更改

== Quantum operations ==

== Quantum operations量子计算 ==

[[File:Bloch Sphere.svg|thumb|The [[Bloch sphere]] is a representation of a [[qubit]], the fundamental building block of quantum computers.]]

[[File:Bloch Sphere.svg|thumb|The [[Bloch sphere]] is a representation of a [[qubit]], the fundamental building block of quantum computers.]]

The [[Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers.]]

The [[Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers.]]

布洛赫球体是量子计算机的基本构件——量子比特的表示。]

[[布洛赫球体是量子计算机的基本构件——量子比特的表示。]]

A memory consisting of <math display="inline">n</math> bits of information has <math display="inline">2^n</math> possible states. A vector representing all memory states thus has <math display="inline">2^n</math> entries (one for each state). This vector is viewed as a probability vector and represents the fact that the memory is to be found in a particular state.

A memory consisting of <math display="inline">n</math> bits of information has <math display="inline">2^n</math> possible states. A vector representing all memory states thus has <math display="inline">2^n</math> entries (one for each state). This vector is viewed as a probability vector and represents the fact that the memory is to be found in a particular state.

一个由 < math display = " inline" > n </math > 比特信息组成的内存有 < math display = " inline" > 2 ^ n </math > 可能的状态。因此，一个代表所有内存状态的向量具有 < math display = " inline" > 2 ^ n </math > 条目(每个状态一个)。这个向量被看作是一个概率向量，它代表了一个事实，即内存将在一个特定的状态下被找到。

一个由<math display="inline">n</math> 比特信息组成的内存有 <math display="inline">2^n</math> 可能的状态。因此，一个代表所有内存状态的向量具有 <math display="inline">2^n</math> 条目(每个状态一个)。这个向量被看作是一个概率向量，它代表了一个事实，即内存将在一个特定的状态下被找到。

In quantum mechanics, probability vectors are generalized to density operators. This is the technically rigorous mathematical foundation for quantum logic gates, but the intermediate quantum state vector formalism is usually introduced first because it is conceptually simpler. This article focuses on the quantum state vector formalism for simplicity.

In quantum mechanics, probability vectors are generalized to density operators. This is the technically rigorous mathematical foundation for quantum logic gates, but the intermediate quantum state vector formalism is usually introduced first because it is conceptually simpler. This article focuses on the quantum state vector formalism for simplicity.

We begin by considering a simple memory consisting of only one bit. This memory may be found in one of two states: the zero state or the one state. We may represent the state of this memory using Dirac notation so that

We begin by considering a simple memory consisting of only one bit. This memory may be found in one of two states: the zero state or the one state. We may represent the state of this memory using Dirac notation so that

我们首先考虑一个只包含一个位的简单内存。这种记忆可以在两种状态中的一种中找到: 零状态或一种状态。我们可以用狄拉克符号来表示这段记忆的状态

我们首先考虑一个只包含一个位的简单内存。这种记忆可以在两种状态中的一种中找到: 零状态或一种状态。我们可以用'''<font color="#ff8000"> 狄拉克符号Dirac notation</font>'''来表示这段记忆的状态

<math display="block">

<math display="block">

In general, the coefficients <math display="inline">\alpha</math> and <math display="inline">\beta</math> are complex numbers. In this scenario, one qubit of information is said to be encoded into the quantum memory. The state <math display="inline">|\psi\rangle</math> is not itself a probability vector but can be connected with a probability vector via a measurement operation. If the quantum memory is measured to determine if the state is <math display="inline">|0\rangle</math> or <math display="inline">|1\rangle</math> (this is known as a computational basis measurement), the zero state would be observed with probability <math display="inline">|\alpha|^2</math> and the one state with probability <math display="inline">|\beta|^2</math>. The numbers <math display="inline">\alpha</math> and <math display="inline">\beta</math> are called quantum amplitudes.

In general, the coefficients <math display="inline">\alpha</math> and <math display="inline">\beta</math> are complex numbers. In this scenario, one qubit of information is said to be encoded into the quantum memory. The state <math display="inline">|\psi\rangle</math> is not itself a probability vector but can be connected with a probability vector via a measurement operation. If the quantum memory is measured to determine if the state is <math display="inline">|0\rangle</math> or <math display="inline">|1\rangle</math> (this is known as a computational basis measurement), the zero state would be observed with probability <math display="inline">|\alpha|^2</math> and the one state with probability <math display="inline">|\beta|^2</math>. The numbers <math display="inline">\alpha</math> and <math display="inline">\beta</math> are called quantum amplitudes.

一般来说，系数 < math display = " inline" > alpha </math > 和 < math display = " inline" > beta </math > 都是复数。在这种情况下，一个量子位的信息被称为被编码到量子存储器中。状态“ inline” > | psi rangle </math > 本身不是一个概率向量，但可以通过一个测量操作与一个概率向量相连。如果量子内存被测量以确定其状态是否为 < math display = " inline" > | 0 rangle </math > 或 < math display = " inline" > | 1 rangle </math > (这被称为计算基础测量) ，那么零状态将被观察到概率 < math display = " " | | ^ 2 </math > 和概率 < math display = " " > beta | ^ 2 inline" </math > 。数字 < math display = " inline" > alpha </math > 和 < math display = " inline" > beta </math > 被称为量子幅值。

一般来说，系数 <math display="inline">\alpha</math> 和 <math display="inline">\beta</math>都是复数。在这种情况下，一个量子比特的信息被称为被编码到量子存储器中。状态<math display="inline">|\psi\rangle</math>本身不是一个概率向量，但可以通过一个测量操作与一个概率向量相连。如果量子内存被测量以确定其状态是否为 <math display="inline">|0\rangle</math> 或<math display="inline">|1\rangle</math>(这被称为计算基础测量) ，那么零状态将被观察到概率 <math display="inline">|\alpha|^2</math>和概率 <math display="inline">|\beta|^2</math> 。数字 <math display="inline">\alpha</math> 和 <math display="inline">\beta</math>被称为量子幅值。

The state of this one-qubit quantum memory can be manipulated by applying quantum logic gates, analogous to how classical memory can be manipulated with classical logic gates. One important gate for both classical and quantum computation is the NOT gate, which can be represented by a matrix

The state of this one-qubit quantum memory can be manipulated by applying quantum logic gates, analogous to how classical memory can be manipulated with classical logic gates. One important gate for both classical and quantum computation is the NOT gate, which can be represented by a matrix

<math display="block">X := \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.</math>

<math display="block">X := \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}.</math>

Mathematically, the application of such a logic gate to a quantum state vector is modelled with matrix multiplication. Thus <math display="inline">X|0\rangle = |1\rangle</math> and <math display="inline">X|1\rangle = |0\rangle</math>.

Mathematically, the application of such a logic gate to a quantum state vector is modelled with matrix multiplication. Thus <math display="inline">X|0\rangle = |1\rangle</math> and <math display="inline">X|1\rangle = |0\rangle</math>.

在数学上，这种逻辑门应用于量子态矢量是用矩阵乘法模型来建模的。因此 < math display = " inline" > x | 0 rangle = | 1 rangle </math > and < math display = " inline" > x | 1 rangle = | 0 rangle </math > 。

在数学上，这种逻辑门应用于量子态矢量是用矩阵乘法模型来建模的。因此 <math display="inline">X|0\rangle = |1\rangle</math> 和 <math display="inline">X|1\rangle = |0\rangle</math>。

The CNOT gate can then be represented using the following matrix:

The CNOT gate can then be represented using the following matrix:

然后，CNOT 门可以用以下矩阵表示:

然后，'''<font color="#ff8000"> 量子受控非门CNOT gate</font>'''可以用以下矩阵表示:

<math display="block">

<math display="block">

As a mathematical consequence of this definition, <math display="inline">CNOT|00\rangle = |00\rangle</math>, <math display="inline">CNOT|01\rangle = |01\rangle</math>, <math display="inline">CNOT|10\rangle = |11\rangle</math>, and <math display="inline">CNOT|11\rangle = |10\rangle</math>. In other words, the CNOT applies a NOT gate (<math display="inline">X</math> from before) to the second qubit if and only if the first qubit is in the state <math display="inline">|1\rangle</math>. If the first qubit is <math display="inline">|0\rangle</math>, nothing is done to either qubit.

As a mathematical consequence of this definition, <math display="inline">CNOT|00\rangle = |00\rangle</math>, <math display="inline">CNOT|01\rangle = |01\rangle</math>, <math display="inline">CNOT|10\rangle = |11\rangle</math>, and <math display="inline">CNOT|11\rangle = |10\rangle</math>. In other words, the CNOT applies a NOT gate (<math display="inline">X</math> from before) to the second qubit if and only if the first qubit is in the state <math display="inline">|1\rangle</math>. If the first qubit is <math display="inline">|0\rangle</math>, nothing is done to either qubit.

作为这个定义的数学推论，< math display = " inline" > CNOT | 00 rangle = | 00 rangle </math > ，< math display = " inline" > CNOT | 01 rangle = | 01 rangle </math > ，< math display = " " > CNOT | 10 rangle = | 11 rangle </math > ，和 math < display = " " > CNOT | 11 rangle = | 10 inline </math > 。换句话说，当且仅当第一个量子位处于状态 < math display = " inline" > | 1 rangle </math > 时，CNOT 对第二个量子位应用 NOT 门(< math display = " inline" > x </math >)。如果第一个量子位是 < math display = " inline" > | 0 rangle </math > ，那么任何一个量子位都不会被处理。

作为这个定义的数学推论，<math display="inline">CNOT|00\rangle = |00\rangle</math>, <math display="inline">CNOT|01\rangle = |01\rangle</math>, <math display="inline">CNOT|10\rangle = |11\rangle</math>, 和<math display="inline">CNOT|11\rangle = |10\rangle</math>。换句话说，当且仅当第一个量子位处于状态 <math display="inline">|1\rangle</math> 时，CNOT 对第二个量子位应用 NOT 门(<math display="inline">X</math>)。如果第一个量子位是 <math display="inline">|0\rangle</math>，则对任何一个量子位都不做处理。

In summary, a quantum computation can be described as a network of quantum logic gates and measurements. Any measurement can be deferred to the end of a quantum computation, though this deferment may come at a computational cost. Because of this possibility of deferring a measurement, most quantum circuits depict a network consisting only of quantum logic gates and no measurements. More information can be found in the following articles: universal quantum computer, Shor's algorithm, Grover's algorithm, Deutsch–Jozsa algorithm, amplitude amplification, quantum Fourier transform, quantum gate, quantum adiabatic algorithm and quantum error correction.

In summary, a quantum computation can be described as a network of quantum logic gates and measurements. Any measurement can be deferred to the end of a quantum computation, though this deferment may come at a computational cost. Because of this possibility of deferring a measurement, most quantum circuits depict a network consisting only of quantum logic gates and no measurements. More information can be found in the following articles: universal quantum computer, Shor's algorithm, Grover's algorithm, Deutsch–Jozsa algorithm, amplitude amplification, quantum Fourier transform, quantum gate, quantum adiabatic algorithm and quantum error correction.

总之，量子计算可以描述为一个由量子逻辑门和测量组成的网络。任何测量都可以推迟到量子计算结束时进行，尽管这种推迟可能会带来计算成本。由于这种延迟测量的可能性，大多数量子电路描述的网络只有量子逻辑门而没有测量。更多信息可以在以下文章中找到: 通用量子计算机，Shor 算法，Grover 算法，Deutsch-Jozsa 算法，振幅放大，量子傅里叶变换，量子门，量子绝热算法和量子误差修正。

总之，'''<font color="#ff8000"> 量子计算</font>'''可以描述为一个由量子逻辑门和测量组成的网络。任何测量都可以推迟到'''<font color="#ff8000"> 量子计算</font>'''结束时进行，尽管这种推迟可能会带来计算成本。由于这种延迟测量的可能性，大多数量子电路描述的网络只有量子逻辑门而没有测量。更多信息可以在以下文章中找到: '''<font color="#ff8000"> 通用量子计算机，Shor 算法，Grover 算法，Deutsch-Jozsa 算法，振幅放大，量子傅里叶变换Quantum Fourier transform，量子门，量子绝热算法和量子误差修正Quantum error correction</font>'''。

Any quantum computation can be represented as a network of quantum logic gates from a fairly small family of gates. A choice of gate family that enables this construction is known as a universal gate set. One common such set includes all single-qubit gates as well as the CNOT gate from above. This means any quantum computation can be performed by executing a sequence of single-qubit gates together with CNOT gates. Though this gate set is infinite, it can be replaced with a finite gate set by appealing to the Solovay-Kitaev theorem. The representation of multiple qubits can be shown as Qsphere.  

Any quantum computation can be represented as a network of quantum logic gates from a fairly small family of gates. A choice of gate family that enables this construction is known as a universal gate set. One common such set includes all single-qubit gates as well as the CNOT gate from above. This means any quantum computation can be performed by executing a sequence of single-qubit gates together with CNOT gates. Though this gate set is infinite, it can be replaced with a finite gate set by appealing to the Solovay-Kitaev theorem. The representation of multiple qubits can be shown as Qsphere.  

任何量子计算都可以表示为一个量子逻辑门网络，它来自一个相当小的量子门家族。使这种结构成为可能的门系列的选择被称为通用门系列。一个常见的这样的集合包括所有的单量子比特门以及上面的 CNOT 门。这意味着任何量子计算都可以通过执行一系列带有 CNOT 门的单量子比特门来完成。虽然这个门集合是无限的，但是它可以通过引用 Solovay-Kitaev 定理用一个有限的门集合来代替。多个量子位的表示可以用 Qsphere 来表示。

任何'''<font color="#ff8000"> 量子计算</font>'''都可以表示为一个量子逻辑门网络，它来自一个相当小的量子门家族。使这种结构成为可能的门系列的选择被称为通用门系列。一个常见的这样的集合包括所有的单量子比特门以及上面的 量子受控非门CNOT 门。这意味着任何量子计算都可以通过执行一系列带有 量子受控非门CNOT 门的单量子比特门来完成。虽然这个门集合是无限的，但是它可以通过引用 Solovay-Kitaev 定理用一个有限的门集合来代替。多个量子位的表示可以用 Qsphere 来表示。

 

 

== Potential applications ==

== Potential applications ==