We combine the multigrid (MG) method with state-of-the-art concepts from the variational formulation of the numerical renormalization group. The resulting MG renormalization (MGR) method is a natural generalization of the MG method for solving partial differential equations. When the solution on a grid of $N$ points is sought, our MGR method has a computational cost scaling as $\mathcal{O}(\log(N))$, as opposed to $\mathcal{O}(N)$ for the best standard MG method. Therefore MGR can exponentially speed up standard MG computations. To illustrate our method, we develop a novel algorithm for the ground state computation of the nonlinear Schr\"{o}dinger equation. Our algorithm acts variationally on tensor products and updates the tensors one after another by solving a local nonlinear optimization problem. We compare several different methods for the nonlinear tensor update and find that the Newton method is the most efficient as well as precise. The combination of MGR with our nonlinear ground state algorithm produces accurate results for the nonlinear Schr\"{o}dinger equation on $N = 10^{18}$ grid points in three spatial dimensions.

The modular valued operator $\widehat{V}_m$ of the von Neumann interaction operator for a projector is defined. The properties of $\widehat{V}_m$ are discussed and contrasted with those of the standard modular value of a projector. The associated notion of a faux qubit is introduced and its possible utility in quantum computation is noted. An experimental implementation of $\widehat{V}_m$ is also highlighted.

Minimally twisted bilayer graphene exhibits a lattice of AB and BA stacked regions. At small carrier densities and large displacement field, topological channels emerge and form a network. We fabricate small-angle twisted bilayer graphene and tune it with local gates. In our transport measurements we observe Fabry-P\'erot and Aharanov-Bohm oscillations which are robust in magnetic fields ranging from 0 to 8T. The Fabry-P\'erot trajectories in the bulk of the system cannot be bent by the Lorentz force. By extracting the enclosed length and area we find that the major contribution originates from trajectories encircling one row of AB/BA regions. The robustness in magnetic field and the linear spacing in density testifies to the fact that charge carriers flow in one-dimensional, topologically protected channels.

We present an algorithm that extends existing quantum algorithms for simulating fermion systems in quantum chemistry and condensed matter physics to include phonons. The phonon degrees of freedom are represented with exponential accuracy on a truncated Hilbert space with a size that increases linearly with the cutoff of the maximum phonon number. The additional number of qubits required by the presence of phonons scales linearly with the size of the system. The additional circuit depth is constant for systems with finite-range electron-phonon and phonon-phonon interactions and linear for long-range electron-phonon interactions. Our algorithm for a Holstein polaron problem was implemented on an Atos Quantum Learning Machine (QLM) quantum simulator employing the Quantum Phase Estimation method. The energy and the phonon number distribution of the polaron state agree with exact diagonalization results for weak, intermediate and strong electron-phonon coupling regimes.

In the paper, a value assignment for projection operators relating to a quantum system is equated with assignment of truth-values to the propositions associated with these operators. In consequence, the Kochen-Specker theorem (its localized variant, to be exact) can be treated as the statement that a logic of those projection operators does not obey the principle of bivalence. This implies that such a logic has a gappy (partial) semantics or many-valued semantics.

Let V be a linear subspace of M_n(C) which contains the identity matrix and is stable under Hermitian transpose. A "quantum k-clique" for V is a rank k orthogonal projection P in M_n(C) for which dim(PVP) = k^2, and a "quantum k-anticlique" is a rank k orthogonal projection for which dim(PVP) = 1. We give upper and lower bounds both for the largest dimension of V which would ensure the existence of a quantum k-anticlique, and for the smallest dimension of V which would ensure the existence of a quantum k-clique.

Devices relying on microwave circuitry form a cornerstone of many classical and emerging quantum technologies. A capability to provide in-situ, noninvasive and direct imaging of the microwave fields above such devices would be a powerful tool for their function and failure analysis. In this work, we build on recent achievements in magnetometry using ensembles of nitrogen vacancy centres in diamond, to present a widefield microwave microscope with few-micron resolution over a millimeter-scale field of view, 130nT/sqrt-Hz microwave amplitude sensitivity, a dynamic range of 48 dB, and sub-ms temporal resolution. We use our microscope to image the microwave field a few microns above a range of microwave circuitry components, and to characterise a novel atom chip design. Our results open the way to high-throughput characterisation and debugging of complex, multi-component microwave devices, including real-time exploration of device operation.

The No Low-Energy Trivial States (NLTS) conjecture of Freedman and Hastings (Quantum Information and Computation 2014), which asserts the existence of local Hamiltonians whose low energy states cannot generated by constant depth quantum circuits, identifies a fundamental obstacle to resolving the quantum PCP conjecture. Progress towards the NLTS conjecture was made by Eldar and Harrow (Foundations of Computer Science 2017), who proved a closely related theorem called No Low-Error Trivial States (NLETS). In this paper, we give a much simpler proof of the NLETS theorem, and use the same technique to establish superpolynomial circuit size lower bounds for noisy ground states of local Hamiltonians (assuming $\mathsf{QCMA} \neq \mathsf{QMA}$), resolving an open question of Eldar and Harrow. We discuss the new light our results cast on the relationship between NLTS and NLETS.

Finally, our techniques imply the existence of \emph{approximate quantum low-weight check (qLWC) codes} with linear rate, linear distance, and constant weight checks. These codes are similar to quantum LDPC codes except (1) each particle may participate in a large number of checks, and (2) errors only need to be corrected up to fidelity $1 - 1/\mathsf{poly}(n)$. This stands in contrast to the best-known stabilizer LDPC codes due to Freedman, Meyer, and Luo which achieve a distance of $O(\sqrt{n \log n})$.

The principal technique used in our results is to leverage the Feynman-Kitaev clock construction to approximately embed a subspace of states defined by a circuit as the ground space of a local Hamiltonian.

We report on the sensing stability of quantum nanosensors in aqueous buffer solutions for the two detection schemes of quantum decoherence spectroscopy and nanoscale thermometry. The electron spin properties of single nitrogen-vacancy (NV) centers in 25-nm-sized nanodiamonds have been characterized by tracking individual nanodiamonds during a continuous change in pH from 4 to 11. We have determined the stability of the NV quantum sensors during the pH change, which provides the fluctuations of $\pm$13\% and $\pm$0.3 MHz for $T_2$ and $\omega_0$ of their mean values. The observed fluctuations are significant when performing quantum decoherence spectroscopy and thermometry in various biological contexts. We discuss how the present observation affects the measurement scheme of nanodiamond-based NV quantum sensing.

In this contribution, we introduce a technique to freeze the parameters which describe the accelerated states between two users to be used in the context of quantum cryptography and quantum teleportation. It is assumed that, the two users share different dimension sizes of particles, where we consider a qubit-qutrit system. This technique depends on local operations, where it is allowed that each particle interacts locally with a noisy phase channel. We show that, the possibility of freezing the information of quantum channel between the users depends on the initial state setting parameters, the initial acceleration parameter strength of the phase channel. It is shown that, one may increase the possibility of freezing the estimation degree of the parameters if only the larger dimension system or both particles pass through the noisy phase channel. Moreover, at small values of initial acceleration and large values of the channel strength, the size of freezing estimation areas increases. The results may be helpful in the context of quantum teleportation and quantum coding.

In this paper, we study a quantum harmonic oscillator in a Mach-Zehnder-type interferometer which interacts with an environment, including electromagnetic oscillators. By solving the Lindblad master equation, we calculate the resulted interference pattern of the system. Interestingly, we show that even if one considers the decoherence effect, the system will keep some of its quantum properties. Indeed, the thermalization process does not completely leave the system in a classical state and the system keeps some of its coherency. Such an effect can be detected, when the frequency of the central system is high and the temperature is low, even with zero phase angle. This observation makes the quantum-to-classical transition remain as a vague notion in decoherence theory. By introducing an entropy measure, we express the influence of the bath as a maximization of system's entropy instead of classicalization of the state.

We address in this paper the notion of relative phase shift for mixed quantum systems. We study the Pancharatnam-Sjoeqvist phase shift for metaplectic isotopies acting on Gaussian mixed states. We complete and generalize previous results obtained by one of us while giving rigorous proofs. This gives us the opportunity to review and complement the theory of the Conley-Zehnder index which plays an essential role in the determination of phase shifts.

We propose a quantum optimal control algorithm that performs a gradient descent in a reduced basis named GRadient Optimization Using Parametrization (GROUP). We compare this optimization algorithm to the other state-of-the-art algorithms in quantum control namely, Gradient-Ascent Pulse Engieering (GRAPE), Krotov's method and Nelder-Mead using Chopped Random Basis (CRAB). We find that GROUP converges much faster than Nelder-Mead with CRAB and achieves better results than GRAPE and Krotov's method on the control problem presented here.

We propose a Global-Local optimization algorithm for quantum control that combines standard local search methodologies with evolutionary algorithms. This allows us to find faster solutions to a set of problems relating to ultracold control of Bose-Einstein condensates.

We analyze the energy spectrum and eigenstates of cold atoms in a tilted brick-wall optical lattice. When the tilt is applied, the system exhibits a sequence of topological phase transitions reflected in an abrupt change of the eigenstates. It is demonstrated that these topological phase transitions can be easily detected in a laboratory experiment by observing Bloch oscillations of cold atoms.

In the interface between general relativity and relativistic quantum mechanics, we analyse rotating effects on the scalar field subject to a hard-wall confining potential. We consider three different scenarios of general relativity given by the cosmic string spacetime, the spacetime with space-like dislocation and the spacetime with a spiral dislocation. Then, by searching for a discrete spectrum of energy, we analyse analogues effects of the Aharonov-Bohm effect for bound states and the Sagnac effect.

Bhat characterizes the family of linear maps defined on $B(\mathcal{H})$ which preserve unitary conjugation. We generalize this idea and study the maps with a similar equivariance property on finite-dimensional matrix algebras. We show that the maps with equivariance property are significant to study $k$-positivity of linear maps defined on finite-dimensional matrix algebras. Choi showed that $n$-positivity is different from $(n-1)$-positivity for the linear maps defined on $n$ by $n$ matrix algebras. In this paper, we present a parametric family of linear maps $\Phi_{\alpha, \beta,n} : M_{n}(\mathbb{C}) \rightarrow M_{n^{2}}(\mathbb{C})$ and study the properties of positivity, completely positivity, decomposability etc. We determine values of parameters $\alpha$ and $\beta$ for which the family of maps $\Phi_{\alpha, \beta,n}$ is positive for any natural number $n \geq 3$. We focus on the case of $n=3,$ that is, $\Phi_{\alpha, \beta,3}$ and study the properties of $2$-positivity, completely positivity and decomposability. In particular, we give values of parameters $\alpha$ and $\beta$ for which the family of maps $\Phi_{\alpha, \beta,3}$ is $2$-positive and not completely positive.

Secure communication is of paramount importance in modern society. Asymmetric cryptography methods such as the widely used RSA method allow secure exchange of information between parties who have not shared secret keys. However, the existing asymmetric cryptographic schemes rely on unproven mathematical assumptions for security. Further, the digital keys used in their implementation are susceptible to copying that might remain unnoticed. Here we introduce a secure communication method that overcomes these two limitations by employing Physical Unclonable Keys (PUKs). Using optical PUKs realized in opaque scattering materials and employing off-the-shelf equipment, we transmit messages in an error-corrected way. Information is transmitted as patterned wavefronts of few-photon wavepackets which can be successfully decrypted only with the receiver's PUK. The security of PUK-Enabled Asymmetric Communication (PEAC) is not based on any stored secret but on the hardness of distinguishing between different few-photon wavefronts.

We derive an attainable bound on the precision of quantum state estimation for finite dimensional systems, providing a construction for the asymptotically optimal measurement. Our results hold under an assumption called local asymptotic covariance, which is weaker than unbiasedness or local unbiasedness. The derivation is based on an analysis of the limiting distribution of the estimator's deviation from the true value of the parameter, and takes advantage of quantum local asymptotic normality, a duality between sequences of identically prepared states and Gaussian states of continuous variable systems. We first prove our results for the mean square error of a special class of models, called D-invariant, and then extend the results to arbitrary models, generic cost functions, and global state estimation, where the unknown parameter is not restricted to a local neighbourhood of the true value. The extension includes a treatment of nuisance parameters, namely parameters that are not of interest to the experimenter but nevertheless affect the estimation. As an illustration of the general approach, we provide the optimal estimation strategies for the joint measurement of two qubit observables, for the estimation of qubit states in the presence of amplitude damping noise, and for noisy multiphase estimation.

We study unextendible maximally entangled bases (UMEBs) in \(\mathbb {C}^{d}\otimes \mathbb {C}^{d^{\prime}}\) ($d<d'$). An operational method to construct UMEBs containing $d(d^{\prime}-1)$ maximally entangled vectors is established, and two UMEBs in \(\mathbb {C}^{5}\otimes \mathbb {C}^{6}\) and \(\mathbb {C}^{5}\otimes \mathbb {C}^{12}\) are given as examples. Furthermore, a systematic way of constructing UMEBs containing $d(d^{\prime}-r)$ maximally entangled vectors in \(\mathbb {C}^{d}\otimes \mathbb {C}^{d^{\prime}}\) is presented for $r=1,2,\cdots, d-1$. Correspondingly, two UMEBs in \(\mathbb {C}^{3}\otimes \mathbb {C}^{10}\) are obtained.