Quantum Computing

A physical implementation of linear algebra, in a simplistic description.

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topological quantum computer with notes

hamming code for adding parity bits for error correction is a classical version

Notes from Microsoft Research 2018

  • built on logic gates just like traditional machines

Bit wise vectors

0

(10)\left(\begin{array}{l} 1 \\ 0 \end{array}\right)

matrix multiplication

(abcd)(xy)=(ax+bycx+dy)\left(\begin{array}{ll} a & b \\ c & d \end{array}\right)\left(\begin{array}{l} x \\ y \end{array}\right)=\left(\begin{array}{l} a x+b y \\ c x+d y \end{array}\right)

rank 3

(abcdefghi)(xyz)=(ax+by+czdx+ey+fzgx+hy+iz)\left(\begin{array}{lll} a & b & c \\ d & e & f \\ g & h & i \end{array}\right)\left(\begin{array}{l} x \\ y \\ z \end{array}\right)=\left(\begin{array}{l} a x+b y+c z \\ d x+e y+f z \\ g x+h y+i z \end{array}\right)

(abcd)(wxyz)=(aw+byax+bzcw+dycx+dz)\left(\begin{array}{ll} a & b \\ c & d \end{array}\right)\left(\begin{array}{ll} w & x \\ y & z \end{array}\right)=\left(\begin{array}{ll} a w+b y & a x+b z \\ c w+d y & c x+d z \end{array}\right)

[1000010000100001][abcd]=[abcd]\left[ \begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]\left[\begin{array}{l} a \\ b \\ c \\ d \end{array}\right]=\left[\begin{array}{l} a \\ b \\ c \\ d \end{array}\right]

(1000001001000001)(0100)=(0010)\left(\begin{array}{llll} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)\left(\begin{array}{l} 0 \\ 1 \\ 0 \\ 0 \end{array}\right)=\left(\begin{array}{l} 0 \\ 0 \\ 1 \\ 0 \end{array}\right)

The bit flipping is important. The four operations of a function are identity, negation, constant 0 and constant 1

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