2.concept.ppt

1、Quantum Concept 2009.03.16,Quantum Computation and Quantum Cryptography,Good question,量子和模糊数学有什么关系吗？ 模糊数学：用于控制论。 健康人的集合- 你在此集合中吗？ 远远大于1的数 25？ 亦此亦彼。不确定！ 概率现象也是不确定的。 量子计算可以将模糊数学包含在内。,outline,Quantum Concept q-bit ,amplitudes,bloch sphere Quantum Measurement Quantum Logic gate,Quantum bit 量子比特,Quantum

2、concept,Classical State Descriptions Entropy-bit Quantum State Descriptions Qubitquantum bit Mathematically objects Dirac notation,Information theory,Claude Shannon 1948, publish a remarkbal pair of papers laying the information for mordern theory of information communication Key step: mathematicall

3、y define the concept of information, Eg,entropy,经典信息论,香农简介 香农(1916-2001),生于美国密执安州的加洛德。1940年获得麻省理工学院数学博士学位和电子工程硕士学位。1941年他加入了贝尔实验室数学部，在此工作了15年。 他在信息论的领域中钻研了8年之久，终于在1949年在贝尔系统技术杂志发表了244页的长篇论著-通信的数学理论。次年，他又在同一杂志上发表了另一篇名著-噪声下的通信。,量子信息论,1995,Ben Schumacher 证明了与Shannon无噪声编码定理类似的结果，并定义了“量子比特” 量子信息论主要有以下几个方

4、面的应用 (1)量子纠错码，例如超密编码 (2)分布式量子计算 (3)网络化量子信息论,Claasical and quantum,Classical quantum Bit qubit 0 1 ?,Superpositions -linear combinations of states,Quantum concept-a qubit,A qubits state is a unit vector in a two-dimension complex vector space computational basis states an orthonormal basis for this v

5、ector space can be in state orther thanor,basis,量子比特的状态是二维复向量空间中的向量。 计算基态，正交基 量子比特的状态可以落在|0和|1之外。,Examples:,qubit,Quantum conceptqubit,P probability of system being in state 0,1-p probability of system being in state 1,vector of amplitudes,complex numbers,A qubit,Quantum State Descriptions,Classical

6、 State Descriptions,Bloch sphere,Multiple qubits- two qubit system,Classical quantum Two Bits two qubits 00 01 10 11 ?,eg,Bell state or EPR pair,John Bell,Multiple qubits,Consider a system of nqubits The computational basis states of this system are of the form， And so a quantum state of such a syst

7、em is specified by amplitudes. Hilbert space is a big place. -carlton Caves,Postulate 1: Associated with any isolated physical system is a complex vector space with inner product known as the state space of the system. The system is completely described by its state vector, which is a unit vector in

8、 the systems state space.,physical system,complex vector space,Quantum State,假设1：任一孤立物理系统都有一个称为系统状态空间的复内积向量空间（即Hilbert空间）与之相联系，系统状态完全由状态向量所描述，这个向量是系统状态空间的一个单位向量。,state of system,physical system,Measurment 测量,Quantum State Measurement Measurement changes the state of a qubit, collapsing it from its s

9、uperposition of |0 and |1. The measurement of a qubit will give only either 0 or 1. Why does this type of collapse occur? Nobody knows! This behavior is simply one of the fundamental postulates of quantum mechanics.,How to determine a qubit,Postulate 3: Quantum measurements are described by a collec

10、tion of measurement operators. These are operators acting on the state space of the system being measured. The index refers to The measurement outcomes that may occur in the experiment. If the state of the quantum system is immediately before the measurement, then the probability of result is given

11、by and the state of the system after the measurement is The measurement operators satisfy,Quantum Measurement,Qubit measurement,量子测量假设： 量子测量由一组测量算子描述，这些算子作用在被测系统状态空间上且测量后系统的状态满足概率为1的事实，即测量算子满足完备性方程。,From a single measurement one obtains only a single bit of information about the state of qubit.,How

12、to determine a qubit,Only if infinitely many identically prepared qubits were measured would one be able to detremine of a qubit.,How to determine a qubit?,测量会改变量子状态，从叠加态坍缩到特定状态。 这是量子力学的基本原理之一。 对于量子比特的一次测量，只得到有关量子比特状态的一比特信息。 那么，如何确定量子比特的状态呢？ 事实上： 只有测量无穷多完全相同的量子比特才能确定,Multiple qubits- two qubit syste

13、m,The measurement result x=(00,01,10,11) occures with probability ,with the state of the qubits after the measurement being,The first qubit alone gives 0 with probability?,If we measure just one qubit of a two qubit system,what will happen?,Post-measurement state?,Multiple qubits-two qubit system,Re

14、-normalized,eg: Bell state or EPR pair 测后状态为|00或|11。,John Bell,对n-qubit系统做测量，若第1个分量测量结果为x，x取自集合0,1，则测后状态为： 重新归一化！,Quantum Logic gate 量子逻辑门,量子计算机是由包含连线和基本量子门排列起来、形成处理量子信息的量子线路建造的。 逻辑门负责处理信息：把信息从一种形式转移为另一种。 quantum NOT gate Z gate Hadamard gate,1. Quantum Logic Gate,A quantum gate or quantum logic gat

15、e is a basic quantum circuit operating on a small number of qubits. They are the analogues for quantum computers to classical logic gates for conventional digital computers. Quantum logic gates are reversible, unlike many classical logic gates. Some universal classical logic gates, such as the C-NOT

16、 gate, provide reversibility and can be directly mapped onto quantum logic gates. Quantum logic gates are represented by unitary matrices.,With Quantum logic gates, we can understand the quantum evolution and quantum measurement more easily.,Quantum computation,Are there any constraints on what matr

17、ices may be used as a quantum gate? Yes!,The matrix U describing the sigle qubit gate be unitary,that is,is the adjoint of U (transposing and complex conjugating),Quantum computation,Sigle qubit gates Eg1:quantum NOT gate,A singal bit logic gate,Eg2:Z gate Eg3:Hadamard gate,X,X,Y,Z,Y,NOT,1. Quantum

18、Logic Gate,H,e.g.,e.g.,Hadamard,eg,Please prove the gate U is unitary,and compute U|0,U|1 and,多量子比特门,受控非门：CNOT 此门有两个输入量子比特，分别是 控制量子比特和目标量子比特。 该门的作用描述如下： 若控制量子比特设置为0，则目标量子比特将保持不变，否则，目标量子比特将翻转。即： |00|00， |01|01， |10|11， |11|10 |A,B|A,AB 表示模2加。,Controlled NOT (CNOT),CNOT is an unlocal gate, which can b

19、e used generate entangled state,Two Qubit Gates,Quantum Logic Gate,Multiple qubit gates,Classical gates: And ,or, Xor (exclusive-OR),NAND,Not gate NAND is known as a universal gate. Any function on bits can be computed from the composition of NAND gates alon.,qubit gates : Controlled -Not (CNOT gate

20、) Three describes of CNOT Any multiple qubit logic gate may be composed from CNOT and single qubit gates.,多量子比特门,经典多比特门: 与, 或, 异或,与非,或非 与非门被称为“通用门”。 重要的理论结果： 比特上的任意函数可以仅用与非门的复合来计算。,多量子比特门: 受控非 三种描述 任意的多量子比特门都可由受控非门和单量子比特门复合而成。,Multiple qubit gates,Other classical gates : For example:NAND can be unde

21、rstood as unitary gate? No! Inreversible There is an irretrievable loss of information.,经典NAND(与非门) 对应着一个量子酉门吗？ 不! 经典NAND(与非门)具有本质的 不可逆性。 这种不可逆性带来了信息的损失。,No-cloning theroem,Qubit copying circuit Can we copy a qubit by C-NOT gate?,No cloning theorem Qubits cannot be copied.,Example Operation - Multip

22、lication By 2,Carry Bit,Input,Output,Ones Bit,We can build a reversible logic circuit to calculate multiplication by 2 using CNOT gates arranged in the following manner:,An example for quantum CNOT gate,CNOT,H,Quantum Logic Gate,Quantum computation 量子计算,(1)Basic model for quantum algorithm The algor

23、ithm on classical computer may not be reversible transformation, but it should be reversible on quantum computer. Therefore, we should transfer classical algorithm f(x) to reversible transformation. This transition can be achieved as follows: Suppose ,the process f(x) is equivalent to the following

24、process: This is a reversible transformation and also an involutory transformation. Where, we ignore the intermediate variables, and usually take b = 0.,2. Model for quantum computation,According to definition，different x is orthogonal with corresponding (x,b), different corresponding x are orthogon

25、al with each other，therefore F transfer one orthogonal vector to another orthogonal vector. From linear algebra, F can be expanded to one unitary transformation on . This shows that any classical algorithm can be transferd to a quantum algorithm.,2. Model for quantum computation,量子计算算法的数学模型 （1）量子计算算

26、法的基本模型 经典计算机上运行的算法未必是可逆变换，但量子计算机却要求量子算法是可逆变换。为此，必须将经典算法f(x)转换为可逆变换。这个转换可用下述方法实现： 设 ,则计算f(x)的过程等价于下述过程： 这就是一个可逆变换,而且它还是对合变换. 这里我们忽略了中间变量，通常我们取 b=0。 根据定义，不同x对应的(x,b)相互正交；不同x对应的 相互正交，因而F将正交向量变换为正交向量。由线性代数知,F可扩充为 上的一个酉变换.这说明 任何一个经典算法都可转化为一个量子计算算法,Quantum Parallelism is a fundamental feature of many quan

27、tum algorithms. Quantum Parallelism allows quantum computers to evaluate a function f(x) for many different values of x simultaneously. limitation： It requires the ability to extract information about more than one value of f(x) from supersition states.,量子并行性是量子算法的一个基本特征。 量子并行性使量子计算机可以同时计算函数f(x)在许多不

28、同的x处的值。 但是它的局限性在于： 它要求从叠加态中得到不止一个f(x)值的 信息抽取能力。,Parallel computation capability of quantum algorithm Let , and b=0, 0m is m-dimensional 0 vector, the superimposed state is Since F is linear transformation and , that is Therefore, It is obvious that, F(A) is superposition of 2n classical states. As l

29、ong as are all not 0, then F(A) contains information of every classical vector.,Model for quantum computation,In other words, after transformation on quantum vector A, the quantum vector F(A) contains the transforming results on all the classical vectors, therefore, performing once computation is eq

30、uivalent to performing parallel computation on all the classical vectors.,Model for quantum computation,Review：Fundamentals of Quantum Computation The calculation to be completed on quantum computer using quantum mechanism is known as quantum computation. Its fundamentals as follows: 1. One qubit ca

31、n denote 0 and 1 at the same time, so n qubits can represent 2n possible inputs. 2. The calculation of n qubits is equivalent to computation of 2n inputs on electronic computer simultaneously, so it has powerful parallel computing capacity. When read the outputs, each result is possible, but the possibility is determinate. 3. If we can find one quantum algorithm and make the helpful result(s) appear with high possibility, we will achieve the objective of quantum computation.,Model for quantum transformati