香港中文大学：《Topics in Theoretical Computer Science》课程教学资源（PPT课件讲稿）量子计算 Quantum computing

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

Postulate 2: operation Evolution: The evolution of a closed quantum system is described by a unitary transformation That is, if a system is in state 11 at time ti and in state l 2 at time t2 then there is a unitary transformation U s t ly2)=U/y1) Unitary transformation Ut=u-1 Recall: Ut=(U)*, transpose complex conjugate You can think of it as a rotation operation

Postulate 2: operation • Evolution: The evolution of a closed quantum system is described by a unitary transformation. • That is, if a system is in state 𝜓1 at time 𝑡1, and in state 𝜓2 at time 𝑡2, then there is a unitary transformation 𝑈 s.t. 𝜓2 = 𝑈 𝜓1 . • Unitary transformation: 𝑈 † = 𝑈 −1 – Recall: 𝑈 † = 𝑈 𝑇 ∗ , transpose + complex conjugate – You can think of it as a rotation operation

Postulate 3: measurement Measurement: We can only observe a quantum system by measuring it The outcome of the measurement is random And the system is changed by the measuremen

Postulate 3: measurement • Measurement: We can only observe a quantum system by measuring it. • The outcome of the measurement is random. • And the system is changed by the measurement

If we measure qubit a[0)+B 1 in the computational basis 10),|1), then outcome“0” 0)+B|1) occurs with prob lal, and outcome 1"occurs with prob 1B12 The system becomes 0)if outcome“ 0 occurs,and A quantum bit becomes|l) f outcome“1” (qubit occurs The system collapses

• If we measure qubit 𝛼 0 + 𝛽 1 in the computational basis 0 , 1 , then outcome “0” occurs with prob. 𝛼 2 , and outcome “1” occurs with prob. 𝛽 2 . • The system becomes 0 if outcome “0” occurs, and becomes 1 if outcome “1” occurs. – The system collapses. A quantum bit (qubit) 𝛼 0 + 𝛽 1 0 1

Measurement on general states In general, an orthogonal measurement of a d-dim state is given by an orthonormal bass{ψ1)…ya) If we measure state )in basis {y1)…ybd), then outcome i∈{1,…,l occurs with prob. I(lvill2 The system collapses to l i if outcome i occurs

Measurement on general states • In general, an orthogonal measurement of a 𝑑-dim state is given by an orthonormal basis 𝜓1 ,… 𝜓𝑑 . • If we measure state 𝜙 in basis 𝜓1 ,… 𝜓𝑑 , then outcome 𝑖 ∈ 1,… , 𝑑 occurs with prob. 𝜙|𝜓𝑖 2 . • The system collapses to 𝜓𝑖 if outcome 𝑖 occurs

Postulate 4: composition Composition: The state of the joint system (S1, S2, where S, is in state l 1 and s2 in y2),is|y1)②|v2) tensor product of vectors (a1,a2)⑧(b1,b2b3)=(a1b1,a1b2,a1b3,a2b1,a2b2a2b3) dim( 1&1 2))=dim( 1)). dimdya2)) size (l idly b2))=size (ly1))+ size(l b2)) size: number of qubits Notation:|0)8n=|0)⑧…|0), m times

Postulate 4: composition • Composition: The state of the joint system (𝑆1, 𝑆2), where 𝑆1 is in state 𝜓1 and 𝑆2 in 𝜓2 , is 𝜓1 ⊗ 𝜓2 . • ⊗: tensor product of vectors. – 𝑎1, 𝑎2 ⊗ 𝑏1, 𝑏2, 𝑏3 = (𝑎1𝑏1, 𝑎1𝑏2, 𝑎1𝑏3, 𝑎2𝑏1, 𝑎2𝑏2, 𝑎2𝑏3). – dim 𝜓1 ⊗ 𝜓2 = dim 𝜓1 ⋅ dim 𝜓2 – size 𝜓1 ⊗ 𝜓2 = size 𝜓1 + size 𝜓2 • size: number of qubits. • Notation: 0 ⊗𝑛 = 0 ⊗ ⋯ ⊗ 0 , 𝑛 times