Random quantum circuits have played a central role in establishing the computational advantages of near-term quantum computers over their conventional counterparts. Here, we use ensembles of low-depth random circuits with local connectivity in $D\ge 1$ spatial dimensions to generate quantum error-correcting codes. For random stabilizer codes and the erasure channel, we find strong evidence that a depth $O(\log N)$ random circuit is necessary and sufficient to converge (with high probability) to zero failure probability for any finite amount below the optimal erasure threshold, set by the channel capacity, for any $D$. Previous results on random circuits have only shown that $O(N^{1/D})$ depth suffices or that $O({\log }^{3}N)$ depth suffices for all-to-all connectivity ($D\to \infty$). We then study the critical behavior of the erasure threshold in the so-called moderate deviation limit, where both the failure probability and the distance to the optimal threshold converge to zero with $N$. We find that the requisite depth scales like $O(\log N)$ only for dimensions $D\ge 2$ and that random circuits require $O(\sqrt{N})$ depth for $D=1$. Finally, we introduce an “expurgation” algorithm that uses quantum measurements to remove logical operators that cause the code to fail by turning them into either additional stabilizers or into gauge operators in a subsystem code. With such targeted measurements, we can achieve sublogarithmic depth in $D\ge 2$ spatial dimensions below capacity without increasing the maximum weight of the check operators. We find that for any rate beneath the capacity, high-performing codes with thousands of logical qubits are achievable with depth 4–8 expurgated random circuits in $D=2$ dimensions. These results indicate that finite-rate quantum codes are practically relevant for near-term devices and may significantly reduce the resource requirements to achieve fault tolerance for near-term applications.

Active fluids exhibit complex turbulentlike flows at low Reynolds number. Recent work predicted that 2D active nematic turbulence follows scaling laws with universal exponents. However, experimentally testing these predictions is conditioned by the coupling to the 3D environment. Here, we measure the spectrum of the kinetic energy $E(q)$ in an active nematic film in contact with a passive oil layer. At small and intermediate scales, we find the scaling regimes $E(q)\sim q^{-4}$ and $E(q)\sim q^{-1}$, respectively, in agreement with the theoretical prediction for 2D active nematics. At large scales, however, we find a new scaling $E(q)\sim q$, which emerges when the dissipation is dominated by the 3D oil layer. In addition, we derive an explicit expression for the spectrum that spans all length scales, thus explaining and connecting the different scaling regimes. This allows us to fit the data and extract the length scale that controls the crossover to the new large-scale regime, which we tune by varying the oil viscosity. Overall, our work experimentally demonstrates the emergence of scaling laws with universal exponents in active turbulence, and it establishes how the spectrum is affected by external dissipation.

We provide a general theory of nonlinear electronic circuits subjected to thermal noise. The devices constituting the circuit can have arbitrary $I\text{−}V$ curves but must display shot noise. This theory includes tunnel junctions, diodes, and MOS transistors in subthreshold operation, among others. The stochastic nonequilibrium thermodynamics of these circuits is also established. The irreversible entropy production is expressed in terms of thermodynamic potentials and forces, and its fluctuations satisfy fluctuation theorems. Our theory is ideally suited to formulate a thermodynamics of computing with realistic architectures, where the reduction in transistor size and operating voltages make thermal fluctuations increasingly important. We demonstrate this point in two ways: first, by proposing a stochastic model of a CMOS inverter whose actual transfer function deviates from the deterministic one due to nonequilibrium fluctuations, and, second, by proposing a low-power full-CMOS design for a probabilistic bit (or binary stochastic neuron) exploiting intrinsic noise.

How fast must an oriented collection of extensile swimmers swim to escape the instability of viscous active suspensions? We show that the answer lies in the dimensionless combination $R=\rho v_0^2/2{\sigma }_a$, where $\rho$ is the suspension mass density, $v_0$ the swim speed, and ${\sigma }_a$ the active stress. Linear stability analysis shows that, for small $R$, disturbances grow at a rate linear in their wave number $q$ and that the dominant instability mode involves twist. The resulting steady state in our numerical studies is isotropic hedgehog-defect turbulence. Past a first threshold $R$ of order unity, we find a slower growth rate, of $O(q^2)$; the numerically observed steady state is phase turbulent: noisy but aligned on average. We present numerical evidence in three and two dimensions that this inertia-driven flocking transition is continuous, with a correlation length that grows on approaching the transition. For much larger $R$, we find an aligned state linearly stable to perturbations at all $q$. Our predictions should be testable in suspensions of mesoscale swimmers [, Soft Matter 15, 8946 (2019)].

Quantum phase transitions are central to our understanding of why matter at very low temperatures can exhibit starkly different properties upon small changes of microscopic parameters. Accurately locating those transitions is challenging experimentally and theoretically. Here, we show that the antithetic strategy of forcing systems out of equilibrium via sudden quenches provides a route to locate quantum phase transitions. Specifically, we show that such transitions imprint distinctive features in the intermediate-time dynamics, and results after equilibration, of local observables in quantum chaotic spin chains. Furthermore, we show that the effective temperature in the expected thermal-like states after equilibration can exhibit minima in the vicinity of the quantum critical points. We discuss how to test our results in experiments with Rydberg atoms and explore nonequilibrium signatures of quantum critical points in models with topological transitions.

We report coherent association of atoms into a single weakly bound NaCs molecule in an optical tweezer through an optical Raman transition. The Raman technique uses a deeply bound electronic excited intermediate state to achieve a large transition dipole moment while reducing photon scattering. Starting from two atoms in their relative motional ground state, we achieve an optical transfer efficiency of 69%. The molecules have a binding energy of 770.2 MHz at 8.83(2) G. This technique does not rely on Feshbach resonances or narrow excited-state lines and may allow a wide range of molecular species to be assembled atom by atom.

We propose and validate a novel experimental technique to measure two-point statistics of turbulent flows. It consists of spreading rigid fibers in the flow and tracking their position and orientation in time and is therefore named “fiber tracking velocimetry.” By choosing different fiber lengths, i.e., within the inertial or dissipative range of scales, the statistics of turbulence fluctuations at the selected length scale can be probed accurately by simply measuring the fiber velocity at its two ends and projecting it along the transverse-to-fiber direction. By means of fully resolved direct numerical simulations and experiments, we show that these fiber-based transverse velocity increments are statistically equivalent to the (unperturbed) flow transverse velocity increments. Moreover, we show that the turbulent energy-dissipation rate can be accurately measured exploiting sufficiently short fibers. The technique is tested against standard particle tracking velocimetry (PTV) of flow tracers with excellent agreement. Our technique overcomes the well-known problem of PTV to probe two-point statistics reliably because of the fast relative diffusion in turbulence that prevents the mutual distance between particles to remain constant at the length scale of interest. This problem, making it difficult to obtain converged statistics for a fixed separation distance, is even more dramatic for natural flows in open domains. A prominent example is oceanic currents, where drifters (i.e., the tracer-particle counterpart used in field measurements) disperse quickly, but at the same time their number has to be limited to save costs. Inspired by our laboratory experiments, we propose pairs of connected drifters as a viable option to solve the issue.

The groundbreaking success of deep learning in many real-world tasks has triggered an intense effort to theoretically understand the power and limitations of deep learning in the training and generalization of complex tasks, so far with limited progress. In this work, we study the statistical mechanics of learning in deep linear neural networks (DLNNs) in which the input-output function of an individual unit is linear. Despite the linearity of the units, learning in DLNNs is highly nonlinear; hence, studying its properties reveals some of the essential features of nonlinear deep neural networks (DNNs). Importantly, we exactly solve the network properties following supervised learning using an equilibrium Gibbs distribution in the weight space. To do this, we introduce the backpropagating kernel renormalization (BPKR), which allows for the incremental integration of the network weights layer by layer starting from the network output layer and progressing backward until the first layer’s weights are integrated out. This procedure allows us to evaluate important network properties, such as its generalization error, the role of network width and depth, the impact of the size of the training set, and the effects of weight regularization and learning stochasticity. BPKR does not assume specific statistics of the input or the task’s output. Furthermore, by performing partial integration of the layers, the BPKR allows us to compute the emergent properties of the neural representations across the different hidden layers. We propose a heuristic extension of the BPKR to nonlinear DNNs with rectified linear units (ReLU). Surprisingly, our numerical simulations reveal that despite the nonlinearity, the predictions of our theory are largely shared by ReLU networks of modest depth, in a wide regime of parameters. Our work is the first exact statistical mechanical study of learning in a family of deep neural networks, and the first successful theory of learning through the successive integration of degrees of freedom in the learned weight space.

Multiple aspects of an economy can be regulated, tampered with, or left to chance. Economic actors can exploit these degrees of freedom, at a cost, to bend the flow of wealth in their favor. If intervention becomes widespread, microeconomic strategies of different actors can build into emergent macroeconomic effects. How viable is a “rigged” economy? How do growing economic complexity and wealth affect it? We study rigged economies with a toy model. In it, economic degrees of freedom progress from minority to coordination games as intervention increases. Growing economic complexity spontaneously defuses cartels. But excessive complexity leads to large-fluctuations regimes, threatening the system’s stability. Simulations suggest that wealth must grow faster than linearly with economic complexity to avoid this regime and keep economies viable in the long run. We discuss a real-case scenario of multiple economic actors coordinated to result in an emergent upset of the stock market.

Suppressing noise in physical systems is of fundamental importance. As quantum computers mature, quantum error correcting codes (QECs) will be adopted in order to suppress errors to any desired level. However in the noisy, intermediate-scale quantum (NISQ) era, the complexity and scale required to adopt even the smallest QEC is prohibitive: a single logical qubit needs to be encoded into many thousands of physical qubits. Here we show that, for the crucial case of estimating expectation values of observables (key to almost all NISQ algorithms) one can indeed achieve an effective exponential suppression. We take $n$ independently prepared circuit outputs to create a state whose symmetries prevent errors from contributing bias to the expected value. The approach is very well suited for current and near-term quantum devices as it is modular in the main computation and requires only a shallow circuit that bridges the $n$ copies immediately prior to measurement. Using no more than four circuit copies, we confirm error suppression below ${10}^{-6}$ for circuits consisting of several hundred noisy gates (2-qubit gate error 0.5%) in numerical simulations validating our approach.

Molecular matter-wave interferometry enables novel strategies for manipulating the internal mechanical motion of complex molecules. Here, we show how chiral molecules can be prepared in a quantum superposition of two enantiomers by far-field matter-wave diffraction and how the resulting tunneling dynamics can be observed. We determine the impact of rovibrational phase averaging and propose a setup for sensing enantiomer-dependent forces, parity-violating weak interactions, and environment-induced superselection of handedness, as suggested to resolve Hund’s paradox. Using ab initio tunneling calculations, we identify [4]-helicene derivatives as promising candidates to implement the proposal with state-of-the-art techniques. This work opens the door for quantum sensing with chiral molecules.

The emergence of evermore complex entities from prebiotic building blocks is a key aspect of origins of life research. The RNA-world hypothesis posits that RNA oligomers known as ribozymes acted as the first self-replicating entities. However, the mechanisms governing the self-assembly of complex informational polymers from the shortest prebiotic building blocks were unclear. One open issue concerns the relation between concentration and oligonucleotide length, usually assumed to be exponentially decreasing. Here, we show that a competition of timescales in the self-assembly of informational polymers by templated ligation generically leads to nonmonotonic strand-length distributions with two distinct length scales. The first length scale characterizes the onset of a strongly nonequilibrium regime and is visible as a local minimum. Dynamically, this regime is governed by extension cascades, where the elongation of a “primer” with a short building block is more likely than its dehybridization. The second length scale appears as a local concentration maximum and reflects a balance between degradation and dehybridization of completely hybridized double strands in a heterocatalytic extension-reassembly process. Analytical arguments and extensive numerical simulations within a sequence-independent model allowed us to predict and control these emergent length scales. Nonmonotonic strand-length distributions confirming our theory were obtained in thermocycler experiments using random DNA sequences from a binary alphabet. Our work emphasizes the role of structure-forming processes already for the earliest stages of prebiotic evolution. The accumulation of strands with a typical length reveals a possible starting point for higher-order self-organization events that ultimately lead to a self-replicating, evolving system.

Generalizing the prior work of P. W. Anderson and E. R. Huggins, we show that a “detailed Josephson-Anderson relation” holds for drag on a finite body held at rest in a classical incompressible fluid flowing with velocity $\mathbf{V}$. The relation asserts an exact equality between the instantaneous power consumption by the drag $-\mathbf{F}·\mathbf{V}$ and the vorticity flux across the potential mass current $-(1/2)\int dJ\int {\epsilon }_{ijk}{\mathrm{\Sigma }}_{ij}d{\ell }_k$. Here, ${\mathrm{\Sigma }}_{ij}$ is the flux in the $i$th coordinate direction of the conserved $j$th component of vorticity, and the line integrals over $\mathbf{\ell }$ are taken along streamlines of the potential-flow solution ${\mathbf{u}}_{\varphi }=\mathbf{\nabla }\varphi$ of the ideal Euler equation, carrying mass flux $dJ=\rho {\mathbf{u}}_{\varphi }·d\mathbf{A}$. Drag and dissipation are thus associated with the motion of vorticity relative to this background ideal potential flow solving Euler’s equation. The results generalize the theories of M. J. Lighthill for flow past a body and, in particular, the steady-state relation $(1/2){\epsilon }_{ijk}⟨{\mathrm{\Sigma }}_{jk}⟩={\partial }_i⟨h⟩$, where $h=p+(1/2)|\mathbf{u}{|}^2$ is the generalized enthalpy or total pressure, extends Lighthill’s theory of vorticity generation at solid walls into the interior of the flow. We use these results to explain drag on the body in terms of vortex dynamics, unifying the theories for classical fluids and for quantum superfluids. The results offer a new solution to the “d’Alembert paradox” at infinite Reynolds numbers, provide an explanation for a long-standing puzzle about the experimental conditions required for anomalous turbulent energy dissipation, and imply the necessary and sufficient conditions for turbulent drag reduction.

Dynamical backaction resulting from radiation pressure forces in optomechanical systems has proven to be a versatile tool for manipulating mechanical vibrations. Notably, dynamical backaction has resulted in the cooling of a mechanical resonator to its ground state, driving phonon lasing, the generation of entangled states, and observation of the optical-spring effect. In certain magnetic materials, mechanical vibrations can interact with magnetic excitations (magnons) via the magnetostrictive interaction, resulting in an analogous magnon-induced dynamical backaction. In this article, we directly observe the impact of magnon-induced dynamical backaction on a spherical magnetic sample’s mechanical vibrations. Moreover, dynamical backaction effects play a crucial role in many recent theoretical proposals; thus, our work provides the foundation for future experimental work pursuing many of these theoretical proposals.

Information-to-energy conversion with feedback measurement stands as one of the most intriguing aspects of the thermodynamics of information in the nanoscale. To date, experiments have focused on feedback protocols for work extraction. Here we address the novel case of dissipation reduction in nonequilibrium systems with feedback. We perform pulling experiments on DNA hairpins with optical tweezers, with a general feedback protocol based on multiple measurements that includes either discrete-time or continuous-time feedback. While feedback can reduce dissipation, it remains unanswered whether it also improves free-energy determination (information-to-measurement conversion). We define thermodynamic information $\mathrm{\Upsilon }$ as the natural logarithm of the feedback efficacy, a quantitative measure of the efficiency of information-to-energy and information-to-measurement conversion in feedback protocols. We find that discrete- and continuous-time feedback reduces dissipation by roughly $k_{B}T\mathrm{\Upsilon }$ without improvement in free-energy determination. Remarkably, a feedback strategy (defined as a correlated sequence of feedback protocols) further reduces dissipation, enhancing information-to-measurement efficiency. Our study underlines the role of temporal correlations to develop feedback strategies for efficient information-to-measurement conversion in small systems.

Many biological systems synchronize their movement through physical interactions. By far, the most well-studied examples concern physical interactions through a fluid: Beating cilia, swimming sperm and worms, and flapping wings all display synchronization behavior through fluid mechanical interactions. However, as the density of a collective increases, individuals may also interact with each other through physical contact. In the field of “active matter” systems, it is well known that inelastic contact between individuals can produce long-range correlations in position, orientation, and velocity. In this work, we demonstrate that contact interactions between undulating robots yield novel phase dynamics such as synchronized motions. We consider undulatory systems in which rhythmic motion emerges from time-independent oscillators that sense and respond to an undulatory bending angle and speed. In pair experiments, we demonstrate that robot joints will synchronize to in-phase and antiphase oscillations through collisions, and a phase-oscillator model describes the stability of these modes. To understand how contact interactions influence the phase dynamics of larger groups, we perform simulations and experiments of simple three-link undulatory robots that interact only through contact. Collectives synchronize their movements through contact as predicted by the theory, and when the robots can adjust their position in response to contact, we no longer observe antiphase synchronization. Lastly, we demonstrate that synchronization dramatically reduces the interaction forces within confined groups of undulatory robots, indicating significant energetic and safety benefits from group synchronization. The theory and experiments in this study illustrate how contact interactions in undulatory active matter can lead to novel collective motion and synchronization.

The combination of nontrivial band topology and symmetry-breaking phases gives rise to novel quantum states and phenomena such as topological superconductivity, quantum anomalous Hall effect, and axion electrodynamics. Evidence of intertwined charge density wave (CDW) and superconducting order parameters has recently been observed in a novel kagome material ${A\mathrm{V}}_3{\mathrm{Sb}}_5$ ($A=\mathrm{K}$, Rb, Cs) that features a ${\mathbb{Z}}_2$ topological invariant in the electronic structure. However, the origin of the CDW and its intricate interplay with the topological state has yet to be determined. Here, using hard-x-ray scattering, we demonstrate a three-dimensional CDW with $2\times 2\times 2$ superstructure in $(\mathrm{Rb},\mathrm{Cs}){\mathrm{V}}_3{\mathrm{Sb}}_5$. Unexpectedly, we find that the CDW fails to induce acoustic phonon anomalies at the CDW wave vector but yields a novel Raman mode that quickly damps into a broad continuum below the CDW transition temperature. Our observations exclude strong electron-phonon-coupling-driven CDW in ${A\mathrm{V}}_3{\mathrm{Sb}}_5$ and support an unconventional CDW that was proposed in the kagome lattice at van Hove filling.

Quantum computers are on the brink of surpassing the capabilities of even the most powerful classical computers, which naturally raises the question of how one can trust the results of a quantum computer when they cannot be compared to classical simulation Here, we present a cross-verification technique that exploits the principles of measurement-based quantum computation to link quantum circuits of different input size, depth, and structure. Our technique enables consistency checks of quantum computations between independent devices, as well as within a single device. We showcase our protocol by applying it to five state-of-the-art quantum processors, based on four distinct physical architectures: nuclear magnetic resonance, superconducting circuits, trapped ions, and photonics, with up to six qubits and up to 200 distinct circuits.

The possibility of suddenly ionized molecules undergoing extremely fast electron hole (or hole) dynamics prior to significant structural change was first recognized more than 20 years ago and termed charge migration. The accurate probing of ultrafast electron hole dynamics requires measurements that have both sufficient temporal resolution and can detect the localization of a specific hole within the molecule. We report an investigation of the dynamics of inner valence hole states in isopropanol where we use an x-ray pump–x-ray probe experiment, with site and state-specific probing of a transient hole state localized near the oxygen atom in the molecule, together with an ab initio theoretical treatment. We record the signature of transient hole dynamics and make the first tentative observation of dynamics driven by frustrated Auger-Meitner transitions. We verify that the effective hole lifetime is consistent with our theoretical prediction. This state-specific measurement paves the way to widespread application for observations of transient hole dynamics localized in space and time in molecules and thus to charge transfer phenomena that are fundamental in chemical and material physics.

The search for topological spin excitations in recently discovered two-dimensional (2D) van der Waals (vdW) magnetic materials is important because of their potential applications in dissipationless spintronics. In the 2D vdW ferromagnetic (FM) honeycomb lattice ${\mathrm{CrI}}_3$ ($T_C=61\mathrm{K}$), acoustic and optical spin waves are found to be separated by a gap at the Dirac points. The presence of such a gap is a signature of topological spin excitations if it arises from the next-nearest-neighbor (NNN) Dzyaloshinskii-Moriya (DM) or bond-angle-dependent Kitaev interactions within the Cr honeycomb lattice. Alternatively, the gap is suggested to arise from an electron correlation effect not associated with topological spin excitations. Here, we use inelastic neutron scattering to conclusively demonstrate that the Kitaev interactions and electron correlation effects cannot describe spin waves, Dirac gaps, and their in-plane magnetic field dependence. Our results support the idea that the DM interactions are the microscopic origin of the observed Dirac gap. Moreover, we find that the nearest-neighbor (NN) magnetic exchange interactions along the $c$ axis are antiferromagnetic (AF), and the NNN interactions are FM. Therefore, our results unveil the origin of the observed $c$-axis AF order in thin layers of ${\mathrm{CrI}}_3$, firmly determine the microscopic spin interactions in bulk ${\mathrm{CrI}}_3$, and provide a new understanding of topology-driven spin excitations in 2D vdW magnets.