In another work [S. Huang and K. R. Brown, Phys. Rev. Lett. 127, 090505 (2021)] we introduced syndrome extraction methods for Calderbank-Shor-Steane quantum error-correction codes that interpolate between the well-known Shor and Steane syndrome extraction methods. Here we provide detailed proofs of the main theorems and show that up to gate ordering there is a one-to-one correspondence between extraction gadgets and partitions of the parity check matrix. Operationally, all the circuits in our framework can be obtained by merging ancilla qubits of Shor error correction with certain rules, which enables us to design fault-tolerant syndrome extraction circuits for given specific codes. We then apply our construction to the toric code and provide a detailed analysis of the time overhead of fault tolerance. In particular, we construct a syndrome extraction family whose time overhead smoothly varies from Shor to Steane error correction. To understand the potential advantage, we consider two simplified error models: no errors on the ancilla block and uncorrelated errors on the ancilla block. We study the threshold behavior of the family for both error models and show its potential advantage for quantum architectures with long coherence times.

Linear regression is a basic and widely used methodology in data analysis. It is known that some quantum algorithms efficiently perform least-squares linear regression of an exponentially large dataset. However, if we obtain values of the regression coefficients as classical data, the complexity of the existing quantum algorithms can be larger than the classical method. This is because it depends strongly on the tolerance error $\epsilon$: the best one among the existing proposals is $O\left({\epsilon }^{-2}\right)$. In this paper, we propose a quantum algorithm for linear regression, which has a complexity of $O\left({\epsilon }^{-1}\right)$ and keeps a logarithmic dependence on the number of data points $N_D$. In this method, we overcome bottleneck parts in the calculation, which take the form of the sum over data points and therefore have a complexity proportional to $N_D$, using quantum amplitude estimation, and other parts classically. Additionally, we generalize our method to some class of convex optimization problems.

The quest for improved sampling methods to solve statistical mechanics problems of physical and chemical interest has proceeded with renewed efforts since the invention of the Metropolis algorithm, in 1953. In particular, the understanding of thermally activated rare-event processes between long-lived metastable states, such as protein folding, is still elusive. In this case, one needs both the finite-temperature canonical distribution function and the reaction current between the reactant and product states to completely characterize the dynamic. Here we show how to tackle this problem using a quantum computer. We use the connection between a classical stochastic dynamics and the Schrödinger equation, also known as stochastic quantization, to variationally prepare quantum states, allowing us to unbiasedly sample from a Boltzmann distribution. Similarly, reaction rate constants can be computed as ground-state energies of suitably transformed operators, following the supersymmetric extension of the formalism. Finally, we propose a hybrid quantum-classical sampling scheme to escape local minima and explore the configuration space in both real-space and spin Hamiltonians. We indicate how to realize the quantum algorithms constructively, without assuming the existence of oracles or quantum walk operators. The quantum advantage concerning the above applications is discussed.

We present a theoretical investigation of the radiative recombination of twisted Bessel electrons with initially hydrogen-like (finally helium-like) heavy ions. In our study, we focus especially on the linear polarization of x-ray photons emitted in the electron capture into the ground $1s_{1/2}^2$ ionic state. Particular emphasis is placed on the question of how this polarization is affected if incident hydrogen-like ions are themselves spin-polarized. To explore such a “polarization transfer” we apply the density matrix theory and derive the Stokes parameters of recombination x rays for the realistic case of collisions between macroscopic electron and ion beams. Based on the developed general approach two scenarios are discussed that are of interest for the planned experiments at ion storage rings. First, we demonstrate how the use of twisted electrons can empower the known method for the diagnostics of spin-polarized ion beams, based on the rotation of the linear polarization of recombination light. In the second scenario we show how the internal structure of ions beams with inhomogeneous intensity and spin patterns can be probed by the capture of Bessel electrons, carrying different values of angular momentum.

We show that the time-dependent Schrödinger equation describing in the dipole approximation the interaction of a one-electron atom with a monochromatic circularly polarized electromagnetic field can be reduced to a stationary Schrödinger equation in the form which allows to separate variables in the asymptotic region and explicitly formulate outgoing-wave boundary conditions. The solutions to this equation are called atomic Siegert states (SSs) in a rotating electric field. We develop the theory of such states and propose an efficient method to construct them using powerful techniques of stationary scattering theory. The method yields not only the SS eigenvalue defining the Stark-shifted energy and total ionization rate of the state, but also the SS eigenfunction defining partial ionization amplitudes and the photoelectron momentum distribution. The theory is illustrated by calculations for a model potential.

For a system of bosons that interact through a class of general memory kernels, a recurrence relation for the partition function is derived within the path-integral formalism. This approach provides a generalization to previously known treatments in the literature of harmonically coupled systems of identical particles. As an example the result is applied to the specific heat of a simplified model of an open quantum system of bosons, harmonically coupled to a reservoir of distinguishable fictitious masses.

An equation is derived for analyzing the self-action of wave packets with few optical cycles in multicore fibers (MCFs). A class of stable out-of-phase spatiotemporal solitons with few-cycle durations in the MCF with cores located in a ring is found and analyzed. The stability boundary of the obtained solutions is determined. As an example of using such solitons, we considered the problem of their self-compression in the process of multisoliton dynamics in the MCF. The formation of laser pulses with a duration of a few optical cycles at the output of a 10-core MCF is shown.

In this paper, we present a thorough investigation for a spontaneous parametric four-wave mixing process in third-order nonlinear waveguides with various continuous tapering patterns. It has been previously shown that these devices can quasi-phase-match the four-wave mixing process and enhance its conversion efficiency by orders of magnitude. By altering the tapering profile curve we found that these devices can enable single-photon sources with either narrow or broadband spectral widths at on-demand frequencies. Using our model, we were also able to identify the waveguide length at which the single-photon spectral purity is maximized.

We use numerical simulations to demonstrate third-harmonic generation with near-unity nonlinear circular dichroism (CD) and high conversion efficiency $\left({10}^{-2}{\text{W}}^{-2}\right)$ in asymmetric Si-on-$\mathrm{Si}{\mathrm{O}}_2$ metasurfaces. The working principle relies on the spin-selective excitation of a quasibound state in the continuum, characterized by a very high $(>{10}^5)$ quality factor. By tuning multimode interference with the variation of the metasurface geometrical parameters, we show the possibility to control both linear CD and nonlinear CD. Our results pave the way for the development of all-dielectric metasurfaces for nonlinear chiro-optical devices with high conversion efficiency.

Squeezing of collective atomic spins has been shown to improve the sensitivity of atomic clocks and magnetometers to levels significantly below the standard quantum limit. In most cases the requisite atom-atom entanglement has been generated by dispersive interaction with a quantized probe field or by state-dependent collisions in a quantum gas. Such experiments typically use complex multilevel atoms like Rb or Cs, with the relevant interactions designed so that atoms behave like pseudospin-$\frac{1}{2}$ particles. We demonstrate the viability of spin squeezing for collective spins composed of the physical angular momenta of ${10}^6$ Cs atoms, each in an internal spin-4 hyperfine state. A peak metrological squeezing of at least 5dB is generated by quantum backaction from a dispersive quantum nondemolition (QND) measurement, implemented using a two-color optical probe that minimizes tensor light shifts without sacrificing measurement strength. Other significant developments include the successful application of composite pulse techniques for accurate dynamical control of the collective spin, enabled by broadband suppression of background magnetic fields inside a state-of-the-art magnetic shield. The absence of classical noise allows us to compare the observed quantum projection noise and squeezing to a theoretical model that properly accounts for both the relevant atomic physics and the spatial mode of the collective spin, finding good quantitative agreement and thereby validating its use in other contexts. Our work sets the stage for experiments on quantum feedback, deterministic squeezing, and closed-loop magnetometry. The implementation of real-time feedback may also create an opportunity for new types of quantum simulation, wherein the evolution of a quantum system is conditioned on the outcome of a time-continuous QND measurement. Such a scheme has the potential to access new regimes near the quantum-classical boundary, with opportunities to study long-standing issues related to quantum-classical correspondence in chaotic systems.

Ionization of atoms and molecules in laser fields can lead to various interesting interference structures in the photoelectron spectrum. For the case of a superintense extreme ultraviolet laser pulse, we identify a novel petal-like interference structure in the electron momentum distribution along the direction of the laser field propagation. We show that this structure is quite general and can be attributed to the Fresnel diffraction of the electronic wave packet by the nucleus. Our results are demonstrated by numerically solving the time-dependent Schrödinger equation of the atomic hydrogen beyond the dipole approximation. By building an analytical model, we find that the electron displacement determines the aforementioned interference pattern. In addition, we establish the physical picture of laser-induced electron Fresnel diffraction which is reinforced by both quantum and semiclassical models.

The Fock-Krylov formalism for the calculation of survival probabilities of unstable states is revisited paying particular attention to the mathematical constraints on the density of states, the Fourier transform of which gives the survival amplitude. We show that it is not possible to construct a density of states corresponding to a purely exponential survival amplitude. The survival probability $P\left(t\right)$ and the autocorrelation function of the density of states are shown to form a pair of cosine Fourier transforms. This result is a particular case of the Wiener-Khinchin theorem and forces $P\left(t\right)$ to be an even function of time which in turn forces the density of states to contain a form factor which vanishes at large energies. Subtle features of the transition regions from the nonexponential to the exponential at small times and the exponential to the power-law decay at large times are discussed by expressing $P\left(t\right)$ as a function of the number of oscillations $n$ performed by it. The transition at short times is shown to occur when the survival probability has completed one oscillation. The number of oscillations depends on the properties of the resonant state and a complete description of the evolution of the unstable state is provided by determining the limits on the number of oscillations in each region.

A striking feature of non-Hermitian tight-binding Hamiltonians is the high sensitivity of both spectrum and eigenstates to boundary conditions. Indeed, if the spectrum under periodic boundary conditions is point gapped by opening the lattice the non-Hermitian skin effect will necessarily occur. Finding the exact skin eigenstates may be demanding in general, and many methods in the literature are based on recurrence equations for the eigenstates' components. Here we devise a general procedure based on the Green's function method to calculate the eigenstates of non-Hermitian tight-binding Hamiltonians under open boundary conditions. We apply it to the Hatano-Nelson and non-Hermitian Su-Schrieffer-Heeger models and finally we contrast the edge states localization with that of bulk states.

This paper examines the performance of spatial search where the Grover diffusion operator is replaced by continuous-time quantum walks on a class of interdependent networks. We prove that, for a set of optimal quantum walk times and marked vertex phase shifts, a deterministic algorithm for structured spatial search is established that finds the marked vertex with 100% probability. This improves on the Childs and Goldstone spatial search algorithm on the same class of graphs, which we show can only amplify to 50% probability. Our method uses $\lceil \frac{\pi }{2\sqrt{2}}\sqrt{N}\rceil$ marked vertex phase shifts for an $N$-vertex graph, making it comparable to Grover's algorithm for an unstructured search. It is expected that this framework can be readily extended to deterministic spatial search on other families of graph structures.

We describe a quantum algorithm to estimate the $\alpha$-Renyi entropy of an unknown density matrix $\rho \in {\mathbb{C}}^{d\times d}$ for $\alpha \ne 1$ by combining the recent technique of quantum singular value transformations with the method of estimating normalized traces in the one clean qubit model. We consider an oracular input model where the input state is prepared via a quantum oracle that outputs a purified version of the state, assumed to be nonsingular. Our method outputs an estimate of the $\alpha$-Renyi entropy to additive precision $\epsilon$, using an expected total number $O\left(\frac{1}{{\left(x\epsilon \right)}^2}\right)$ of independent applications of a quantum circuit which coherently queries the input unitary $O(\frac{1}{\delta }log\frac{d}{\epsilon })$ times, in each case measuring a single output qubit. Here $\delta$ is a lower cutoff on the smallest eigenvalue of $\rho$ and $x=\frac{1}{d}\mathrm{Tr}\left({\rho }^{\alpha }\right)$. The expected number of measurements made in this method can be compared to results in the sample complexity model that generally require $\mathrm{\Theta }\left(\frac{d^2}{{\epsilon }^2}\right)$ samples. Furthermore, we also show that multiplicative approximations can be obtained by iteratively using additive approximations, with an overhead logarithmic in the dimension $d$.

Recent interest in the physics of nonperturbative light-matter coupling led to the development of solid-state cavity quantum electrodynamics setups in which the interaction energies are comparable with the bare ones. In such a regime, the ground state of the coupled system becomes interaction-dependent and is predicted to contain a population of virtual excitations, which, despite being the object of many investigations, remain unobserved. In this paper, we investigate how virtual electronic excitations in quantum wells modify the ground-state charge distribution, and we propose two methods to measure such a cavity-induced perturbation. The first approach is based on spectroscopic mapping of the electronic population at a specific location in the quantum well using localized defect states. The second approach exploits instead the photonic equivalent of a Kelvin probe to measure the average charge distribution across the quantum well. We find both effects observable with present-day or near-future technology. Our results thus provide a route toward a demonstration of cavity-induced modulation of ground-state electronic properties.

We study a two-dimensional spin-$\frac{1}{2}$ interacting Bose-Einstein condensate with a Rashba-Dresselhaus spin-orbit coupling. The analytical expression of the superfluid density is derived from the linear response theory and Bogoliubov approximation. We find that the superfluid density can be divided into three terms, which originate from the condensate, the interaction between the condensate and the Bogoliubov excitations, and only the excitations. We show that the condensate-excitation interaction changes the bare mass in the superfluid density into the effective mass. In the isotropic spin-orbit coupling limit, the effective mass diverges, and the superfluid density is predicted to be zero within previous studies; however, our work shows that the excitation-induced superfluid density remains nonzero, which means that quantum fluctuations rescue the superfluidity. Our results show the importance of the quantum fluctuations in understanding the superfluid properties of spin-orbit-coupled Bose-Einstein condensates.

We study zero-energy collisions between three identical bosons with spin $f=1$ interacting via pairwise potentials. We quantify the corresponding three-body scattering hypervolumes, which parametrize the effective three-body interaction strengths in a many-body description of spin-1 Bose-Einstein condensates. Our results demonstrate universal behavior of the scattering hypervolumes for strong $s$- and $p$-wave two-body interactions. At weak interactions, we find that the real parts of the scattering hypervolumes are predominantly determined by hard-hyperspherelike collisions, which we characterize by a simple formula. With this universal result, we estimate that spin mixing via three-body collisions starts to dominate over two-body spin mixing at a typical particle density of ${10}^{17}{\mathrm{cm}}^{-3}$ for ${}^{23}\mathrm{Na}$ and ${}^{41}\mathrm{K}$ spinor condensates. This density can be reduced by tuning the two-body interactions to an $s$- or $p$-wave dimer resonance or to a point where two-body spin mixing effectively vanishes. Another possibility to observe the effects of three-body spin mixing involves the application of weak magnetic fields to cancel out the effective two-body interaction strength in the characteristic timescale describing the spin dynamics.

The standard quantum limit (SQL) of a classical mechanical force detection results from quantum backaction impinged by the meter on a probe mechanical transducer perturbed by the force of interest. In this paper we introduce a technique of continuous broadband backaction, avoiding measurements for the case of a resonant signal force acting on a linear mechanical oscillator supporting one of the mirrors of an optical Michelson-Sagnac interferometer (MSI). The interferometer with the movable mirror is an optomechanical transducer able to support the polychromatic probe field. The method involves a dichromatic optical probe resonant with the MSI modes and having frequency separation equal to the mechanical frequency. We show that analyzing each of the harmonics of the probe reflected from the mechanical system separately and postprocessing the measurement results allows excluding the backaction in a broad frequency band and measuring the force with sensitivity better than the SQL.

An explicit formula for a type of beams, which in this work are called the “special” hyperbolic Bessel-Gaussian (SHBG) beams, has been derived, using the method of the Hankel transform formulated in our previous work [T. Radożycki, arXiv:2103.06988]. The fundamental properties of these beams are analyzed. The parameters that define the beam shape have been identified and related to those of the fundamental Gaussian beam. The analytical expressions for the SHBG beams include an additional parameter $\gamma$, which allows the beam's shape to be modified to some extent. In the plane perpendicular to the propagation direction, these beams exhibit the annular nature. Interestingly, initially (i.e., near the beam's spot) a single ring splits into a number of rings as one is moving along the beam. This is especially apparent for $\gamma$ close to unity, as this effect then appears for values of $z$ relatively small compared to the Rayleigh length, i.e., where the energy concentration in the beam is still high. The phase of the wave, whose behavior is in certain aspects typical of modes having the vortex character, is also studied in this paper.