Quantum Computation

Linear Algebra Vector: $|\psi\rangle$ Row vector: $\langle\psi|=(\overline{|\psi\rangle})^T$ Inner product: $\langle\psi|\varphi\rangle=\sum i\overline{u} iv i$ $\langle\varphi|A|\...

Postulates 1. Associated to any isolated physical system is a complex vector space with inner product known as state space of the system. The system is completely describled by its...

Query value of function $U f:|x,y\rangle\rightarrow|x,y\oplus f(x)\rangle$ $f:\{0,1\}^n\rightarrow\{0,1\}$ 对应的矩阵为置换阵 $2^n\times 2^n$ Quantum Parallelism: $U f H^{\oplus n}|0^n\rang...

Polynomial method T query algorithm: $|\psi x^t\rangle=U tO xU {t 1}O x\cdots U 2O xU 1O x|\psi\rangle=\sum {x,i}\alpha {x,i}^{(t)}|i\rangle|\psi {x,i}^t\rangle$ $\alpha {x,i}^t$ i...

Distance Measures over distribution Given two distribution $\{p x\},\{q x\}$ Trace distance (Total variance): $D(p x,q x) = \frac{1}{2}\sum {x\in X}|p x q x|$ Fidelity: $F(p x,q x)...

Classical linear codes CSS codes Modified Lo Chau

Vernam cipher/ One time pad: 绝对安全（信息论安全），Eve 做任何计算 加密前后概率分布需一样 Theorem: the secret key has to be as long as the message 密码学安全：Eve 在多项式时间内无法得到任何有效信息 DH Key Exchange: 离散对数问题，量子计算机下不安...

matrix = operator = gate Single qubit operator Pauli operators $I=(1,0;0,1)$ $X=(0,1;1,0)$ (非门) $X|0\rangle=|1\rangle,X|1\rangle=|0\rangle$ $Y=(0, i;i,0)$ $Y=iXZ,XY= YX,XZ= ZX,YZ=...