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There are several models of quantum computing, including the quantum circuit model, quantum Turing machine, adiabatic quantum computer, one-way quantum computer, and various quantum cellular automata. The most widely used model is the quantum circuit. Quantum circuits are based on the quantum bit, or "qubit", which is somewhat analogous to the bit in classical computation. Qubits can be in a 1 or 0 quantum state, or they can be in a superposition of the 1 and 0 states. However, when qubits are measured the result of the measurement is always either a 0 or a 1; the probabilities of these two outcomes depend on the quantum state that the qubits were in immediately prior to the measurement. Computation is performed by manipulating qubits with quantum logic gates, which are somewhat analogous to classical logic gates.

There are several models of quantum computing, including the quantum circuit model, quantum Turing machine, adiabatic quantum computer, one-way quantum computer, and various quantum cellular automata. The most widely used model is the quantum circuit. Quantum circuits are based on the quantum bit, or "qubit", which is somewhat analogous to the bit in classical computation. Qubits can be in a 1 or 0 quantum state, or they can be in a superposition of the 1 and 0 states. However, when qubits are measured the result of the measurement is always either a 0 or a 1; the probabilities of these two outcomes depend on the quantum state that the qubits were in immediately prior to the measurement. Computation is performed by manipulating qubits with quantum logic gates, which are somewhat analogous to classical logic gates.

'''<font color="#ff8000"> 量子计算</font>'''有几种模型，包括'''<font color="#ff8000">量子电路模型、量子图灵机、绝热量子计算机、单向量子计算机和各种量子细胞自动机</font>'''。使用最广泛的模型是'''<font color="#ff8000"> 量子电路Quantum circuits </font>'''。量子电路是基于量子比特或'''<font color="#ff8000"> “量子位”"qubit"</font>'''的，它在某种程度上类似于经典计算中的'''<font color="#ff8000"> “比特”"bit"</font>'''。'''<font color="#ff8000"> 量子比特</font>'''可以处于1或0的量子态，也可以处于1和0的叠加态。然而，当量子比特被测量时，测量结果总是0或1; 这两种结果发生的概率取决于量子比特在被测量之前所处的量子状态。计算是通过'''<font color="#ff8000"> 靠量子逻辑门Quantum logic gates</font>'''操纵量子比特来完成的，这在某种程度上类似于经典逻辑门。

'''<font color="#ff8000"> 量子计算</font>'''有几种模型，包括'''<font color="#ff8000">量子电路模型、量子图灵机、绝热量子计算机、单向量子计算机和各种量子细胞自动机</font>'''。使用最广泛的模型是'''<font color="#ff8000"> 量子电路Quantum circuits </font>'''。量子电路是基于量子比特或'''<font color="#ff8000"> “量子位”"qubit"</font>'''的，它在某种程度上类似于经典计算中的'''<font color="#ff8000"> “比特”"bit"</font>'''。'''<font color="#ff8000"> 量子比特</font>'''可以处于1或0的量子态，也可以处于1和0的叠加态。然而，当<font color="#ff8000"> 量子比特</font>被测量时，测量结果总是0或1; 这两种结果发生的概率取决于量子比特在被测量之前所处的量子状态。计算是通过'''<font color="#ff8000"> 量子逻辑门Quantum logic gates</font>'''操纵量子比特来完成的，这在某种程度上类似于经典逻辑门。

Any computational problem that can be solved by a classical computer can also, in principle, be solved by a quantum computer. Conversely, quantum computers obey the Church–Turing thesis; that is, any computational problem that can be solved by a quantum computer can also be solved by a classical computer. While this means that quantum computers provide no additional advantages over classical computers in terms of computability, they do in theory enable the design of algorithms for certain problems that have significantly lower time complexities than known classical algorithms. Notably, quantum computers are believed to be able to quickly solve certain problems that no classical computer could solve in any feasible amount of time—a feat known as "quantum supremacy." The study of the computational complexity of problems with respect to quantum computers is known as quantum complexity theory.

Any computational problem that can be solved by a classical computer can also, in principle, be solved by a quantum computer. Conversely, quantum computers obey the Church–Turing thesis; that is, any computational problem that can be solved by a quantum computer can also be solved by a classical computer. While this means that quantum computers provide no additional advantages over classical computers in terms of computability, they do in theory enable the design of algorithms for certain problems that have significantly lower time complexities than known classical algorithms. Notably, quantum computers are believed to be able to quickly solve certain problems that no classical computer could solve in any feasible amount of time—a feat known as "quantum supremacy." The study of the computational complexity of problems with respect to quantum computers is known as quantum complexity theory.

任何可以由经典计算机解决的'''<font color="#ff8000"> 可计算问题</font>'''，原则上也可以由量子计算机解决。相反，量子计算机遵循 Church-Turing 理论; 也就是说，任何可以由量子计算机解决的计算问题也可以由经典计算机解决。虽然这意味着量子计算机在可计算性方面没有比传统计算机提供额外的优势，但在理论上，它们确实能够为某些问题设计算法，这些问题的时间复杂性明显低于已知的经典算法。值得注意的是，人们相信量子计算机能够快速解决某些问题，而这些问题是任何传统计算机都无法在可行的时间内解决的——这一壮举被称为'''<font color="#ff8000"> “量子优势”"quantum supremacy"</font>'''。研究量子计算机问题的计算复杂性被称为'''<font color="#ff8000"> 量子复杂性理论Quantum complexity theory</font>'''。

任何可以由经典计算机解决的'''<font color="#ff8000"> 计算问题</font>'''，原则上也可以由量子计算机解决。相反，量子计算机遵循 Church-Turing 理论; 也就是说，任何可以由量子计算机解决的计算问题也可以由经典计算机解决。虽然这意味着量子计算机在可计算性方面没有比传统计算机多提供额外的优势，但在理论上，它们确实能够为某些问题设计算法，这些算法的时间复杂性明显低于已知的经典算法。值得注意的是，人们相信量子计算机能够快速解决某些问题，而这些问题是任何传统计算机都无法在可行的时间内解决的——这一壮举被称为'''<font color="#ff8000"> “量子优势”"quantum supremacy"</font>'''。量子计算机问题的计算复杂性研究被称为'''<font color="#ff8000"> 量子复杂性理论Quantum complexity theory</font>'''。

The prevailing model of quantum computation describes the computation in terms of a network of quantum logic gates.

The prevailing model of quantum computation describes the computation in terms of a network of quantum logic gates.

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> 种可能的状态。因此，一个代表所有内存状态的向量具有 <math display="inline">2^n</math> 个条目(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——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> 种可能的状态。因此，一个代表所有内存状态的向量具有 <math display="inline">2^n</math> 个条目(每个状态一个)。这个向量被看作是一个概率向量，它表达这样一个事实——内存总是被发现处在一个特定的状态下。

In the classical view, one entry would have a value of 1 (i.e. a 100% probability of being in this state) and all other entries would be zero.   

In the classical view, one entry would have a value of 1 (i.e. a 100% probability of being in this state) and all other entries would be zero.   

在经典的观点中，一个条目的值为1(即。100% 的概率处于这种状态) ，所有其他条目都是零。

在经典的观点中，一个条目的值为1(即有100% 的概率处于这种状态) ，所有其他条目都是零。

In quantum mechanics, probability vectors are generalized to [[Density matrix|density operators]]. This is the technically rigorous [[Mathematical formulation of quantum mechanics|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 matrix|density operators]]. This is the technically rigorous [[Mathematical formulation of quantum mechanics|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.

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.

在量子力学中，概率向量被推广到'''<font color="#ff8000"> 密度算子Density operators</font>'''。这是技术上严格的'''<font color="#ff8000"> 量子逻辑门的数学基础</font>'''，但中间量子态向量形式通常首先介绍，因为它在概念上比较简单。为了简单起见，本文着重讨论量子态向量形式。

在<font color="#ff8000"> 量子力学</font>中，概率向量被推广到'''<font color="#ff8000"> 密度算子Density operators</font>'''。这是技术上严格的'''<font color="#ff8000"> 量子逻辑门的数学基础</font>'''，但中间量子态向量形式通常首先被介绍，因为它在概念上比较简单。为了简单起见，本文着重讨论量子态向量形式。

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'''<font color="#ff8000"> 狄拉克符号Dirac notation</font>'''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，因此

<math display="block">

<math display="block">

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'''<font color="#ff8000">量子态矢量</font>'''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>。

在数学上，这种逻辑门应用于'''<font color="#ff8000">量子态向量</font>'''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>。

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'''<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。

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 <font color="#ff8000"> 量子受控非门CNOT 门</font>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 潜在应用==

== Potential applications 潜在应用==

Integer factorization, which underpins the security of public key cryptographic systems, is believed to be computationally infeasible with an ordinary computer for large integers if they are the product of few prime numbers (e.g., products of two 300-digit primes). By comparison, a quantum computer could efficiently solve this problem using Shor's algorithm to find its factors. This ability would allow a quantum computer to break many of the cryptographic systems in use today, in the sense that there would be a polynomial time (in the number of digits of the integer) algorithm for solving the problem. In particular, most of the popular public key ciphers are based on the difficulty of factoring integers or the discrete logarithm problem, both of which can be solved by Shor's algorithm. In particular, the RSA, Diffie–Hellman, and elliptic curve Diffie–Hellman algorithms could be broken. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic privacy and security.

Integer factorization, which underpins the security of public key cryptographic systems, is believed to be computationally infeasible with an ordinary computer for large integers if they are the product of few prime numbers (e.g., products of two 300-digit primes). By comparison, a quantum computer could efficiently solve this problem using Shor's algorithm to find its factors. This ability would allow a quantum computer to break many of the cryptographic systems in use today, in the sense that there would be a polynomial time (in the number of digits of the integer) algorithm for solving the problem. In particular, most of the popular public key ciphers are based on the difficulty of factoring integers or the discrete logarithm problem, both of which can be solved by Shor's algorithm. In particular, the RSA, Diffie–Hellman, and elliptic curve Diffie–Hellman algorithms could be broken. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic privacy and security.

'''<font color="#ff8000"> 整数因式分解Integer factorization</font>'''是'''<font color="#ff8000"> 公钥密码系统Public key cryptographic systems</font>'''安全性的基础，如果大整数是几个素数的乘积（例如，两个300位素数的乘积），那么在普通计算机上计算大整数是不可行的。相比之下，量子计算机可以有效地解决这个问题，使用'''<font color="#ff8000"> 肖尔Shor算法</font>'''来寻找它的因子。这种能力将使量子计算机能够破解目前使用的许多密码系统，在这个意义上，将有一个多项式时间（整数位数）算法来解决这个问题。特别是目前流行的公钥密码算法大多是基于整数的因式分解困难或离散对数问题，这两个问题都可以用'''<font color="#ff8000"> 肖尔Shor算法</font>'''来解决。特别是'''<font color="#ff8000"> RSA、Diffie-Hellman和椭圆曲线Diffie-Hellman算法</font>'''可能会被打破。它们用于保护安全网页、加密电子邮件和许多其他类型的数据。破坏这些将对电子隐私和安全产生重大影响。

'''<font color="#ff8000"> 整数因式分解Integer factorization</font>'''是'''<font color="#ff8000"> 公钥密码系统Public key cryptographic systems</font>'''安全性的基础，如果大的整数是几个素数的乘积（例如，两个300位素数的乘积），那么在普通计算机上计算大整数是做不到的。相比之下，量子计算机可以有效地解决这个问题，使用'''<font color="#ff8000"> 肖尔Shor算法</font>'''来寻找它的因子。这种能力将使量子计算机能够破解目前使用的许多密码系统，在这个意义上，将有一个<font color="#ff8000">多项式时间（整数位数）算法</font>来解决这个问题。特别是目前流行的公钥密码算法大多是基于整数的因式分解困难或离散对数问题，这两个问题都可以用'''<font color="#ff8000"> 肖尔Shor算法</font>'''来解决。特别是'''<font color="#ff8000"> RSA、Diffie-Hellman和椭圆曲线Diffie-Hellman算法</font>'''可能会被打破。它们用于保护安全网页、加密电子邮件和许多其他类型的数据。破坏这些将对电子隐私和安全产生重大影响。

However, other cryptographic algorithms do not appear to be broken by those algorithms. Some public-key algorithms are based on problems other than the integer factorization and discrete logarithm problems to which Shor's algorithm applies, like the McEliece cryptosystem based on a problem in coding theory. Lattice-based cryptosystems are also not known to be broken by quantum computers, and finding a polynomial time algorithm for solving the dihedral hidden subgroup problem, which would break many lattice based cryptosystems, is a well-studied open problem. It has been proven that applying Grover's algorithm to break a symmetric (secret key) algorithm by brute force requires time equal to roughly 2^n/2 invocations of the underlying cryptographic algorithm, compared with roughly 2ⁿ in the classical case, meaning that symmetric key lengths are effectively halved: AES-256 would have the same security against an attack using Grover's algorithm that AES-128 has against classical brute-force search (see Key size).

However, other cryptographic algorithms do not appear to be broken by those algorithms. Some public-key algorithms are based on problems other than the integer factorization and discrete logarithm problems to which Shor's algorithm applies, like the McEliece cryptosystem based on a problem in coding theory. Lattice-based cryptosystems are also not known to be broken by quantum computers, and finding a polynomial time algorithm for solving the dihedral hidden subgroup problem, which would break many lattice based cryptosystems, is a well-studied open problem. It has been proven that applying Grover's algorithm to break a symmetric (secret key) algorithm by brute force requires time equal to roughly 2^n/2 invocations of the underlying cryptographic algorithm, compared with roughly 2ⁿ in the classical case, meaning that symmetric key lengths are effectively halved: AES-256 would have the same security against an attack using Grover's algorithm that AES-128 has against classical brute-force search (see Key size).

然而，其他密码算法似乎并没有被这些算法破解。有些公钥算法是基于除整数分解和离散对数问题以外的问题，'''<font color="#ff8000"> 肖尔Shor算法</font>'''适用于这些问题，例如McEliece密码体制基于编码理论中的一个问题。基于格的密码体制也不被量子计算机破解，寻找一个多项式时间算法来解决二面体隐子群问题，这将打破许多基于格的密码体制，是一个研究得很好的开放性问题。已经证明，应用'''<font color="#ff8000"> Grover算法</font>'''以暴力破解对称（密钥）算法所需的时间大约相当于基础加密算法的2次调用^n/2，而在经典情况下大约需要2ⁿ，这意味着对称密钥长度有效地减半：AES-256对于使用'''<font color="#ff8000"> Grover算法</font>'''的攻击的安全性与AES-128针对经典暴力搜索的安全性相同（参见密钥大小）。

然而，其他密码算法似乎并没有被这些算法破解。有些公钥算法是基于除整数分解和离散对数问题以外的问题，'''<font color="#ff8000"> 肖尔Shor算法</font>'''适用于这些问题，例如McEliece密码体制基于编码理论中的一个问题。基于格的密码体制也不能被量子计算机破解，寻找一个多项式时间算法来解决<font color="#ff8000"> 二面体隐子群问题Dihedral hidden subgroup problem</font>，这将打破许多基于<font color="#ff8000"> 格</font>的密码体制，是一个研究得很好的开放性问题。已经证明，应用'''<font color="#ff8000"> Grover算法</font>'''以暴力破解对称（密钥）算法所需的时间大约相当于基础加密算法的2次调用^n/2，而在经典情况下大约需要2ⁿ，这意味着对称密钥长度有效地减半：AES-256对于使用'''<font color="#ff8000"> Grover算法</font>'''的攻击的安全性与AES-128针对经典暴力搜索的安全性相同（参见密钥大小）。