Efficiently estimating properties of large and strongly coupled quantum systems is a central focus in many-body physics and quantum information theory. While quantum computers promise speedups for many of these tasks, near-term devices are prone to noise that will generally reduce the accuracy of such estimates. Here, we propose a sample-efficient and noise-resilient protocol for learning properties of quantum states building on the shadow estimation scheme [Huang et al., Nature Physics 16, 1050–1057 (2020)]. By introducing an experimentally friendly calibration procedure, our protocol can efficiently characterize and mitigate noises in the shadow estimation scheme, given only minimal assumptions on the experimental conditions. When the strength of noises can be bounded, our protocol approximately retains the same order of sample efficiency as the standard shadow estimation scheme, while also possessing a provable noise resilience. We give rigorous bounds on the sample complexity of our protocol and demonstrate its performance with several numerical experiments, including estimations of quantum fidelity, correlation functions and energy expectations, etc., which highlight a wide spectrum of potential applications of our protocol on near-term devices.

The entanglement properties of random pure states are relevant to a variety of problems ranging from chaotic quantum dynamics to black-hole physics. The averaged bipartite entanglement entropy of such states admits a volume law and upon increasing the subregion size follows the Page curve. In this paper, we generalize this setup to random mixed states by coupling the system to a bath and use the partial transpose to study their entanglement properties. We develop a diagrammatic method to incorporate partial transpose within random matrix theory and formulate a perturbation theory in $1/L$, the inverse of the Hilbert-space dimension. We compute several quantities including the spectral density of partial transpose (or entanglement negativity spectrum), two-point correlator of eigenvalues, and the logarithmic negativity. As long as the bath is smaller than the system, we find that upon sweeping the subregion size, the logarithmic negativity shows an initial increase and a final decrease similar to the Page curve, while it admits a plateau in the intermediate regime where the logarithmic negativity depends only on the size of the system and of the bath but not on how the system is partitioned. This intermediate phase has no analog in random pure states, and is separated from the two other regimes by a critical point. We further show that when the bath is larger than the system by at least two extra qubits the logarithmic negativity is identically zero, which implies that there is no distillable entanglement. Using the diagrammatic approach, we provide a simple derivation of the semicircle law of the entanglement negativity spectrum in the latter two regimes. We show that despite the appearance of a semicircle distribution, reminiscent of Gaussian unitary ensemble (GUE), the higher-order corrections to the negativity spectrum and two-point correlator deviate from those of GUE.

Recent progress in the realm of noisy intermediate-scale quantum (NISQ) devices [J. Preskill, Quantum 2, 79 (2018)] represents an exciting opportunity for many-body physics by introducing new laboratory platforms with unprecedented control and measurement capabilities. We explore the implications of NISQ platforms for many-body physics in a practical sense: we ask which physical phenomena, in the domain of quantum statistical mechanics, they may realize more readily than traditional experimental platforms. While a universal quantum computer can simulate any system, the eponymous noise inherent to NISQ devices practically favors certain simulation tasks over others in the near term. As a particularly well-suited target, we identify discrete time crystals (DTCs), novel nonequilibrium states of matter that break time translation symmetry. These can only be realized in the intrinsically out-of-equilibrium setting of periodically driven quantum systems stabilized by disorder-induced many-body localization. While promising precursors of the DTC have been observed across a variety of experimental platforms—ranging from trapped ions to nitrogen-vacancy centers to NMR crystals—none have all the necessary ingredients for realizing a fully fledged incarnation of this phase, and for detecting its signature long-range spatiotemporal order. We show that a new generation of quantum simulators can be programmed to realize the DTC phase and to experimentally detect its dynamical properties, a task requiring extensive capabilities for programmability, initialization, and readout. Specifically, the architecture of Google’s Sycamore processor is a remarkably close match for the task at hand. We also discuss the effects of environmental decoherence, and how they can be distinguished from ‘internal’ decoherence coming from closed-system thermalization dynamics. Already with existing technology and noise levels, we find that DTC spatiotemporal order would be observable over hundreds of periods, with parametric improvements to come as the hardware advances.

The field of materials for quantum information science is rapidly growing with a focus on scaling and integrating new solid-state qubits. However, despite the extraordinary progress, major challenges must still be overcome for scaling. Specifically, there is a critical need for efficient coupling of tens to hundreds of solid-state qubits and multiqubit error correction to mitigate environmental interactions. This Perspective looks forward to the challenges ahead in the realization of multiqubit operations and collective phenomena with a focus on solid-state quantum materials. We provide a theorists’ point of view on the modeling and rational design of these multiqubit systems. Our Perspective identifies a path for bridging the gap between the model Hamiltonians used to develop quantum algorithms and control sequences and the ab initio calculations used to understand and characterize single solid-state-based qubits.

The development of robust architectures capable of large-scale fault-tolerant quantum computation should consider both their quantum error-correcting codes and the underlying physical qubits upon which they are built, in tandem. Following this design principle, we demonstrate remarkable error-correction performance by concatenating the XZZX surface code with Kerr-cat qubits. We contrast several variants of fault-tolerant systems undergoing different circuit-noise models that reflect the physics of Kerr-cat qubits. Our simulations show that our system is scalable below a threshold gate infidelity of $p_{\mathrm{CX}}\sim 6.5\mathrm{\%}$ within a physically reasonable parameter regime, where $p_{\mathrm{CX}}$ is the infidelity of the noisiest gate of our system, the controlled-not gate. This threshold can be reached in a superconducting-circuit architecture with a Kerr nonlinearity of $10\mathrm{MHz}$, an approximately $6.25$-photon cat qubit, single-photon lifetime of $\gtrsim 64\mu \mathrm{s}$, and a thermal photon population $\lesssim 8\mathrm{\%}$. Such parameters are routinely achieved in superconducting circuits.

We study the reduced dynamics of open quantum spin chains of arbitrary length $N$ with nearest-neighbor $XX$ interactions, immersed within an external constant magnetic field along the $z$ direction, the end spins of which are weakly coupled to heat baths at different temperatures, via energy-preserving couplings. We find the analytic expression of the unique stationary state of the master equation obtained in the so-called global approach based on the spectralization of the full-chain Hamiltonian. Hinging upon the explicit stationary state, we reveal the presence of sink and source terms in the spin-flow continuity equation and compare their behavior with that of the stationary heat flow. Moreover, we also obtain analytic expressions for the steady-state two-spin reduced density matrices and for their concurrence. We then set up an algorithm suited to compute the stationary bipartite entanglement along the chain and to study its dependence on the Hamiltonian parameters and on the bath temperatures.

Optimal control theory is a powerful mathematical tool, which has known a rapid development since the 1950s, mainly for engineering applications. More recently, it has become a widely used method to improve process performance in quantum technologies by means of highly efficient control of quantum dynamics. This tutorial aims at providing an introduction to key concepts of optimal control theory that is accessible to physicists and engineers working in quantum control or in related fields. The different mathematical results are introduced intuitively, before being rigorously stated. This tutorial describes modern aspects of optimal control theory, with a particular focus on the Pontryagin maximum principle, which is the main tool for determining open-loop control laws without experimental feedback. The different steps to solve an optimal control problem are discussed, before moving on to more advanced topics such as the existence of optimal solutions or the definition of the different types of extremals, namely normal, abnormal, and singular. The tutorial covers various quantum control issues and describes their mathematical formulation suitable for optimal control. The connection between the Pontryagin maximum principle and gradient-based optimization algorithms used for high-dimensional quantum systems is described. The optimal solution of different low-dimensional quantum systems is presented in detail, illustrating how the mathematical tools are applied in a practical way.

Continuous-variable (CV) cluster states are a universal quantum computing platform that has experimentally outscaled qubit platforms by orders of magnitude. Room-temperature implementation of CV cluster states has been achieved with quantum optics by using multimode squeezed Gaussian states. It has also been proven that fault tolerance thresholds for CV quantum computing can be reached at realistic squeezing levels. In this paper, we show that standard approaches to design and characterize CV cluster states can miss entanglement present in the system. Such hidden entanglement may be used to increase the power of a quantum computer but it can also, if undetected, hinder the successful implementation of a quantum algorithm. By a detailed analysis of the structure of Gaussian states, we derive an algorithm that reveals hidden entanglement in an arbitrary Gaussian state and optimizes its use for one-way quantum computing.

We present a quantum algorithm for the dynamical simulation of time-dependent Hamiltonians. Our method involves expanding the interaction-picture Hamiltonian as a sum of generalized permutations, which leads to an integral-free Dyson series of the time-evolution operator. Under this representation, we perform a quantum simulation for the time-evolution operator by means of the linear combination of unitaries technique. We optimize the time steps of the evolution based on the Hamiltonian’s dynamical characteristics, leading to a gate count that scales with an $L^1$-norm-like scaling with respect only to the norm of the interaction Hamiltonian, rather than that of the total Hamiltonian. We demonstrate that the cost of the algorithm is independent of the Hamiltonian’s frequencies, implying its advantage for systems with highly oscillating components, and for time-decaying systems the cost does not scale with the total evolution time asymptotically. In addition, our algorithm retains the near optimal $\log (1/ϵ)/\log \log (1/ϵ)$ scaling with simulation error $ϵ$.

We propose a superconducting quantum circuit based on a general symmetry principle—combinatorial gauge symmetry—designed to emulate topologically ordered quantum liquids and serve as a foundation for the construction of topological qubits. The proposed circuit exhibits rich features: in the classical limit of large capacitances its ground state consists of two superimposed loop structures; one is a crystal of small loops containing disordered $U(1)$ degrees of freedom, and the other is a gas of loops of all sizes associated with ${\mathbb{Z}}_2$ topological order. We show that these classical results carry over to the quantum case, where phase fluctuations arise from the presence of finite capacitances, yielding ${\mathbb{Z}}_2$ quantum topological order. A key feature of the exact gauge symmetry is that amplitudes connecting different ${\mathbb{Z}}_2$ loop states arise from paths having zero classical energy cost. As a result, these amplitudes are controlled by dimensional confinement rather than tunneling through energy barriers. We argue that this effect may lead to larger energy gaps than previous proposals that are limited by such barriers, potentially making it more likely for a topological phase to be experimentally observable. Finally, we discuss how our superconducting circuit realization of combinatorial gauge symmetry can be implemented in practice.

We propose a way to use optical tools from quantum imaging and quantum communication to search for physics beyond the standard model. Spontaneous parametric down-conversion (SPDC) is a commonly used source of entangled photons in which pump photons convert to a signal-idler pair. We propose to search for “dark-SPDC” (dSPDC) events in which a new dark-sector particle replaces the idler. Though it does not interact, the presence of a dark particle can be inferred by the properties of the signal photon. Examples of dark states include axionlike particles and dark photons. We show that the presence of an optical medium opens the phase space of the down-conversion process, or decay, which would be forbidden in a vacuum. Search schemes are proposed that employ optical imaging and/or spectroscopy of the signal photons. The signal rates in our proposal scales with the second power of the small coupling to new physics, as opposed to light-shining-through-wall experiments, the signal of which scales with coupling to the fourth power. We analyze the characteristics of the optical media needed to enhance dSPDC and estimate the rate.

A unitary $t$-design is a powerful tool in quantum information science and fundamental physics. Despite its usefulness, only approximate implementations were known for general $t$. In this paper, we provide quantum circuits that generate exact unitary $t$-designs for any $t$ on an arbitrary number of qubits. Our construction is inductive and is of practical use in small systems. We then introduce a $t\mathrm{th}$-order generalization of randomized benchmarking ($t$-RB) as an application of exact $2t$-designs. We particularly study the $2$-RB in detail and show that it reveals self-adjointness of quantum noise, a metric related to the feasibility of quantum error correction (QEC). We numerically demonstrate that the $2$-RB in one- and two-qubit systems is feasible, and experimentally characterize background noise of a superconducting qubit by the $2$-RB. It is shown from the experiment that interactions with adjacent qubits induce the noise that may result in an obstacle toward a realization of QEC.

Artificial atoms realized by superconducting circuits offer unique opportunities to store and process quantum information with high fidelity. Among them, implementations of circuits that harness intrinsic noise protection have been rapidly developed in recent years. These noise-protected devices constitute a new class of qubits in which the computational states are largely decoupled from local noise channels. The main challenges in engineering such systems are simultaneously guarding against both bit- and phase-flip errors, and also ensuring high-fidelity qubit control. Although partial noise protection is possible in superconducting circuits relying on a single quantum degree of freedom, the promise of complete protection can only be fulfilled by implementing multimode or hybrid circuits. This Perspective reviews the theoretical principles at the heart of these new qubits, describes recent experiments, and highlights the potential of robust encoding of quantum information in superconducting qubits.

Mistrustful cryptography includes important tasks like bit commitment, oblivious transfer, coin flipping, secure computations, position authentication, digital signatures and secure unforgeable tokens. Practical quantum implementations presently use photonic setups. In many such implementations, Alice sends photon pulses encoding quantum states and Bob chooses measurements on these states. In practice, Bob generally uses single-photon threshold detectors, which cannot distinguish the number of photons in detected pulses. Also, losses and other imperfections require Bob to report the detected pulses. Thus, malicious Alice can send and track multiphoton pulses and thereby gain information about Bob’s measurement choices, violating the protocols’ security. Here, we provide a theoretical framework for analyzing such multiphoton attacks, and present known and new attacks. We illustrate the power of these attacks with an experiment, and study their application to earlier experimental demonstrations of mistrustful quantum cryptography. We analyze countermeasures based on selective reporting and prove them inadequate. We also discuss side-channel attacks where Alice controls further degrees of freedom or sends other physical systems.

We evaluate the usefulness of holographic stabilizer codes for fault-tolerant quantum computing by studying their allowed sets of fault-tolerantly implementable gates. We treat them as subsystem codes and show that the set of transversally implementable logical operations is contained in the Clifford group for sufficiently localized logical subsystems. As well as proving this concretely for several specific codes, we argue that this restriction naturally arises in any stabilizer subsystem code that comes close to capturing certain properties of holography. We extend these results to approximate encodings, locality-preserving gates, and certain codes whose logical algebras have nontrivial centers, and discuss cases where restrictions can be made to other levels of the Clifford hierarchy. A few auxiliary results may also be of interest, including a general definition of an entanglement wedge map for any subsystem code, and a thorough classification of different correctability properties for regions in a subsystem code.

Inspired by biological molecular machines, we explore the idea of an active quantum robot the purpose of which is to delay decoherence. A conceptual model capable of partially protecting arbitrary logical qubit states against single-physical-qubit errors is presented. Implementation of an instance of that model—the entanglement qubot—is proposed using laser-dressed Rydberg atoms. The dynamics of the system are studied using stochastic wave-function methods.

Quantum supermaps are transformations that map quantum operations to quantum operations. It is known that quantum supermaps which respect a definite, predefined causal order between their input operations correspond to fixed-order quantum circuits, also called quantum combs. A systematic understanding of the physical interpretation of more general types of quantum supermaps—in particular, those incompatible with a definite causal structure—is however lacking. In this paper, we identify two types of circuits that naturally generalize the fixed-order case and that likewise correspond to distinct classes of quantum supermaps, which we fully characterize. We first introduce “quantum circuits with classical control of causal order,” in which the order of operations is still well defined, but not necessarily fixed in advance: it can, in particular, be established dynamically, in a classically controlled manner, as the circuit is being used. We then consider “quantum circuits with quantum control of causal order,” in which the order of operations is controlled coherently. The supermaps described by these classes of circuits are physically realizable, and the latter encompasses all known examples of physically realizable processes with indefinite causal order, including the celebrated “quantum switch.” Interestingly, it also contains other examples arising from the combination of dynamical and coherent control of causal order, and we detail explicitly one such process. Nevertheless, we show that quantum circuits with quantum control of causal order can only generate “causal” correlations, compatible with a well-defined causal order. We furthermore extend our considerations to probabilistic circuits that produce also classical outcomes, and we demonstrate by an example how the characterizations derived in this work allow us to identify advantages for quantum information processing tasks that could be demonstrated in practice.

Quantum computing is in its greatest upswing, with so-called noisy-intermediate-scale-quantum devices heralding the computational power to be expected in the near future. While the field is progressing toward quantum advantage, quantum computers already have the potential to tackle classically intractable problems. Here, we consider gauge theories describing fundamental-particle interactions. On the way to their full-fledged quantum simulations, the challenge of limited resources on near-term quantum devices has to be overcome. We propose an experimental quantum simulation scheme to study ground-state properties in two-dimensional quantum electrodynamics (2D QED) using existing quantum technology. Our protocols can be adapted to larger lattices and offer the perspective to connect the lattice simulation to low-energy observable quantities, e.g., the hadron spectrum, in the continuum theory. By including both dynamical matter and a nonminimal gauge-field truncation, we provide the novel opportunity to observe 2D effects on present-day quantum hardware. More specifically, we present two variational-quantum-eigensolver- (VQE) based protocols for the study of magnetic field effects and for taking an important first step toward computing the running coupling of QED. For both instances, we include variational quantum circuits for qubit-based hardware. We simulate the proposed VQE experiments classically to calculate the required measurement budget under realistic conditions. While this feasibility analysis is done for trapped ions, our approach can be directly adapted to other platforms. The techniques presented here, combined with advancements in quantum hardware, pave the way for reaching beyond the capabilities of classical simulations.

In holonomic quantum computation, quantum gates are performed using driving protocols that trace out closed loops on the Bloch sphere, making them robust to certain pulse errors. However, dephasing noise that is transverse to the drive, which is significant in many qubit platforms, lies outside the family of correctable errors. Here, we present a general procedure that combines two types of geometry—holonomy loops on the Bloch sphere and geometric space curves in three dimensions—to design gates that simultaneously suppress pulse errors and transverse noise errors. We demonstrate this doubly geometric control technique by designing explicit examples of single-qubit and two-qubit dynamically corrected holonomic gates.

Quantum many-body systems are characterized by patterns of correlations defining highly nontrivial manifolds when interpreted as data structures. Physical properties of phases and phase transitions are typically retrieved via correlation functions, that are related to observable response functions. Recent experiments have demonstrated capabilities to fully characterize quantum many-body systems via wave-function snapshots, opening new possibilities to analyze quantum phenomena. Here, we introduce a method to data mine the correlation structure of quantum partition functions via their path integral (or equivalently, stochastic series expansion) manifold. We characterize path-integral manifolds generated via state-of-the-art quantum Monte Carlo methods utilizing the intrinsic dimension (ID) and the variance of distances between nearest-neighbor (NN) configurations: the former is related to data-set complexity, while the latter is able to diagnose connectivity features of points in configuration space. We show how these properties feature universal patterns in the vicinity of quantum criticality, that reveal how data structures simplify systematically at quantum phase transitions. This is further reflected by the fact that both ID and variance of NN distances exhibit universal scaling behavior in the vicinity of second-order and Berezinskii-Kosterlitz-Thouless critical points. Finally, we show how non-Abelian symmetries dramatically influence quantum data sets, due to the nature of (noncommuting) conserved charges in the quantum case. Complementary to neural-network representations, our approach represents a first elementary step towards a systematic characterization of path-integral manifolds before any dimensional reduction is taken, that is informative about universal behavior and complexity, and can find immediate application to both experiments and Monte Carlo simulations.