Consider a square lattice \(\mathcal{L}\) in \(D\) spatial dimensions with \(N\) sites, where each site \(s_{1},...,s_{N} \in \mathcal{L}\) is a complex vector space \(\mathbb{V}\) with finite dimension \(\chi\), termed the bond dimension. Site \(s\) can be described by a pure state \(|\Psi\rangle \in \mathbb{V}^{\otimes N}\), which has a reduced density matrix \(\rho^{[s]}=\operatorname{tr}_{\bar{s}}(|\Psi\rangle\langle\Psi|)\). State \(|\Psi\rangle\) can be generated by a quantum circut \(\mathcal{C}\) with depth \(\Theta \equiv 2 \log _2(N)-1\)... read more

Let \(\mathscr{N}\) denote noise, \(\mathscr{R}\) denote the recovery process, and \(L(H)\) denote the space of linear operators on \(H\). The noise and recovery processes can be described as quantum operations \(\mathscr{N}, \mathscr{R}: L(H) \rightarrow L(H)\), from where the mapping can be represented as an operator-sum such as:

\(\mathscr{N}(\rho)=\sum_i E_i \rho E_i^{\dagger},\) \(\quad E_i \in L(H),\) \(\quad \mathscr{N}=\left\{E_i\right\}\)

Given the quantum code \(C_{Q}\), a subspace of Hilbert space \(H\), there exists a recovery ...read more

The formula in the case of \(SU(2)\) is as follows. Choose a polynomial \(Q\). For each \(k \in \mathbb{Z}_{+}\), let \(\xi(k, t)\) be the formal power series in \(t\) that represents a unique critical point of the function in \(\xi\).

\(F(\xi ; h, k, t)\) \(:=\frac{1}{2}(\xi-k)^2+t \cdot \frac{h}{2 \pi^2} \cdot Q(\pi \xi / h)\)
where, \(t\) is treated as a formal variable. \(\xi(k,t)\) undergoes \(t\)-series expansion near the minimum \(\xi(k,0)=k\), formulated as:

\(\displaystyle{ \int_{F(\Sigma ; G)} \exp \left\{h \omega+t \cdot[\Sigma] \backslash Q\left(u^* \phi_2\right)\right\} }\) \(\displaystyle{ =\# Z(G) \cdot h^{3 g-3} \operatorname{vol}(G)^{2 g-2} }\) \(\displaystyle{ \sum_{k>0}\left[\frac{1+\frac{t}{2 h} Q^{\prime \prime}(\xi(k, t))}{\xi(t, k)^2}\right]^{g-1} }\)
such that, \(\#Z(G)=2\) for \(SU(2)\)... read more

A Riemannian manifold is a smooth manifold \(M\) with a Riemannian metric denoted \((M,g)\). For every point \(p\in M\) the tangent bundle of \(M\) assigns a vector space \(T_{p}M\), termed the tangent space of \(M\) at \(p\). The Riemanniam metric defines a positive-definite inner product:

\(g_p: T_p M \times T_p M \rightarrow \mathbb{R}\)
with a norm \(|\cdot|_p: T_p M \rightarrow \mathbb{R}\) defined as:

\(|v|_p=\sqrt{g_p(v, v)}\)
...read more

The thalamic nuclei are paired structures of the thalamus divided into three main groups: the lateral nuclear, medial nuclear, and anterior nuclear groups. The internal medullary lamina, a Y-shaped structure that splits these groups, is present on each side of the thalamus. A midline, thin thalamic nuclei, adjacent to the... read more