We introduce a single-number metric, quantum volume, that can be measured using a concrete protocol on near-term quantum computers of modest size ($n\lesssim 50$), and measure it on several state-of-the-art transmon devices, finding values as high as 16. The quantum volume is linked to system error rates, and is empirically reduced by uncontrolled interactions within the system. It quantifies the largest random circuit of equal width and depth that the computer successfully implements. Quantum computing systems with high-fidelity operations, high connectivity, large calibrated gate sets, and circuit rewriting toolchains are expected to have higher quantum volumes. The quantum volume is a pragmatic way to measure and compare progress toward improved system-wide gate error rates for near-term quantum computation and error-correction experiments.

In this paper, we present the general methodologies and framework to evaluate the time efficiency of an experimental realization of quantum algorithms. We do so by describing the factorization of $N=15$ in an NMR quantum computer (using Shor's algorithm) as an example. We began by simulating a quantum computer which performs the algorithm. Using this simulation, we devised a Monte Carlo algorithm to calculate the expected time in which a theoretical quantum computer could perform this calculation under the same energy conditions as current working quantum computers. We found that experimentally, a nuclear magnetic resonance quantum computer would take $1.59\pm 0.04\text{s}$ to perform our simulated computation, whereas the expected optimal time under the same energy conditions is $0.955\pm 0.004\mathrm{ms}$. Moreover, we found that the expected time is inversely proportional to the energy variance of our qubit states (as expected). Finally, we propose this theoretical method for analyzing the time efficiency of future quantum computing experiments.

The leading computational QED uncertainty of the Lamb shift in the $1s$ state in the hydrogen atom and helium ion is due to two-loop pure self-energy contributions. An exact evaluation in $Z\alpha$ has not yet been performed for $Z=1,2$, while a calculation based on the $Z\alpha$ expansion has a limited accuracy. To improve it, we calculate a leading logarithmic contribution in order ${\alpha }^2{\left(Z\alpha \right)}^{8}m$ and estimate the nonleading subterms in orders ${\alpha }^2{\left(Z\alpha \right)}^{7}m$ and ${\alpha }^2{\left(Z\alpha \right)}^{8}m$. To further improve the accuracy, we take advantage of a combined evaluation of the numerical results of the exact calculations in $Z\alpha$ performed at $Z=10–30$, and the results of the $Z\alpha$ expansion in order to reach the most accurate result, which can be done by fitting the data. The quality of the fit, including its reliability and accuracy, strongly depends on a theoretical input, which is the main target of this paper. Strictly speaking, we develop not a fit, but a theoretical approximation for the pure self-energy contribution, which may serve as a base for a robust fit. It is also important to estimate a deviation of the truncated function from the true one. For this purpose, we calculate the leading logarithmic term and estimate the subleading ones in order ${\alpha }^2{\left(Z\alpha \right)}^{9}m$. That allows us to find a reliable result which is more accurate than both fits over the numerical data and the results from the $Z\alpha$ expansion, which have been previously considered in literature. Performing the fit procedures, we focus our attention on the theoretical input which plays an essential role in the reliability and accuracy of the fit. Our results for the two-loop pure self-energy contribution to the $1s$ Lamb shift are expressed in a standard notation as $B_{60}=-72\left(7\right),G_{60}(Z=1)=-80\left(6\right)$, and $G_{60}(Z=2)=-83\left(5\right)$. That significantly improves the theoretical accuracy of the $1s$ Lamb shift in hydrogen, deuterium, and in the helium ions.

We report an electron momentum spectroscopy study on valence shake-up satellites of neon. A symmetric noncoplanar $(e,2e)$ experiment was performed at an incident electron energy of 1.2 keV, and the recoil-ion-momentum–dependent $(e,2e)$ cross sections or momentum profiles of the shake-up satellites have been obtained. Furthermore, theoretical momentum profiles have been calculated on the basis of the distorted-wave Born approximation using configuration interaction wave functions of the initial neutral and final ionic states. Comparison between experiment and theory has revealed that contrary to a general expectation, electron correlation in the initial neutral state has a significant influence on the shapes of momentum profiles for neon satellites. It has been shown that the Dyson orbitals of transitions to the $2p^4\left({}^1\mathrm{S}\right)3s{}^2\mathrm{S}$ and $2p^4\left({}^1\mathrm{D}\right)3p{}^2\mathrm{P}$ ionic states are more localized than those of the associated primary ionizations due to electron-correlation effects.

We study numerically the dynamical instabilities and splitting of singly and doubly quantized composite vortices in two-component Bose-Einstein condensates harmonically confined to quasi two dimensions. In this system, the vortices become pointlike composite defects that can be classified in terms of an integer pair $({\kappa }_1,{\kappa }_2)$ of phase winding numbers. Our simulations based on zero-temperature mean-field theory reveal several vortex splitting behaviors that stem from the multicomponent nature of the system and do not have direct counterparts in single-component condensates. By calculating the Bogoliubov excitations of stationary axisymmetric composite vortices, we find nonreal excitation frequencies (i.e., dynamical instabilities) for the singly quantized $(1,1)$ and $(1,-1)$ vortices and for all variants of doubly quantized vortices, which we define by the condition ${max}_{j=1,2}\left|{\kappa }_j\right|=2$. While the short-time predictions of the linear Bogoliubov analysis are confirmed by direct time integration of the Gross-Pitaevskii equations of motion, the time integration also reveals intricate long-time decay behavior not captured by the linearized dynamics. First, the $(1,\pm 1)$ vortex is found to be unstable against splitting into a $(1,0)$ vortex and a $(0,\pm 1)$ vortex. Second, the $(2,1)$ vortex exhibits a two-step decay process in which its initial splitting into a $(2,0)$ vortex and a $(0,1)$ vortex is followed by the off-axis splitting of the $(2,0)$ vortex into two $(1,0)$ vortices. Third, the $(2,-2)$ vortex is observed to split into a $(-1,1)$ vortex, three $(1,0)$ vortices, and three $(0,-1)$ vortices. Each of these splitting processes is the dominant decay mechanism of the respective stationary composite vortex for a wide range of intercomponent interaction strengths and relative populations of the two condensate components and should be amenable to experimental detection. Our results contribute to a better understanding of vortex physics, hydrodynamic instabilities, and two-dimensional quantum turbulence in multicomponent superfluids.

We find a solution of the wave equation in the paraxial approximation that describes attosecond pulses with spatiotemporal helical structure in the phase and in the intensity recently generated by means of highly nonlinear optical processes driven by visible or infrared femtosecond vortex pulses. Having a simple analytical model for these helical pulses will greatly facilitate the study of their predicted applications, particularly their interaction with matter after their generation. It also follows from our analysis that the topological charge dispersion inherent to helical pulses allows one to beat the minimum duration to which a pulsed vortex without charge dispersion is limited.

We discuss general properties of the equilibrium state of parametric down-conversion in superconducting quantum circuits with detunings and Kerr anharmonicities, in the strongly nonlinear regime. By comparing moments of the steady state and those of a Schrödinger cat, we show that true Schrödinger cats cannot survive in the steady state if there is any single-photon loss. A delta-function “catlike” steady-state distribution can be formed, but this only exists in the limit of an extremely large nonlinearity. The steady state is a mixed state, which is more complex than a mixture or linear combination of delta functions, and the purity of which is reduced by driving. We expect this general behavior to occur in other driven, dissipative quantum subharmonic nonequilibrium open systems.

We show that the mutually exclusive nature of classical and quantum correlations distributed in multiparty quantum systems can be characterized in terms of $q$-expectation value. Using Tsallis $q$ entropy and $q$-expectation value, we first provide generalized definitions of classical and quantum correlations, and establish their trade-off relations in three-party quantum systems of arbitrary dimension with respect to $q$-expectation value for $q\ge 1$. We also provide equivalence conditions for monogamy and polygamy inequalities of quantum entanglement and quantum discord distributed in three-party quantum systems of arbitrary dimension with respect to $q$-expectation value for $q\ge 1$.

A simple analytical formula for partial wave cross section describing low-energy elastic scattering in the presence of a shape resonance located near the elastic threshold is proposed. The formula is based on the extension of the regularized method of analytical continuation in the coupling constant (RAC) recently proposed by Horáček and co-workers. The RAC method provides the complex energies of resonances and virtual states requiring as input information a set of real bound state energies of a perturbed system for which standard highly sophisticated numerical codes may be employed. No scattering calculations are needed. These parameters define the low-energy partial wave resonance scattering cross section. Cross sections are calculated for the $2s^2\varepsilon p{}^{2}P^0$ state of beryllium and $3s^2\varepsilon p{}^{2}P^0$ of magnesium. The results compare well with recently published data.

Time-resolved x-ray scattering (TRXS) measures internuclear separations in a molecule following laser-induced photoexcitation. The molecular dynamics induced by the excitation laser may lie on one or several bound or dissociative electronic states. Time-resolved x-ray scattering from these states can be difficult to isolate because they generally overlap in the angle-resolved x-ray scattering pattern $I(x,y,\tau )$, where $\tau$ is the pump-probe delay and $(x,y)$ are the physical pixel positions. Here we show how standard transform methods can isolate the dynamics from individual states. We form the temporal Fourier transform $\tilde{I}(x,y,\omega )={\int }_{-\infty }^{+\infty }d\tau e^{-i\omega \tau }I(x,y,\tau )$. This frequency-resolved x-ray scattering (FRXS) signal segregates the bound states according to their vibrational frequencies ${\omega }_i$ and also displays dissociative states along straight lines $\omega =vQ$, where the slope $v$ is the rate of increase of the internuclear distance and $Q$ is the momentum transfer between the incident and scattered x-ray photon. We derive this relation and use FRXS to extract state-specific dynamics from experimental TRXS from molecular iodine following a 520-nm pump. Dynamics observed include one- and two-photon dissociation of the ${}^1{\mathrm{\Pi }}_u$ and ${}^1{\mathrm{\Sigma }}_g{}^{+}$ excited states and vibrational wave packets on the $B{}^3{\mathrm{\Pi }}_{0u}{}^{+}$ state.

The theory of rovibronic spectroscopy of light-dressed molecules is presented within the framework of quantum mechanically treated molecules interacting with classical light fields. The numerical application involves the homonuclear diatomic molecule ${\mathrm{Na}}_2$, for which the general formulas can be simplified considerably and the physical processes leading to the light-dressed spectra can be understood straightforwardly. The physical origin of different peaks in the light-dressed spectrum of ${\mathrm{Na}}_2$ is given and the light-dressed spectrum is investigated in terms of its dependence on the dressing field's intensity and wavelength, the turn-on time of the dressing field, and the temperature. The important implications of light-dressed spectroscopy on deriving field-free spectroscopic quantities are also discussed.

In a cold atom gas subject to a two-dimensional spin-dependent optical lattice potential with hexagonal symmetry, trapped atoms execute circular motion around the potential minima. Such atoms are elementary quantum rotors. The theory of such quantum rotors is developed. Wave functions, energies, and degeneracies are determined for both bosonic and fermionic atoms, and magnetic dipole transitions between quantum rotor states are elucidated. Quantum rotors in optical lattices with precisely one atom per unit cell can be used as extremely high sensitivity rotation sensors, accelerometers, and magnetometers.

The analysis of twisted (vortex) paraxial photons and electrons is fulfilled in the framework of relativistic quantum mechanics. The use of the Foldy-Wouthuysen representation radically simplifies the description of relativistic electrons and clarifies the fundamental properties of twisted particles. It is demonstrated that the twisted and other structured photons are luminal. Their subluminality apparently takes place because the photon energy is also contributed by a hidden motion. This motion is vanished by averaging and disappears in the semiclassical description based on expectation values of the momentum and velocity operators. It is proven that semiclassical quanta of structured light are subluminal and massive. The quantum-mechanical and semiclassical descriptions of twisted and other structured electrons lead to similar results. The effect of a quantization of the velocity and the effective mass of the structured photon and electron is predicted. This effect is observable for the photon. The twisted and untwisted semiclassical photons and electrons modeled by the centroids are considered in the accelerated and rotating noninertial frame. The coincidence of their inertial masses with kinematic ones is shown. The orbital magnetic moment of the Laguerre-Gauss electron does not depend on the radial quantum number.

To analyze the joint measurability of given measurements, we introduce a Hermitian operator-valued measure, called $W$ measure, such that it has marginals of positive operator-valued measures. We prove that $W$ measure is a positive operator-valued measure (POVM) if and only if its marginal POVMs are jointly measurable. The proof suggests employing a negative $W$ measure as an indicator of nonjoint measurability. This translates the joint measurability problem to unconstrained convex minimization of nondifferentiable function, which is solvable using a subgradient method. By applying triangle inequalities to the negativity, we derive the analytic joint measurability criteria for dichotomic and trichotomic variables. Moreover, we propose an operational test for joint measurability in a sequential measurement scenario.

We consider the problem of quantum phase estimation with access to arbitrary measurements in a single suboptimal basis. The achievable sensitivity limit in this case is determined by the classical Cramér-Rao bound with respect to the fixed basis. Here, we show that the sensitivity can be enhanced beyond this limit if knowledge about the energy expectation value is available. The combined information is shown to be equivalent to a direct measurement of an optimal linear combination of the basis projectors and the phase-imprinting Hamiltonian. Application to an atomic clock with oversqueezed spin states yields a sensitivity gain that scales linearly with the number of atoms. Our analysis further reveals that small modifications of the observable can have a strong impact on the sensitivity.

We analyze the quantum phase transitions taking place in a one-dimensional transverse-field Ising model with long-range couplings that decay algebraically with distance. We are interested in the Kibble-Zurek universal scaling laws emerging in nonequilibrium dynamics and in the potential for the unambiguous observation of such behavior in a realistic experimental setup based on trapped ions. To this end, we determine the phase diagram of the model and the critical exponents characterizing its quantum phase transitions by means of density-matrix renormalization group calculations and finite-size scaling theory, which allows us to obtain good estimates for different range of ferromagnetic and antiferromagnetic interactions. Beyond critical equilibrium properties, we tackle a nonequilibrium scenario in which quantum Kibble-Zurek scaling laws may be retrieved. Here, it is found that the predicted nonequilibrium universal behavior, i.e., the scaling laws as a function of the quench rate and critical exponents, can be observed in systems comprising an experimentally feasible number of spins. Finally, a scheme is introduced to simulate the algebraically decaying couplings accurately by means of a digital quantum simulation with trapped ions. Our results suggest that quantum Kibble-Zurek physics can be explored and observed in state-of-the-art experiments with trapped ions realizing long-range Ising models.

Identifying the classical-quantum divide and quantifying nonclassicality (quantumness) are fundamental issues in quantum exploration. By use of the Wigner-Yanase skew information [E. P. Wigner and M. M. Yanase, Proc. Natl. Acad. Sci. USA 49, 910 (1963)], which is a version of quantum Fisher information with a well-established interpretation as quantum uncertainty, we introduce a simple and effective quantifier for nonclassicality of bosonic field states. This is achieved by simultaneously yet approximately measuring the canonical conjugate quadrature pair (equivalently, measuring the non-Hermitian annihilation operator in a generalized sense) from an information-theoretic perspective. This quantifier, apart from capturing the idea of nonclassicality as quantum uncertainty in a succinct way, is mathematically computable, physically intuitive, and possesses operational significance. We reveal its fundamental properties and illustrate its physical applications in quantifying nonclassicality of light. We compare it with various existent quantifiers for nonclassicality and illustrate its merits. A related quantifier for classicality, which is in some sense dual to nonclassicality, is also introduced. This pair of classicality and nonclassicality provides insight into the classical-quantum interplay.

As an analogy of the fully entangled fraction in the framework of entanglement theory, we have introduced the notion of quantum coherence fraction $C_{\mathcal{F}}$, which quantifies the closeness between a given state and the set of maximally coherent states. By providing an alternative formulation of the robustness of coherence $C_{\mathcal{R}}$, we have elucidated the relationship between the quantum coherence fraction and the normalized version of $C_{\mathcal{R}}$, i.e., ${\overline{C}}_{\mathcal{R}}$, where the role of genuinely incoherent operations is highlighted. Numerical simulation shows that, though as expected $C_{\mathcal{F}}$ is upper bounded by ${\overline{C}}_{\mathcal{R}}$, $C_{\mathcal{F}}$ constitutes a good approximation to ${\overline{C}}_{\mathcal{R}}$ especially in low-dimensional Hilbert spaces. Even more intriguingly, we can analytically prove that $C_{\mathcal{F}}$ is exactly equivalent to ${\overline{C}}_{\mathcal{R}}$ for qubit and qutrit states. Moreover, some intuitive properties and implications of $C_{\mathcal{F}}$ are also indicated.

Leakage errors take qubits out of the computational subspace and will accumulate if not addressed. A leaked qubit will reduce the effectiveness of quantum error correction protocols due to the cost of implementing leakage reduction circuits and the harm caused by interacting leaked states with qubit states. Ion trap qubits driven by Raman gates have a natural choice between qubits encoded in magnetically insensitive hyperfine states that can leak and qubits encoded in magnetically sensitive Zeeman states of the electron spin that cannot leak. In our previous work, we compared these two qubits in the context of the toric code with a depolarizing leakage error model and found that for magnetic-field noise with a standard deviation less than 32 $\mu \mathrm{G}$ that the ${}^{174}\mathrm{Yb}^{+}$ Zeeman qubit outperforms the ${}^{171}\mathrm{Yb}^{+}$ hyperfine qubit. Here we examine a physically motivated leakage error model based on ions interacting via the Mølmer-Sørenson gate. We find that this greatly improves the performance of hyperfine qubits but the Zeeman qubits are more effective for magnetic-field noise with a standard deviation less than 10 $\mu \mathrm{G}$. At these low magnetic fields, we find that the best choice is a mixed qubit scheme where the hyperfine qubits are the syndrome qubits and the leakage is handled without the need of an additional leakage reduction circuit.