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.

The ability to shuttle coherently individual electron spins in arrays of quantum dots is a key procedure for the development of scalable quantum information platforms. It allows the use of sparsely populated electron spin arrays, envisioned to efficiently tackle the one- and two-qubit gate challenges. When the electrons are displaced in an array, they are exposed to site-dependent environment interactions such as hyperfine coupling with substrate nuclear spins. Here, we demonstrate that the electron multidirectional displacement in a $3\times 3$ array of tunnel-coupled quantum dots enhances the spin-coherence time via the motional narrowing phenomenon. More specifically, up to ten charge configurations are explored by the electrons to study the impact of the displacement on spin dynamics. An increase of the coherence time by a factor up to 10 is observed in the case of fast and repetitive displacement. A simple model quantitatively captures the physical mechanism underlying this enhancement of the spin-coherence time induced by displacement. The implications on spin-coherence properties during the electron displacement are discussed in the context of large-scale quantum circuits.

Deriving the laws of thermodynamics from a microscopic picture is a central quest of statistical mechanics. This tutorial focuses on the derivation of the first and second law for isolated and open quantum systems far from equilibrium, where such foundational questions also become practically relevant for emergent nanotechnologies. The derivation is based on a microscopic definition of five essential quantities: internal energy, thermodynamic entropy, work, heat, and temperature. These definitions are shown to satisfy the phenomenological laws of nonequilibrium thermodynamics for a large class of states and processes. The consistency with previous results is demonstrated. The framework applies to multiple baths including particle transport and accounts for processes with, e.g., a changing temperature of the bath, which is determined microscopically. An integral fluctuation theorem for entropy production is satisfied. In summary, this tutorial introduces a consistent and versatile framework to understand and apply the laws of thermodynamics in the quantum regime and beyond.

If gravitational perturbations are quantized into gravitons in analogy with the electromagnetic field and photons, the resulting graviton interactions should lead to an entangling interaction between massive objects. We suggest a test of this prediction. To do this, we introduce the concept of interactive quantum information sensing. This novel sensing protocol is tailored to provable verification of weak dynamical entanglement generation between a pair of systems. We show that this protocol is highly robust to typical thermal noise sources. Moreover, the sensitivity can be increased both using an initial thermal state and/or an initial phase of entangling via a nongravitational interaction. We outline a concrete implementation testing the ability of the gravitational field to generate entanglement between an atomic interferometer and a mechanical oscillator. Preliminary numerical estimates suggest that near-term devices could feasibly be used to perform the experiment.

Violating Bell’s inequalities (BIs) allows one to certify the preparation of entangled states from minimal assumptions—in a device-independent manner. Finding BIs tailored to many-body correlations as prepared in present-day quantum computers and simulators is however a highly challenging endeavor. In this work, we focus on BIs violated by very coarse-grain features of the system: two-body correlations averaged over all permutations of the parties. For two-outcome measurements, specific BIs of this form have been theoretically and experimentally studied in the past, but it is practically impossible to explicitly test all such BIs. Data-driven methods—reconstructing a violated BI from the data themselves—have therefore been considered. Here, inspired by statistical physics, we develop a novel data-driven approach specifically tailored to such coarse-grain data. Our approach offers two main improvements over the existing literature: (1) it is directly designed for any number of outcomes and settings; (2) the obtained BIs are quadratic in the data, offering a fundamental scaling advantage for the precision required in experiments. This very flexible method, whose complexity does not scale with the system size, allows us to systematically improve over all previously known Bell inequalities robustly violated by ensembles of quantum spin $1/2$; and to discover novel families of Bell inequalities, tailored to spin-squeezed states and many-body spin singlets of arbitrary spin-$j$ ensembles.

Characterizingthe noise in the set of gate operations that form the building blocks of a quantum computational device is a necessity for assessing the quality of the device. Here, we introduce randomized linear gate-set tomography, an easy-to-implement gate-set-tomography procedure that combines the idea of state-preparation-and-measurement-error-free characterization of standard gate-set tomography with no-design randomized tomographic circuits and computational ease brought about by an appropriate linear approximation. We demonstrate the performance of our scheme through simulated examples as well as experiments done on the IBM Quantum Experience Platform. In each case, we see that the performance of our procedure is comparable with that of standard gate-set tomography, while requiring no complicated tomographic circuit design and taking much less computational time in deducing the estimate of the noise parameters. This allows for straightforward on-the-fly characterization of the gate operations in an experiment.

We show that a ${\mathbb{Z}}_3$ quantum double can be realized in an array of superconducting wires coupled via Josephson junctions. With a suitably chosen magnetic flux threading the system, the interwire Josephson couplings take the form of a complex Hadamard matrix, which possesses combinatorial gauge symmetry—a local ${\mathbb{Z}}_3$ symmetry involving permutations and shifts by $\pm 2\pi /3$ of the superconducting phases. The sign of the star potential resulting from the Josephson energy is inverted in this physical realization, leading to a massive degeneracy in the nonzero flux sectors. A dimerization pattern encoded in the capacitances of the array lifts up these degeneracies, resulting in a ${\mathbb{Z}}_3$ topologically ordered state. Moreover, this dimerization pattern leads to a larger effective vison gap as compared to the canonical case with the usual (uninverted) star term. We further show that our model maps to a quantum three-state Potts model under a duality transformation. We argue, using a combination of bosonization and mean field theory, that altering the dimerization pattern of the capacitances leads to a transition from the ${\mathbb{Z}}_3$ topological phase into a quantum $XY$-ordered phase. Our work highlights that combinatorial gauge symmetry can serve as a design principle to build quantum double models using systems with realistic interactions.

Novel quantum technologies and devices place unprecedented demands on the performance of experimental components, while their widespread deployment beyond the laboratory necessitates increased robustness and fast affordable production. We show how the use of additive manufacturing, together with mathematical optimization techniques and innovative designs, allows the production of compact lightweight components with greatly enhanced performance. We use such components to produce a magneto-optical trap that captures approximately $2\times {10}^8$ rubidium atoms, employing for this purpose a compact and highly stable device for spectroscopy and optical power distribution, optimized neodymium magnet arrays for magnetic field generation, and a lightweight additively manufactured ultrahigh-vacuum chamber. We show how the use of additive manufacturing enables substantial weight reduction and stability enhancement, while also illustrating the transferability of our approach to experiments and devices across the quantum technology sector and beyond.

Continuous-variable measurement-based quantum computation on cluster states has in recent years shown great potential for scalable, universal, and fault-tolerant quantum computation when combined with the Gottesman-Kitaev-Preskill (GKP) code and quantum error correction. However, no complete fault-tolerant architecture exists that includes everything from cluster-state generation with finite squeezing to gate implementations with realistic noise and error correction. In this work, we propose a simple architecture for the preparation of a cluster state in three dimensions in which gates can be efficiently implemented by gate teleportation. To accommodate scalability, we propose architectures that allow both spatial and temporal multiplexing, with the temporally encoded version requiring as little as two squeezed light sources. Because of its three-dimensional structure, the architecture supports topological qubit error correction, while GKP error correction is efficiently realized within the architecture by teleportation. To validate fault tolerance, the architecture is simulated using surface-GKP codes, including noise from GKP states as well as gate noise caused by finite squeezing in the cluster state. We find a fault-tolerant squeezing threshold of 12.7 dB, with room for further improvement.

We consider metric-aware quantum algorithms that use a quantum computer to efficiently estimate both a matrix and a vector object. For example, the recently introduced quantum natural gradient approach uses the Fisher matrix as a metric tensor to correct the gradient vector for the codependence of the circuit parameters. We rigorously characterize and upper bound the number of measurements required to determine an iteration step to a fixed precision, and propose a general approach for optimally distributing samples between matrix and vector entries. Finally, we establish that the number of circuit repetitions needed for estimating the quantum Fisher information matrix is asymptotically negligible for an increasing number of iterations and qubits.

Quasiparticle poisoning, expected to arise during the measurement of the Majorana zero-mode state, poses a fundamental problem for the realization of Majorana-based quantum computation. Parafermions, a natural generalization of Majorana fermions, can encode topological qudits immune to quasiparticle poisoning. While parafermions are expected to emerge in superconducting fractional quantum Hall systems, they are not yet attainable with current technology. To bypass this problem, we employ a photonic quantum simulator to experimentally demonstrate the key components of parafermion-based universal quantum computation. Our contributions in this paper are twofold. First, by manipulating the photonic states, we realize Clifford-operator Berry phases that correspond to braiding statistics of parafermions. Second, we investigate the quantum contextuality in a topological system for the first time by demonstrating the contextuality of parafermion-encoded qudit states. Importantly, we find that the topologically encoded contextuality opens the way to magic state distillation, while both the contextuality and the braiding-induced Clifford gates are resilient against local noise. By introducing contextuality, our photonic quantum simulation provides the first step toward a physically robust methodology for realizing topological quantum computation.

Rydberg-atom arrays are a leading platform for quantum computing and simulation, combining strong interactions with highly coherent operations and flexible geometries. However, the achievable fidelities are limited by the finite lifetime of the Rydberg states, as well as by technical imperfections such as atomic motion. In this work, we propose a novel approach to Rydberg-atom arrays using long-lived circular Rydberg states in optical traps. Based on the extremely long lifetime of these states, exceeding seconds in cryogenic microwave cavities that suppress radiative transitions, and gate protocols that are robust to finite atomic temperature, we project that arrays of hundreds of circular Rydberg atoms with two-qubit gate errors around ${10}^{-5}$ can be realized using current technology. This approach combines several key elements, including a quantum-nondemolition detection technique for circular Rydberg states, local manipulation using the ponderomotive potential of focused optical beams, a gate protocol using multiple circular levels to encode qubits, and robust dynamical-decoupling sequences to suppress unwanted interactions and errors from atomic motion. This represents a significant improvement on the current state of the art in quantum computing and simulation with neutral atoms.

Microwave photons are used to wire up modular quantum processors, but mitigating the effects of loss between modules remains a crucial challenge. We use a low-loss bus resonator to couple bosonic qubits across a superconducting network with protocols made robust to photon loss in the bus. We transfer a multiphoton qubit and track loss events, improving the fidelity to the break-even point with respect to the best uncorrectable encoding. We also demonstrate a entanglement protocol using Hong-Ou-Mandel interference and error detection to prepare a two-photon Bell state with fidelity 94% and success probability 0.79, halving the error obtained with a single photon. This network link also presents new opportunities for resource-efficient direct gates between modules.

We theoretically and experimentally investigate quantum features of an interacting light-matter system from a multidisciplinary perspective, combining approaches from semiconductor physics, quantum optics, and quantum-information science. To this end, we quantify the amount of quantum coherence that results from the quantum superposition of Fock states, constituting a measure of the resourcefulness of the produced state for modern quantum protocols. This notion of quantum coherence from quantum-information theory is distinct from other quantifiers of nonclassicality that have previously been applied to condensed-matter systems. As an archetypal example of a hybrid light-matter interface, we study a polariton condensate and implement a numerical model to predict its properties. Our simulation is confirmed by our proof-of-concept experiment in which we measure and analyze the phase-space distributions of the emitted light. Specifically, we drive a polariton microcavity across the condensation threshold and observe the transition from an incoherent thermal state to a coherent state in the emission, thus confirming the buildup of quantum coherence in the condensate itself.

A quantum random access memory (qRAM) is considered an essential computing unit to enable polynomial speedups in quantum information processing. Proposed implementations include the use of neutral atoms and superconducting circuits to construct a binary tree but these systems still require demonstrations of the elementary components. Here, we propose a photonic-integrated-circuit (PIC) architecture integrated with solid-state memories as a viable platform for constructing a qRAM. We also present an alternative scheme based on quantum teleportation and extend it to the context of quantum networks. Both implementations realize the two key qRAM operations, (1) quantum state transfer and (2) quantum routing, with already demonstrated components: electro-optic modulators, a Mach-Zehnder interferometer (MZI) network, and nanocavities coupled to artificial atoms for spin-based memory writing and retrieval. Our approaches furthermore benefit from built-in error detection based on photon heralding. Detailed theoretical analysis of the qRAM efficiency and query fidelity shows that our proposal presents viable near-term designs for a general qRAM.

The Hamiltonian operator plays a central role in quantum theory being a generator of unitary quantum dynamics. Its expectation value describes the energy of a quantum system. Typically being a nonunitary operator, the action of the Hamiltonian is either encoded using complex ancilla-based circuits, or implemented effectively as a sum of Pauli string terms. Here, we show how to approximate the Hamiltonian operator as a sum of propagators using a differential representation. The proposed approach, named the Hamiltonian operator approximation (HOA), is designed to benefit analog quantum simulators, where one has direct access to simulation of quantum dynamics, but measuring separate circuits is not possible. We describe how to use this strategy in the hybrid quantum-classical workflow for performing energy measurements. Benchmarking the measurement scheme, we discuss the relevance of the discretization step size, stencil order, number of shots, and noise. We also use HOA to prepare ground states of complex material science models with direct iteration and quantum filter diagonalization, finding the lowest energy for the 12-qubit Hamiltonian of hydrogen chain ${\mathrm{H}}_6$ with ${10}^{-5}$ Hartree precision using $11$ time-evolved reference states. The approach is compared to the variational quantum eigensolver, proving that HOA is beneficial for systems at increasing size, corresponding to noisy large-scale quantum devices. We find that, for the Heisenberg model with 12 or more spins, our approach may outperform variational methods, both in terms of the gate depth and the total number of measurements.

Geometrically frustrated spin-chain compounds such as ${\mathrm{Ca}}_3{\mathrm{Co}}_2{\mathrm{O}}_6$ exhibit extremely slow relaxation under a changing magnetic field. Consequently, both low-temperature laboratory experiments and Monte Carlo simulations have shown peculiar out-of-equilibrium magnetization curves, which arise from trapping in metastable configurations. In this work, we simulate this phenomenon in a superconducting quantum annealing processor, allowing us to probe the impact of quantum fluctuations on both the equilibrium and dynamics of the system. Increasing the quantum fluctuations with a transverse field reduces the impact of metastable traps in out-of-equilibrium samples and aids the development of three-sublattice ferrimagnetic (up-up-down) long-range order with magnetization 1/3. At equilibrium, we identify a finite-temperature shoulder in the 1/3-to-saturated phase transition, promoted by quantum fluctuations but with an entropic origin. This work demonstrates the viability of dynamical as well as equilibrium studies of frustrated magnetism using large-scale programmable quantum systems and is therefore an important step toward programmable simulation of dynamics in materials using quantum hardware.