量子计算

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唐糖糖（讨论 | 贡献）2021年8月1日 (日) 12:49的版本
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量子计算 Quantum computing是利用量子现象(如叠加和纠缠Superposition and Entanglement)来执行计算。执行量子计算的计算机被称为量子计算机。[1]量子计算机能够从根本上比传统计算机更快地解决比如整数分解(RSA 加密的基础)这类特定的计算问题。量子计算是量子信息科学Quantum information science的一个分支。


< ! -- 历史 -- >

量子计算始于20世纪80年代早期，当时物理学家保罗 · 贝尼奥夫Paul Benioff提出了图灵机Turing machine的量子力学模型。[2]理查德 · 费曼Richard Feynman和尤里 · 曼宁Yuri Manin后来提出，量子计算机有潜力去模拟传统计算机所无法模拟的东西。[3][4]1994年，Peter Shor 开发了一种量子算法，用于分解整数，这种算法有可能解密 rsa 加密的通信。[5]尽管自20世纪90年代后期以来，实验取得了进展，但大多数研究人员认为，“容错量子计算机仍然是一个相当遥远的梦想。”[6]近年来，量子计算研究的投资在公共和私营部门都有所增加。[7][8]2019年10月23日，谷歌AI与美国宇航局U.S. National Aeronautics and Space Administration (NASA)合作，声称已经完成了在任何传统计算机上都不可能完成的量子计算。[9]


< ! -- 量子计算的基本原理 -- >


量子计算有几种模型，包括量子电路模型、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。使用最广泛的模型是量子电路Quantum circuits 。量子电路是基于量子比特或“量子位”"qubit"的，它在某种程度上类似于经典计算中的“比特”"bit"。量子比特可以处于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。计算是通过量子逻辑门Quantum logic gates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。


< ! -- 物理实现 -- >

目前实现量子计算机主要有两种方法: 模拟和数字。模拟方法进一步分为量子模拟、量子退火模拟和绝热量子计算。数字量子计算机使用量子逻辑门进行计算。两种方法都使用量子比特。[1]:2–13 There are currently a number of significant obstacles in the way of constructing useful quantum computers. In particular, it is difficult to maintain the quantum states of qubits as they are prone to quantum decoherence, and quantum computers require significant error correction as they are far more prone to errors than classical computers.[10][11]


< ! -- 可计算性和复杂性 -- >

任何可以由经典计算机解决的计算问题，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“量子优势”"quantum supremacy"。量子计算机问题的计算复杂性研究被称为量子复杂性理论Quantum complexity theory。


Quantum operations量子计算

布洛赫球体是量子计算机的基本构件——量子比特的表示模型。


当前流行的量子计算模型用量子逻辑门网络来描述计算。


一个由[math]\displaystyle{ n }[/math] 比特信息组成的内存有 [math]\displaystyle{ 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]\displaystyle{ 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，它代表这样一个事实——内存总是在某个特定的状态下被访问。


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。


在量子力学中，概率向量被推广到密度算子Density operators。它是技术上严格的量子逻辑门的数学基础，但介绍的时候通常首先引入中间量子态的向量形式，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。


我们首先考虑一个只有1位的简单内存。这种内存只有0或1两种状态。我们可以用狄拉克符号Dirac notation来表示这段内存的状态，因此

[math]\displaystyle{ |0\rangle := \begin{pmatrix} 1 \\ 0 \end{pmatrix};\quad |1\rangle := \begin{pmatrix} 0 \\ 1 \end{pmatrix} }[/math]


一个量子内存可能处在两种经典状态的量子叠加态中的任意一种状态:

[math]\displaystyle{ |\psi\rangle := \alpha\,|0\rangle + \beta\,|1\rangle |\psi\rangle := \alpha\,|0\rangle + \beta\,|1\rangle | psi rangle: = alpha，| 0 rangle + beta，| 1 rangle = \begin{pmatrix} \alpha \\ \beta \end{pmatrix};\quad = \begin{pmatrix} \alpha \\ \beta \end{pmatrix};\quad 2.1.2.2.2.2.3.3.3.3.3.3.3.3.3.3.3.3.3.4.3.3.3.4.3.3.3.3.3.3.3.3.3.4.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3.3 |\alpha|^2 + |\beta|^2 = 1. |\alpha|^2 + |\beta|^2 = 1. | alpha | ^ 2 + | beta | ^ 2 = 1. }[/math]


一般来说，系数 [math]\displaystyle{ \alpha }[/math] 和 [math]\displaystyle{ \beta }[/math]都是复数。在这种情况下，信息的一个量子比特被编码到量子内存中。状态[math]\displaystyle{ |\psi\rangle }[/math]本身不是一个概率向量，但可以通过测量操作与概率向量相连。如果量子内存被测量以确定其状态是 [math]\displaystyle{ |0\rangle }[/math] 还是[math]\displaystyle{ |1\rangle }[/math](这被称为计算基础测量) ，那么0状态将以概率 [math]\displaystyle{ |\alpha|^2 }[/math]被观测到，而1状态将以概率 [math]\displaystyle{ |\beta|^2 }[/math]被观测到。数字 [math]\displaystyle{ \alpha }[/math] 和 [math]\displaystyle{ \beta }[/math]被称为量子幅值Quantum amplitudes。


这种单比特量子存储器的状态可以通过量子逻辑门来控制，类似于用经典逻辑门来控制经典存储器。对经典和量子计算都很重要的门是非门NOT gate，它可以用矩阵表示

[math]\displaystyle{ X := \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}. }[/math]

[math]\displaystyle{ X := \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}. }[/math]

数学显示 = “ block” > x: = begin { pmatrix }0 & 11 & 0 end { pmatrix }


在数学上，逻辑门作用于量子态向量可以建模成矩阵乘法。因此 [math]\displaystyle{ X|0\rangle = |1\rangle }[/math] 和 [math]\displaystyle{ X|1\rangle = |0\rangle }[/math]。


单个量子比特门的数学可以通过两种重要的方式扩展到对多量子比特量子存储器的操作。一种方法是简单地选择一个量子位并将该门应用于目标量子位，同时不影响其余的内存。另一种方法是，只有当内存的另一部分处于被需要状态时，才将门应用于目标量子位。这两种选择可以用另一个例子来说明。两比特量子存储器的可能状态包括

[math]\displaystyle{ |00\rangle := \begin{pmatrix} 1 \\ 0 \\ 0 \\ 0 \end{pmatrix};\quad 1000 end { pmatrix } ; quad |01\rangle := \begin{pmatrix} 0 \\ 1 \\ 0 \\ 0 \end{pmatrix};\quad |10\rangle := \begin{pmatrix} 0 \\ 0 \\ 1 \\ 0 \end{pmatrix};\quad |11\rangle := \begin{pmatrix} 0 \\ 0 \\ 0 \\ 1 \end{pmatrix}. }[/math]


然后，量子受控非门CNOT gate可以用以下矩阵表示:

[math]\displaystyle{ CNOT := CNOT := = \begin{pmatrix} \begin{pmatrix} 开始{ pmatrix } 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 0 & 0 & 1 & 0 0 & 0 & 1 & 0 \end{pmatrix}. \end{pmatrix}. 结束{ pmatrix }。 }[/math]

作为这个定义的数学推论，[math]\displaystyle{ CNOT|00\rangle = |00\rangle }[/math], [math]\displaystyle{ CNOT|01\rangle = |01\rangle }[/math], [math]\displaystyle{ CNOT|10\rangle = |11\rangle }[/math], 和[math]\displaystyle{ 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]\displaystyle{ |1\rangle }[/math] 时，CNOT 才对第二个量子位应用 非门([math]\displaystyle{ X }[/math])。如果第一个量子位是 [math]\displaystyle{ |0\rangle }[/math]，则对任何一个量子位都不做处理。


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。更多信息可以参考以下文章: 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。


任何量子计算都可以表示为一个量子逻辑门网络，量子逻辑门是门中的一个小类。使这种结构成为可能的一类门的被称为通用门集合。常见的这种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 定理被一个有限的门集合来代替。多个量子位可以用 Qsphere 来表示。

Potential applications 潜在应用

Cryptography 密码学

Main article: Quantum cryptography


整数因式分解Integer factorization是公钥密码系统Public key cryptographic systems安全性的基础，如果一个大整数是几个素数的乘积（例如，两个300位素数的乘积）[12]，那么在普通计算机上计算是不可行的。相比之下，量子计算机可以有效地解决这个问题，使用肖尔Shor算法来寻找它的因子。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，也就是说，可以用多项式时间（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大整数因式分解或离散对数问题的困难性，而这两个问题都可以用肖尔Shor算法来解决。尤其是RSA、Diffie-Hellman和椭圆曲线Diffie-Hellman算法可能会被破解，它们一般用于保护安全网页、加密电子邮件和许多其他类型的数据。破解这些算法将对电子隐私和安全产生重大影响。



然而，其他密码算法似乎并没有被那些算法破解。[13][14]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，肖尔Shor算法并不适用于这些问题，例如McEliece密码体制基于编码理论中的一个问题。[13][15]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二面体隐子群问题Dihedral hidden subgroup problem，将打破许多基于格的密码体制，这是一个充分研究的开放性问题。[16]已经证明，用Grover算法来暴力破解对称（密钥）算法所需的时间大约相当于基础加密算法的2n/2次调用，而在经典情况下大约需要2n，[17]这意味着对称密钥长度将有效地减半：AES-256应对使用Grover算法的攻击的安全性与AES-128应对经典暴力搜索的安全性相同（参见密钥大小）。


量子密码学Quantum cryptography可以实现公开密钥加密的一些功能。因此，面对量子黑客攻击时，基于量子的加密系统可能比传统系统更安全。[18]


Quantum search 量子搜索

Besides factorization and discrete logarithms, quantum algorithms offering a more than polynomial speedup over the best known classical algorithm have been found for several problems, including the simulation of quantum physical processes from chemistry and solid state physics, the approximation of Jones polynomials, and solving Pell's equation. No mathematical proof has been found that shows that an equally fast classical algorithm cannot be discovered, although this is considered unlikely.引用错误：没有找到与</ref>对应的<ref>标签其中包括化学和固态物理方面的量子物理过程仿真，琼斯多项式Jones polynomials的近似，以及佩尔方程Pell's equation的求解。目前还没有从数学上证明同样快速的经典算法无法被发现，尽管这被认为是不太可能的。然而，量子计算机为某些问题提供了多项式加速。最著名的例子是量子数据库搜索，它可以通过格罗夫Grover算法 来解决，比经典算法所需的数据库查询次数少二次方。在这种情况下，这种优势不仅是可证明的，而且是最优的，已经证明Grover的算法为任何数量的oracle查找提供了找到所需元素的最大可能概率。随后又发现了其他一些为查询问题进行可证明的量子加速的例子，例如在两对一函数中寻找碰撞和评估NAND树。


|title=Quantum Computers|isbn=9781439243497

|date=2009-06-19

Problems that can be addressed with Grover's algorithm have the following properties:[citation needed]

可以通过格罗夫 Grover算法解决的问题有以下属性:

}}模板:Self-published inline</ref> However, quantum computers offer polynomial speedup for some problems. The most well-known example of this is quantum database search, which can be solved by Grover's algorithm using quadratically fewer queries to the database than that are required by classical algorithms. In this case, the advantage is not only provable but also optimal, it has been shown that Grover's algorithm gives the maximal possible probability of finding the desired element for any number of oracle lookups. Several other examples of provable quantum speedups for query problems have subsequently been discovered, such as for finding collisions in two-to-one functions and evaluating NAND trees.[citation needed]

}}模板:Self-published inline</ref>然而，量子计算机为某些问题提供了多项式加速。这方面最著名的例子是“量子数据库搜索”，它可以通过格罗夫算法使用比经典算法要求的查询次数少二次方的查询次数来解决。在这种情况下，这种优势不仅是可证明的，而且是最优的，已经证明Grover的算法为任何数量的oracle查询提供了找到所需元素的最大可能概率。随后又发现了其他几个可证明的量子加速的例子，例如在两对一函数中寻找碰撞和评估NAND树。[citation needed]

在可能的答案的集合中没有可搜索的结构,

要检查的可能答案的数量与算法的输入数量相同，以及


存在一个布尔函数，它评估每个输入，并确定它是否是正确的答案

  1. 在可能的答案集合中没有可搜索的结构，
  2. 要检查的可能答案的数量与算法的输入数量相同，并且

对于具有以上所有性质的问题，Grover算法在量子计算机上的运行时间将按输入（或数据库中元素）数量的平方根来缩放，而不是经典算法的线性缩放。Grover算法可以应用的一类问题是布尔可满足性问题。在此例中，算法迭代使用的数据库是所有可能答案的数据库。这方面的作为例子的（也是可能的）应用是一个密码破解程序，它试图猜测加密文件或系统的密码或密钥。对称密码如三重DES和AES特别容易受到这种攻击。量子计算的这一应用是政府机构主要感兴趣的。

  1. 存在一个布尔函数，它计算每个输入并确定它是否是正确的答案


For problems with all these properties, the running time of Grover's algorithm on a quantum computer will scale as the square root of the number of inputs (or elements in the database), as opposed to the linear scaling of classical algorithms. A general class of problems to which Grover's algorithm can be applied is Boolean satisfiability problem. In this instance, the database through which the algorithm is iterating is that of all possible answers. An example (and possible) application of this is a password cracker that attempts to guess the password or secret key for an encrypted file or system. Symmetric ciphers such as Triple DES and AES are particularly vulnerable to this kind of attack.[citation needed] This application of quantum computing is a major interest of government agencies.[19]

对于具有所有这些性质的问题，Grover算法在量子计算机上的运行时间将按输入（或数据库中元素）数量的平方根进行缩放，而不是经典算法的线性缩放。一类可以应用Grover算法的一般问题是[20]布尔可满足性问题。在本例中，算法迭代使用的“数据库”是所有可能答案的数据库。这方面的一个例子（也是可能的）应用是一个密码破解器试图猜测加密文件或系统的密码或密钥。对称密码例如Triple DES和 AES特别容易受到此类攻击。{{引文需要{日期=2019年11月}}量子计算的这一应用是政府机构主要感兴趣的。


Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, many believe quantum simulation will be one of the most important applications of quantum computing. Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider.

由于化学和纳米技术依赖于对量子系统的理解，而这样的系统是不可能以有效的经典方式进行模拟的，许多人相信量子模拟将是量子计算最重要的应用之一。量子模拟也可以用来模拟原子和粒子在非正常条件下的行为，比如对撞机内部的反应。

Quantum simulation 量子模拟

Main article: Quantum simulator

Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, many believe quantum simulation will be one of the most important applications of quantum computing.

由于化学和纳米技术依赖于对量子系统的理解，而这种系统不可能以经典的方式进行有效的模拟，许多人相信量子模拟将是量子计算最重要的应用之一。量子模拟也可以用来模拟原子和粒子在异常条件下的行为，例如对撞机内部的反应。[21] Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider.[22]


Quantum annealing or Adiabatic quantum computation relies on the adiabatic theorem to undertake calculations. A system is placed in the ground state for a simple Hamiltonian, which is slowly evolved to a more complicated Hamiltonian whose ground state represents the solution to the problem in question. The adiabatic theorem states that if the evolution is slow enough the system will stay in its ground state at all times through the process.

量子退火或绝热量子计算依赖于绝热定理进行计算。在一个简单的哈密顿体系中，系统处于基态，这个哈密顿体系慢慢演化成一个更复杂的哈密顿体系，它的基态代表问题的解。绝热定理指出，如果演化足够慢，系统在整个演化过程中将始终处于基态。

Quantum annealing and adiabatic optimization量子退火与绝热优化

Quantum annealing or Adiabatic quantum computation relies on the adiabatic theorem to undertake calculations. A system is placed in the ground state for a simple Hamiltonian, which is slowly evolved to a more complicated Hamiltonian whose ground state represents the solution to the problem in question. The adiabatic theorem states that if the evolution is slow enough the system will stay in its ground state at all times through the process.

量子退火或绝热量子计算依赖绝热定理进行计算。在一个简单的哈密顿体系中，系统处于基态，这个哈密顿体系慢慢演化成一个更复杂的哈密顿体系，其基态代表问题的解决方案。绝热定理指出，如果演化足够慢，系统将在整个过程中始终保持在基态。

The Quantum algorithm for linear systems of equations, or "HHL Algorithm", named after its discoverers Harrow, Hassidim, and Lloyd, is expected to provide speedup over classical counterparts.

以其发现者 哈罗Harrow，哈西迪姆Hassidim和劳埃德Lloyd命名的线性方程组的量子算法，或称“ HHL 算法”，有望提供比经典算法更快的速度。

Solving linear equations 求解线性方程

The Quantum algorithm for linear systems of equations, or "HHL Algorithm", named after its discoverers Harrow, Hassidim, and Lloyd, is expected to provide speedup over classical counterparts.[23]

以其发现者哈罗、哈西迪姆和劳埃德命名的用于线性方程组的量子算法或“HHL算法”，有望提供比经典算法更快的速度。

John Preskill has introduced the term quantum supremacy to refer to the hypothetical speedup advantage that a quantum computer would have over a classical computer in a certain field. Google announced in 2017 that it expected to achieve quantum supremacy by the end of the year though that did not happen. IBM said in 2018 that the best classical computers will be beaten on some practical task within about five years and views the quantum supremacy test only as a potential future benchmark. Although skeptics like Gil Kalai doubt that quantum supremacy will ever be achieved, in October 2019, a Sycamore processor created in conjunction with Google AI Quantum was reported to have achieved quantum supremacy, with calculations more than 3,000,000 times as fast as those of Summit, generally considered the world's fastest computer. Bill Unruh doubted the practicality of quantum computers in a paper published back in 1994. Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle.

约翰 · 普雷斯基尔提出了量子优势Quantum supremacy这一术语，指的是量子计算机在特定领域相对于经典计算机的设想加速优势。谷歌在2017年宣布，它希望在今年年底前实现量子优势，尽管这一目标没有实现。IBM 在2018年表示，最好的经典计算机将在大约5年内在某些实际任务上被击败，并将量子优势测试视为未来的潜在基准。尽管像吉尔 · 卡莱这样的怀疑者对量子优势的实现持怀疑态度，但在2019年10月，据报道，与谷歌人工智能量子公司合作开发的 Sycamore 处理器已经取得了量子优势，其计算速度是最高级计算机的300万倍以上，后者通常被认为是世界上最快的计算机。比尔 · 安鲁在1994年发表的一篇论文中对量子计算机的实用性表示怀疑。保罗·戴维斯认为一台400量子位的计算机甚至会与全息原理暗示的宇宙学信息限制发生冲突。

Quantum supremacy 量子至上

Main article: Quantum supremacy

John Preskill has introduced the term quantum supremacy to refer to the hypothetical speedup advantage that a quantum computer would have over a classical computer in a certain field. Google announced in 2017 that it expected to achieve quantum supremacy by the end of the year though that did not happen. IBM said in 2018 that the best classical computers will be beaten on some practical task within about five years and views the quantum supremacy test only as a potential future benchmark. Although skeptics like Gil Kalai doubt that quantum supremacy will ever be achieved, in October 2019, a Sycamore processor created in conjunction with Google AI Quantum was reported to have achieved quantum supremacy, with calculations more than 3,000,000 times as fast as those of Summit, generally considered the world's fastest computer. Bill Unruh doubted the practicality of quantum computers in a paper published back in 1994. Paul Davies argued that a 400-qubit computer would even come into conflict with the cosmological information bound implied by the holographic principle.

John Preskill引入了“量子至上”一词来指量子计算机在某一领域相对于经典计算机所具有的假设加速优势。[24]Google在2017年宣布，它预计将在今年年底实现量子霸权，但这并没有实现。IBM在2018年表示，最好的经典计算机将在大约五年内完成一些实际任务，并将量子优势测试视为未来潜在的基准。[25]尽管像Gil Kalai这样的怀疑论者怀疑量子霸权是否会实现，[26][27]据报道，2019年10月，与谷歌AI Quantum联合创建的Sycamore processor实现了量子优势[28]，它的计算速度是世界上最快的计算机 Summit的300多万倍。[29]比尔 · 安鲁在1994年发表的一篇论文中对量子计算机的实用性表示怀疑。[30]Paul Davies认为，一台400 量子比特的计算机甚至会与[[全息原理]所隐含的宇宙信息界发生冲突。[31]

There are a number of technical challenges in building a large-scale quantum computer. Physicist David DiVincenzo has listed the following requirements for a practical quantum computer:

建造大型量子计算机存在许多技术挑战。物理学家 David DiVincenzo 列出了实用量子计算机的以下要求:

Obstacles 阻碍

There are a number of technical challenges in building a large-scale quantum computer. Physicist David DiVincenzo has listed the following requirements for a practical quantum computer:

建造大型量子计算机面临许多技术挑战。[32]物理学家 David DiVincenzo为一台实用的量子计算机列出了以下要求 ：[33]

  • Scalable physically to increase the number of qubits
  • 物理上可扩展以增加量子比特的数量
  • Qubits that can be initialized to arbitrary values
  • 可以初始化为随机值的量子位
  • Quantum gates that are faster than decoherence time
  • 比退相干时间快的量子门
  • Universal gate set
  • 通用门组

Sourcing parts for quantum computers is also very difficult. Many quantum computers, like those constructed by Google and IBM, need Helium-3, a nuclear research byproduct, and special superconducting cables that are only made by the Japanese company Coax Co.

寻找量子计算机的零部件也非常困难。许多量子计算机，比如谷歌和 IBM 制造的计算机，需要核研究的副产品氦 -3，以及只有日本 Coax 公司制造的特殊超导电缆。

  • Qubits that can be read easily
  • 易于读取的量子位


The control of multi qubit systems requires the generation and coordination of a large number of electrical signals with tight and deterministic timing resolution. This had led to the development of quantum controllers which enable interfacing the qubit. Scaling these systems to support a growing number of qubits is an additional challenge in the scaling of quantum computers.

多量子比特系统的控制需要产生和协调大量的电信号并保证严格和确定的时序分辨率。这导致了量子控制器的发展，它能够接通量子比特。扩展这些系统以支持越来越多的量子比特是量子计算机扩展的额外挑战。

Sourcing parts for quantum computers is also very difficult. Many quantum computers, like those constructed by Google and IBM, need Helium-3, a nuclear research byproduct, and special superconducting cables that are only made by the Japanese company Coax Co.[34]

量子计算机的零件采购也非常困难。许多量子计算机，比如由Google和IBM建造的量子计算机，需要Hemien-3， Nuclear研究副产品，以及只由日本Coax Co.公司制造的超导电缆。


The control of multi qubit systems requires the generation and coordination of a large number of electrical signals with tight and deterministic timing resolution. This had led to the development of quantum controllers which enable interfacing the qubit. Scaling these systems to support a growing number of qubits is an additional challenge in the scaling of quantum computers.[citation needed]

多量子比特系统的控制需要产生和协调大量的电信号并保证严格和确定的时序分辨率。这导致了量子控制器的发展，它能够接通量子比特。扩展这些系统以支持越来越多的量子比特是量子计算机扩展的额外挑战。模板:需要引文

Quantum decoherence 量子退相干

One of the greatest challenges involved with constructing quantum computers is controlling or removing quantum decoherence. This usually means isolating the system from its environment as interactions with the external world cause the system to decohere. However, other sources of decoherence also exist. Examples include the quantum gates, and the lattice vibrations and background thermonuclear spin of the physical system used to implement the qubits. Decoherence is irreversible, as it is effectively non-unitary, and is usually something that should be highly controlled, if not avoided. Decoherence times for candidate systems in particular, the transverse relaxation time T2 (for NMR and MRI technology, also called the dephasing time), typically range between nanoseconds and seconds at low temperature. Currently, some quantum computers require their qubits to be cooled to 20 millikelvins in order to prevent significant decoherence. A 2020 study argues that ionizing radiation such as cosmic rays can nevertheless cause certain systems to decohere within millisections.

构建量子计算机的最大挑战之一是控制或消除量子退相干Quantum decoherence。这通常意味着将系统与其环境隔离，因为与外部世界的交互会导致系统退相干。然而，也存在其他的退相干源。例如量子门，晶格振动和用于实现量子比特的物理系统的背景热核自旋。退相干是不可逆的，因为它实际上是非酉Non-unitary的，如果不能避免的话，通常也应该高度控制。候选系统的退相干时间，特别是横向弛豫时间T2（对于核磁共振和磁共振成像技术，也称为“去相位时间”），在低温下通常处于纳秒和秒之间。目前，一些量子计算机要求将量子比特冷却到20毫开尔文，以防止严重的退相干。2020年的一项研究认为，尽管如此，诸如宇宙射线这样的电离辐射仍能导致某些系统在毫秒范围内退凝。

Main article: Quantum decoherence


As a result, time-consuming tasks may render some quantum algorithms inoperable, as maintaining the state of qubits for a long enough duration will eventually corrupt the superpositions.

因此，耗时的任务可能会使一些量子算法无法操作，因为维持量子位的状态足够长的时间最终会破坏这些叠加。

One of the greatest challenges involved with constructing quantum computers is controlling or removing quantum decoherence. This usually means isolating the system from its environment as interactions with the external world cause the system to decohere. However, other sources of decoherence also exist. Examples include the quantum gates, and the lattice vibrations and background thermonuclear spin of the physical system used to implement the qubits. Decoherence is irreversible, as it is effectively non-unitary, and is usually something that should be highly controlled, if not avoided. Decoherence times for candidate systems in particular, the transverse relaxation time T2 (for NMR and MRI technology, also called the dephasing time), typically range between nanoseconds and seconds at low temperature.[35] Currently, some quantum computers require their qubits to be cooled to 20 millikelvins in order to prevent significant decoherence.[36] A 2020 study argues that ionizing radiation such as cosmic rays can nevertheless cause certain systems to decohere within millisections.[37]

构建量子计算机的最大挑战之一是控制或消除量子退相干。这通常意味着将系统与其环境隔离，因为与外部世界的交互会导致系统退相干。然而，也存在其他的退相干源。例如量子门，晶格振动和用于实现量子比特的物理系统的背景热核自旋。退相干是不可逆的，因为它实际上是非酉的，如果不能避免的话，通常也应该高度控制。候选系统的退相干时间，特别是横向弛豫时间“T”2（对于 NMR和MRI技术，也称为“去相位时间”），在低温下通常在纳秒到秒之间。.[35]目前，一些量子计算机要求将量子比特冷却到20毫开尔文，以防止严重的退相干。[38]2020年的一项研究认为，尽管如此，诸如[宇宙射线]]这样的电离辐射仍能导致某些系统在毫秒范围内退凝。[39]

These issues are more difficult for optical approaches as the timescales are orders of magnitude shorter and an often-cited approach to overcoming them is optical pulse shaping. Error rates are typically proportional to the ratio of operating time to decoherence time, hence any operation must be completed much more quickly than the decoherence time.

这些问题对于光学方法来说更加困难，因为光学方法的时间数量级更小，而克服这些问题的常用方法是光脉冲整形。错误率通常和操作时间与去相干时间的比率成正比，因此任何操作都必须比退相干时间快得多。

As a result, time-consuming tasks may render some quantum algorithms inoperable, as maintaining the state of qubits for a long enough duration will eventually corrupt the superpositions.

因此，耗时的任务可能会使一些量子算法无法操作，因为维持量子位的状态足够长的时间最终会破坏这些叠加。[40]

As described in the Quantum threshold theorem, if the error rate is small enough, it is thought to be possible to use quantum error correction to suppress errors and decoherence. This allows the total calculation time to be longer than the decoherence time if the error correction scheme can correct errors faster than decoherence introduces them. An often cited figure for the required error rate in each gate for fault-tolerant computation is 10−3, assuming the noise is depolarizing.

正如量子阈值定理所描述的那样，如果误差率足够小，则可以利用量子误差修正来抑制误差和退相干。如果纠错方案能够比消相干引入的误差更快地纠正误差，则会使得总计算时间比消相干时间更长。假设噪声是去极化的，则容错计算中每个门所需的错误率经常引用的数字是10−3。

These issues are more difficult for optical approaches as the timescales are orders of magnitude shorter and an often-cited approach to overcoming them is optical pulse shaping. Error rates are typically proportional to the ratio of operating time to decoherence time, hence any operation must be completed much more quickly than the decoherence time.

这些问题对于光学方法来说更为困难，因为光学方法的时间数量级更小，而克服这些问题的常用方法是光脉冲整形。错误率通常和操作时间与退相干时间的比率成正比，因此任何操作都必须比退相干时间快得多。

Meeting this scalability condition is possible for a wide range of systems. However, the use of error correction brings with it the cost of a greatly increased number of required qubits. The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between L and L2, where L is the number of qubits in the number to be factored; error correction algorithms would inflate this figure by an additional factor of L. For a 1000-bit number, this implies a need for about 104 bits without error correction. With error correction, the figure would rise to about 107 bits. Computation time is about L2 or about 107 steps and at 1 MHz, about 10 seconds.

满足这种可伸缩性条件对于各种系统都是可能的。然而，纠错的使用带来了大量增加所需量子比特的代价。使用Shor算法对整数进行因子运算所需的数量级仍然是多项式的，并且被认为在L和L2之间，其中L是要被分解的数量中的量子位数；纠错算法将使这个数字膨胀一个额外的系数L。对于1000位的数字，这意味着需要大约104位没有纠错。通过纠错，这个数字将上升到大约107位。计算时间约为L2 或约 107步，在主频为1兆赫时，大约10秒。


As described in the Quantum threshold theorem, if the error rate is small enough, it is thought to be possible to use quantum error correction to suppress errors and decoherence. This allows the total calculation time to be longer than the decoherence time if the error correction scheme can correct errors faster than decoherence introduces them. An often cited figure for the required error rate in each gate for fault-tolerant computation is 10−3, assuming the noise is depolarizing.

如量子阈值定理所述，如果错误率足够小，则可以利用量子纠错来抑制误差和退相干。如果纠错方案能够比退相干引入的错误更快地纠正错误，则会使得总计算时间比退相干时间长。假设噪声是去极化的，则容错计算中每个门所需的错误率经常引用的数字是10-3。

A very different approach to the stability-decoherence problem is to create a topological quantum computer with anyons, quasi-particles used as threads and relying on braid theory to form stable logic gates.

稳定性退相干问题的另一种不同的方法是用任意子、准粒子Anyons, Quasi-particles作为线程，依靠辫子理论Braid theory形成稳定的逻辑门，创建一个拓扑量子计算机。

Meeting this scalability condition is possible for a wide range of systems. However, the use of error correction brings with it the cost of a greatly increased number of required qubits. The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between L and L2, where L is the number of qubits in the number to be factored; error correction algorithms would inflate this figure by an additional factor of L. For a 1000-bit number, this implies a need for about 104 bits without error correction. With error correction, the figure would rise to about 107 bits. Computation time is about L2 or about 107 steps and at 1 MHz, about 10 seconds.

满足这种可伸缩性条件对于各种系统都是可能的。然而，纠错的使用带来了大量增加所需量子比特的代价。使用Shor算法对整数进行因子运算所需的数量级仍然是多项式的，并且被认为在L和L2之间，其中L是要被分解的数量中的量子位数；纠错算法将使这个数字膨胀一个额外的系数L。对于1000位的数字，这意味着需要大约104位没有纠错。[41]通过纠错，这个数字将上升到大约107位。计算时间约为L2 或约 107步，在主频为1兆赫时，大约10秒。


Physicist Mikhail Dyakonov has expressed skepticism of quantum computing as follows:

物理学家米哈伊尔 · 迪亚科诺夫对量子计算表示怀疑，他说:

A very different approach to the stability-decoherence problem is to create a topological quantum computer with anyons, quasi-particles used as threads and relying on braid theory to form stable logic gates.

解决稳定退相干问题的一个非同寻常的方法是用任意子s，准粒子s作为线程，依靠辫子理论来形成稳定的逻辑门[42][43]


Physicist Mikhail Dyakonov has expressed skepticism of quantum computing as follows:

物理学家Mikhail Dyakonov对量子计算表示了以下怀疑：

"So the number of continuous parameters describing the state of such a useful quantum computer at any given moment must be... about 10300... Could we ever learn to control the more than 10300 continuously variable parameters defining the quantum state of such a system? My answer is simple. No, never."[44][45]

：“因此，描述这样一个有用的量子计算机在任何给定时刻的状态的连续参数的数量必须是...大约10300... 我们能不能学会控制定义这样一个系统量子态的超过10300个连续可变的参数？我的回答很简单“不，永远不会”。

Developments 发展

Quantum computing models 量子计算模型

There are a number of quantum computing models, distinguished by the basic elements in which the computation is decomposed. The four main models of practical importance are:

有许多量子计算模型，其区别在于计算被分解时的基本元素。具有实践重要性的四种主要模式是：

[46]

Conversely, any problem solvable by a quantum computer is also solvable by a classical computer; or more formally, any quantum computer can be simulated by a Turing machine. In other words, quantum computers provide no additional power over classical computers in terms of computability. This means that quantum computers cannot solve undecidable problems like the halting problem and the existence of quantum computers does not disprove the Church–Turing thesis.

相反，任何量子计算机可以解决的问题也可以用经典计算机来解决; 或者更正式地说，任何量子计算机都可以用图灵机来模拟。换句话说，就可计算性而言，量子计算机并不比传统计算机提供额外的能力。这意味着量子计算机不能解决不可判定的问题，例如停机问题，而且量子计算机的存在并不能否定丘奇-图灵论点Church–Turing thesis。

The quantum Turing machine is theoretically important but the physical implementation of this model is not feasible. All four models of computation have been shown to be equivalent; each can simulate the other with no more than polynomial overhead.

量子图灵机理论上很重要，但是这个模型的物理实现是不可行的。所有四种计算模型被证明是等价的; 每种模型只需要不超过多项式的开销就可以模拟另一种模型。

As of yet, quantum computers do not satisfy the strong Church thesis. While hypothetical machines have been realized, a universal quantum computer has yet to be physically constructed. The strong version of Church's thesis requires a physical computer, and therefore there is no quantum computer that yet satisfies the strong Church thesis.

到目前为止，量子计算机还不能满足强丘奇理论。虽然假想的机器已经被实现，但是通用的量子计算机还有待物理构造。切奇论点的更强版本需要一台物理计算机实体，因此现在没有量子计算机能够满足强丘奇理论。

Physical realizations 物理实现

For physically implementing a quantum computer, many different candidates are being pursued, among them (distinguished by the physical system used to realize the qubits):

对于量子计算机的物理实现，人们正在寻找许多不同的候选方案，其中包括（以实现量子比特的物理系统为区别）：

< ! -- 量子计算机的能力和极限 -- >

While quantum computers cannot solve any problems that classical computers cannot already solve, it is suspected that they can solve many problems faster than classical computers. For instance, it is known that quantum computers can efficiently factor integers, while this is not believed to be the case for classical computers. However, the capacity of quantum computers to accelerate classical algorithms has rigid upper bounds, and the overwhelming majority of classical calculations cannot be accelerated by the use of quantum computers.

虽然量子计算机不能解决任何传统计算机已经不能解决的问题，人们怀疑它们能比传统计算机更快地解决许多问题。例如，众所周知量子计算机可以高效地对整数进行因子分解，而经典计算机则不然。可是，量子计算机加速经典算法的能力具有严格的上限，绝大多数经典计算不能被量子计算机加速。

  • Neutral atoms in Optical lattices (qubit implemented by internal states of neutral atoms trapped in an optical lattice)[51][52]
  • 光学晶格s中的中性原子（由被困在光学晶格中的中性原子的内部状态实现的量子比特）[53][54]


< ! -- BQP 的基本定义 -- >

  • Quantum dot computer, spatial-based (qubit given by electron position in double quantum dot)[57]
  • 基于空间的量子点计算机（由双量子点中的电子位置给出的量子比特）[58]

The class of problems that can be efficiently solved by a quantum computer with bounded error is called BQP, for "bounded error, quantum, polynomial time". More formally, BQP is the class of problems that can be solved by a polynomial-time quantum Turing machine with error probability of at most 1/3. As a class of probabilistic problems, BQP is the quantum counterpart to BPP ("bounded error, probabilistic, polynomial time"), the class of problems that can be solved by polynomial-time probabilistic Turing machines with bounded error. It is known that BPP[math]\displaystyle{ \subseteq }[/math]BQP and is widely suspected that BQP[math]\displaystyle{ \nsubseteq }[/math]BPP, which intuitively would mean that quantum computers are more powerful than classical computers in terms of time complexity.

误差有界的量子计算机能高效解决的一类问题称为BQP，即“有界误差，量子，多项式时间”。更正式地说，BQP是一类可以用多项式时间量子图灵机求解（其错误概率最大为1/3）的问题。作为一类概率问题，BQP是BPP（“有界误差，概率，多项式时间”）的量子对应物，BPP是一类可由误差有界的多项式时间概率图灵机求解的问题。众所周知，BPP[math]\displaystyle{ \subseteq }[/math]BQP，并被广泛怀疑为BQP[math]\displaystyle{ \nsubseteq }[/math]BPP，这直观地意味着量子计算机在时间复杂度方面比经典计算机更强大。


  • Quantum computing using engineered quantum wells, which could in principle enable the construction of quantum computers that operate at room temperature[59][60]
  • 使用工程量子阱进行量子计算，原则上可以建造在室温下工作的量子计算机[61][62]


< ! -- BQP 与基本复杂类的关系 -- >

The suspected relationship of BQP to several classical complexity classes.

BQP 与几个经典复杂性类的可疑关系。

The exact relationship of BQP to P, NP, and PSPACE is not known. However, it is known that P[math]\displaystyle{ \subseteq }[/math]BQP[math]\displaystyle{ \subseteq }[/math]PSPACE; that is, all problems that can be efficiently solved by a deterministic classical computer can also be efficiently solved by a quantum computer, and all problems that can be efficiently solved by a quantum computer can also be solved by a deterministic classical computer with polynomial space resources. It is further suspected that BQP is a strict superset of P, meaning there are problems that are efficiently solvable by quantum computers that are not efficiently solvable by deterministic classical computers. For instance, integer factorization and the discrete logarithm problem are known to be in BQP and are suspected to be outside of P. On the relationship of BQP to NP, little is known beyond the fact that some NP problems that are believed not to be in P are also in BQP (integer factorization and the discrete logarithm problem are both in NP, for example). It is suspected that NP[math]\displaystyle{ \nsubseteq }[/math]BQP; that is, it is believed that there are efficiently checkable problems that are not efficiently solvable by a quantum computer. As a direct consequence of this belief, it is also suspected that BQP is disjoint from the class of NP-complete problems (if an NP-complete problem were in BQP, then it would follow from NP-hardness that all problems in NP are in BQP).

BQP与P、NP和PSPACE的确切关系尚不清楚。然而，众所周知P[math]\displaystyle{ \subseteq }[/math]BQP[math]\displaystyle{ \subseteq }[/math]PSPACE，即所有能被确定性经典计算机高效解决的问题也能被量子计算机高效地解决，所有能被量子计算机有效解决的问题，也能用有多项式空间资源的确定性经典计算机来求解。人们进一步怀疑BQP是P的严格超集，这意味着一些问题能被量子计算机有效地解决，但无法靠确定性经典计算机有效地解决。例如，整数因式分解和离散对数问题属于BQP，但被怀疑不属于P。关于BQP与NP的关系，除了知道一些NP问题不在P中但在BQP中（比如整数因式分解和离散对数问题都属于NP）之外，人们知之甚少。有人怀疑NP[math]\displaystyle{ \nsubseteq }[/math]BQP；也就是说，人们相信存在着量子计算机无法有效解决的有效可检查问题。作为这种观点的直接结果，人们还怀疑BQP与NP完全问题类不相交（如果一个NP完全问题在BQP中，那么从NP困难问题NP-hardness可以看出NP中的所有问题都在BQP中）。


  • Electrons-on-helium quantum computers (qubit is the electron spin)
  • 量子计算机上的电子（量子比特是电子的自旋）
  • Cavity quantum electrodynamics (CQED) (qubit provided by the internal state of trapped atoms coupled to high-finesse cavities)
  • 腔量子电动力学（CQED）（由与高精细腔耦合的被俘获原子的内部状态提供的量子比特）

< ! -- 本质复杂性类的关系总结 -- >

The relationship of BQP to the basic classical complexity classes can be summarized as follows:

BQP 与基本经典复杂性类的关系可以概括如下:

[math]\displaystyle{ \mathsf{P \subseteq BPP \subseteq BQP \subseteq PP \subseteq PSPACE} }[/math]

{ p subseteq BPP subseteq subseteq PP subseteq PSPACE } </math >

It is also known that BQP is contained in the complexity class #P (or more precisely in the associated class of decision problems P#P), Theories of quantum gravity, such as M-theory and loop quantum gravity, may allow even faster computers to be built. However, defining computation in these theories is an open problem due to the problem of time; that is, within these physical theories there is currently no obvious way to describe what it means for an observer to submit input to a computer at one point in time and then receive output at a later point in time.

众所周知，BQP属于复杂性类P（或者更准确地说，属于决策问题类P#P的相关类），量子引力理论，如M理论和环路量子引力，可以使计算机的速度更快。然而，由于时间问题，在这些理论中定义计算是一个未解决问题；也就是说，在这些物理理论中，目前没有明显的方法来描述观察者在一个时间点向计算机提交输入，然后在以后的时间点接收输出意味着什么。

  • Transistor-based quantum computer – string quantum computers with entrainment of positive holes using an electrostatic trap
  • 基于晶体管的量子计算机——使用静电阱夹带正空穴的串量子计算机
  • Rare-earth-metal-ion-doped inorganic crystal based quantum computers引用错误：没有找到与</ref>对应的<ref>标签[80] (qubit realized by the internal electronic state of dopants in optical fibers)
  • Metallic-like carbon nanospheres based quantum computers[81]
  • 基于类金属碳纳米球的量子计算机[81]


A large number of candidates demonstrates that quantum computing, despite rapid progress, is still in its infancy.[citation needed] 大量的候选方案表明，尽管量子计算技术发展迅速，但仍处于初级阶段。[citation needed]

Relation to computability and complexity theory与可计算性和复杂性理论的关系

Computability theory可计算性理论

See also: Computability theory

模板:另见可计算性理论

Any computational problem solvable by a classical computer is also solvable by a quantum computer.[82] Intuitively, this is because it is believed that all physical phenomena, including the operation of classical computers, can be described using quantum mechanics, which underlies the operation of quantum computers.

经典计算机可以解决的任何计算问题也可以由量子计算机解决。[83]直觉上，这是因为人们相信，所有物理现象（包括经典计算机的运行），都可以用量子力学来描述，而这是量子计算机操作的基础。

Category:Models of computation

类别: 计算模型


Category:Quantum cryptography

类别: 量子密码学

Conversely, any problem solvable by a quantum computer is also solvable by a classical computer; or more formally, any quantum computer can be simulated by a Turing machine. In other words, quantum computers provide no additional power over classical computers in terms of computability. This means that quantum computers cannot solve undecidable problems like the halting problem and the existence of quantum computers does not disprove the Church–Turing thesis.[84]

相反，量子计算机可以解决的任何问题也可以用经典计算机来解决；或者更正式地说，任何量子计算机都可以用图灵机<来模拟！--增加关于量子虚拟机的介绍，它可以在经典的量子计算机-->上模拟量子计算机。换句话说，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。这意味着量子计算机不能解决像停止问题一样的不可判定问题，量子计算机的存在也并不能否定丘奇-图灵论点。

Category:Information theory

范畴: 信息论


Category:Computational complexity theory

类别: 计算复杂性理论

As of yet, quantum computers do not satisfy the strong Church thesis. While hypothetical machines have been realized, a universal quantum computer has yet to be physically constructed. The strong version of Church's thesis requires a physical computer, and therefore there is no quantum computer that yet satisfies the strong Church thesis.

到目前为止，量子计算机还不能满足强丘奇理论。虽然假想的机器已经实现，但一个通用的量子计算机还没有被物理构造出来。丘奇理论的强版本需要一台物理计算机实体，所以还没有一台量子计算机能够满足强大的丘奇理论。

Category:Classes of computers

类别: 电脑类别


Category:Theoretical computer science

类别: 理论计算机科学

Quantum complexity theory量子复杂性理论

Category:Open problems

类别: 开放性问题

Main article: Quantum complexity theory

Category:Computer-related introductions in 1980

类别: 1980年与计算机有关的介绍


Category:Emerging technologies

类别: 新兴技术

编辑推荐

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  42. ↑ {{cite journal "So the number of continuous parameters describing the state of such a useful quantum computer at any given moment must be... about 10300... Could we ever learn to control the more than 10300 continuously variable parameters defining the quantum state of such a system? My answer is simple. No, never." “因此，描述这样一个有用的量子计算机在任何给定时刻的状态的连续参数的数量必须是... ... 大约10300 ... ... 我们能否学会控制定义这样一个系统的量子态的超过10300 个连续可变参数？我的答案很简单。不，永远不会。” | last1 = Freedman | first1 = Michael H. | author1-link = Michael Freedman | last2 = Kitaev | first2 = Alexei | author2-link = Alexei Kitaev | last3 = Larsen | first3 = Michael J. | author3-link = Michael J. Larsen | last4 = Wang | first4 = Zhenghan There are a number of quantum computing models, distinguished by the basic elements in which the computation is decomposed. The four main models of practical importance are: 有许多量子计算模型，以计算被分解的基本元素来区分。具有实际重要性的四个主要模式是: | arxiv = quant-ph/0101025 | doi = 10.1090/S0273-0979-02-00964-3 | issue = 1 | journal = Bulletin of the American Mathematical Society | mr = 1943131 The quantum Turing machine is theoretically important but the physical implementation of this model is not feasible. All four models of computation have been shown to be equivalent; each can simulate the other with no more than polynomial overhead. 量子图灵机在理论上是重要的，但是这个模型的物理实现是不可行的。所有四种计算模型被证明是等价的; 每种模型只需要不超过多项式的开销就可以模拟另一种模型。 | pages = 31–38 | title = Topological quantum computation | volume = 40 For physically implementing a quantum computer, many different candidates are being pursued, among them (distinguished by the physical system used to realize the qubits): 对于物理实现量子计算机，人们正在寻找许多不同的候选方案，其中包括（以实现量子比特的物理系统为区别）： | year = 2003}}
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  47. ↑ Nayak, Chetan; Simon, Steven; Stern, Ady A large number of candidates demonstrates that quantum computing, despite rapid progress, is still in its infancy. 大量的候选方案表明，量子计算尽管进展迅速，但仍然处于初级阶段。; Das Sarma, Sankar (2008). "Nonabelian Anyons and Quantum Computation". Reviews of Modern Physics. 80 (3): 1083–1159. arXiv:0707.1889. Bibcode:[https://ui.adsabs.harvard.edu/abs/2008RvMP...80.1083N Any computational problem solvable by a classical computer is also solvable by a quantum computer. Intuitively, this is because it is believed that all physical phenomena, including the operation of classical computers, can be described using quantum mechanics, which underlies the operation of quantum computers. 任何经典计算机可解的计算问题也可以用量子计算机来解。直观地说，这是因为人们相信所有的物理现象（包括经典计算机的运算），都可以用量子计算机运算的基础----量子力学来描述。 2008RvMP...80.1083N Any computational problem solvable by a classical computer is also solvable by a quantum computer. Intuitively, this is because it is believed that all physical phenomena, including the operation of classical computers, can be described using quantum mechanics, which underlies the operation of quantum computers. 任何经典计算机可解的计算问题也可以用量子计算机来解。直观地说，这是因为人们相信所有的物理现象（包括经典计算机的运算），都可以用量子计算机运算的基础----量子力学来描述。]. doi:10.1103/RevModPhys.80.1083. S2CID 119628297. {{cite journal}}: Check |bibcode= length (help); line feed character in |bibcode= at position 20 (help); line feed character in |first3= at position 4 (help)
  48. ↑ 48.0 48.1 Clarke, John; Wilhelm, Frank K. (18 June 2008). "Superconducting quantum bits". Nature. 453 (7198): 1031–1042. Bibcode:2008Natur.453.1031C. doi:10.1038/nature07128. PMID 18563154. S2CID 125213662.
  49. ↑ Kaminsky, William M.; Lloyd, Seth; Orlando, Terry P. (12 March 2004). "Scalable Superconducting Architecture for Adiabatic Quantum Computation". arXiv:quant-ph/0403090. Bibcode:2004quant.ph..3090K. {{cite journal}}: Cite journal requires |journal= (help)
  50. ↑ Kaminsky, William M.; Lloyd, Seth; Orlando, Terry P. (12 March 2004). "Scalable Superconducting Architecture for Adiabatic Quantum Computation". arXiv:quant-ph/0403090. Bibcode:2004quant.ph..3090K. {{cite journal}}: Cite journal requires |journal= (help)
  51. ↑ Khazali, Mohammadsadegh; Mølmer, Klaus (2020-06-11). "Fast Multiqubit Gates by Adiabatic Evolution in Interacting Excited-State Manifolds of Rydberg Atoms and Superconducting Circuits". Physical Review X. 10 (2): 021054. Bibcode:2020PhRvX..10b1054K. doi:10.1103/PhysRevX.10.021054.
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  82. ↑ Nielsen, p. 29
  83. ↑ Nielsen, p. 29
  84. ↑ Nielsen, p. 126
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