Standard quantum theory was formulated with complex-valued Schrödinger equations, wave functions, operators, and Hilbert spaces. Previous work attempted to simulate quantum systems using only real numbers by exploiting an enlarged Hilbert space. A fundamental question arises: are the complex numbers really necessary in the standard formalism of quantum theory? To answer this question, a quantum game has been developed to distinguish standard quantum theory from its real-number analog, by revealing a contradiction between a high-fidelity multiqubit quantum experiment and players using only real-number quantum theory. Here, using superconducting qubits, we faithfully realize the quantum game based on deterministic entanglement swapping with a state-of-the-art fidelity of 0.952. Our experimental results violate the real-number bound of 7.66 by 43 standard deviations. Our results disprove the real-number formulation and establish the indispensable role of complex numbers in the standard quantum theory.

In this Letter, we deploy for the first time reinforcement-learning algorithms in the context of the conformal-bootstrap program to obtain numerical solutions of conformal field theories (CFTs). As an illustration, we use a soft actor-critic algorithm and find approximate solutions to the truncated crossing equations of two-dimensional CFTs, successfully identifying well-known theories like the 2D Ising model and the 2D CFT of a compactified scalar. Our methods can perform efficient high-dimensional searches that can be used to study arbitrary (unitary or nonunitary) CFTs in any spacetime dimension.

Silica, water, and hydrogen are known to be the major components of celestial bodies, and have significant influence on the formation and evolution of giant planets, such as Uranus and Neptune. Thus, it is of fundamental importance to investigate their states and possible reactions under the planetary conditions. Here, using advanced crystal structure searches and first-principles calculations in the $\mathrm{Si}\text{−}\mathrm{O}\text{−}\mathrm{H}$ system, we find that a silica-water compound ${({\mathrm{SiO}}_2)}_2({\mathrm{H}}_2\mathrm{O})$ and a silica-hydrogen compound ${\mathrm{SiO}}_2{\mathrm{H}}_2$ can exist under high pressures above 450 and 650 GPa, respectively. Further simulations reveal that, at high pressure and high temperature conditions corresponding to the interiors of Uranus and Neptune, these compounds exhibit superionic behavior, in which protons diffuse freely like liquid while the silicon and oxygen framework is fixed as solid. Therefore, these superionic silica-water and silica-hydrogen compounds could be regarded as important components of the deep mantle or core of giants, which also provides an alternative origin for their anomalous magnetic fields. These unexpected physical and chemical properties of the most common natural materials at high pressure offer key clues to understand some abstruse issues including demixing and erosion of the core in giant planets, and shed light on building reliable models for solar giants and exoplanets.

The origin of water on the Earth is a long-standing mystery, requiring a comprehensive search for hydrous compounds, stable at conditions of the deep Earth and made of Earth-abundant elements. Previous studies usually focused on the current range of pressure-temperature conditions in the Earth’s mantle and ignored a possible difference in the past, such as the stage of the core-mantle separation. Here, using ab initio evolutionary structure prediction, we find that only two magnesium hydrosilicate phases are stable at megabar pressures, $\alpha \text{−}{\mathrm{Mg}}_2{\mathrm{SiO}}_5{\mathrm{H}}_2$ and $\beta \text{−}{\mathrm{Mg}}_2{\mathrm{SiO}}_5{\mathrm{H}}_2$, stable at 262–338 GPa and $>338\mathrm{GPa}$, respectively (all these pressures now lie within the Earth’s iron core). Both are superionic conductors with quasi-one-dimensional proton diffusion at relevant conditions. In the first 30 million years of Earth’s history, before the Earth’s core was formed, these must have existed in the Earth, hosting much of Earth’s water. As dense iron alloys segregated to form the Earth’s core, ${\mathrm{Mg}}_2{\mathrm{SiO}}_5{\mathrm{H}}_2$ phases decomposed and released water. Thus, now-extinct ${\mathrm{Mg}}_2{\mathrm{SiO}}_5{\mathrm{H}}_2$ phases have likely contributed in a major way to the evolution of our planet.

We generalize the flux insertion argument due to Laughlin, Niu-Thouless-Tao-Wu, and Avron-Seiler-Zograf to the case of fractional quantum Hall states on a higher-genus surface. We propose this setting as a test to characterize the robustness, or topologicity, of the quantum state of matter and apply our test to the Laughlin states. Laughlin states form a vector bundle, the Laughlin bundle, over the Jacobian—the space of Aharonov-Bohm fluxes through the holes of the surface. The rank of the Laughlin bundle is the degeneracy of Laughlin states or, in the presence of quasiholes, the dimension of the corresponding full many-body Hilbert space; its slope, which is the first Chern class divided by the rank, is the Hall conductance. We compute the rank and all the Chern classes of Laughlin bundles for any genus and any number of quasiholes, settling, in particular, the Wen-Niu conjecture. Then we show that Laughlin bundles with nonlocalized quasiholes are not projectively flat and that the Hall current is precisely quantized only for the states with localized quasiholes. Hence our test distinguishes these states from the full many-body Hilbert space.

If primordial black holes (PBHs) contribute more than 10% of the dark matter (DM) density, their energy density today is of the same order as that of the baryons. Such a cosmic coincidence might hint at a mutual origin for the formation scenario of PBHs and the baryon asymmetry of the Universe. Baryogenesis can be triggered by a sharp transition of the rolling rate of inflaton from slow-roll to (nearly) ultraslow-roll phases that produce large curvature perturbations for PBH formation in single-field inflationary models. We show that the baryogenesis requirement drives the PBH contribution to DM, along with the inferred PBH mass range, the resulting stochastic gravitational wave background frequency window, and the associated cosmic microwave background tensor-to-scalar ratio amplitude, into potentially observable regimes.

Stimulated by the recent experiment [F. Ando et al., Nature (London) 584, 373 (2020).], we propose an intrinsic mechanism to cause the superconducting diode effect (SDE). SDE refers to the nonreciprocity of the critical current for the metal-superconductor transition. Among various mechanisms for the critical current, the depairing current is known to be intrinsic to each material and has recently been observed in several superconducting systems. We clarify the temperature scaling of the nonreciprocal depairing current near the critical temperature and point out its significant enhancement at low temperatures. It is also found that the nonreciprocal critical current shows sign reversals upon increasing the magnetic field. These behaviors are understood by the nonreciprocity of the Landau critical momentum and the change in the nature of the helical superconductivity. The intrinsic SDE unveils the rich phase diagram and functionalities of noncentrosymmetric superconductors.

We propose a tensor network approach to compute amplitudes and probabilities for a large number of correlated bitstrings in the final state of a quantum circuit. As an application, we study Google’s Sycamore circuits, which are believed to be beyond the reach of classical supercomputers and have been used to demonstrate quantum supremacy. By employing a small computational cluster containing 60 graphical processing units (GPUs), we compute exact amplitudes and probabilities of $2\times {10}^6$ correlated bitstrings with some entries fixed (which span a subspace of the output probability distribution) for the Sycamore circuit with 53 qubits and 20 cycles. The obtained results verify the Porter-Thomas distribution of the large and deep quantum circuits of Google, provide datasets and benchmarks for developing approximate simulation methods, and can be used for spoofing the linear cross entropy benchmark of quantum supremacy. Then we extend the proposed big-batch method to a full-amplitude simulation approach that is more efficient than the existing Schrödinger method on shallow circuits and the Schrödinger-Feynman method in general, enabling us to obtain the state vector of Google’s simplifiable circuit with $n=43$ qubits and $m=14$ cycles using only one GPU. We also manage to obtain the state vector for Google’s simplifiable circuits with $n=50$ qubits and $m=14$ cycles using a small GPU cluster, breaking the previous record on the number of qubits in full-amplitude simulations. Our method is general in computing bitstring probabilities for a broad class of quantum circuits and can find applications in the verification of quantum computers. We anticipate that our method will pave the way for combining tensor network–based classical computations and near-term quantum computations for solving challenging problems in the real world.

Quasisymmetry is an unusual symmetry that can be present in toroidal magnetic fields, enabling the confinement of charged particles and plasma. Here it is shown that both quasiaxisymmetry and quasihelical symmetry can be achieved to a much higher precision than previously thought over a significant volume, resulting in exceptional confinement. For a 1 Tesla mean field far from axisymmetry (vacuum rotational transform $>0.4$), symmetry-breaking mode amplitudes throughout a volume of aspect ratio 6 can be made as small as the typical $\sim 50\mu \mathrm{T}$ geomagnetic field.

Verifying the correct functioning of quantum gates is a crucial step toward reliable quantum information processing, but it becomes an overwhelming challenge as the system size grows due to the dimensionality curse. Recent theoretical breakthroughs show that it is possible to verify various important quantum gates with the optimal sample complexity of $O(1/\epsilon )$ using local operations only, where $\epsilon$ is the estimation precision. In this Letter, we propose a variant of quantum gate verification (QGV) that is robust to practical gate imperfections and experimentally realize efficient QGV on a 2-qubit controlled-not gate and a 3-qubit Toffoli gate using only local state preparations and measurements. The experimental results show that, by using only 1600 and 2600 measurements on average, we can verify with 95% confidence level that the implemented controlled-not gate and Toffoli gate have fidelities of at least 99% and 97%, respectively. Demonstrating the superior low sample complexity and experimental feasibility of QGV, our work promises a solution to the dimensionality curse in verifying large quantum devices in the quantum era.

A global network of optical atomic clocks will enable unprecedented measurement precision in fields including tests of fundamental physics, dark matter searches, geodesy, and navigation. Free-space laser links through the turbulent atmosphere are needed to fully exploit this global network, by enabling comparisons to airborne and spaceborne clocks. We demonstrate frequency transfer over a 2.4 km atmospheric link with turbulence comparable to that of a ground-to-space link, achieving a fractional frequency stability of $6.1\times {10}^{-21}$ in 300 s of integration time. We also show that clock comparison between ground and low Earth orbit will be limited by the stability of the clocks themselves after only a few seconds of integration. This significantly advances the technologies needed to realize a global timescale network of optical atomic clocks.

We experimentally and theoretically investigate the influence of the magnetic component of an electromagnetic field on high-order above-threshold ionization of xenon atoms driven by ultrashort femtosecond laser pulses. The nondipole shift of the electron momentum distribution along the light-propagation direction for high energy electrons beyond the $2U_p$ classical cutoff is found to be vastly different from that below this cutoff, where $U_p$ is the ponderomotive potential of the driving laser field. A local minimum structure in the momentum dependence of the nondipole shift above the cutoff is identified for the first time. With the help of classical and quantum-orbit analysis, we show that large-angle rescattering of the electrons strongly alters the partitioning of the photon momentum between electron and ion. The sensitivity of the observed nondipole shift to the electronic structure of the target atom is confirmed by three-dimensional time-dependent Schrödinger equation simulations for different model potentials. Our work paves the way toward understanding the physics of extreme light-matter interactions at long wavelengths and high electron kinetic energies.

Confined active nematics exhibit rich dynamical behavior, including spontaneous flows, periodic defect dynamics, and chaotic “active turbulence.” Here, we study these phenomena using the framework of exact coherent structures, which has been successful in characterizing the routes to high Reynolds number turbulence of passive fluids. Exact coherent structures are stationary, periodic, quasiperiodic, or traveling wave solutions of the hydrodynamic equations that, together with their invariant manifolds, serve as an organizing template of the dynamics. We compute the dominant exact coherent structures and connecting orbits in a preturbulent active nematic channel flow, which enables a fully nonlinear but highly reduced-order description in terms of a directed graph. Using this reduced representation, we compute instantaneous perturbations that switch the system between disparate spatiotemporal states occupying distant regions of the infinite-dimensional phase space. Our results lay the groundwork for a systematic means of understanding and controlling active nematic flows in the moderate- to high-activity regime.

The breakup and coalescence of drops are elementary topological transitions in interfacial flows. The breakup of a drop changes dramatically when polymers are added to the fluid. With the strong elongation of the polymers during the process, long threads connecting the two droplets appear prior to their eventual pinch-off. Here, we demonstrate how elasticity affects drop coalescence, the complement of the much studied drop pinch-off. We reveal the emergence of an elastic singularity, characterized by a diverging interface curvature at the point of coalescence. Intriguingly, while the polymers dictate the spatial features of coalescence, they hardly affect the temporal evolution of the bridge. These results are explained using a novel viscoelastic similarity analysis and are relevant for drops created in biofluids, coating sprays, and inkjet printing.

We present a novel framework to solve simultaneously the electroweak-hierarchy problem and the strong-$CP$ problem. A small but finite Higgs vacuum expectation value and a small $\theta$ angle are selected after the QCD phase transition, without relying on the Peccei-Quinn mechanism or other traditional solutions. We predict a distinctive pattern of correlated signals at hadronic EDM, fuzzy dark matter, and axion experiments.

Fluctuation-induced forces are observed in numerous physical systems spanning from quantum to macroscopic scale. However, there is as yet no experimental report of their existence in hydrodynamic turbulence. Here, we present evidence of an attraction force mediated via turbulent fluctuations by using two walls locally confining 2D turbulence. This long-range interaction is a function of the wall separation and the energy injection rate in the turbulent flow. As the wall spacing decreases, the confined flow becomes less energetic and more anisotropic in the bounded domain, producing stronger attraction. The mechanism of force generation is rooted in a nontrivial fluid-wall coupling where coherent flow structures are guided by the cavity walls. For the narrowest cavities studied, a resonance phenomenon at the flow forcing scale leads to a complex short-range interaction. The results could be relevant to problems encountered in a range of fields from industrial multiphase flows to modeling of planetary formation.

We present an analytical derivation of the volume fractions for random close packing (RCP) in both $d=3$ and $d=2$, based on the same methodology. Using suitably modified nearest neighbor statistics for hard spheres, we obtain ${\varphi }_{\mathrm{RCP}}=0.65896$ in $d=3$ and ${\varphi }_{\mathrm{RCP}}=0.88648$ in $d=2$. These values are well within the interval of values reported in the literature using different methods (experiments and numerical simulations) and protocols. This statistical derivation suggests some considerations related to the nature of RCP: (i) RCP corresponds to the onset of mechanical rigidity where the finite shear modulus emerges, (ii) the onset of mechanical rigidity marks the maximally random jammed state and dictates ${\varphi }_{\mathrm{RCP}}$ via the coordination number $z$, (iii) disordered packings with $\varphi >{\varphi }_{\mathrm{RCP}}$ are possible at the expense of creating some order, and $z=12$ at the fcc limit acts as a boundary condition.

Recent experiments demonstrate how a soluble body placed in a fluid spontaneously forms a dissolution pinnacle—a slender, upward pointing shape that resembles naturally occurring karst pinnacles found in stone forests. This unique shape results from the interplay between interface motion and the natural convective flows driven by the descent of relatively heavy solute. Previous investigations suggest these structures to be associated with shock formation in the underlying evolution equations, with the regularizing Gibbs-Thomson effect required for finite tip curvature. Here, we find a class of exact solutions that act as attractors for the shape dynamics in two and three dimensions. Intriguingly, the solutions exhibit large but finite tip curvature without any regularization, and they agree remarkably well with experimental measurements. The relationship between the dimensions of the initial shape and the final state of dissolution may offer a principle for estimating the age and environmental conditions of geological structures.

Non-Maxwellian electron velocity distribution functions composed of a warm bulk population and a cold beam are directly measured during electron-only reconnection with a strong out-of-plane (guide) magnetic field in a laboratory plasma. Electron heating is localized to the separatrix, and the electron temperature increases continuously along the separatrix. The measured gain in enthalpy flux is 70% of the incoming Poynting flux. The electron beams are oppositely directed on either side of the $X$ point, and their velocities are comparable to, and scale with, the electron Alfvén speed. Particle-in-cell simulations are consistent with the measurements. The experimental results are consistent with, and go beyond, recent observations in the magnetosheath.

The calculation of turbulence statistics is considered the key unsolved problem of fluid mechanics, i.e., precisely the computation of arbitrary statistical velocity moments from first principles alone. Using symmetry theory, we derive turbulent scaling laws for moments of arbitrary order in two regions of a turbulent channel flow. Besides the classical scaling symmetries of space and time, the key symmetries for the present work reflect the two well-known characteristics of turbulent flows: non-Gaussianity and intermittency. To validate the new scaling laws we made a new simulation at an unprecedented friction Reynolds number of 10 000, large enough to test the new scaling laws. Two key results appear as an application of symmetry theory, which allowed us to generate symmetry invariant solutions for arbitrary orders of moments for the underlying infinite set of moment equations. First, we show that in the sense of the generalization of the deficit law all moments of the streamwise velocity in the channel center follow a power-law scaling, with exponents depending on the first and second moments alone. Second, we show that the logarithmic law of the mean streamwise velocity in wall-bounded flows is indeed a valid solution of the moment equations, and further, all higher moments in this region follow a power law, where the scaling exponent of the second moment determines all higher moments. With this we give a first complete mathematical framework for all moments in the log region, which was first discovered about 100 years ago.