quantum computation. Although basic technologies for Gaussian gates and non-Gaussian state generation have long been developed, these building blocks have not yet been integrated in a scalable fashion. Here, we integrate them to develop a scalable and programmable optical quantum computing platform that can sequentially perform an essential Gaussian gate, the squeezing gate, on a non-Gaussian input state. The key enablers are a loop-based optical circuit with dynamical and programmable controllability and its time synchronization with the probabilistic non-Gaussian state generation. We verify the deterministic, programmable, and repeatable quantum gates on a typical non-Gaussian state by implementing up to three-step gates. The gates implemented are of such high quality that strong evidence of the nonclassicalities of the states, the negativities of the Wigner functions, are preserved even after multistep gates. This platform is compatible with other non-Gaussian states and can in principle realize large-scale universal quantum computing by incorporating other existing processing technologies.

Midcircuit measurements (MCMs) are crucial ingredients in the development of fault-tolerant quantum computation. While there have been rapid experimental progresses in realizing MCMs, a systematic method for characterizing noisy MCMs is still under exploration. In this work, we develop a cycle benchmarking (CB)-type algorithm to characterize noisy MCMs. The key idea is to use a joint Fourier transform on the classical and quantum registers and then estimate parameters in the Fourier space, analogous to Pauli fidelities used in CB-type algorithms for characterizing the Pauli-noise channel of Clifford gates. Furthermore, we develop a theory of the noise learnability of MCMs, which determines what information can be learned about the noise model (in the presence of state preparation and terminating measurement noise) and what cannot, which shows that all learnable information can be learned using our algorithm. As an application, we show how to use the learned information to test the independence between measurement noise and state-preparation noise in an MCM. Finally, we conduct numerical simulations to illustrate the practical applicability of the algorithm. Similar to other CB-type algorithms, we expect the algorithm to provide a useful toolkit that is of experimental interest.

The second law of thermodynamics states that the entropy of an isolated system can only increase over time. This appears to conflict with the reversible evolution of isolated quantum systems under the Schrödinger equation, which preserves the von Neumann entropy. Nonetheless, one finds that with respect to many observables, expectation values approach a fixed value—their equilibrium value. This ultimately raises the question: in what sense does the entropy of an isolated quantum system increase over time? For classical systems, one introduces the assumption of a low-entropy initial state along with the concept of ignorance about the microscopic details of the physical system, leading to a statistical interpretation of the second law. By considering the observables through which we examine quantum systems, both these assumptions can be incorporated, building upon recent studies of the equilibration on average of observables. While the statistical behavior of observable expectation values is well established, a quantitative connection to entropy increase has been lacking so far. In deriving novel bounds for the equilibration of observables, and considering the entropy of the system relative to observables, we recover a variant of the second law: the entropy with respect to a given observable tends toward its equilibrium value in the course of the system’s unitary evolution. These results also support recent findings that question the necessity of nonintegrability for equilibration in quantum systems. We further illustrate our bounds using numerical results from the paradigmatic example of a quantum Ising model on a chain of spins. There, we observe entropy increasing up to equilibrium values, as well as fluctuations, which expose the underlying reversible evolution in accordance with the derived bounds.

Andreev (or superconducting) spin qubits (ASQs) have recently emerged as a promising qubit platform that combines superconducting circuits with semiconductor spin degrees of freedom. While recent experiments have successfully coupled two ASQs, how to realize a scalable architecture for extending this coupling to multiple distant qubits remains an open question. In this work, we resolve this challenge by introducing an architecture that achieves all-to-all connectivity between multiple remote ASQs. Our approach enables selective connectivity between any qubit pair while keeping all other qubit pairs uncoupled. Furthermore, we demonstrate the feasibility of efficient readout using circuit quantum electrodynamics techniques and compare different readout configurations. Our architecture shows promise both for gate-based quantum computing and for analog quantum simulation applications by offering higher qubit connectivity than alternative solid-state platforms.

Quantum measurements are a fundamental component of quantum computing. However, on present-day quantum computers, measurements can be more error prone than quantum gates and are susceptible to nonunital errors as well as nonlocal correlations due to measurement crosstalk. While readout errors can be mitigated in postprocessing, this is inefficient in the number of qubits due to a combinatorially large number of possible states that need to be characterized. In this work, we show that measurement errors can be tailored into a simple stochastic error model using randomized compiling, enabling the efficient mitigation of readout errors via quasiprobability distributions reconstructed from the measurement of a single preparation state in an exponentially large confusion matrix. We demonstrate the scalability and power of this approach by correcting readout errors without matrix inversion on a large number of different preparation states applied to a register of eight superconducting transmon qubits. Moreover, we show that this method can be extended to midcircuit measurements used for active feedback via quasiprobabilistic error cancellation, and we demonstrate the correction of measurement errors on an ancilla qubit used to detect and actively correct bit-flip errors on an entangled memory qubit. Our approach enables the correction of readout errors on large numbers of qubits and offers a strategy for correcting readout errors in adaptive circuits in which the results of midcircuit measurements are used to perform conditional operations on nonlocal qubits in real time.

Geometric locality is an important theoretical and practical factor for quantum low-density parity-check (qLDPC) codes that affects code performance and ease of physical realization. For device architectures restricted to two-dimensional (2D) local gates, naively implementing the high-rate codes suitable for low-overhead fault-tolerant quantum computing incurs prohibitive overhead. In this work, we present an error-correction protocol built on a bilayer architecture that aims to reduce operational overheads when restricted to 2D local gates by measuring some generators less frequently than others. We investigate the family of bivariate-bicycle qLDPC codes and show that they are well suited for a parallel syndrome-measurement scheme using fast routing with local operations and classical communication (LOCC). Through circuit-level simulations, we find that in some parameter regimes, bivariate-bicycle codes implemented with this protocol have logical error rates comparable to the surface code while using fewer physical qubits.

Photon loss is the dominant noise mechanism in photonic quantum technologies. Designing fault-tolerant schemes with high tolerance to loss is thus a central challenge in scaling photonic quantum information processors. Concatenation of a fault-tolerant construction with a code able to efficiently correct loss is a promising approach to achieve this but practical ways to implement code concatenation with photons have been lacking. We propose schemes for generating concatenated graph codes using multiphoton emission from two quantum emitters or a single quantum emitter coupled to a memory—capabilities available in several photonic platforms. We show that these schemes enable fault-tolerant fusion-based quantum computation in practical regimes with high photon loss and standard fusion gates without the need for auxiliary photons.

Quantum metrology offers the potential to surpass its classical counterpart, pushing the boundaries of measurement precision toward the ultimate Heisenberg limit. This enhanced precision is normally attained by utilizing large squeezed states or multiparticle entangled quantum states, both of which are often challenging to implement and prone to decoherence in real quantum devices. In this work, we present a versatile and on-demand protocol for deterministic parameter estimation that leverages two efficient state-transfer operations on a single bosonic mode. Specifically, we demonstrate this protocol in the context of phase estimation using the superposition of coherent states in the bosonic circuit quantum electrodynamics (cQED) platform. With low average photon numbers of only up to 1.76, we achieve quantum enhanced precision approaching the Heisenberg scaling, reaching a metrological gain of $7.5(6)$ dB. Importantly, we show that the gain or sensitivity range can be further enhanced on the fly by tailoring the input states, with different superposition weights, based on specific system constraints. The realization of this versatile and efficient scheme affords a promising path toward practical quantum enhanced sensing, not only for bosonic cQED hardware but also readily extensible to other continuous-variable platforms.

Quantum state reconstruction is an essential element in quantum information processing. However, efficient and reliable reconstruction of nontrivial quantum states in the presence of hardware imperfections can be challenging. This task is particularly demanding for high-dimensional states encoded in continuous-variable (CV) systems, where a large number of grid-based measurements are often used to adequately sample relevant regions of phase space. In this work, we introduce an efficient and robust technique of optimized reconstruction with excitation-number sampling (ORENS) based on the idea of the generalized $Q$ function. We use a standard bosonic circuit quantum electrodynamics (cQED) setup to experimentally demonstrate effective state reconstruction using the theoretically minimum number of measurements. Our investigation highlights that ORENS is naturally free of parasitic system dynamics and resilient to decoherence effects in the hardware, enabling it to outperform the conventional reconstruction techniques in cQED such as Wigner tomography. Finally, ORENS relies only on the ability to accurately measure the excitation number of a given CV state, making it a versatile and accessible tool for a wide range of CV platforms and readily scalable to multimode systems. Thus, our work provides a crucial and valuable primitive for practical quantum information processing using bosonic modes.

Long-term quantum storage for quantum communication applications will require optically accessible states with coherence times exceeding the desired storage time. Ground-state hyperfine levels in $\mathrm{Eu}$${}^{3+}$:$\mathrm{Y}$${}_2$$\mathrm{Si}\mathrm{O}$${}_5$ have previously shown coherence times as long as 6 h at 1.4 K when operated at a zero-first-order-Zeeman (ZEFOZ) point. Here, we have characterized the limiting decoherence mechanisms acting in this regime, which shows that the coherence time is limited by the presence of spin impurities. This dephasing effect has been suppressed by cooling the sample to 125 mK, resulting in an extension in the coherence time to over 13 h. In the absence of these impurities, the dominant source of dephasing is the field inhomogeneity of the ZEFOZ point arising from the spin inhomogeneous broadening. We have shown that coherence times over 18 h can be achieved by operating using a spectrally narrow subensemble selected from within the inhomogeneous line. The observed coherence time is longer than that of any other system suitable for an optical quantum memory. Furthermore, at a much higher temperature of 6 K, we have realized a 6-h coherence time, which has been obtainable using single-stage cryocoolers. These results imply advanced engineering feasibility in designing and operating future quantum information-storage devices for a wide range of applications, including in satellites.

Neutral-atom arrays have emerged as a versatile platform toward scalable quantum computation and optimization. In this paper, we present demonstrations of solving maximum-weighted independent-set problems on a Rydberg-atom array using annealing with local light shifts. We verify the ability to prepare weighted graphs in one-dimensional (1D) and two-dimensional (2D) arrays, including embedding a five-vertex nonunit-disk graph using nine physical qubits and demonstration of a simple crossing gadget. We find common annealing ramps leading to preparation of the target ground state robustly over a substantial range of different graph weightings. This work provides a route to exploring large-scale optimization of nonplanar weighted graphs relevant for solving relevant real-world problems.

Successful implementations of quantum technologies require protocols and algorithms that use as few quantum resources as possible. However, many important quantum operations, such as continuous rotation gates in quantum computing or broadband pulses in NMR or magnetic resonance imaging (MRI) applications, can only be implemented approximately using finite quantum resources. This work develops an approach that, on average—at the cost of a modestly increased measurement repetition rate—enables exact implementations. One proceeds by first building a library of a large number of different approximations to the desired gate operation; by randomly selecting these operations according to a preoptimized probability distribution, one can on average implement the desired operation with a rigorously controllable approximation error. The approach relies on sophisticated tools from convex optimization to efficiently find optimal probability distributions. A diverse spectrum of applications are demonstrated as (a) exactly synthesizing rotations in fault-tolerant quantum computers using only low $T$-count circuits and (b) synthesizing broadband and band-selective pulses of superior performance in quantum optimal control with (c) further applications in NMR or MRI. The approach is very general and a broad spectrum of practical applications in quantum technologies are explicitly demonstrated.

Quantum correlations, like entanglement, represent the characteristic trait of quantum mechanics and pose essential issues and challenges to the interpretation of this pillar of modern physics. Although quantum correlations are largely acknowledged as a major resource to achieve quantum advantage in many tasks of quantum technologies, their full quantitative description and the axiomatic basis underlying them are still under investigation. Previous works have suggested that the origin of nonlocal correlations is grounded in principles capturing (from outside the quantum formalism) the essence of quantum uncertainty. In particular, the recently introduced principle of relativistic independence has given rise to a new bound intertwining local and nonlocal correlations. Here, we test such a bound by realizing together sequential and joint weak measurements on entangled photon pairs, allowing us to simultaneously quantify both local and nonlocal correlations by measuring incompatible observables on the same quantum system without collapsing its state, a task typically forbidden in the traditional (projective) quantum measurement framework. Our results demonstrate the existence of a fundamental limit on the extent of quantum correlations, shedding light on the profound role of uncertainty in both enabling and balancing them.

Indistinguishability of identical particles is a resource for quantum information processing and has been utilized to generate entanglement from independent particles that spatially overlap only at the detection stage. Here, we introduce a controllable scheme capable of generating via three steps, i.e., initialization, deformation, and postselection, different classes of multipartite entangled states starting from a product state of $N$ spatially distinguishable identical qubits. While our scheme is generalizable to any class of entangled bosonic and fermionic systems, we provide an explicit recipe for the generation of $W$, Dicke, GHZ, and cluster states, which are resource states for quantum information processing. Using graph-based representations within the framework of spatially localized operations and classical communication (sLOCC), we mathematically demonstrate a direct translation of the generation schemes of specific entangled states into colored, complex, and weighted digraphs, each corresponding to a given experimental setup. We also show that this graph-theoretical approach allows for the optimization of the generation efficiency of specific multipartite entangled states by exploring a variety of generation schemes. The presented theoretical approach, while already implementable with current linear optics architectures, has the potential to bring clear advantages over existing technologies, such as in quantum computing search algorithms and in the design of new experiments in quantum optics or other platforms.