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.

The concept of quantum complexity has far-reaching implications spanning theoretical computer science, quantum many-body physics, and high-energy physics. The quantum complexity of a unitary transformation or quantum state is defined as the size of the shortest quantum computation that executes the unitary or prepares the state. It is reasonable to expect that the complexity of a quantum state governed by a chaotic many-body Hamiltonian grows linearly with time for a time that is exponential in the system size; however, because it is hard to rule out a shortcut that improves the efficiency of a computation, it is notoriously difficult to derive lower bounds on quantum complexity for particular unitaries or states without making additional assumptions. To go further, one may study more generic models of complexity growth. We provide a rigorous connection between complexity growth and unitary $k$-designs, ensembles that capture the randomness of the unitary group. This connection allows us to leverage existing results about design growth to draw conclusions about the growth of complexity. We prove that local random quantum circuits generate unitary transformations whose complexity grows linearly for a long time, mirroring the behavior one expects in chaotic quantum systems and verifying conjectures by Brown and Susskind. Moreover, our results apply under a strong definition of quantum complexity based on optimal distinguishing measurements.

We introduce a theoretical framework for resource-efficient characterization and control of non-Markovian open quantum systems, which naturally allows for the integration of given, experimentally motivated, control capabilities and constraints. This is achieved by developing a transfer filter-function formalism based on the general notion of a frame and by appropriately tying the choice of frame to the available control. While recovering the standard frequency-based filter-function formalism as a special instance, this control-adapted generalization affords intrinsic flexibility and, crucially, it permits an efficient representation of the relevant control matrix elements and dynamical integrals if an appropriate finite-size frame condition is obeyed. Our frame-based formulation overcomes important limitations of existing approaches. In particular, we show how to implement quantum noise spectroscopy in the presence of nonstationary noise sources, and how to effectively achieve control-driven model reduction for noise-optimized prediction and quantum gate design.

Leakage outside of the qubit computational subspace poses a threatening challenge to quantum error correction (QEC). We propose a scheme using two leakage-reduction units (LRUs) that mitigate these issues for a transmon-based surface code, without requiring an overhead in terms of hardware or QEC-cycle time as in previous proposals. For data qubits, we consider a microwave drive to transfer leakage to the readout resonator, where it quickly decays, ensuring that this negligibly disturbs the computational states for realistic system parameters. For ancilla qubits, we apply a $|1⟩\leftrightarrow |2⟩\pi$ pulse conditioned on the measurement outcome. Using density-matrix simulations of the distance-3 surface code, we show that the average leakage lifetime is reduced to almost one QEC cycle, even when the LRUs are implemented with limited fidelity. Furthermore, we show that this leads to a significant reduction of the logical error rate. This LRU scheme opens the prospect for near-term scalable QEC demonstrations.

We propose entanglement negativity as a fine-grained probe of measurement-induced criticality. We motivate this proposal in stabilizer states, where for two disjoint subregions, comparing their “mutual negativity” and their mutual information leads to a precise distinction between bipartite and multipartite entanglement. In a measurement-only stabilizer circuit that maps exactly to two-dimensional critical percolation, we show that the mutual information and the mutual negativity are governed by boundary conformal fields of different scaling dimensions at long distances. We then consider a class of “hybrid” circuit models obtained by perturbing the measurement-only circuit with unitary gates of progressive levels of complexity. While other critical exponents vary appreciably for different choices of unitary gate ensembles at their respective critical points, the mutual negativity has scaling dimension $3$ across remarkably many of the hybrid circuits, which is notably different from that in percolation. We contrast our results with limiting cases where a geometrical minimal-cut picture is available.

One of the major application areas of interest for both near-term and fault-tolerant quantum computers is the optimization of classical objective functions. In this work, we develop intuitive constructions for a large class of these algorithms based on connections to simple dynamics of quantum systems, quantum walks, and classical continuous relaxations. We focus on developing a language and tools connected with kinetic energy on a graph for understanding the physical mechanisms of success and failure to guide algorithmic improvement. This physical language, in combination with uniqueness results related to unitarity, allow us to identify some potential pitfalls from kinetic energy fundamentally opposing the goal of optimization. This is connected to effects from wavefunction confinement, phase randomization, and shadow defects lurking in the objective far away from the ideal solution. As an example, we explore the surprising deficiency of many quantum methods in solving uncoupled spin problems and how this is both predictive of performance on some more complex systems while immediately suggesting simple resolutions. Further examination of canonical problems like the Hamming ramp or bush of implications show that entanglement can be strictly detrimental to performance results from the underlying mechanism of solution in approaches like the quantum approximate optimization algorithm (QAOA). Kinetic energy and graph Laplacian perspectives provide new insights to common initialization and optimal solutions in QAOA as well as new methods for more effective layerwise training. Connections to classical methods of continuous extensions, homotopy methods, and iterated rounding suggest new directions for research in quantum optimization. Throughout, we unveil many pitfalls and mechanisms in quantum optimization using a physical perspective, which aim to spur the development of novel quantum optimization algorithms and refinements.