We construct quantum MDS codes for quantum systems of dimension $q$ of length

$q^2+1$ and minimum distance $d$ for all $d \leqslant q+1$, $d \neq q$. These

codes are shown to exist by proving that there are classical generalised

Reed-Solomon codes which are contained in their Hermitian-dual. These

constructions include many constructions which were previously known but in

some cases these codes appear to be new. We go on to prove that if $d\geqslant

q+2$ then there in no generalised Reed-Solomon code which is contained in its

# All

In this manuscript, we investigate the exact bound state solution of the

Klein Gordon equation with an energy-dependent Coulomb-like potential energy in

the presence of position-energy dependent mass. First, we examine the case

where the mixed vector and scalar potential energy possess equal magnitude and

equal sign. Then, we extend the investigation with the cases where the mixed

potential energies have equal magnitude and opposite sign. Furthermore, we

We present and analyze a proposal for a macroscopic quantum delayed-choice

experiment with massive mechanical resonators. In our approach, the electronic

spin of a single nitrogen-vacancy impurity is employed to control the coherent

coupling between the mechanical modes of two carbon nanotubes. We demonstrate

that a mechanical phonon can be in a coherent superposition of wave and

particle, thus exhibiting both behaviors at the same time. We also discuss the

Fixed node diffusion quantum Monte Carlo (FN-DMC) is an increasingly used

computational approach for investigating the electronic structure of molecules,

solids, and surfaces with controllable accuracy. It stands out among equally

accurate electronic structure approaches for its favorable cubic scaling with

system size, which often makes FN-DMC the only computationally affordable

high-quality method in large condensed phase systems with more than 100 atoms.

In such systems FN-DMC deploys pseudopotentials to substantially improve

Kippenhahn's Theorem asserts that the numerical range of a matrix is the

convex hull of a certain algebraic curve. Here, we show that the joint

numerical range of finitely many hermitian matrices is similarly the convex

hull of a semi-algebraic set. We discuss an analogous statement regarding the

dual convex cone to a hyperbolicity cone and prove that the class of convex

bases of these dual cones is closed under linear operations. The result offers

a new geometric method to analyze quantum states.

We present a secure multi-party quantum summation protocol based on quantum

teleportation, in which a malicious, but non-collusive, third party (TP) helps

compute the summation. In our protocol, TP is in charge of entanglement

distribution and Bell states are shared between participants. Users encode the

qubits in their hand according to their private bits and perform Bell-state

measurements. After obtaining participants' measurement results, TP can figure

Hybrid quantum/classical variational algorithms can be implemented on noisy

intermediate-scale quantum computers and can be used to find solutions for

combinatorial optimization problems. Approaches discussed in the literature

minimize the expectation of the problem Hamiltonian for a parameterized trial

quantum state. The expectation is estimated as the sample mean of a set of

measurement outcomes, while the parameters of the trial state are optimized

classically. This procedure is fully justified for quantum mechanical

Imperfect measurement can degrade a quantum error correction scheme. A

solution that restores fault tolerance is to add redundancy to the process of

syndrome extraction. In this work, we show how to optimize this process for an

arbitrary ratio of data qubit error probability to measurement error

probability. The key is to design the measurements so that syndromes that

correspond to different errors are separated by the maximum distance in the

signal space, in close analogy to classical error correction codes. We find

Entanglement entropy in free scalar field theory at its ground state is

dominated by an area law term. However, when mixed states are considered this

property ceases to exist. We show that in such cases the mutual information

obeys an "area law". The proportionality constant connecting the area to the

mutual information has an interesting dependence on the temperature. At

infinite temperature it tends to a finite value which coincides with the

classical calculation.

In classical mechanics, external constraints on the dynamical variables can

be easily implemented within the Lagrangian formulation. Conversely, the

extension of this idea to the quantum realm, which dates back to Dirac, has

proven notoriously difficult due to the noncommutativity of observables.

Motivated by recent progress in the experimental control of quantum systems, we

propose a framework for the implementation of quantum constraints based on the

idea of work protocols, which are dynamically engineered to enforce the