Following our work [Phys. Rev. Lett. 125, 020401 (2020)], we discuss a semiclassical description of one-dimensional quantum tunneling through multibarrier potentials in terms of complex time. We start by defining a complex-extended continuum level density of unbound systems and show its relation to a complex time shift of the transmitted wave. While the real part of the level density and time shift describes the passage of the particle through classically allowed coordinate regions, the imaginary part is connected with an instanton-like picture of the tunneling through forbidden regions. We describe singularities in the real and imaginary parts of the level density and time shift caused by stationary points of the tunneling potential, and show that they represent a dual extension of excited-state quantum phase transitions from bound to continuum systems. Using the complex scaling method, we numerically verify the predicted effects in several tunneling potentials.

In the last few years, quantum computing and machine learning fostered rapid developments in their respective areas of application, introducing new perspectives on how information processing systems can be realized and programmed. The rapidly growing field of Quantum Machine Learning aims at bringing together these two ongoing revolutions. Here we first review a series of recent works describing the implementation of artificial neurons and feed-forward neural networks on quantum processors. We then present an original realization of efficient individual quantum nodes based on variational unsampling protocols. We investigate different learning strategies involving global and local layer-wise cost functions, and we assess their performances also in the presence of statistical measurement noise. While keeping full compatibility with the overall memory-efficient feed-forward architecture, our constructions effectively reduce the quantum circuit depth required to determine the activation probability of single neurons upon input of the relevant data-encoding quantum states. This suggests a viable approach towards the use of quantum neural networks for pattern classification on near-term quantum hardware.

It was shown recently that stochastic quantization can be made into a well defined quantization scheme on (pseudo-)Riemannian manifolds using second order differential geometry, which is an extension of the commonly used first order differential geometry. In this letter, we show that restrictions to relativistic theories can be obtained from this theory by imposing a stochastic energy-momentum relation. In the process, we derive non-perturbative quantum corrections to the line element as measured by scalar particles. Furthermore, we extend the framework of stochastic quantization to massless scalar particles.

This work completes the program started in \cite{bb1,bb2,bb3} to derive the Heisenberg uncertainty relation for relativistic particles. Sharp uncertainty relations for massive relativistic particles with spin 0 and spin 1 are derived. The main conclusion is that the uncertainty relations for relativistic bosons are markedly different from those for relativistic fermions. The uncertainty relations for bosons are based on the energy density. It is shown that the uncertainty relations based on the time-component of the four-current, as was have done in \cite{bb3} for electrons, are untenable because they lead to contradictions.

We give an algebraic derivation of the energy eigenvalues for the two-dimensional(2D) quantum harmonic oscillator on the sphere and the hyperbolic plane in the context of the method proposed by Daskaloyannis for the 2D quadratically superintegrable systems. We provide the special form of the quadratic Poisson algebra for the classical harmonic oscillator system and deformed it into the quantum associative algebra of the operators. We express the corresponding Casimir operator for this algebra in terms of the Hamiltonian and provide the finite-dimensional representations for this quantum associative algebra by using the deformed parafermionic oscillator technique. The calculation of the energy eigen-values is then reduced to finding the solution of the two algebraic equations whose form is universal for all the 2D quadratically superintegrable systems. The result derived algebraically agrees with the energy eigenvalues obtained by solving the Schrodinger equation.

Two-photon absorption (TPA) and other nonlinear interactions of molecules with time-frequency-entangled photon pairs (EPP) has been predicted to display a variety of fascinating effects. Therefore, their potential use in practical quantum-enhanced molecular spectroscopy requires close examination. This paper presents in tutorial style a detailed theoretical study of one- and two-photon absorption by molecules, focusing on how to treat the quantum nature of light. We review some basic quantum optics theory, then we review the density-matrix (Liouville) derivation of molecular optical response, emphasizing how to incorporate quantum states of light into the treatment. For illustration we treat in detail the TPA of photon pairs created by spontaneous parametric down conversion, with an emphasis on how quantum light TPA differs from that with classical light. In particular, we treat the question of how much enhancement of the TPA rate can be achieved using entangled states.

Dirac-Frenkel variational method with Davydov D2 trial wavefunction is extended by introducing a thermalization algorithm and applied to simulate dynamics of a general open quantum system. The algorithm allows to control temperature variations of a harmonic finite size bath, when in contact with the quantum system. Thermalization of the bath vibrational modes is realised via stochastic scatterings, implemented as a discrete-time Bernoulli process with Poisson statistics. It controls bath temperature by steering vibrational modes' evolution towards their canonical thermal equilibrium. Numerical analysis of the exciton relaxation dynamics in a small molecular cluster reveals that thermalization additionally provides significant calculation speed up due to reduced number of vibrational modes needed to obtain the convergence.

The Wilson action for Euclidean lattice gauge theory defines a positive-definite transfer matrix that corresponds to a unitary lattice gauge theory time-evolution operator if analytically continued to real time. Hoshina, Fujii, and Kikukawa (HFK) recently pointed out that applying the Wilson action discretization to continuum real-time gauge theory does not lead to this, or any other, unitary theory and proposed an alternate real-time lattice gauge theory action that does result in a unitary real-time transfer matrix. The character expansion defining the HFK action is divergent, and in this work we apply a path integral contour deformation to obtain a convergent representation for U(1) HFK path integrals suitable for numerical Monte Carlo calculations. We also introduce a class of real-time lattice gauge theory actions based on analytic continuation of the Euclidean heat-kernel action. Similar divergent sums are involved in defining these actions, but for one action in this class this divergence takes a particularly simple form, allowing construction of a path integral contour deformation that provides absolutely convergent representations for U(1) and SU(N) real-time lattice gauge theory path integrals. We perform proof-of-principle Monte Carlo calculations of real-time U(1) and SU(3) lattice gauge theory and verify that exact results for unitary time evolution of static quark-antiquark pairs in (1 + 1)D are reproduced.

We determine analytically the phase diagram of the toric code model in a parallel magnetic field which displays three distinct regions. Our study relies on two high-order perturbative expansions in the strong- and weak-field limit, as well as a large-spin analysis. Calculations in the topological phase establish a quasiparticle picture for the anyonic excitations. We obtain two second-order transition lines that merge with a first-order line giving rise to a multicritical point as recently suggested by numerical simulations. We compute the values of the corresponding critical fields and exponents that drive the closure of the gap. We also give the one-particle dispersions of the anyonic quasiparticles inside the topological phase.

This paper presents a systematic method to analyze stability and robustness of uncertain Quantum Input-Output Networks (QIONs). A general form of uncertainty is introduced into quantum networks in the SLH formalism. Results of this paper are built up on the notion of uncertainty decomposition wherein the quantum network is decomposed into nominal (certain) and uncertain sub-networks in cascade connection. Sufficient conditions for robust stability are derived using two different methods. In the first approach, a generalized small-gain theorem is presented and in the second approach, robust stability is analyzed within the framework of Lyapunov theory. In the second method, the robust stability problem is reformulated as feasibility of a Linear Matrix Inequality (LMI), which can be examined using the well-established systematic methods in the literature.

This paper presents a detailed Lyapunov-based theory to control and stabilize continuously-measured quantum systems, which are driven by Stochastic Schrodinger Equation (SSE). Initially, equivalent classes of states of a quantum system are defined and their properties are presented. With the help of equivalence classes of states, we are able to consider global phase invariance of quantum states in our mathematical analysis. As the second mathematical modelling tool, the conventional Ito formula is further extended to non-differentiable complex functions. Based on this extended Ito formula, a detailed stochastic stability theory is developed to stabilize the SSE. Main results of this proposed theory are sufficient conditions for stochastic stability and asymptotic stochastic stability of the SSE. Based on the main results, a solid mathematical framework is provided for controlling and analyzing quantum system under continuous measurement, which is the first step towards implementing weak continuous feedback control for quantum computing purposes.

We study two-qubit circuits over the Clifford+CS gate set, which consists of the Clifford gates together with the controlled-phase gate CS=diag(1,1,1,i). The Clifford+CS gate set is universal for quantum computation and its elements can be implemented fault-tolerantly in most error-correcting schemes through magic state distillation. Since non-Clifford gates are typically more expensive to perform in a fault-tolerant manner, it is often desirable to construct circuits that use few CS gates. In the present paper, we introduce an efficient and optimal synthesis algorithm for two-qubit Clifford+CS operators. Our algorithm inputs a Clifford+CS operator U and outputs a Clifford+CS circuit for U, which uses the least possible number of CS gates. Because the algorithm is deterministic, the circuit it associates to a Clifford+CS operator can be viewed as a normal form for that operator. We give an explicit description of these normal forms and use this description to derive a worst-case lower bound of 5log(1/epsilon)+O(1) on the number of CS gates required to epsilon-approximate elements of SU(4). Our work leverages a wide variety of mathematical tools that may find further applications in the study of fault-tolerant quantum circuits.

Given $x,y\in\{0,1\}^n$, Set Disjointness consists in deciding whether $x_i=y_i=1$ for some index $i \in [n]$. We study the problem of computing this function in a distributed computing scenario in which the inputs $x$ and $y$ are given to the processors at the two extremities of a path of length $d$. Set Disjointness on a Line was introduced by Le Gall and Magniez (PODC 2018) for proving lower bounds on the quantum distributed complexity of computing the diameter of an arbitrary network in the CONGEST model.

In this work, we prove an unconditional lower bound of $\widetilde{\Omega}(\sqrt[3]{n d^2}+\sqrt{n} )$ rounds for Set Disjointness on a Line. This is the first non-trivial lower bound when there is no restriction on the memory used by the processors. The result gives us a new lower bound of $\widetilde{\Omega} (\sqrt[3]{n\delta^2}+\sqrt{n} )$ on the number of rounds required for computing the diameter $\delta$ of any $n$-node network with quantum messages of size $O(\log n)$ in the CONGEST model.

We draw a connection between the distributed computing scenario above and a new model of query complexity. In this model, an algorithm computing a bi-variate function $f$ has access to the inputs $x$ and $y$ through two separate oracles $O_x$ and $O_y$, respectively. The restriction is that the algorithm is required to alternately make $d$ queries to $O_x$ and $d$ queries to $O_y$. The technique we use for deriving the round lower bound for Set Disjointness on a Line also applies to the number of rounds in this query model. We provide an algorithm for Set Disjointness in this query model with round complexity that matches the round lower bound stated above, up to a polylogarithmic factor. In this sense, the round lower bound we show for Set Disjointness on a Line is optimal.

Analog quantum simulation is expected to be a significant application of near-term quantum devices. Verification of these devices without comparison to known simulation results will be an important task as the system size grows beyond the regime that can be simulated classically. We introduce a set of experimentally-motivated verification protocols for analog quantum simulators, discussing their sensitivity to a variety of error sources and their scalability to larger system sizes. We demonstrate these protocols experimentally using a two-qubit trapped-ion analog quantum simulator and numerically using models of up to five qubits.

We study ergodicity breaking in the clean Bose-Hubbard chain for small hopping strength. We see the existence of a non-ergodic regime by means of indicators as the half-chain entanglement entropy of the eigenstates, the average level spacing ratio, {the properties of the eigenstate-expectation distribution of the correlation and the scaling of the Inverse Participation Ratio averages.} We find that this ergodicity breaking {is different from many-body localization} because the average half-chain entanglement entropy of the eigenstates obeys volume law. This ergodicity breaking appears unrelated to the spectrum being organized in quasidegenerate multiplets at small hopping and finite system sizes, so in principle it can survive also for larger system sizes. We find that some imbalance oscillations in time which could mark the existence of a glassy behaviour in space are well described by the dynamics of a single symmetry-breaking doublet and {quantitatively} captured by a perturbative effective XXZ model. We show that the amplitude of these oscillations vanishes in the large-size limit. {Our findings are numerically obtained for systems with $L < 12$. Extrapolations of our scalings to larger system sizes should be taken with care, as discussed in the paper.

A key challenge in quantum computation is the implementation of fast and local qubit control while simultaneously maintaining coherence. Qubits based on hole spins offer, through their strong spin-orbit interaction, a way to implement fast quantum gates. Strikingly, for hole spins in one-dimensional germanium and silicon devices, the spin-orbit interaction has been predicted to be exceptionally strong yet highly tunable with gate voltages. Such electrical control would make it possible to switch on demand between qubit idling and manipulation modes. Here, we demonstrate ultrafast and universal quantum control of a hole spin qubit in a germanium/silicon core/shell nanowire, with Rabi frequencies of several hundreds of megahertz, corresponding to spin-flipping times as short as ~1 ns - a new record for a single-spin qubit. Next, we show a large degree of electrical control over the Rabi frequency, Zeeman energy, and coherence time - thus implementing a switch toggling from a rapid qubit manipulation mode to a more coherent idling mode. We identify an exceptionally strong but gate-tunable spin-orbit interaction as the underlying mechanism, with a short associated spin-orbit length that can be tuned over a large range down to 3 nm for holes of heavy-hole mass. Our work demonstrates a spin-orbit qubit switch and establishes hole spin qubits defined in one-dimensional germanium/silicon nanostructures as a fast and highly tunable platform for quantum computation.

The theory of quantum scarring -- a remarkable violation of quantum unique ergodicity -- rests on two complementary pillars: the existence of unstable classical periodic orbits and the so-called quasimodes, i.e., the non-ergodic states that strongly overlap with a small number of the system's eigenstates. Recently, interest in quantum scars has been revived in a many-body setting of Rydberg atom chains. While previous theoretical works have identified periodic orbits for such systems using time-dependent variational principle (TDVP), the link between periodic orbits and quasimodes has been missing. Here we provide a conceptually simple analytic construction of quasimodes for the non-integrable Rydberg atom model, and prove that they arise from a "requantisation" of previously established periodic orbits when quantum fluctuations are restored to all orders. Our results shed light on the TDVP classical system simultaneously playing the role of both the mean-field approximation and the system's classical limit, thus allowing us to firm up the analogy between the eigenstate scarring in the Rydberg atom chains and the single-particle quantum systems.

We propose a complete architecture for deterministic generation of entangled multiphoton states. Our approach utilizes periodic driving of a quantum-dot emitter and an efficient light-matter interface enabled by a photonic crystal waveguide. We assess the quality of the photonic states produced from a real system by including all intrinsic experimental imperfections. Importantly, the protocol is robust against the nuclear spin bath dynamics due to a naturally built-in refocussing method reminiscent to spin echo. We demonstrate the feasibility of producing Greenberger-Horne-Zeilinger and one-dimensional cluster states with fidelities and generation rates exceeding those achieved with conventional 'fusion' methods in current state-of-the-art experiments. The proposed hardware constitutes a scalable and resource-efficient approach towards implementation of measurement-based quantum communication and computing.

We devise a mathematical framework for assessing the fidelity of multi-photon entangled states generated by a single solid-state quantum emitter, such as a quantum dot or a nitrogen-vacancy center. Within this formalism, we theoretically study the role of imperfections present in real systems on the generation of time-bin encoded Greenberger-Horne-Zeilinger and one-dimensional cluster states. We consider both fundamental limitations, such as the effect of phonon-induced dephasing, interaction with the nuclear spin bath, and second-order emissions, as well as technological imperfections, such as branching effects, non-perfect filtering, and photon losses. In a companion paper, we consider a particular physical implementation based on a quantum dot emitter embedded in a photonic crystal waveguide and apply our theoretical formalism to assess the fidelities achievable with current technologies.

We study the time evolution of a PT-symmetric, non-Hermitian quantum system for which the associated phase space is compact. We focus on the simplest non-trivial example of such a Hamiltonian, which is linear in the angular momentum operators. In order to describe the evolution of the system, we use a particular disentangling decomposition of the evolution operator, which remains numerically accurate even in the vicinity of the Exceptional Point. We then analyze how the non-Hermitian part of the Hamiltonian affects the time evolution of two archetypical quantum states, coherent and Dicke states. For that purpose we calculate the Husimi distribution or Q function and study its evolution in phase space. For coherent states, the characteristics of the evolution equation of the Husimi function agree with the trajectories of the corresponding angular momentum expectation values. This allows to consider these curves as the trajectories of a classical system. For other types of quantum states, e.g. Dicke states, the equivalence of characteristics and trajectories of expectation values is lost.