CMSC5706 Topics in Theoretical Computer Science Week 12： Quantum computing Instructor: Shengyu Zhang 1
Instructor: Shengyu Zhang 1
Roadmap Intro to math model of quantum mechanics Review of quantum algorithms The power of quantum computers Quantum games
Roadmap • Intro to math model of quantum mechanics • Review of quantum algorithms • The power of quantum computers. • Quantum games
Postulate 1 States State space: Every isolated physical system corresponds to a unit vector in a complex vector space Unit vector t-norm is 1 Such states are called pure states We use a weird“ket" notation|·) to denote such a state
Postulate 1: States • State space: Every isolated physical system corresponds to a unit vector in a complex vector space. – Unit vector: ℓ2 -norm is 1. • Such states are called pure states. • We use a weird “ket” notation ⋅ to denote such a state
Ket notation Mathematically, I )is a column vector And is a row vector (yllo) is the inner product between the vectors|φ)and|y) ·吵|Mly) is just the quadratic form yb Myl
Ket notation • Mathematically, ⋅ is a column vector. • And ⋅ is a row vector. • 𝜓 𝜙 is the inner product between the vectors 𝜙 and 𝜓 . • 𝜓 𝑀 𝜓 is just the quadratic form 𝜓 𝑇𝑀𝜓
A quantum bit, or qubit, is a state of the form a|0)+1 a0)+B|1) Where a,B∈ are called amplitudes, satisfying that a|2+|62=1. So a qubit can sit anywhere between 0 and 1(on the unit A quantum bit circle) (qubit We say that the state is in superposition of 0)and 1)
• A quantum bit, or qubit, is a state of the form 𝛼 0 + 𝛽 1 where 𝛼, 𝛽 ∈ ℂ are called amplitudes, satisfying that 𝛼 2 + 𝛽 2 = 1. • So a qubit can sit anywhere between 0 and 1 (on the unit circle). • We say that the state is in superposition of 0 and 1 . A quantum bit (qubit) 𝛼 0 + 𝛽 1 0 1