\( \def\ket#1{{ \left| #1 \right> }} \def\bra#1{{ \left< #1 \right| }} \def\bracket#1{{ \left( #1 \right) }} \) \( \DeclareMathOperator{\tr}{tr} \) 1) Encoding into higher dimensional space In QEC, one needs to encode information into a higher dimensional Hilbert space, which can be a system of multiple qubits (like conventional repetition codes, Shor's code, Steane's code,...) or a $d$-level sy..
\( \def\ket#1{{ \left| #1 \right> }} \) \( \def\bracket#1{{ \left( #1 \right) }} \) As discussed in the QFT note, given a frequency $\varphi \; \left( 0 \le \varphi < 1\right)$ that satisfies \[ k \equiv 2^n \varphi \in \mathbb{N}, \label{condition} \] the matrix representation of the following vector is the $k^{\text{th}}$-column of DFT matrix \[ \ket{\widetilde{\varphi}} = \frac{1}{2^{n/2}} \s..
Quick summary: Except a special class of quantum states where quantum Fourier transform can exponentially speedup, in general it is not faster than classical Fourier transform. Definition of classical Fourier transform \( \def\ket#1{{ \left| #1 \right> }} \) \( \def\bracket#1{{ \left( #1 \right) }} \) A discrete Fourier transform (DFT) of a $2^n$-length vector $x= \left( x_0, x_1, \ldots, x_{2^n..
Honeycomb lattices Honeycomb lattice (or hexagonal lattice) is realized by graphene. The lattice consists of two carbon atoms (hereafter we call them $A$ and $B$ sites) per unit cell. The lattice vectors can be written as \[ \vec{a}_1 = \frac{a}{2} (3, \sqrt{3}), \quad \vec{a}_2 = \frac{a}{2} (3, -\sqrt{3}) ,\] where $a$ denotes carbon-carbon distance. The corresponding reciprocal lattice vector..
Physics Corner 2022. 7. 17. 11:37
\( \def\ket#1{{ \left| #1 \right> }} \def\bra#1{{ \left< #1 \right| }} \def\bracket#1{{ \left( #1 \right) }} \) \( \DeclareMathOperator{\tr}{tr} \) 1) Encoding into higher dimensional space In QEC, one needs to encode information into a higher dimensional Hilbert space, which can be a system of multiple qubits (like conventional repetition codes, Shor's code, Steane's code,...) or a $d$-level sy..
Physics Corner 2022. 7. 17. 10:59
\( \def\ket#1{{ \left| #1 \right> }} \) \( \def\bracket#1{{ \left( #1 \right) }} \) \( \def\bra#1{{ \left< #1 \right| }} \) $\left[7,1 \right]$ Steane code is an ubiquitous example of Calderbank-Shor-Steane (CSS) class of quantum error correction codes. Steane code was constructed from classical $\left[7,4,3 \right]$ Hamming code whose parity-check matrix takes the following form \begin{equation..
Physics Corner 2022. 7. 17. 10:23
\( \def\ket#1{{ \left| #1 \right> }} \) \( \def\bracket#1{{ \left( #1 \right) }} \) As discussed in the QFT note, given a frequency $\varphi \; \left( 0 \le \varphi < 1\right)$ that satisfies \[ k \equiv 2^n \varphi \in \mathbb{N}, \label{condition} \] the matrix representation of the following vector is the $k^{\text{th}}$-column of DFT matrix \[ \ket{\widetilde{\varphi}} = \frac{1}{2^{n/2}} \s..
Physics Corner 2022. 7. 16. 22:46
Quick summary: Except a special class of quantum states where quantum Fourier transform can exponentially speedup, in general it is not faster than classical Fourier transform. Definition of classical Fourier transform \( \def\ket#1{{ \left| #1 \right> }} \) \( \def\bracket#1{{ \left( #1 \right) }} \) A discrete Fourier transform (DFT) of a $2^n$-length vector $x= \left( x_0, x_1, \ldots, x_{2^n..
Physics Corner 2022. 7. 16. 17:13
Honeycomb lattices Honeycomb lattice (or hexagonal lattice) is realized by graphene. The lattice consists of two carbon atoms (hereafter we call them $A$ and $B$ sites) per unit cell. The lattice vectors can be written as \[ \vec{a}_1 = \frac{a}{2} (3, \sqrt{3}), \quad \vec{a}_2 = \frac{a}{2} (3, -\sqrt{3}) ,\] where $a$ denotes carbon-carbon distance. The corresponding reciprocal lattice vector..