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1

An elementary way to proceed is as follows. Let's put an explicit factor of $\epsilon$ in $B$. The problem is then to solve the following equation for the matrix $X$: $$A + \epsilon B = X^{2}$$ We want to do this perturbatively, so we assume that $X$ can be represented as: $$X = \sum_{n=0}^{\infty}\epsilon^{n}X^{(n)}$$ We can then write: $$ ...


2

Hints: The square root function has a Taylor expansion around $a>0$ $$\tag{1} \sqrt{a+b}~=~\sum_{n=0}^{\infty} \begin{pmatrix}\frac{1}{2} \cr n\end{pmatrix}a^{\frac{1}{2}-n}b^n, \qquad |b| ~<~a. $$ One may show that a possible non-commutative generalization reads $$ \sqrt{A+B}~=~\sqrt{A}+\sum_{n=1}^{\infty} \begin{pmatrix}\frac{1}{2} \cr ...


0

This is probably related to the fact that in GW method the potential is an effective screened potential while there is no screening in the other methods.


-1

The closed orbits are those classical trajectories that can never leave the system and these closed orbits are then correlate with Quantun interference. To under this closed orbit theory read following articles: PRL 58, 1731 (1987) PRA 38, 1896 (1988) PRA 38, 1913 (1988)


2

The standard DFT (LDA & GGA) could not properly capture the exchange-correlation of a quantum system being investigated, it sort of lacks interaction/quantum property resulting to smaller gaps (inaccurate result in other words). A better method that will give bigger gaps (closely accurate results/closer to experiments) should be performed. This is why ...


0

The effect you describe is unphysical. The energy for a randomized, undriven system should never rise when the system moves into equilibrium. If your energy rises (and especially, if it diverges!) there is a problem in your code. This is most likely due to numerical instability in your method for integrating the equations of motion. This is easy to check by ...


0

This is a bit of a cop-out answer, but it's relatively easy to find a set of gates that can be substituted for NOT gates. For example: $$[Y][Z]/i = [\text{NOT}] $$ Where $[Y]$ and $[Z]$ are Pauli gates. It's already known that quantum computing can do universal computations, using only a specific sets of gates. One such set is the Hadmard gate, a phase ...


1

You're basically looking for a smoothing algorithm, something that takes a collection of points and turns it into a density. This can be done with the help of a kernel, a weighting function $K(u)$ that satisfies the following two conditions: \begin{align} \int_{-\infty}^{+\infty} K(u)\,du=1 \\ K(u)=K(-u)\quad\forall u \end{align} The Wikipedia article I link ...


3

The entanglement of any region in a matrix product state of bond dimension $D$ is bounded by $S\le 2\log D$. Thus, in order to simulate a system with a lot of entanglement, the bond dimension (and thus the memory and time of the computation) will grow exponentially with the entropy. Conversely, we know that if for a state $\vert\psi\rangle$ the ...


5

Enzo is fundamentally a grid-based finite-volume hydrodynamics code. That is, the domain is divided into cells, each is assigned various fluid properties (density, velocity, etc.), and at each timestep fluxes of those quantities across the interfaces between cells are used to update the quantities in the cells. It has a choice of particular methods for ...



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