量子计算与信息基础导论

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"《量子计算与信息初步》2018版 Bernard Zygelman" 《量子计算与信息初步》是Bernard Zygelman撰写的一本关于量子计算与信息的入门书籍，旨在向读者介绍这个领域的基础知识。这本书的核心内容涵盖了量子力学的基本工具、量子位（qubits）的概念以及二进制算术等关键概念。 1.1 量子力学工具箱在计算和信息处理的基础中，比特（bits）扮演着核心角色，它代表二值信息，如是/否答案、开关的开/关状态或停止/继续的决策。比特通常用整数0和1表示，所有数字计算机都是由这些不可分割的比特构建的。例如，灯泡的开和关状态就可以存储一个比特的信息。 1.1.1 量子位（Qubits）量子位（qubits）是量子计算中的基本单位，它扩展了比特的概念。虽然它们在功能上类似，但qubits拥有量子力学的特性，如叠加态和纠缠态。在不深入讨论这些复杂的特性时，我们可以简单地通过符号区分qubits和bits，将0和1分别替换为量子态符号|0⟩和|1⟩。这两个符号表示qubit的两种可能状态。 1.1.2 二进制算术比特构成了二进制数系统的基础，就像字母组成我们的语言一样。在这个系统中，0和1可以进行基本的算术运算，如加法、减法、乘法和除法。这种运算方式在传统计算机中至关重要，而在量子计算中，由于qubits的叠加性，计算过程变得更加复杂且潜力巨大。 1.2 量子计算的优势量子计算机利用qubits的量子态叠加和纠缠，能够并行处理大量信息，理论上能显著提高计算效率，解决某些问题的速度远超经典计算机。例如，量子计算机在分解大素数（如用于RSA加密）和模拟量子物理系统方面具有潜在优势。 1.3 量子信息处理量子信息处理包括编码、传输和解码信息的方式，这些在量子通信和量子密码学中尤为关键。量子隐形传态和量子密钥分发利用了量子态的不可克隆性和测量的破坏性，实现了理论上无法被破解的安全通信。总结来说，《量子计算与信息初步》这本书是理解量子计算原理及其对信息技术影响的起点，它揭示了量子世界如何颠覆我们对计算和信息的传统理解，并展示了这一领域未来可能的突破性进展。

2 1 A Quantum Mechanic’s Toolbox

Bits are also used to represent numbers. Obviously, the integer 0 is represented

by the bit whose state is 0 and the number 1 by the bit whose value is 1. How about

the integers 2, 3 ...? Consider the string

1100010

which is interpreted by the following algorithm,

1100010 → 1 ×2

+ 1 ×2

+ 0 ×2

+ 1 ×2

+ 0 ×2

and has the value 98 in the base ten system. Each entry in the string represents a

coefﬁcient of consecutive powers of 2

, where n is an integer. Note that

98 = 0 ×10

+ 9 ×10

+ 8 ×10

conforms to this algorithm but powers of 10

replace 2

, and the coefﬁcients of each

term are the symbols 0, 1, 2 ...9.

Mathematica Notebook 1.1: The binary system and operations within it http://

www.physics.unlv.edu/%7Ebernard/MATH_book/Chap1/chap1_link.html

Imagine a set, or register, of ﬁve light bulbs. As each light bulb can store a bit

of information, a little thought shows that this register can store one of 2

= 32

integers at any single instant of time. For example, reverting to qubit notation, the

number 26 is represented by the physical state

|11010.

In this notation, the ﬁrst light bulb, starting from the right-hand side (r.h.s), is off,

the second one is on, the third is off, while the fourth and ﬁfth bulbs are in the on

position. Consider the following expression

|11010+|00101. (1.2)

What should the + symbol signify here and how can we interpret this construct?

At ﬁrst sight, it might seem reasonable to deﬁne it as the arithmetic operation of

addition so that |11010+|00101is equal to |11111which represents the integer 31

in binary form. It is apparent, under scrutiny, that this deﬁnition is unsatisfactory for

the following reason. We agreed that |11010represents a physical state in which the

light bulbs have the corresponding on-off values at a single instance of time. Thus

the expression |11010+|01001 suggests a register of light bulbs simultaneously

in two different conﬁgurations at the same time, an absurd proposition and so

construct (1.2) appears to be meaningless.

1.2 A Short Introduction to Linear Vector Spaces 3

Let’s posit a ﬁve-qubit register comprised of the quantum mechanical analog

of a light bulb, that I call a qbulb. Because atoms/ions obey the laws of quantum

mechanics (QM), it is plausible to deﬁne a qbulb register as an array of ﬁve atoms,

each of which can be toggled on and off, in analogy with a light bulb.

Amore

precise deﬁnition is forthcoming in later chapters. As a quantum system, this array

of atoms/ions must obey the postulates of the quantum theory. Within this theory,

we will show how to make sense of expression (1.2). Before elaborating on this

statement, let’s ﬁrst embark on a short mathematical detour.

1.2 A Short Introduction to Linear Vector Spaces

Consider a set of objects α, β, γ ···. We say that these objects belong to a linear

vector space V provided that

(i) There exists an operation, which we denote by the + sign, so that if α, γ are

any two members of the vector space V then so is the quantity α + γ .

(ii) For scalar c, there exists a scalar multiplication operation deﬁned so that if β

is a vector in V then so is cβ = βc.Ifa, b are scalars abβ = a(b β).

(iii) Scalar multiplication is distributive, i.e. c(α + β) = cα + cβ , also for scalar

a, b, (a + b)α = aα + bα.

(iv) The + operation is associative, i.e. α + (β +γ)= (α + β) + γ.

(v) The + operation is commutative, i.e. α +β = β +α.

(vi) There exists a null vector 0 which has the property 0 + α = α for very vector

α in V .

(vii) For every α in V there exists an inverse vector −α that has the property

α +−

α = 0.

Numerous mathematical structures form vector spaces. Perhaps the most familiar

are the vectors that deﬁne a direction in space. Convince yourself that the set of

numbers on the real number line form a vector space. Do the set of all integers

constitute a vector space?

Armed with these deﬁnitions, we are ready to tackle, in the context of the ﬁve-

qbulb register, the postulates of quantum mechanics. Because our discussion here is

limited to the ﬁve-qbulb register, we stress the tentative nature of these postulates

by labeling them as meta-postulates or m-postulates for short.

m-Postulate I

Following a measurement (observation) of this quantum register,

we observe only one out of the 32 possible qbulb-on/off conﬁgura-

tions. Immediately after that measurement the register is found in

the state corresponding to the values measured.

David Wineland [7], shows that such a notion is not as fanciful as might ﬁrst appear.

4 1 A Quantum Mechanic’s Toolbox

For example, if the measurement resulted in the ﬁrst qbulb on, the second off,

the last three on, the state corresponding to this measurement is |11101, and the

postulate asserts that the system occupies this state immediately after measurement.

The verity of this postulate appears to be self-evident. It almost seems not

worthwhile stating as it surely applies to the classical counterpart of this system.

After introducing the second postulate we will appreciate its signiﬁcance and

implications.

m-Postulate II a

The possible 32 states of our qubit register are vectors in a

linear vector space.

According to this postulate, and property (i) of the itemized list above, expres-

sion (1.2) must also be a vector in this space. Indeed so is

|Φ≡

|00000+|00001+|00010+|00011+|00100+|00101+|00110+|00111

|01000+|01001+|01010+|01011+|01100+|01101 +|01110+|01111

|10000+|10001+|10010+|10011+|10100+|10101+|10110+|10111

|11000+|11001+|11010+|11011+|11100+|11101+|11110+|11111,

(1.3)

as is any combination deﬁned by the +operation. This property, that a quantum state



can be expressed as a linear combination of other quantum states, is sometimes

called the superposition principle (Fig. 1.1).

m-Postulate II b

A complete physical description of this quantum register is

encapsulated by a vector |Ψ  in this vector space.

What is a complete physical description? A classical register requires an itemization

of bulbs that are on/off to characterize its physical state. For the quantum version

of this system, m-Postulate II states that an abstract quantity, a vector |Ψ , that is a

linear combination of states in which the system ﬁnds itself after a measurement is

made, deﬁnes the system. For example, suppose the complete physical description

of our quantum register is given by the vector (state) (1.3), i.e. |Φ. Construct |Φ

suggests that all possible on/off conﬁgurations exist at the same time, a strange

proposition that counters everyday experience. If we are going to make sense of

this theory, the necessity for m-Postulate I becomes apparent. If |Φ does exist, m-

Postulate I insures that we observe only one of the possible qbulb conﬁgurations.

In addition, immediately after that observation |Φ “collapses” into that particular

state. The above scenario should stimulate lots of questions, perhaps even healthy

skepticism. m-Postulate II argues for a “reality” in which all possible outcomes exist

at the same time. m-Postulate I grounds us because it prevents direct observation of

1.3 Hilbert Space 5

Fig. 1.1 The

|11010+|01001

superposition state prior to

measurement (left panel). The

measured register value

11010 results in the collapse

of the amplitude into state

|11010. The normalization

constant is not shown in this

ﬁgure. The yellow bulbs are

in the on-state, denoted by the

binary digit 1

this “reality”. The theory as expressed by these postulates reminds us of the child’s

conundrum; is the moon there when we are not watching it? [5].

We have not yet addressed the question; which of the possible 32 conﬁgurations

do we observe after a measurement on a system described by |Φ? m-Postulate I

guarantees that one of the conﬁgurations is found but says nothing about which one.

To answer this question we look toward another postulate, sometimes called the

Born rule,afterMax Born one of the founders of the quantum theory. But ﬁrst we

need to provide additional structure to our linear vector space. That discussion leads

us to consider a special type of vector space called a Hilbert space.

1.3 Hilbert Space

We pointed out that if |α is a vector so is c |α where c is a scalar quantity. In

Hilbert space the scalar c is generally a complex number. Consider the following

linear combination of n vectors

|α

+c

|α

+...c

|α

. (1.4)

If this sum equates to the null vector 0 only if c

= c

··· = c

= 0 then the

set of vectors |α

, |α

,... |α

 are said to be linearly independent. A space that

admits n linear independent vectors, but not n+1, is called an n-dimensional space.

In general, a Hilbert space allows inﬁnite dimension [2, 4] but we are primarily

concerned, in this text, with Hilbert spaces spanned by a ﬁnite and denumerable set

of basis vectors. The Hilbert space of the ﬁve-qbulb system has dimension 32.

1.3.1 Dirac’s Bra-Ket Notation

The inner, or scalar product is a Hilbert space structure that provides a measure

of the degree of “overlap” between two vectors. The dot product of two vectors

in ordinary three-dimensional Euclidian space is an example of the latter. Here we

6 1 A Quantum Mechanic’s Toolbox

will be a little bit more abstract and introduce the bra-ket notation to deﬁne an

inner product. Indeed, we already made use of this notation in the previous sections

where we employed the symbol |... to denote a vector in Hilbert space. It is called

a ket and was introduced by the brilliant twentieth century physicist Paul Dirac,

an architect of the modern quantum theory. Throughout this text, we will employ

ket notation to describe quantum states. For example, ket |Ψ  is a vector in Hilbert

space where the symbol Ψ nested within the ket is an identiﬁcation parameter.

Having established ket notation, we introduce a new linear vector space called

dual space. For our purposes, it is convenient to think of dual space as a mirror

image of the vectors (or kets) in Hilbert space. For example, the ket |α has a mirror

counterpart in dual space. The label α should also parameterize it, but we cannot use

ket notation as this vector “lives” in a different linear vector space. Dirac suggested

the notation α| to denote it and called the symbol ...| a bra. Keeping in mind that

every ket |β has a corresponding bra β|, we require that the ket

|Ψ =c

|α

+c

|α

+...c

|α

 (1.5)

has a bra counterpart. The rule for generating a bra Ψ | from ket |Ψ  is

Ψ |=c

∗

α

|+c

∗

α

|,...c

∗

α

|. (1.6)

The expansion coefﬁcients for the bra vector are complex conjugates of the

corresponding coefﬁcients in ket space. Kets and bras are added to each of their

kind, so the following expression

|Ψ +Φ|

is meaningless, but we are allowed to build additional structures by certain

combinations of kets and bras. Two constructs which use kets and bras as its building

blocks are expressions that have the form

Φ|Ψ  (1.7)

and

|Ψ Φ|. (1.8)

The former (1.7), denotes an inner product and evaluates to a complex number. The

latter expression (1.8) is called an outer product, and it is neither a scalar or vector.

Later we show how it can be interpreted as an operator. Both structures enable

various transformations and manipulations of vectors in Hilbert space. Let’s ﬁrst

discuss the inner, or scalar, product. The scalar quantity deﬁned by expression (1.7)

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