Landauer’s principle asserts that any computation has an unavoidable energy cost that grows proportionally to its degree of logical irreversibility. But even a logically reversible operation, when run on a physical processor that operates on different energy levels, requires energy. Here we quantify this energy requirement, providing upper and lower bounds that coincide up to a constant factor. We derive these bounds from a general quantum resource-theoretic argument, which implies that the initial resource requirement for implementing a unitary operation within an error $\epsilon$ grows like $1/\sqrt{\epsilon }$ times the amount of resource generated by the operation. Applying these results to quantum circuits, we find that their energy requirement can, by an appropriate design, be made independent of their time complexity.

Periodically driven coherent conductors provide a universal platform for the development of quantum transport devices. Here, we lay down a comprehensive theory to describe the thermodynamics of these systems. We first focus on moderate thermoelectrical biases and low driving frequencies. For this linear response regime, we establish generalized Onsager-Casimir relations and an extended fluctuation-dissipation theorem. Furthermore, we derive a family of thermodynamic bounds proving that any local matter or heat current puts a nontrivial lower limit on the overall dissipation rate of a coherent transport process. These bounds do not depend on system-specific parameters, are robust against dephasing, and involve only experimentally accessible quantities. They thus provide powerful tools to optimize the performance of mesoscopic devices and for thermodynamic inference, as we demonstrate by working out three specific applications. We then show that physically transparent extensions of our bounds hold also for strong biases and high frequencies. These generalized bounds imply a thermodynamic uncertainty relation that fully accounts for quantum effects and periodic driving. Moreover, they lead to a universal and operationally accessible bound on entropy production that can be readily used for thermodynamic inference and device engineering far from equilibrium. Connecting a broad variety of topics that range from thermodynamic geometry over thermodynamic uncertainty relations to quantum engineering, our work provides a unifying thermodynamic theory of coherent transport that can be tested and utilized with current technologies.

The dynamics of quantum information in strongly interacting systems, known as quantum information scrambling, has recently become a common thread in our understanding of black holes, transport in exotic non-Fermi liquids, and many-body analogs of quantum chaos. To date, verified experimental implementations of scrambling have focused on systems composed of two-level qubits. Higher-dimensional quantum systems, however, may exhibit different scrambling modalities and are predicted to saturate conjectured speed limits on the rate of quantum information scrambling. We take the first steps toward accessing such phenomena, by realizing a quantum processor based on superconducting qutrits (three-level quantum systems). We demonstrate the implementation of universal two-qutrit scrambling operations and embed them in a five-qutrit quantum teleportation protocol. Measured teleportation fidelities $F_{\mathrm{avg}}=0.568\pm 0.001$ confirm the presence of scrambling even in the presence of experimental imperfections and decoherence. Our teleportation protocol, which connects to recent proposals for studying traversable wormholes in the laboratory, demonstrates how quantum technology that encodes information in higher-dimensional systems can exploit a larger and more connected state space to achieve the resource efficient encoding of complex quantum circuits.

In canonical optomechanical systems, mechanical vibrations are dynamically encoded on an optical probe field, which reciprocally exerts a backaction force. Because of the weak single-photon coupling strength achieved with macroscopic oscillators, most of the existing experiments were conducted with large photon numbers to achieve sizable effects, thereby hiding the original optomechanical nonlinearity. To increase the optomechanical interaction, we make use of subwavelength-sized ultrasensitive suspended nanowires inserted in the mode volume of a fiber-based microcavity. By scanning the nanowire within the cavity mode volume and measuring its impact on the cavity mode, we obtain a map of the 2D optomechanical interaction. Then, by using the toolbox of nanowire-based force-sensing protocols, we explore the backaction of the optomechanical interaction and map the optical force field experienced by the nanowire. These measurements also allow us to demonstrate the possibility to detect variations of the mean intracavity photon number smaller than unity. This implementation should also allow us to enter the promising regime of cavity optomechanics, where a single intracavity photon can displace the oscillator by more than its zero-point fluctuations, which will open novel perspectives in the field.

We consider a clean quantum system subject to strong periodic driving. The existence of a dominant energy scale, $h_D^x$, can generate considerable structure in an effective description of a system that, in the absence of the drive, is nonintegrable and interacting, and does not host localization. In particular, we uncover points of freezing in the space of drive parameters (frequency and amplitude). At those points, the dynamics is severely constrained due to the emergence of an almost exact, local conserved quantity, which scars the entire Floquet spectrum by preventing the system from heating up ergodically, starting from any generic state, even though it delocalizes over an appropriate subspace. At large drive frequencies, where a naïve Magnus expansion would predict a vanishing effective (average) drive, we devise instead a strong-drive Magnus expansion in a moving frame. There, the emergent conservation law is reflected in the appearance of the “integrability” of an effective Hamiltonian. These results hold for a wide variety of Hamiltonians, including the Ising model in a transverse field in any dimension and for any form of Ising interaction. The phenomenon is also shown to be robust in the presence of two-body Heisenberg interactions with any arbitrary choice of couplings. Furthermore, we construct a real-time perturbation theory that captures resonance phenomena where the conservation breaks down, giving way to unbounded heating. This approach opens a window on the low-frequency regime where the Magnus expansion fails.

Quantum catalysis is a fascinating concept that demonstrates how certain transformations can only become possible when given access to a specific resource that has to be returned unaffected. It was first discovered in the context of entanglement theory, and since then, it has been applied in a number of resource-theoretic frameworks, including quantum thermodynamics. Although, in that case, the necessary (and sometimes also sufficient) conditions on the existence of a catalyst are known, almost nothing is known about the precise form of the catalyst state required by the transformation. In particular, it is not clear whether it has to have some special properties or be finely tuned to the desired transformation. In this work, we describe a surprising property of multicopy states: We show that in resource theories governed by majorization, all resourceful states are catalysts for all allowed transformations. In quantum thermodynamics, this means that the so-called “second laws of thermodynamics” do not require a fine-tuned catalyst; rather, any state, given sufficiently many copies, can serve as a useful catalyst. These analytic results are accompanied by several numerical investigations that indicate that neither a multicopy form nor a very-large-dimension catalyst is required to activate most allowed transformations catalytically.

One of the most fundamental problems in quantum many-body physics is the characterization of correlations among thermal states. Of particular relevance is the thermal area law, which justifies the tensor network approximations to thermal states with a bond dimension growing polynomially with the system size. In the regime of sufficiently low temperatures, which is crucially important for practical applications, the existing techniques do not yield optimal bounds. Here, we propose a new thermal area law that holds for generic many-body systems on lattices. We improve the temperature dependence from the original $\mathcal{O}(\beta )$ to $\mathcal{O}({\beta }^{2/3})$ up to a logarithmic factor, thereby suggesting subballistic propagation of entanglement by imaginary-time evolution. This qualitatively differs from the real-time evolution, which usually induces linear growth of entanglement. We also prove analogous bounds for the Rényi entanglement of purification and the entanglement of formation. Our analysis is based on a polynomial approximation to the exponential function which provides a relationship between the imaginary-time evolution and random walks. Moreover, for one-dimensional (1D) systems with $n$ spins, we prove that the Gibbs state is well approximated by a matrix product operator with a sublinear bond dimension for $\beta =o[\log (n)]$. This proof allows us to rigorously establish, for the first time, a quasilinear time classical algorithm for constructing a matrix product state representation of 1D quantum Gibbs states at arbitrary temperatures of $\beta =o[\log (n)]$. Our new technical ingredient is a block decomposition of the Gibbs state that bears a resemblance to the decomposition of real-time evolution given by Haah et al. [Proceedings of the 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, New York, 2018), pp. 350–360].

According to thermodynamics, the inevitable increase of entropy allows the past to be distinguished from the future. From this perspective, any clock must incorporate an irreversible process that allows this flow of entropy to be tracked. In addition, an integral part of a clock is a clockwork, that is, a system whose purpose is to temporally concentrate the irreversible events that drive this entropic flow, thereby increasing the accuracy of the resulting clock ticks compared to counting purely random equilibration events. In this article, we formalize the task of autonomous temporal probability concentration as the inherent goal of any clockwork based on thermal gradients. Within this framework, we show that a perfect clockwork can be approximated arbitrarily well by increasing its complexity. Furthermore, we combine such an idealized clockwork model, comprised of many qubits, with an irreversible decay mechanism to showcase the ultimate thermodynamic limits to the measurement of time.

Agile cryptography allows for a resource-efficient swap of a cryptographic core in case the security of an underlying classical cryptographic algorithm becomes compromised. Conversely, versatile cryptography allows the user to switch the cryptographic task without requiring any knowledge of its inner workings. In this paper, we suggest how these related principles can be applied to the field of quantum cryptography by explicitly demonstrating two quantum cryptographic protocols, quantum digital signatures (QDS) and quantum secret sharing (QSS), on the same hardware sender and receiver platform. Crucially, the protocols differ only in their classical postprocessing. The system is also suitable for quantum key distribution (QKD) and is highly compatible with deployed telecommunication infrastructures, since it uses standard quadrature phase-shift keying encoding and heterodyne detection. For the first time, QDS protocols are modified to allow for postselection at the receiver, enhancing protocol performance. The cryptographic primitives QDS and QSS are inherently multipartite, and we prove that they are secure not only when a player internal to the task is dishonest, but also when (external) eavesdropping on the quantum channel is allowed. In our first proof-of-principle demonstration of an agile and versatile quantum communication system, the quantum states are distributed at GHz rates. A 1-bit message may be securely signed using our QDS protocols in less than 0.05 ms over a 2-km fiber link and in less than 0.2 s over a 20-km fiber link. To our knowledge, this also marks the first demonstration of a continuous-variable direct QSS protocol.

Phase transitions share the universal feature of enhanced fluctuations near the transition point. Here, we show that density fluctuations reveal how a Bose-Einstein condensate of dipolar atoms spontaneously breaks its translation symmetry and enters the supersolid state of matter—a phase that combines superfluidity with crystalline order. We report on the first direct in situ measurement of density fluctuations across the superfluid-supersolid phase transition. This measurement allows us to introduce a general and straightforward way to extract the static structure factor, estimate the spectrum of elementary excitations, and image the dominant fluctuation patterns. We observe a strong response in the static structure factor and infer a distinct roton minimum in the dispersion relation. Furthermore, we show that the characteristic fluctuations correspond to elementary excitations such as the roton modes, which are theoretically predicted to be dominant at the quantum critical point, and that the supersolid state supports both superfluid as well as crystal phonons.

Transforming an initial quantum state into a target state through the fastest possible route—a quantum brachistochrone—is a fundamental challenge for many technologies based on quantum mechanics. In two-level systems, the quantum brachistochrone solutions are long known. These solutions, however, are not applicable to larger systems, especially when the target state cannot be reached through a local transformation. Here, we demonstrate fast coherent transport of an atomic wave packet over a distance of 15 times its size—a paradigmatic case of quantum processes going beyond the two-level system. Our measurements of the transport fidelity reveal the existence of a minimum duration—a quantum speed limit—for the coherent splitting and recombination of matter waves. We obtain physical insight into this limit by relying on a geometric interpretation of quantum state dynamics. These results shed light on a fundamental limit of quantum state dynamics and are expected to find relevant applications in quantum sensing and quantum computing.

We present a circuit design composed of two Josephson junctions coupled by a nonreciprocal element, the gyrator, whose ground space is doubly degenerate. The ground states are approximate code words of the Gottesman-Kitaev-Preskill code. We determine the low-energy dynamics of the circuit by working out the equivalence of this system to the problem of a single electron in a crystal, confined to a two-dimensional plane, and subjected to a strong, homogeneous magnetic field. We find that the circuit is naturally protected against the common noise channels in superconducting circuits, such as charge and flux noise, implying that it can be used for passive quantum error correction. We also propose realistic design parameters for an experimental realization, and we describe possible protocols to perform logical one- and two-qubit gates, state preparation, and readout.

The estimation of multiple parameters in quantum metrology is important for a vast array of applications in quantum information processing. However, the unattainability of fundamental precision bounds for incompatible observables greatly diminishes the applicability of estimation theory in many practical implementations. The Holevo Cramér-Rao bound (HCRB) provides the most fundamental, simultaneously attainable bound for multiparameter estimation problems. A general closed form for the HCRB is not known given that it requires a complex optimization over multiple variables. In this work, we develop an analytic approach to solving the HCRB for two parameters. Our analysis reveals the role of the HCRB and its interplay with alternative bounds in estimation theory. For more parameters, we generate a lower bound to the HCRB. Our work greatly reduces the complexity of determining the HCRB to solving a set of linear equations that even numerically permits a quadratic speedup over previous state-of-the-art approaches. We apply our results to compare the performance of different probe states in magnetic field sensing and characterize the performance of state tomography on the code space of noisy bosonic error-correcting codes. The sensitivity of state tomography on noisy binomial code states can be improved by tuning two coding parameters that relate to the number of correctable phase and amplitude damping errors. Our work provides fundamental insights and makes significant progress toward the estimation of multiple incompatible observables.

We describe an approach to the high-fidelity measurement of a superconducting qubit using an on-chip microwave photon counter. The protocol relies on the transient response of a dispersively coupled measurement resonator to map the state of the qubit to “bright” and “dark” cavity pointer states that are characterized by a large differential photon occupation. Following this mapping, we photodetect the resonator using the Josephson photomultiplier, which transitions between classically distinguishable flux states when cavity photon occupation exceeds a certain threshold. Our technique provides access to the binary outcome of projective quantum measurement at the millikelvin stage without the need for quantum-limited preamplification and thresholding at room temperature. We achieve raw single-shot measurement fidelity in excess of 98% across multiple samples using this approach in total measurement times under 500 ns. In addition, we show that the backaction and crosstalk associated with our measurement protocol can be mitigated by exploiting the intrinsic damping of the Josephson photomultiplier itself.

It is interesting to observe that all optical materials with a positive refractive index have a value of index that is of order unity. Surprisingly, though, a deep understanding of the mechanisms that lead to this universal behavior seems to be lacking. Moreover, this observation is difficult to reconcile with the fact that a single isolated atom is known to have a giant optical response, as characterized by a resonant scattering cross section that far exceeds its physical size. Here, we theoretically and numerically investigate the evolution of the optical properties of an ensemble of ideal atoms as a function of density, starting from the dilute gas limit, including the effects of multiple scattering and near-field interactions. Interestingly, despite the giant response of an isolated atom, we find that the maximum index does not indefinitely grow with increasing density but rather reaches a limiting value of $n\approx 1.7$. This limit arises purely from electrodynamics, as it occurs at densities far below those where chemical processes become important. We propose an explanation based upon strong-disorder renormalization group theory, in which the near-field interaction combined with random atomic positions results in an inhomogeneous broadening of atomic resonance frequencies. This mechanism ensures that, regardless of the physical atomic density, light at any given frequency only interacts with at most a few near-resonant atoms per cubic wavelength, thus limiting the maximum index attainable. Our work is a promising first step to understand the limits of the refractive index from a bottom-up, atomic physics perspective, and it also introduces the renormalization group as a powerful tool to understand the generally complex problem of multiple scattering of light overall.

Interacting many-body systems with explicitly accessible spatiotemporal correlation functions are extremely rare, especially in the absence of Bethe-ansatz or Yang-Baxter integrability. Recently, we identified a remarkable class of such systems and termed them dual-unitary quantum circuits. These are brickwork-type local quantum circuits whose dynamics are unitary in both time and space. The spatiotemporal correlation functions of these systems turn out to be nontrivial only at the edge of the causal light cone and can be computed in terms of one-dimensional transfer matrices. Dual unitarity, however, requires fine-tuning, and the degree of generality of the observed dynamical features remains unclear. Here, we address this question by studying perturbed dual-unitary quantum circuits. Considering arbitrary perturbations of the local gates, we prove that for a particular class of unperturbed elementary dual-unitary gates the correlation functions are still expressed in terms of one-dimensional transfer matrices. These matrices, however, are now contracted over generic paths connecting the origin to a fixed end point inside the causal light cone. The correlation function is given as a sum over all such paths. Our statement is rigorous in the “dilute limit,” where only a small fraction of the gates is perturbed, and in the presence of random longitudinal fields, but we provide theoretical arguments and stringent numerical checks supporting its validity even in the clean case (no randomness) and when all gates are perturbed. As a by-product of our analysis, in the case of random longitudinal fields—which turns out to be equivalent to certain classical Markov chains—we find four types of non-dual-unitary (and nonintegrable) interacting many-body systems where the correlation functions are exactly solvable and given—without approximations—by the path-sum formula.

Herein we show that nonresonant inelastic x-ray scattering involving an $s$ core level is a powerful spectroscopic method to characterize the excited states of transition metal compounds. The spherical charge distribution of the $s$ core hole allows the orientational dependence of the intensities of the various spectral features to produce a spatial charge image of the associated multiplet states in a straightforward manner, thereby facilitating the identification of their orbital character. In addition, the $s$ core hole does not add an extra orbital angular momentum component to the multiplet structure so that the well-established Sugano-Tanabe-Kamimura diagrams can be used for the analysis of the spectra. For $\alpha$-MnS we observe the spherical charge density corresponding to its high-spin $3d^5$ (${}^6{A}_1$) ground state configuration and we were able to selectively image its excited states and identify them as $t_{2g}$ (${}^5{T}_2$) and $e_g$ (${}^{5}E$) with an energy splitting $10Dq$ of 0.78 eV.

The ability to control and measure the temperature of propagating microwave modes down to very low temperatures is indispensable for quantum information processing and may open opportunities for studies of heat transport at the nanoscale, also in the quantum regime. Here, we propose and experimentally demonstrate primary thermometry of propagating microwaves using a transmon-type superconducting circuit. Our device operates continuously, with a sensitivity down to $4\times {10}^{-4}\mathrm{photons}/\sqrt{\mathrm{Hz}}$ and a bandwidth of 40 MHz. We measure the thermal occupation of the modes of a highly attenuated coaxial cable in a range of 0.001 to 0.4 thermal photons, corresponding to a temperature range from 35 mK to 210 mK at a frequency around 5 GHz. To increase the radiation temperature in a controlled fashion, we either inject calibrated, wideband digital noise, or heat the device and its environment. This thermometry scheme can find applications in benchmarking and characterization of cryogenic microwave setups, temperature measurements in hybrid quantum systems, and quantum thermodynamics.

Blurring the boundary between bosons and fermions lies at the heart of a wide range of intriguing quantum phenomena in multiple disciplines, ranging from condensed matter physics and atomic, molecular, and optical physics to high-energy physics. One such example is a multicomponent Fermi gas with $\mathrm{SU}(N)$ symmetry that is expected to behave like spinless bosons in the large-$N$ limit, where the large number of internal states weakens constraints from the Pauli exclusion principle. However, bosonization in $\mathrm{SU}(N)$ fermions has never been established in high dimensions where exact solutions are absent. Here, we report direct evidence for bosonization in a $\mathrm{SU}(N)$ fermionic ytterbium gas with tunable $N$ in three dimensions (3D). We measure contacts, the central quantity controlling dilute quantum gases, from the momentum distribution and find that the contact per spin approaches a constant with a $1/N$ scaling in the low-fugacity regime consistent with our theoretical prediction. This scaling signifies the vanishing role of the fermionic statistics in thermodynamics and allows us to verify bosonization through measuring a single physical quantity. Our work delivers a highly controllable quantum simulator to exchange the bosonic and fermionic statistics through tuning the internal degrees of freedom in any generic dimensions. It also suggests a new route toward exploring multicomponent quantum systems and their underlying symmetries with contacts.

More than a century after the inception of quantum theory, the question of which traits and phenomena are fundamentally quantum remains under debate. Here, we give an answer to this question for temporal processes that are probed sequentially by means of projective measurements of the same observable. Defining classical processes as those that can, in principle, be simulated by means of classical resources only, we fully characterize the set of such processes. Based on this characterization, we show that for non-Markovian processes (i.e., processes with memory), the absence of coherence does not guarantee the classicality of observed phenomena; furthermore, we derive an experimentally and computationally accessible measure for nonclassicality in the presence of memory. We then provide a direct connection between classicality and the vanishing of quantum discord between the evolving system and its environment. Finally, we demonstrate that—in contrast to the memoryless setting—in the non-Markovian case, there exist processes that are genuinely quantum; i.e., they display nonclassical statistics independent of the measurement scheme that is employed to probe them.