We propose a topological qubit in which braiding and readout are mediated by the $4\pi$ Majorana-Josephson effect. The braidonium device consists of three Majorana nanowires that come together to make a tri-junction; in order to control the superconducting phase differences at the tri-junction the nanowires are enclosed in a ring made of a conventional superconductor; and in order to perform initialization/readout one of the nanowires is coupled to a fluxonium qubit through a topological Josephson junction. We analyze how flux-based control and readout protocols can be used to demonstrate braiding and qubit operation for realistic materials and circuit parameters.

Atomic physics experiments commonly use magnetic fields of between a few and a few hundred Gauss to provide a quantization axis. As atomic transition frequencies depend on the amplitude of this field, many experiments require a stable absolute field. Most setups use electromagnets, which require a power supply stability not usually met by commercially available units. We demonstrate stabilization of a field of 146 G to 43 $\mu$G rms noise (0.29 ppm), compared to noise of $\gtrsim$ 1 mG without any stabilization. The rms noise is measured using a field-dependent hyperfine transition in a single $^{43}$Ca$^+$ ion held in a Paul trap at the centre of the magnetic field coils. For the $^{43}$Ca$^+$ "atomic clock" qubit transition at 146 G, which depends on the field only in second order, this would yield a projected coherence time of many hours. Our system consists of a feedback loop and a feedforward circuit that control the current through the field coils and could easily be adapted to other field amplitudes, making it suitable for other applications such as magneto-optical traps.

A sequential application of the Grover algorithm to solve the iterated search problem has been improved by Ozhigov by parallelizing the application of the oracle. In this work a representation of the parallel Grover as dynamic system of inversion about the mean and Grover operators is given. Within this representation the parallel Grover for $k = 2$ can be interpreted as rotation in three-dimensional space and it can be shown that the sole application of the parallel Grover operator does not lead to a solution for $k > 2$. We propose a solution for $k = 3$ with a number of approximately $1.51\sqrt{N}$ iterations.

The uniformity of the intensity and phase of laser beams is crucial to high-performance atom interferometers. Inhomogeneities in the laser intensity profile cause contrast reductions and systematic effects in interferometers operated with atom sources at micro-Kelvin temperatures, and detrimental diffraction phase shifts in interferometers using large momentum transfer beam splitters. We report on the implementation of a so-called tophat laser beam in a long-interrogation-time cold-atom interferometer to overcome the issue of the inhomogeneous laser intensity encountered when using Gaussian laser beams. We characterize the intensity and relative phase profiles of the tophat beam and demonstrate the associated gain in atom interferometer contrast, in agreement with numerical simulations. We discuss the application of tophat beams to improve the performance of different architectures of atom interferometers.

Semi-transparent mirrors are standard elements in light optics for splitting light beams or creating two versions of the same image. Such mirrors do not exist in electron optics, although they could be beneficial in existing techniques such as electron interferometry and holography and enable novel electron imaging and spectroscopy techniques. We propose a design for an electron beam splitter using the concept of quantum interaction-free measurement (IFM). The design combines an electron resonator with a weak phase grating. Fast switching gates allow electrons to enter and exit the resonator. While in the resonator, the phase grating transfers intensity from the direct beam into one of the weakly diffracted beams at each pass. To make the beam splitter an efficient two-port splitter, the intensity in all other diffracted beams is blocked by an aperture. The IFM principle minimizes the loss of total intensity by this aperture. We use a scattering matrix method to analyze the performance of the beam splitter, including the effects of inelastic scattering in the phase grating. This design can be generalized to beam splitters for not only electrons, but also photons, neutrons, atoms, and other quantum mechanical systems.

The Aharanov-Bohm (AB) effect, which predicts that a magnetic field strongly influences the wave function of an electrically charged particle, is now investigated in a molecular junction setup. The AB effect leads to a non-monotonic dependence of the steady-state current on the gauge phase associated with the molecular ring. This dependence is sensitive to site energy, temperature, and dephasing, and can be explained using the concept of the dark state. Although implications of the phase effect vanish in the steady-state current for strong dephasing, the phase dependence becomes visible in an associated waiting-time distribution, especially at short times. Interesting, the phase rigidity (i.e., the symmetry of the AB phase) observed in the steady-state current is now broken in the waiting-time statistics, which can be explained by the interference between transfer pathways.

Electromagnetic fields carry momentum, which upon reflection on matter, gives rise to the radiation pressure of photons. The radiation pressure has recently been utilized in cavity optomechanics for controlling mechanical motions of macroscopic objects at the quantum limit. However, because of the weakness of the interaction, attempts so far had to use a strong coherent drive to reach the quantum limit. Therefore, the single photon quantum regime, where even the presence of a totally off-resonant single photon alters the quantum state of the mechanical mode significantly, is one of the next milestones in cavity optomechanics. Here we demonstrate an artificial realization of the radiation pressure of microwave photons acting on phonons in a surface acoustic wave resonator. The order-of-magnitude enhancement of the interaction strength originates in the well-tailored strong second-order nonlinearity of a superconducting Josephson-junction circuit. The synthetic radiation pressure interaction adds a key element to the quantum optomechanical toolbox and can be applied to quantum information interfaces between electromagnetic and mechanical degrees of freedom.

In arXiv:1208.0365 entanglement polytopes where introduced as a coarsening of the SLOCC classification of multipartite entanglement. The advantages of classifying entanglement by entanglement polytopes are a finite hierarchy for all dimensions and a number of parameters linear in system size. In arXiv:1208.0365 a method to compute entanglement polytopes using geometric invariant theory is presented. In this thesis we consider alternative methods to compute them. Some geometrical and algebraical tools are presented that can be used to compute inequalities giving an outer approximation of the entanglement polytopes. Furthermore we present a numerical method which, in theory, can compute the entanglement polytope of any given SLOCC class given a representative. Using it we classify the entanglement polytopes of $2 \times 3 \times N$ systems.

Quantum mechanics is inherently probabilistic in light of Born's rule. Using quantum circuits as probabilistic generative models for classical data exploits their superior expressibility and efficient direct sampling ability. However, training of quantum circuits can be more challenging compared to classical neural networks due to lack of efficient differentiable learning algorithm. We devise an adversarial quantum-classical hybrid training scheme via coupling a quantum circuit generator and a classical neural network discriminator together. After training, the quantum circuit generative model can infer missing data with quadratic speed up via amplitude amplification. We numerically simulate the learning and inference of generative adversarial quantum circuit using the prototypical Bars-and-Stripes dataset. Generative adversarial quantum circuits is a fresh approach to machine learning which may enjoy the practically useful quantum advantage on near-term quantum devices.

Quantum nonlocal correlations mainly contain quantum entanglement, Bell nonlocality and Einstein-Podolsky-Rosen steering. Recently, coherence, as an important quantum resource, plays a crucial role in quantum information processing. Hence, revealing the mutual relations between nonlocal correlations and coherence has attracted much attention. Here we investigate the correlations of the tripartite states (the W-class states) based on the bipartite correlation measures, such as entanglement (concurrence), coherence (the degree of coherence) and Bell nonlocality (quantified by the Clauser-Horne-Shimony-Holt (CHSH) inequality maximum violation). The interrelations among the concurrence, the degree of coherence, purity and Bell nonlocality are presented. Furthermore, we discuss the relations between the concurrence and the degree of coherence considering Bell nonlocal and Bell local (satisfied the CHSH inequality) states for the two-qubit subsystems derived from the tripartite states, and derive some analytical formulas with respect to the degree of coherence and concurrence. Moreover, we consider two specific cases (the W-class state under the decoherence channel and a practical system of renormalized spin-1/2 chain), and derive some relations between nonlocal correlations and coherence as well.

In this manuscript, we present analytical solution of the Klein-Gordon equation with the multi-parameter q-deformed Woods-Saxon type potential energy under the spin symmetric limit in $(1+1)$ dimension. In the scattering case, we obtain the reflection and transmission probabilities and prove the conservation of the total probability. Moreover, we analyze the correlation between the potential parameters with the reflection and transmission probabilities. In the bound state case, we use the continuity conditions and derive a quantization scheme. To confirm our results numerically, in both cases we randomly assign values to the potential parameters and find numerical results by using the Newton Raphson method.

We propose a scheme of fast three-qubit Toffoli quantum gate for ultracold neutral-atom qubits. The scheme is based on the Stark-tuned three-body F\"{o}rster resonances, which we have observed in our recent experiment [D.B.Tretyakov et al., Phys.Rev.Lett. 119, 173402 (2017)]. The three-body resonance corresponds to a transition when the three interacting atoms change their states simultaneously, and it occurs at a different dc electric field with respect to the two-body F\"{o}rster resonance. A combined effect of three-body and two-body F\"{o}rster interactions in external electric and magnetic fields near the three-body resonance results in complex coherent behavior of the populations and phases of collective states of a three-atom system. We have found that it is possible to obtain experimental conditions suitable to implement three-qubit Toffoli gate with 96.8\% fidelity and less than 3~$\mu$s duration.

We study algorithms for solving Subtraction games, which sometimes are referred to as one-heap Nim games. We describe a quantum algorithm which is applicable to any game on DAG, and show that its query compexity for solving an arbitrary Subtraction game of $n$ stones is $O(n^{3/2}\log n)$. The best known deterministic algorithms for solving such games are based on the dynamic programming approach. We show that this approach is asymptotically optimal and that classical query complexity for solving a Subtraction game is generally $\Theta(n^2)$. This paper perhaps is the first explicit "quantum" contribution to algorithmic game theory.

With the emergence of the field of quantum communications, the appropriate choice of photonic degrees of freedom used for encoding information is of paramount importance. Highly precise techniques for measuring the polarisation, frequency, and arrival time of a photon have been developed. However, the transverse spatial degree of freedom still lacks a measurement scheme that allows the reconstruction of its full transverse structure with a simple implementation and a high level of accuracy. Here we show a method to measure the azimuthal and radial modes of Laguerre-Gaussian beams with a greater than 99% accuracy, using a single phase screen. We compare our technique with previous commonly used methods and demonstrate the significant improvements it presents for quantum key distribution and state tomography of high-dimensional quantum states of light. Moreover, our technique can be readily extended to any arbitrary family of spatial modes, such as mutually unbiased bases, Hermite-Gauss, and Ince-Gauss. Our scheme will significantly enhance existing quantum and classical communication protocols that use the spatial structure of light, as well as enable fundamental experiments on spatial-mode entanglement to reach their full potential.

For a large class of time-dependent non-Hermitain Hamiltonians expressed in terms linear and bilinear combinations of the generators for an Euclidean Lie-algebra respecting different types of PT-symmetries, we find explicit solutions to the time-dependent Dyson equation. A specific Hermitian model with explicit time-dependence is analyzed further and shown to be quasi-exactly solvable. Technically we constructed the Lewis-Riesenfeld invariants making useof the metric picture, which is an equivalent alternative to the Schr\"{o}dinger, Heisenberg and interaction picture containing the time-dependence in the metric operator that relates the time-dependent Hermitian Hamiltonian to a static non-Hermitian Hamiltonian.

A weak value is an effective description of the influence of a pre and post-selected 'principal' system on another 'meter' system to which it is weakly coupled. Weak values can describe anomalously large deflections of the meter, and deflections in otherwise unperturbed variables: this motivates investigation of the potential benefits of the protocol in precision metrology. We present a visual interpretation of weak value experiments in phase space, enabling an evaluation of the effects of three types of detector noise as 'Fisher information efficiency' functions. These functions depend on the marginal distribution of the Wigner function of the meter, and give a unified view of the weak value protocol as a way of protecting Fisher information from detector imperfections. This approach explains why weak value techniques are more effective for avoiding detector saturation than for mitigating detector jitter or pixelation.

Full counting statistics for a wide class of Luttinger liquid tunnel junctions in a "weak link" regime is considered beyond the lowest orders in tunnel coupling and out of the equilibrium in the time domain. Especially, two important mathematical statements: the FCS-S-theorem and the FCS-S-lemma about exact re-exponentiation of Keldysh-contour-ordered evolution operator are proven. These statements are the generalizations of S-theorem and S-lemma have been recently proven by the author in Ref.[\textit{G.A.Skorobagatko, Phys.Rev.B, 98, 045409 (2018)}]. It is shown that FCS-S-theorem can be treated also as the proof of dynamical Jarzynski equality for tunnel electron transport out of the equilibrium (in time domain). As the result, exact time-dependent cumulant generating functional is derived being valid at arbitrary electron-electron repulsion in the Luttinger liquid leads of the junction, arbitrary temperature and arbitrary bias voltage. Respective general formula in its long-time asymptotics turns into a non-perturbative Luttinger liquid generalization of a well-known Levitov-Lesovik formula for cumulant generating function. Hence, demonstrated proof of FCS-S-theorem can be considered also as the proof of detailed balance theorem for the long-time limit of strongly correlated electron transport in arbitrary tunnel junctions.

Quantum computers can efficiently simulate many-body systems. As a widely used Hamiltonian simulation tool, the Trotter-Suzuki scheme splits the evolution into the number of Trotter steps $N$ and approximates the evolution of each step by a product of exponentials of each individual term of the total Hamiltonian. The algorithmic error due to the approximation can be reduced by increasing $N$, which however requires a longer circuit and hence inevitably introduces more physical errors. In this work, we first study such a trade-off and numerically find the optimal number of Trotter steps $N_{\textrm{opt}}$ given a physical error model in a near-term quantum hardware. Practically, physical errors can be suppressed using recently proposed error mitigation methods. We then extend physical error mitigation methods to suppress the algorithmic error in Hamiltonian simulation. By exploiting the simulation results with different numbers of Trotter steps $N\le N_{\textrm{opt}}$, we can infer the exact simulation result within a higher accuracy and hence mitigate algorithmic errors. We numerically test our scheme with a five qubit system and show significant improvements in the simulation accuracy by applying both physical and algorithmic error mitigations.

The problem of causal inference is to determine if a given probability distribution on observed variables is compatible with some causal structure. The difficult case is when the causal structure includes latent variables. We here introduce the $\textit{inflation technique}$ for tackling this problem. An inflation of a causal structure is a new causal structure that can contain multiple copies of each of the original variables, but where the ancestry of each copy mirrors that of the original. To every distribution of the observed variables that is compatible with the original causal structure, we assign a family of marginal distributions on certain subsets of the copies that are compatible with the inflated causal structure. It follows that compatibility constraints for the inflation can be translated into compatibility constraints for the original causal structure. Even if the constraints at the level of inflation are weak, such as observable statistical independences implied by disjoint causal ancestry, the translated constraints can be strong. We apply this method to derive new inequalities whose violation by a distribution witnesses that distribution's incompatibility with the causal structure (of which Bell inequalities and Pearl's instrumental inequality are prominent examples). We describe an algorithm for deriving all such inequalities for the original causal structure that follow from ancestral independences in the inflation. For three observed binary variables with pairwise common causes, it yields inequalities that are stronger in at least some aspects than those obtainable by existing methods. We also describe an algorithm that derives a weaker set of inequalities but is more efficient. Finally, we discuss which inflations are such that the inequalities one obtains from them remain valid even for quantum (and post-quantum) generalizations of the notion of a causal model.

We propose to use the eigenfunctions of a one-electron model Hamiltonian to perform electron-nucleus mean field configuration interaction (EN-MFCI) calculations. The potential energy of our model Hamiltonian corresponds to the Coulomb potential of an infinite wire with charge $Z$ distributed according to a Gaussian function. The time independent \sch equation for this Hamiltonian is solved perturbationally in the limit of small amplitude vibration (Gaussian function width close to zero).