Quantum++ is a modern general-purpose multi-threaded quantum computing library written in C++11 and composed solely of header files. The library is not restricted to qubit systems or specific quantum information processing tasks, being capable of simulating arbitrary quantum processes. The main design factors taken in consideration were the ease of use, portability, and performance. The library's simulation capabilities are only restricted by the amount of available physical memory. On a typical machine (Intel i5 8Gb RAM) Quantum++ can successfully simulate the evolution of 25 qubits in a pure state or of 12 qubits in a mixed state reasonably fast. The library also includes support for classical reversible logic, being able to simulate classical reversible operations on billions of bits. This latter feature may be useful in testing quantum circuits composed solely of Toffoli gates, such as certain arithmetic circuits.

Starting from the Rabi Hamiltonian, which is useful to get non-perturbative results within the rotating wave approximation, we have obtained the Einstein's B coefficient to be time-dependent, $B(t)\propto|J_0(\omega_\gamma t)|$, for a two-level system (atom or molecule) in the thermal radiation field. Here $\omega_\gamma$ is the corresponding Rabi flopping (angular) frequency and $J_0$ is the zeroth order Bessel function of the first kind. The resulting oscillations in the $B$ coefficient -- even in the limit of very small $\omega_\gamma$ -- drives the system away from the thermodynamic equilibrium at any finite temperature in contrary to Einstein's prediction. The time-dependent generalized $B$ coefficient facilitates a path to go beyond the Pauli's formalism of non-equilibrium statistical mechanics involving the quantum statistical Boltzmann (master) equation. We have obtained the entropy production of the two-level system, in this context, by revising Einstein's rate equations considering the $A$ coefficient to be the original time-independent one and $B$ coefficient to be the time-dependent one.

Informational dependence between statistical or quantum subsystems can be described with Fisher matrix or Fubini-Study metric obtained from variations of the sample/configuration space coordinates. Using these non-covariant objects as macroscopic constraints we consider statistical ensembles over the space of classical probability distributions or quantum wave-functions. The ensembles are covariantized using dual field theories with either complex or real scalar fields identified with complex wave-functions or square root of probabilities. We argue that a full space-time covariance on a field theory side is dual to local computations on the information theory side. We define a fully covariant information-computation tensor and show that it must satisfy conservation equations. Then we switch to a thermodynamic description and argue that the (inverse of) space-time metric tensor is a conjugate thermodynamic variable to the ensemble-averaged information-computation tensor. In the equilibrium the entropy production vanishes and the metric is not dynamical, but away from equilibrium the entropy production gives rise to an emergent dynamics of the metric. This dynamics can be described by expanding the entropy production into products of generalized forces (derivatives of metric) and conjugate fluxes. Near equilibrium these fluxes are given by an Onsager tensor contracted with generalized forces and on the grounds of time-reversal symmetry the Onsager tensor is expected to be symmetric. We show that a particularly simple and highly symmetric form of the Onsager tensor gives rise to the Einstein-Hilbert term. This proves that general relativity is equivalent to a theory of non-equilibrium (thermo)dynamics of the metric which is expected to break down far away from equilibrium where the symmetries of the Onsager tensor are to be broken.

We show that a quantum particle subjected to a positive force in one path of a Mach-Zehnder interferometer and a null force in the other path may receive a negative average momentum transfer when it leaves the interferometer by a particular exit. In this scenario, an ensemble of particles may receive an average momentum in the opposite direction of the applied force due to quantum interference, a behavior with no classical analogue. We discuss some experimental schemes that could verify the effect with current technology, with electrons or neutrons in Mach-Zehnder interferometers in free space and with atoms from a Bose-Einstein condensate.

Linear canonical transformations of bosonic modes correspond to Gaussian unitaries, which comprise passive linear-optical transformations as effected by a multiport passive interferometer and active Bogoliubov transformations as effected by a nonlinear amplification medium. As a consequence of the Bloch-Messiah theorem, any Gaussian unitary can be decomposed into a passive interferometer followed by a layer of single-mode squeezers and another passive interferometer. Here, it is shown how to circumvent the need for active transformations. Namely, we provide a technique to simulate sampling from the joint input and output distributions of any Gaussian circuit with passive interferometry only, provided two-mode squeezed vacuum states are available as a prior resource. At the heart of the procedure, we exploit the fact that a beam splitter under partial time reversal simulates a two-mode squeezer, which gives access to an arbitrary Gaussian circuit without any nonlinear optical medium. This yields, in particular, a procedure for simulating with linear optics an extended boson sampling experiment, where photons jointly propagate through an arbitrary multimode Gaussian circuit, followed by the detection of output photon patterns.

Reservoir engineering enables the robust preparation of pure quantum states in noisy environments. We show how a new family of quantum states of a mechanical oscillator can be stabilized in a cavity that is parametrically coupled to both the mechanical displacement and the displacement squared. The cavity is driven with three tones, on the red sideband, on the cavity resonance and on the second blue sideband. The states so stabilized are (squeezed and displaced) superpositions of a finite number of phonons. They show the unique feature of encompassing two prototypes of nonclassicality for bosonic systems: by adjusting the strength of the drives, one can in fact move from a single-phonon- to a Schrodinger-cat-like state. The scheme is deterministic, supersedes the need for measurement-and-feedback loops and does not require initialization of the oscillator to the ground state. As such, it enables the unconditional preparation of nonclassical states of a macroscopic object.

With the current rate of progress in quantum computing technologies, systems with more than 50 qubits will soon become reality. Computing ideal quantum state amplitudes for devices of such and larger sizes is a fundamental step to assess their fidelity, but memory requirements for such calculations on classical computers grow exponentially. In this study, we present a new approach for this task that extends the boundaries of what can be computed on a classical system. We present results obtained from a calculation of the complete set of output amplitudes of a universal random circuit with depth 27 in a 2D lattice of $7 \times 7$ qubits, and an arbitrarily selected slice of $2^{37}$ amplitudes of a universal random circuit with depth 23 in a 2D lattice of $8 \times 7$ qubits. Combining our methodology with other decomposition techniques found in the literature, we show that we can simulate $7 \times 7$-qubit random circuits to arbitrary depth by leveraging secondary storage. These calculations were thought to be impossible due to memory requirements; our methodology requires memory within the limits of existing classical computers.

The dark states of a group of two-level atoms in the Tavis-Cummings resonator with zero detuning are considered. In these states, atoms can not emit photons, although they have non-zero energy. They are stable and can serve as a controlled energy reservoir from which photons can be extracted by differentiated effects on atoms, for example, their spatial separation. Dark states are the simplest example of a subspace free of decoherence in the form of a photon flight, and therefore are of interest for quantum computing. It is proved that a) the dimension of the subspace of dark states of atoms is the Catalan numbers, b) in the RWA approximation, any dark state is a linear combination of tensor products of singlet-type states and the ground states of individual atoms. For the exact model, in the case of the same force of interaction of atoms with the field, the same decomposition is true, and only singlets participate in the products and the dark states can neither emit a photon nor absorb it. The proof is based on the method of quantization of the amplitude of states of atomic ensembles, in which the roles of individual atoms are interchangeable. In such an ensemble there is a possibility of micro-causality: the trajectory of each quantum of amplitude can be uniquely assigned.

Thermal attenuator channels model the decoherence of quantum systems interacting with a thermal bath, e.g., a two-level system subject to thermal noise and an electromagnetic signal travelling through a fiber or in free-space. Hence determining the quantum capacity of these channels is an outstanding open problem for quantum computation and communication. Here we derive several upper bounds on the quantum capacity of qubit and bosonic thermal attenuators. We introduce an extended version of such channels which is degradable and hence has a single-letter quantum capacity, bounding that of the original thermal attenuators. Another bound for bosonic attenuators is given by the bottleneck inequality applied to a particular channel decomposition. With respect to previously known bounds we report better results in a broad range of attenuation and noise: we can now approximate the quantum capacity up to a negligible uncertainty for most practical applications, e.g., for low thermal noise.

A tenet of time-resolved spectroscopy is -faster laser pulses for shorter timescales- . Here we suggest turning this paradigm around, and slow down the system dynamics via repeated measurements, to do spectroscopy on longer timescales. This is the principle of the quantum Zeno effect. We exemplify our approach with the Auger process, and find that repeated measurements increase the core-hole lifetime, redistribute the kinetic energy of Auger electrons, and alter entanglement formation. We further provide an explicit experimental protocol for atomic Li, to make our proposal concrete.

A revised version of the massively parallel simulator of a universal quantum computer, described in this journal eleven years ago, is used to benchmark various gate-based quantum algorithms on some of the most powerful supercomputers that exist today. Adaptive encoding of the wave function reduces the memory requirement by a factor of eight, making it possible to simulate universal quantum computers with up to 48 qubits on the Sunway TaihuLight and on the K computer. The simulator exhibits close-to-ideal weak-scaling behavior on the Sunway TaihuLight,on the K computer, on an IBM Blue Gene/Q, and on Intel Xeon based clusters, implying that the combination of parallelization and hardware can track the exponential scaling due to the increasing number of qubits. Results of executing simple quantum circuits and Shor's factorization algorithm on quantum computers containing up to 48 qubits are presented.

We use a simple example to illustrate why it is not possible to consider that a measurement reveals an underlying objective reality of a property of a quantum system, that continues the same after the measurement is performed. This kind of incompatibility between realism and quantum mechanics is theoretically demonstrated with an example where sequential spin measurements are performed on a spin-2 quantum particle. We discuss the relation of this result with other investigations about realism and quantum mechanics. In particular, we criticize the realistic view adopted on recent discussions about the reality of the quantum state.

The stabilizer group for an $n$-qubit state $\ket{\phi}$ is the set of all invertible local operators (ILO) $g=g_1\otimes g_2\otimes \cdots\otimes g_n,$ $ g_i\in \mathcal{GL}(2,\mathbb{C})$ such that $\ket{\phi}=g\ket{\phi}.$ Recently, G. Gour $et$ $al.$ \cite{GKW} presented that almost all $n$-qubit state $\ket{\psi}$ own a trivial stabilizer group when $n\ge 5.$ In this article, we consider the case when the stabilizer group of an $n$-qubit symmetric pure state $\ket{\psi}$ is trivial. First we show that the stabilizer group for an n-qubit symmetric pure state $\ket{\phi}$ is nontrivial when $n\le 4$. Then we present a class of $n$-qubit symmetric states $\ket{\phi}$ with the trivial stabilizer group. At last, we propose a conjecture and prove that an $n$-qubit symmetric pure state owns a trivial stabilizer group when its diversity number is bigger than 5 under the conjecture we make, which confirms the main result of \cite{GKW} partly.

We present a non-equilibrium quantum field theory approach to the initial-state dynamics of spin models based on two-particle irreducible (2PI) functional integral techniques. It employs a mapping of spins to Schwinger bosons for arbitrary spin interactions and spin lengths. At next-to-leading order (NLO) in an expansion in the number of field components, a wide range of non-perturbative dynamical phenomena are shown to be captured, including relaxation of magnetizations in a 3D long-range interacting system with quenched disorder, different relaxation behaviour on both sides of a quantum phase transition and the crossover from relaxation to arrest of dynamics in a disordered spin chain previously shown to exhibit many-body-localization. Where applicable, we employ alternative state-of-the-art techniques and find rather good agreement with our 2PI NLO results. As our method can handle large system sizes and converges relatively quickly to its thermodynamic limit, it opens the possibility to study those phenomena in higher dimensions in regimes in which no other efficient methods exist. Furthermore, the approach to classical dynamics can be investigated as the spin length is increased.

To study the trade-off between information and disturbance, we obtain the first and second derivatives of the disturbance with respect to information for a fundamental class of quantum measurements. We focus on measurements lying on the boundaries of the physically allowed regions in four information--disturbance planes, using the derivatives to investigate the slopes and curvatures of these boundaries and hence clarify the shapes of the allowed regions.

Entanglement swapping entangles two particles that have never interacted[1], which implicitly assumes that each particle carries an independent local hidden variable, i.e., the presence of bilocality[2]. Previous experimental studies of bilocal hidden variable models did not fulfill the central requirement that the assumed two local hidden variable models must be mutually independent and hence their conclusions are flawed on the rejection of local realism[3-5]. By harnessing the laser phase randomization[6] rising from the spontaneous emission to the stimulated emission to ensure the independence between entangled photon-pairs created at separate sources and separating relevant events spacelike to satisfy the no-signaling condition, for the first time, we simultaneously close the loopholes of independent source, locality and measurement independence in an entanglement swapping experiment in a network. We measure a bilocal parameter of 1.181$\pm$0.004 and the CHSH game value of 2.652$\pm$0.059, indicating the rejection of bilocal hidden variable models by 45 standard deviations and local hidden variable models by 11 standard deviations. We hence rule out local realism and justify the presence of quantum nonlocality in our network experiment. Our experimental realization constitutes a fundamental block for a large quantum network. Furthermore, we anticipate that it may stimulate novel information processing applications[7,8].

We present a technique for reducing the computational requirements by several orders of magnitude in the evaluation of semidefinite relaxations for bounding the set of quantum correlations arising from finite-dimensional Hilbert spaces. The technique, which we make publicly available through a user-friendly software package, relies on the exploitation of symmetries present in the optimisation problem to reduce the number of variables and the block sizes in semidefinite relaxations. It is widely applicable in problems encountered in quantum information theory and enables computations that were previously too demanding. We demonstrate its advantages and general applicability in several physical problems. In particular, we use it to robustly certify the non-projectiveness of high-dimensional measurements in a black-box scenario based on self-tests of $d$-dimensional symmetric informationally complete POVMs.

According to the Schiff theorem, the atomic electrons completely screen the atomic nucleus from an external static electric field. However, this is not the case if the field is time-dependent. Electronic orbitals in atoms either shield the nucleus from an oscillating electric field when the frequency of the field is off the atomic resonances or enhance this field when its frequency approaches an atomic transition energy. In molecules, not only electronic, but also rotational and vibrational states are responsible for the screening of oscillating electric fields. As will be shown in this paper, the screening of a low-frequency field inside molecules is much weaker than it appears in atoms owing to the molecular ro-vibrational states. We systematically study the screening of oscillating electric fields inside diatomic molecules in different frequency regimes,i.e., when the field's frequency is either of order of ro-vibrational or electronic transition frequencies. In the resonance case, we demonstrate that the microwave-frequency electric field may be enhanced up to six orders in magnitude due to ro-vibrational states. We also derive the general formulae for the screening and resonance enhancement of oscillating electric field in polyatomic molecules. Possible applications of these results include nuclear electric dipole moment measurements and stimulation of nuclear reactions by laser light.

Quantum machine learning is one of the most promising applications of a full-scale quantum computer. Over the past few years, many quantum machine learning algorithms have been proposed that can potentially offer considerable speedups over the corresponding classical algorithms. In this paper, we introduce q-means, a new quantum algorithm for clustering which is a canonical problem in unsupervised machine learning. The $q$-means algorithm has convergence and precision guarantees similar to $k$-means, and it outputs with high probability a good approximation of the $k$ cluster centroids like the classical algorithm. Given a dataset of $N$ $d$-dimensional vectors $v_i$ (seen as a matrix $V \in \mathbb{R}^{N \times d})$ stored in QRAM, the running time of q-means is $\widetilde{O}\left( k d \frac{\eta}{\delta^2}\kappa(V)(\mu(V) + k \frac{\eta}{\delta}) + k^2 \frac{\eta^{1.5}}{\delta^2} \kappa(V)\mu(V) \right)$ per iteration, where $\kappa(V)$ is the condition number, $\mu(V)$ is a parameter that appears in quantum linear algebra procedures and $\eta = \max_{i} ||v_{i}||^{2}$. For a natural notion of well-clusterable datasets, the running time becomes $\widetilde{O}\left( k^2 d \frac{\eta^{2.5}}{\delta^3} + k^{2.5} \frac{\eta^2}{\delta^3} \right)$ per iteration, which is linear in the number of features $d$, and polynomial in the rank $k$, the maximum square norm $\eta$ and the error parameter $\delta$. Both running times are only polylogarithmic in the number of datapoints $N$. Our algorithm provides substantial savings compared to the classical $k$-means algorithm that runs in time $O(kdN)$ per iteration, particularly for the case of large datasets.

We address the nature of spin transport in the integrable XXZ spin chain, focusing on the isotropic Heisenberg limit. We calculate the diffusion constant using a kinetic picture based on generalized hydrodynamics combined with Gaussian fluctuations: we find that it diverges, and show that a self-consistent treatment of this divergence gives superdiffusion, with an effective time-dependent diffusion constant that scales as $D(t) \sim t^{1/3}$. This exponent had previously been observed in large-scale numerical simulations, but had not been theoretically explained. We briefly discuss XXZ models with easy-axis anisotropy $\Delta > 1$. Our method gives closed-form expressions for the diffusion constant $D$ in the infinite-temperature limit for all $\Delta > 1$. We find that $D$ saturates at large anisotropy, and diverges as the Heisenberg limit is approached, as $D \sim (\Delta - 1)^{-1/2}$.