1.1.1 量子计算和量子信息的历史

物理危机

我们的故事始于二十世纪之交，当时一场突如其来的科学革命正在进行。物理学中出现了一系列危机。问题是，当时的物理理论(现在被称为经典物理学)预测了一些荒谬的东西：比如涉及无限能量的“紫外灾难”，还有电子能够旋转进入原子核。

起初，这些问题是通过在经典物理学中加入一些特别假设来解决的，但随着对原子和辐射的进一步了解，这些解释变得越来越复杂。在经历了四分之一个世纪的动荡之后，这场危机在20世纪20年代初达到了顶峰，并导致了现代量子力学理论的诞生。从那以后，量子力学一直是科学中不可或缺的一部分，并在太阳系内外的一切事物上取得了巨大的成功，包括原子的结构、恒星中的核聚变、超导体、DNA的结构和自然界的基本粒子。

什么是量子力学?量子力学是构建物理理论的数学框架或一套规则。例如，有一种被称为量子电动力学的物理理论，它以惊人的精确度描述了原子和光的相互作用。量子电动力学是在量子力学的框架内建立起来的，但它包含了量子力学所不能确定的特定规律。量子力学与特定物理理论(如量子电动力学)之间的关系很像计算机操作系统与特定应用软件之间的关系——操作系统设置了某些基本参数和操作模式，但对应用程序如何完成特定任务保持开放。

量子力学的规则很简单，但即使是专家也会觉得它们违反直觉。最著名的量子力学批评家阿尔伯特·爱因斯坦(Albert Einstein)到死都不认同该理论，即使他帮助发现了它。从那以后，一代又一代的物理学家一直在与量子力学作斗争，努力使其预测更容易被接受。量子计算和量子信息的目标之一是开发工具，使我们对量子力学的直觉更加敏锐，并使其预测对人类思想更加透明。

【以上校对】

For example, in the early 1980s, interest arose in whether it might be possible to use quantum effects to signal faster than light – a big no-no according to Einstein’s theory of relativity. The resolution of this problem turns out to hinge on whether it is possible to clone an unknown quantum state, that is, construct a copy of a quantum state. If cloning were possible, then it would be possible to signal faster than light using quantum effects.

例如，在20世纪80年代早期，人们对是否有可能利用量子效应发出比光速更快的信号产生了兴趣——根据爱因斯坦的相对论，这是一个大禁忌。这个问题的解决取决于是否有可能克隆一个未知的量子态，也就是说，构建一个量子态的副本。如果克隆是可能的，那么就有可能利用量子效应发出比光更快的信号。

However, cloning – so easy to accomplish with classical information (consider the words in front of you, and where they came from!) – turns out not to be possible in general in quantum mechanics. This no-cloning theorem, discovered in the early 1980s, is one of the earliest results of quantum computation and quantum information. Many refinements of the no-cloning theorem have since been developed, and we now have conceptual tools which allow us to understand how well a (necessarily imperfect) quantum cloning device might work. These tools, in turn, have been applied to understand other aspects of quantum mechanics.

然而，克隆——用经典信息很容易完成(考虑一下你面前的单词，以及它们来自哪里!)——在量子力学中是不可能实现的。这个不可克隆定理发现于20世纪80年代初，是量子计算和量子信息最早的结果之一。自那以后，不可克隆定理得到了许多改进，我们现在有了概念性的工具，使我们能够理解量子克隆设备(必然是不完美的)的工作效果如何。这些工具，反过来，已经被应用于理解量子力学的其他方面。

A related historical strand contributing to the development of quantum computation and quantum information is the interest, dating to the 1970s, of obtaining complete control over single quantum systems. Applications of quantum mechanics prior to the 1970s typically involved a gross level of control over a bulk sample containing an enormous number of quantum mechanical systems, none of them directly accessible. For example, superconductivity has a superb quantum mechanical explanation. However, because a superconductor involves a huge (compared to the atomic scale) sample of conducting metal, we can only probe a few aspects of its quantum mechanical nature, with the individual quantum systems constituting the superconductor remaining inaccessible. Systems such as particle accelerators do allow limited access to individual quantum systems, but again provide little control over the constituent systems.

单量子控制

促进量子计算和量子信息发展的一个相关历史链是，可追溯到20世纪70年代，对获得对单量子系统的完全控制的兴趣。在20世纪70年代之前，量子力学的应用通常涉及对包含大量量子力学系统的大量样本的总体控制，这些系统中没有一个是直接可访问的。例如，超导有一个极好的量子力学解释。然而，由于超导体涉及巨大的(与原子尺度相比)导电金属样品，我们只能探索其量子力学性质的几个方面，而构成超导体的单个量子系统仍然无法进入。像粒子加速器这样的系统确实允许对单个量子系统进行有限的访问，但同样对组成系统提供很少的控制。

Since the 1970s many techniques for controlling single quantum systems have been developed. For example, methods have been developed for trapping a single atom in an ‘atom trap’, isolating it from the rest of the world and allowing us to probe many different aspects of its behavior with incredible precision. The scanning tunneling microscope has been used to move single atoms around, creating designer arrays of atoms at will. Electronic devices whose operation involves the transfer of only single electrons have been demonstrated.

自20世纪70年代以来，许多控制单量子系统的技术已经开发出来。例如，已经开发出了将单个原子困在“原子陷阱”中的方法，将其与外界隔离开来，使我们能够以令人难以置信的精度探测其行为的许多不同方面。扫描隧道显微镜已被用于移动单个原子，随意创建设计的原子阵列。只涉及单电子转移操作的电子装置已经问世。

Why all this effort to attain complete control over single quantum systems? Setting aside the many technological reasons and concentrating on pure science, the principal answer is that researchers have done this on a hunch. Often the most profound insights in science come when we develop a method for probing a new regime of Nature. For example, the invention of radio astronomy in the 1930s and 1940s led to a spectacular sequence of discoveries, including the galactic core of the Milky Way galaxy, pulsars, and quasars. Low temperature physics has achieved its amazing successes by finding ways to lower the temperatures of different systems. In a similar way, by obtaining complete control over single quantum systems, we are exploring untouched regimes of Nature in the hope of discovering new and unexpected phenomena. We are just now taking our first steps along these lines, and already a few interesting surprises have been discovered in this regime. What else shall we discover as we obtain more complete control over single quantum systems, and extend it to more complex systems?

为什么所有这些努力都是为了获得对单量子系统的完全控制?撇开许多技术上的原因，把注意力集中在纯科学上，主要的答案是，研究人员是凭直觉做到这一点的。通常，当我们开发出一种探索自然新体制的方法时，科学中最深刻的见解就会出现。例如，20世纪30年代和40年代射电天文学的发明导致了一系列壮观的发现，包括银河系的星系核心、脉冲星和类星体。低温物理学通过寻找降低不同系统温度的方法取得了惊人的成功。同样，通过获得对单量子系统的完全控制，我们正在探索未被触及的自然状态，希望发现新的和意想不到的现象。我们现在正沿着这条路线迈出我们的第一步，并且已经在这一制度中发现了一些有趣的惊喜。当我们获得对单量子系统的更完全的控制，并将其扩展到更复杂的系统时，我们还会发现什么?

Quantum computation and quantum information fit naturally into this program. They provide a useful series of challenges at varied levels of difficulty for people devising methods to better manipulate single quantum systems, and stimulate the development of new experimental techniques and provide guidance as to the most interesting directions in which to take experiment. Conversely, the ability to control single quantum systems is essential if we are to harness the power of quantum mechanics for applications to quantum computation and quantum information.

量子计算和量子信息自然适合这个程序。它们为人们设计更好地操纵单量子系统的方法提供了一系列不同难度的有用挑战，并刺激了新实验技术的发展，并为最有趣的实验方向提供了指导。相反，如果我们要利用量子力学的力量将其应用于量子计算和量子信息，那么控制单个量子系统的能力是必不可少的。

Despite this intense interest, efforts to build quantum information processing systems have resulted in modest success to date. Small quantum computers, capable of doing dozens of operations on a few quantum bits (or qubits) represent the state of the art in quantum computation. Experimental prototypes for doing quantum cryptography – a way of communicating in secret across long distances – have been demonstrated, and are even at the level where they may be useful for some real-world applications. However, it remains a great challenge to physicists and engineers of the future to develop techniques for making large-scale quantum information processing a reality.

尽管有这种强烈的兴趣，但迄今为止，构建量子信息处理系统的努力取得了一定的成功。能够在几个量子比特(或量子位)上进行数十次操作的小型量子计算机代表了量子计算的最新水平。量子密码学是一种远距离秘密通信的方式，它的实验原型已经被证明，甚至可以用于一些现实世界的应用。然而，对于未来的物理学家和工程师来说，开发大规模量子信息处理技术仍然是一个巨大的挑战。

图灵机器

Let us turn our attention from quantum mechanics to another of the great intellectual triumphs of the twentieth century, computer science. The origins of computer science are lost in the depths of history. For example, cuneiform tablets indicate that by the time of Hammurabi (circa 1750 B.C.) the Babylonians had developed some fairly sophisticated algorithmic ideas, and it is likely that many of those ideas date to even earlier times.

让我们把注意力从量子力学转向20世纪另一项伟大的智力成就——计算机科学。计算机科学的起源已经消失在历史的深处。例如，楔形文字碑表明，在汉谟拉比时代(大约公元前1750年)，巴比伦人已经发展出一些相当复杂的算法思想，而且很可能这些思想中的许多可以追溯到更早的时代。

The modern incarnation of computer science was announced by the great mathematician Alan Turing in a remarkable 1936 paper. Turing developed in detail an abstract notion of what we would now call a programmable computer, a model for computation now known as the Turing machine, in his honor. Turing showed that there is a Universal Turing Machine that can be used to simulate any other Turing machine. Furthermore, he claimed that the Universal Turing Machine completely captures what it means to perform a task by algorithmic means. That is, if an algorithm can be performed on any piece of hardware (say, a modern personal computer), then there is an equivalent algorithm for a Universal Turing Machine which performs exactly the same task as the algorithm running on the personal computer. This assertion, known as the Church–Turing thesis in honor of Turing and another pioneer of computer science, Alonzo Church, asserts the equivalence between the physical concept of what class of algorithms can be performed on some physical device with the rigorous mathematical concept of a Universal Turing Machine. The broad acceptance of this thesis laid the foundation for the development of a rich theory of computer science.

计算机科学的现代化身是由伟大的数学家艾伦·图灵在1936年一篇非凡的论文中宣布的。图灵详细地发展了一个抽象概念，我们现在称之为可编程计算机，一个计算模型，现在被称为图灵机，以纪念他。图灵证明了有一个通用图灵机可以用来模拟任何其他的图灵机。此外，他声称通用图灵机完全抓住了通过算法手段执行任务的意义。也就是说，如果一个算法可以在任何硬件上执行(比如，一台现代的个人计算机)，那么通用图灵机就有一个等效的算法，它执行的任务与在个人计算机上运行的算法完全相同。这个论断被称为丘奇-图灵命题，以纪念图灵和另一位计算机科学的先驱阿朗佐·丘奇，它断言，在某种物理设备上可以执行哪类算法的物理概念与通用图灵机的严格数学概念之间是等价的。这篇论文的广泛接受为丰富的计算机科学理论的发展奠定了基础。

Not long after Turing’s paper, the first computers constructed from electronic components were developed. John von Neumann developed a simple theoretical model for how to put together in a practical fashion all the components necessary for a computer to be fully as capable as a Universal Turing Machine. Hardware development truly took off, though, in 1947, when John Bardeen, Walter Brattain, and Will Shockley developed the transistor. Computer hardware has grown in power at an amazing pace ever since, so much so that the growth was codified by Gordon Moore in 1965 in what has come to be known as Moore’s law, which states that computer power will double for constant cost roughly once every two years.

摩尔定律

在图灵的论文发表后不久，第一台由电子元件构成的计算机问世了。约翰·冯·诺伊曼建立了一个简单的理论模型，用来将计算机所需的所有部件以实用的方式组合在一起，使其完全具备通用图灵机的能力。1947年，约翰·巴丁(John Bardeen)、沃尔特·布拉顿(Walter Brattain)和威尔·肖克利(Will Shockley)发明了晶体管，硬件的发展才真正起飞。从那以后，计算机硬件的能力以惊人的速度增长，以至于戈登·摩尔(Gordon Moore)在1965年将这种增长编入了著名的摩尔定律(Moore’s law)，该定律指出，如果成本不变，计算机的能力大约每两年翻一番。

Amazingly enough, Moore’s law has approximately held true in the decades since the 1960s. Nevertheless, most observers expect that this dream run will end some time during the first two decades of the twenty-first century. Conventional approaches to the fabrication of computer technology are beginning to run up against fundamental difficulties of size. Quantum effects are beginning to interfere in the functioning of electronic devices as they are made smaller and smaller.

令人惊讶的是，自20世纪60年代以来的几十年里，摩尔定律几乎一直是正确的。然而，大多数观察家预计，这一梦想之旅将在二十一世纪头二十年的某个时候结束。制造计算机技术的传统方法开始遇到尺寸的基本困难。随着电子设备变得越来越小，量子效应开始干扰它们的功能。

One possible solution to the problem posed by the eventual failure of Moore’s law is to move to a different computing paradigm. One such paradigm is provided by the theory of quantum computation, which is based on the idea of using quantum mechanics to perform computations, instead of classical physics. It turns out that while an ordinary computer can be used to simulate a quantum computer, it appears to be impossible to perform the simulation in an efficient fashion. Thus quantum computers offer an essential speed advantage over classical computers. This speed advantage is so significant that many researchers believe that no conceivable amount of progress in classical computation would be able to overcome the gap between the power of a classical computer and the power of a quantum computer.

对于摩尔定律最终失败所带来的问题，一个可能的解决方案是转向一种不同的计算范式。量子计算理论提供了一个这样的范例，它基于使用量子力学来执行计算的想法，而不是经典物理学。事实证明，虽然普通计算机可以用来模拟量子计算机，但似乎不可能以有效的方式进行模拟。因此，量子计算机提供了比经典计算机更重要的速度优势。这种速度优势是如此显著，以至于许多研究人员认为，经典计算的任何进步都无法克服经典计算机和量子计算机之间的能力差距。

图灵模拟

What do we mean by ‘efficient’ versus ‘inefficient’ simulations of a quantum computer? Many of the key notions needed to answer this question were actually invented before the notion of a quantum computer had even arisen. In particular, the idea of efficient and inefficient algorithms was made mathematically precise by the field of computational complexity. Roughly speaking, an efficient algorithm is one which runs in time polynomial in the size of the problem solved. In contrast, an inefficient algorithm requires superpolynomial (typically exponential) time. What was noticed in the late 1960s and early 1970s was that it seemed as though the Turing machine model of computation was at least as powerful as any other model of computation, in the sense that a problem which could be solved efficiently in some model of computation could also be solved efficiently in the Turing machine model, by using the Turing machine to simulate the other model of computation. This observation was codified into a strengthened version of the Church– Turing thesis:Any algorithmic process can be simulated efficiently using a Turing machine.

我们所说的量子计算机的“高效”与“低效”模拟是什么意思?回答这个问题所需的许多关键概念实际上在量子计算机的概念出现之前就已经发明了。特别是，高效和低效算法的概念通过计算复杂性领域在数学上变得精确。粗略地说，一个有效的算法是在解决问题的大小上以时间多项式运行的算法。相反，低效的算法需要超多项式(通常是指数)的时间。在20世纪60年代末和70年代初，人们注意到，图灵机的计算模型似乎至少和其他任何计算模型一样强大，从某种意义上说，可以在某些计算模型中有效解决的问题，也可以在图灵机模型中有效解决，通过使用图灵机来模拟其他计算模型。这一观察结果被整理成一个强化版的丘奇-图灵论文:任何算法过程都可以用图灵机有效地模拟。

The key strengthening in the strong Church–Turing thesis is the word efficiently. If the strong Church–Turing thesis is correct, then it implies that no matter what type of machine we use to perform our algorithms, that machine can be simulated efficiently using a standard Turing machine. This is an important strengthening, as it implies that for the purposes of analyzing whether a given computational task can be accomplished efficiently, we may restrict ourselves to the analysis of the Turing machine model of computation.

强丘奇-图灵论文中关键的强化是“高效”一词。如果强丘奇-图灵命题是正确的，那么它意味着无论我们使用什么类型的机器来执行我们的算法，该机器都可以使用标准图灵机有效地模拟。这是一个重要的加强，因为它意味着为了分析给定的计算任务是否可以有效地完成，我们可以将自己限制在分析图灵机计算模型上。

One class of challenges to the strong Church–Turing thesis comes from the field of analog computation. In the years since Turing, many different teams of researchers have noticed that certain types of analog computers can efficiently solve problems believed to have no efficient solution on a Turing machine. At first glance these analog computers appear to violate the strong form of the Church–Turing thesis. Unfortunately for analog computation, it turns out that when realistic assumptions about the presence of noise in analog computers are made, their power disappears in all known instances; they cannot efficiently solve problems which are not efficiently solvable on a Turing machine. This lesson – that the effects of realistic noise must be taken into account in evaluating the efficiency of a computational model – was one of the great early challenges of quantum computation and quantum information, a challenge successfully met by the development of a theory of quantum error-correcting codes and fault-tolerant quantum computation. Thus, unlike analog computation, quantum computation can in principle tolerate a finite amount of noise and still retain its computational advantages.

对强丘奇-图灵论文的一类挑战来自模拟计算领域。自图灵以来的几年里，许多不同的研究团队已经注意到，某些类型的模拟计算机可以有效地解决在图灵机上无法有效解决的问题。乍一看，这些模拟计算机似乎违反了丘奇-图灵理论的强形式。不幸的是，对于模拟计算来说，事实证明，当对模拟计算机中存在噪声做出现实假设时，它们的力量在所有已知的情况下都消失了;它们不能有效地解决在图灵机上不能有效解决的问题。这一教训——在评估计算模型的效率时必须考虑到现实噪声的影响——是量子计算和量子信息的早期重大挑战之一，量子纠错码和容错量子计算理论的发展成功地解决了这一挑战。因此，与模拟计算不同，量子计算原则上可以容忍有限数量的噪声，并且仍然保持其计算优势。

图灵机器的挑战——随机模拟

The first major challenge to the strong Church–Turing thesis arose in the mid 1970s, when Robert Solovay and Volker Strassen showed that it is possible to test whether an integer is prime or composite using a randomized algorithm. That is, the Solovay–Strassen test for primality used randomness as an essential part of the algorithm. The algorithm did not determine whether a given integer was prime or composite with certainty. Instead, the algorithm could determine that a number was probably prime or else composite with certainty. By repeating the Solovay–Strassen test a few times it is possible to determine with near certainty whether a number is prime or composite. The Solovay-Strassen test was of especial significance at the time it was proposed as no deterministic test for primality was then known, nor is one known at the time of this writing. Thus, it seemed as though computers with access to a random number generator would be able to efficiently perform computational tasks with no efficient solution on a conventional deterministic Turing machine. This discovery inspired a search for other randomized algorithms which has paid off handsomely, with the field blossoming into a thriving area of research.

对强Church-Turing理论的第一个重大挑战出现在20世纪70年代中期，当时Robert Solovay和Volker Strassen表明，可以使用随机算法来测试整数是素数还是合数。也就是说，Solovay-Strassen素数检验使用随机性作为算法的重要组成部分。该算法不能确定给定的整数是素数还是合数。相反，该算法可以确定一个数字可能是素数或合数。通过多次重复Solovay-Strassen检验，可以几乎肯定地确定一个数是素数还是合数。Solovay-Strassen测试在它被提出的时候具有特别重要的意义，因为当时还没有关于原始性的确定性测试，在撰写本文的时候也没有。因此，似乎可以访问随机数生成器的计算机能够有效地执行计算任务，而在传统的确定性图灵机上没有有效的解决方案。这一发现激发了人们对其他随机算法的探索，这些算法获得了丰厚的回报，使该领域成为一个蓬勃发展的研究领域。

Randomized algorithms pose a challenge to the strong Church–Turing thesis, suggesting that there are efficiently soluble problems which, nevertheless, cannot be efficiently solved on a deterministic Turing machine. This challenge appears to be easily resolved by a simple modification of the strong Church–Turing thesis:Any algorithmic process can be simulated efficiently using a probabilistic Turing machine.

随机算法对强丘奇-图灵理论提出了挑战，表明存在有效可解的问题，但在确定性图灵机上无法有效解决。这个挑战似乎很容易通过对强丘奇-图灵理论的简单修改来解决:任何算法过程都可以使用概率图灵机有效地模拟。

This ad hoc modification of the strong Church–Turing thesis should leave you feeling rather queasy. Might it not turn out at some later date that yet another model of computation allows one to efficiently solve problems that are not efficiently soluble within Turing’s model of computation? Is there any way we can find a single model of computation which is guaranteed to be able to efficiently simulate any other model of computation?

这种对强丘奇-图灵理论的特别修改应该会让你感到相当恶心。在以后的某一天，会不会出现另一种计算模型，让人们能够高效地解决在图灵计算模型中无法有效解决的问题?我们是否有办法找到一个单一的计算模型，保证能够有效地模拟任何其他的计算模型?

Motivated by this question, in 1985 David Deutsch asked whether the laws of physics could be use to derive an even stronger version of the Church–Turing thesis. Instead of adopting ad hoc hypotheses, Deutsch looked to physical theory to provide a foundation for the Church–Turing thesis that would be as secure as the status of that physical theory. In particular, Deutsch attempted to define a computational device that would be capable of efficiently simulating an arbitrary physical system. Because the laws of physics are ultimately quantum mechanical, Deutsch was naturally led to consider computing devices based upon the principles of quantum mechanics. These devices, quantum analogues of the machines defined forty-nine years earlier by Turing, led ultimately to the modern conception of a quantum computer used in this book.

受到这个问题的启发，1985年，大卫·多伊奇(David Deutsch)提出，是否可以利用物理定律推导出一个更强大的丘奇-图灵命题。多伊奇没有采用特别的假设，而是把目光转向物理理论，为丘奇和图灵的论文提供一个基础，这个基础和物理理论的地位一样可靠。特别是，Deutsch试图定义一种能够有效地模拟任意物理系统的计算设备。因为物理定律最终是量子力学的，所以多伊奇很自然地考虑基于量子力学原理的计算设备。这些设备，是图灵49年前定义的机器的量子模拟，最终导致了本书中使用的量子计算机的现代概念。

At the time of writing it is not clear whether Deutsch’s notion of a Universal Quantum Computer is sufficient to efficiently simulate an arbitrary physical system. Proving or refuting this conjecture is one of the great open problems of the field of quantum computation and quantum information. It is possible, for example, that some effect of quantum field theory or an even more esoteric effect based in string theory, quantum gravity or some other physical theory may take us beyond Deutsch’s Universal Quantum Computer, giving us a still more powerful model for computation. At this stage, we simply don’t know.

在撰写本文时，尚不清楚Deutsch关于通用量子计算机的概念是否足以有效地模拟任意物理系统。证明或反驳这一猜想是量子计算和量子信息领域的重大开放问题之一。例如，量子场论的某些效应，或者基于弦理论、量子引力或其他一些物理理论的更深奥的效应，可能会让我们超越多伊奇的通用量子计算机，给我们一个更强大的计算模型。在这个阶段，我们根本不知道。

What Deutsch’s model of a quantum computer did enable was a challenge to the strong form of the Church–Turing thesis. Deutsch asked whether it is possible for a quantum computer to efficiently solve computational problems which have no efficient solution on a classical computer, even a probabilistic Turing machine. He then constructed a simple example suggesting that, indeed, quantum computers might have computational powers exceeding those of classical computers.

多伊奇的量子计算机模型对丘奇-图灵理论的强大形式构成了挑战。Deutsch问道，量子计算机是否有可能有效地解决在经典计算机，甚至是概率图灵机上无法有效解决的计算问题。然后，他构建了一个简单的例子，表明量子计算机的计算能力确实可能超过经典计算机。

This remarkable first step taken by Deutsch was improved in the subsequent decade by many people, culminating in Peter Shor’s 1994 demonstration that two enormously important problems – the problem of finding the prime factors of an integer, and the so-called ‘discrete logarithm’ problem – could be solved efficiently on a quantum computer. This attracted widespread interest because these two problems were and still are widely believed to have no efficient solution on a classical computer. Shor’s results are a powerful indication that quantum computers are more powerful than Turing machines, even probabilistic Turing machines. Further evidence for the power of quantum computers came in 1995 when Lov Grover showed that another important problem – the problem of conducting a search through some unstructured search space – could also be sped up on a quantum computer. While Grover’s algorithm did not provide as spectacular a speedup as Shor’s algorithms, the widespread applicability of search-based methodologies has excited considerable interest in Grover’s algorithm.

多伊奇迈出的这一了不起的第一步在随后的十年中得到了许多人的改进，最终在彼得·肖尔1994年的演示中达到高潮，他证明了两个非常重要的问题——寻找整数的质因数的问题，以及所谓的“离散对数”问题——可以在量子计算机上有效地解决。这引起了广泛的兴趣，因为这两个问题在过去和现在仍然被广泛认为在经典计算机上没有有效的解决方案。肖尔的研究结果有力地表明，量子计算机比图灵机更强大，甚至比概率图灵机更强大。1995年，洛夫·格罗弗(Lov Grover)展示了另一个重要问题——在一些非结构化搜索空间中进行搜索的问题——也可以在量子计算机上加速，这进一步证明了量子计算机的强大。虽然Grover的算法没有提供像Shor算法那样惊人的加速，但基于搜索的方法的广泛适用性激发了对Grover算法的相当大的兴趣。

At about the same time as Shor’s and Grover’s algorithms were discovered, many people were developing an idea Richard Feynman had suggested in 1982. Feynman had pointed out that there seemed to be essential difficulties in simulating quantum mechanical systems on classical computers, and suggested that building computers based on the principles of quantum mechanics would allow us to avoid those difficulties. In the 1990s several teams of researchers began fleshing this idea out, showing that it is indeed possible to use quantum computers to efficiently simulate systems that have no known efficient simulation on a classical computer. It is likely that one of the major applications of quantum computers in the future will be performing simulations of quantum mechanical systems too difficult to simulate on a classical computer, a problem with profound scientific and technological implications.

大约在肖尔和格罗弗算法被发现的同时，许多人正在发展理查德·费曼在1982年提出的一个想法。费曼曾指出，在经典计算机上模拟量子力学系统似乎存在本质上的困难，并建议基于量子力学原理建造计算机将使我们能够避免这些困难。在20世纪90年代，几个研究小组开始充实这个想法，表明确实有可能使用量子计算机来有效地模拟在经典计算机上无法有效模拟的系统。未来量子计算机的主要应用之一很可能是对量子力学系统进行模拟，这在经典计算机上很难模拟，这是一个具有深远科学和技术意义的问题。

What other problems can quantum computers solve more quickly than classical com puters? The short answer is that we don’t know. Coming up with good quantum algo rithms seems to be hard. A pessimist might think that’s because there’s nothing quantum computers are good for other than the applications already discovered! We take a differ ent view. Algorithm design for quantum computers is hard because designers face two difficult problems not faced in the construction of algorithms for classical computers. First, our human intuition is rooted in the classical world. If we use that intuition as an aid to the construction of algorithms, then the algorithmic ideas we come up with will be classical ideas. To design good quantum algorithms one must ‘turn off’ one’s classical intuition for at least part of the design process, using truly quantum effects to achieve the desired algorithmic end. Second, to be truly interesting it is not enough to design an algorithm that is merely quantum mechanical. The algorithm must be better than any existing classical algorithm! Thus, it is possible that one may find an algorithm which makes use of truly quantum aspects of quantum mechanics, that is nevertheless not of widespread interest because classical algorithms with comparable performance charac teristics exist. The combination of these two problems makes the construction of new quantum algorithms a challenging problem for the future.

量子计算机还能比传统计算机更快地解决哪些问题?简短的回答是，我们不知道。想出好的量子算法似乎很难。悲观主义者可能会认为，这是因为除了已经发现的应用之外，量子计算机没有任何好处!我们有不同的看法。量子计算机的算法设计之所以困难，是因为设计人员面临着经典计算机算法构建中没有遇到的两个难题。首先，我们人类的直觉根植于古典世界。如果我们用这种直觉作为构建算法的辅助，那么我们提出的算法思想将是经典的思想。为了设计好的量子算法，我们必须在设计过程中至少部分地“关闭”经典直觉，使用真正的量子效应来实现期望的算法结果。其次，要想真正有趣，仅仅设计一个量子力学的算法是不够的。该算法必须优于任何现有的经典算法!因此，人们有可能找到一种算法，它利用了量子力学的真正量子方面，然而，它并没有引起广泛的兴趣，因为存在具有可比性能特征的经典算法。这两个问题的结合使得未来新的量子算法的构建成为一个具有挑战性的问题。

Even more broadly, we can ask if there are any generalizations we can make about the power of quantum computers versus classical computers. What is it that makes quantum computers more powerful than classical computers – assuming that this is indeed the case? What class of problems can be solved efficiently on a quantum computer, and how does that class compare to the class of problems that can be solved efficiently on a classical computer? One of the most exciting things about quantum computation and quantum information is how little is known about the answers to these questions! It is a great challenge for the future to understand these questions better.

更广泛地说，我们可以问一下，我们是否可以对量子计算机与经典计算机的能力做出任何概括。是什么让量子计算机比传统计算机更强大——假设事实确实如此?哪一类问题可以在量子计算机上有效地解决，这一类问题与在经典计算机上有效地解决的问题相比如何?关于量子计算和量子信息最令人兴奋的事情之一是，我们对这些问题的答案知之甚少!更好地理解这些问题对未来是一个巨大的挑战。

信息论

Having come up to the frontier of quantum computation, let’s switch to the history of another strand of thought contributing to quantum computation and quantum information: information theory. At the same time computer science was exploding in the 1940s, another revolution was taking place in our understanding of communication. In 1948 Claude Shannon published a remarkable pair of papers laying the foundations for the modern theory of information and communication.

在谈到量子计算的前沿之后，让我们转到对量子计算和量子信息做出贡献的另一种思想的历史:信息论。在计算机科学在20世纪40年代爆发的同时，我们对通信的理解也发生了另一场革命。1948年，克劳德·香农(Claude Shannon)发表了两篇引人注目的论文，为现代信息与传播理论奠定了基础。

Perhaps the key step taken by Shannon was to mathematically define the concept of information. In many mathematical sciences there is considerable flexibility in the choice of fundamental definitions. Try thinking naively for a few minutes about the following question: how would you go about mathematically defining the notion of an information source? Several different answers to this problem have found widespread use; however, the definition Shannon came up with seems to be far and away the most fruitful in terms of increased understanding, leading to a plethora of deep results and a theory with a rich structure which seems to accurately reflect many (though not all) real-world communications problems.

也许香农迈出的关键一步是用数学方法定义了信息的概念。在许多数学科学中，在选择基本定义方面有相当大的灵活性。试着花几分钟天真地思考下面的问题:你将如何从数学上定义信息源的概念?对于这个问题，有几个不同的答案被广泛使用;然而，就增进理解而言，香农提出的定义似乎是最富有成效的，导致了大量深刻的结果和一个具有丰富结构的理论，似乎准确地反映了许多(尽管不是全部)现实世界的通信问题。

Shannon was interested in two key questions related to the communication of in formation over a communications channel. First, what resources are required to send information over a communications channel? For example, telephone companies need to know how much information they can reliably transmit over a given telephone cable. Second, can information be transmitted in such a way that it is protected against noise in the communications channel?

香农对通过通信信道进行信息通信的两个关键问题感兴趣。首先，通过通信通道发送信息需要哪些资源?例如，电话公司需要知道在给定的电话电缆上他们可以可靠地传输多少信息。第二，信息的传输是否可以使其免受通信信道中的噪声的影响?

Shannon answered these two questions by proving the two fundamental theorems of information theory. The first, Shannon’s noiseless channel coding theorem, quantifies the physical resources required to store the output from an information source. Shan non’s second fundamental theorem, the noisy channel coding theorem, quantifies how much information it is possible to reliably transmit through a noisy communications channel. To achieve reliable transmission in the presence of noise, Shannon showed that error-correcting codes could be used to protect the information being sent. Shannon’s noisy channel coding theorem gives an upper limit on the protection afforded by error correcting codes. Unfortunately, Shannon’s theorem does not explicitly give a practically useful set of error-correcting codes to achieve that limit. From the time of Shannon’s pa pers until today, researchers have constructed more and better classes of error-correcting codes in their attempts to come closer to the limit set by Shannon’s theorem. A sophisti cated theory of error-correcting codes now exists offering the user a plethora of choices in their quest to design a good error-correcting code. Such codes are used in a multitude of places including, for example, compact disc players, computer modems, and satellite communications systems.

香农通过证明信息论的两个基本定理回答了这两个问题。首先，香农的无噪声信道编码定理量化了存储信息源输出所需的物理资源。单农的第二个基本定理，噪声信道编码定理，量化了通过噪声通信信道可能可靠传输的信息量。为了在有噪声的情况下实现可靠的传输，香农展示了可以使用纠错码来保护被发送的信息。香农噪声信道编码定理给出了纠错码保护的上限。不幸的是，香农定理并没有明确地给出一组实用的纠错码来达到这个极限。从香农的论文到今天，研究人员已经构建了更多更好的纠错码类，试图更接近香农定理设定的极限。现在存在一种复杂的纠错代码理论，为用户提供了大量的选择来设计一个好的纠错代码。这样的代码被用于许多地方，例如，包括激光唱机、计算机调制解调器和卫星通信系统。

Quantum information theory has followed with similar developments. In 1995, Ben Schumacher provided an analogue to Shannon’s noiseless coding theorem, and in the process defined the ‘quantum bit’ or ‘qubit’ as a tangible physical resource. However, no analogue to Shannon’s noisy channel coding theorem is yet known for quantum information. Nevertheless, in analogy to their classical counterparts, a theory of quantum error-correction has been developed which, as already mentioned, allows quantum computers to compute effectively in the presence of noise, and also allows communication over noisy quantum channels to take place reliably.

量子纠错

量子信息理论也有了类似的发展。1995年，本·舒马赫提出了香农无噪声编码定理的类比，并在此过程中将“量子比特”或“量子位”定义为有形的物理资源。然而，对于量子信息，尚没有类似香农的噪声信道编码定理。然而，与经典对应物类似，量子纠错理论已经发展起来，如前所述，它允许量子计算机在存在噪声的情况下有效地进行计算，并且还允许在有噪声的量子信道上可靠地进行通信。

Indeed, classical ideas of error-correction have proved to be enormously important in developing and understanding quantum error-correcting codes. In 1996, two groups working independently, Robert Calderbank and Peter Shor, and Andrew Steane, discov-ered an important class of quantum codes now known as CSS codes after their initials. This work has since been subsumed by the stabilizer codes, independently discovered by Robert Calderbank, Eric Rains, Peter Shor and Neil Sloane, and by Daniel Gottesman. By building upon the basic ideas of classical linear coding theory, these discoveries greatly facilitated a rapid understanding of quantum error-correcting codes and their application to quantum computation and quantum information.

事实上，经典的纠错思想在开发和理解量子纠错码方面已经被证明是非常重要的。1996年，罗伯特·卡尔德班克、彼得·肖尔和安德鲁·斯蒂恩这两个独立工作的小组发现了一类重要的量子密码，现在以其首字母缩写命名为CSS代码。这项工作后来被归入稳定器代码，由Robert Calderbank, Eric Rains, Peter Shor和Neil Sloane以及Daniel Gottesman独立发现。通过建立经典线性编码理论的基本思想，这些发现极大地促进了对量子纠错码的快速理解及其在量子计算和量子信息中的应用。

The theory of quantum error-correcting codes was developed to protect quantum states against noise. What about transmitting ordinary classical information using a quantum channel? How efficiently can this be done? A few surprises have been discovered in this arena. In 1992 Charles Bennett and Stephen Wiesner explained how to transmit two classical bits of information, while only transmitting one quantum bit from sender to receiver, a result dubbed superdense coding.

量子纠错码理论是为了保护量子态不受噪声的影响而发展起来的。用量子信道传输普通的经典信息怎么样?如何有效地做到这一点?在这个舞台上发现了一些惊喜。1992年，查尔斯·贝内特和斯蒂芬·威斯纳解释了如何传输两个经典比特的信息，而从发送者到接收者只传输一个量子比特，这种结果被称为超密集编码。

分布式量子计算

Even more interesting are the results in distributed quantum computation. Imagineyou have two computers networked, trying to solve a particular problem. How muchcommunication is required to solve the problem? Recently it has been shown that quantum computers can require exponentially less communication to solve certain problemsthan would be required if the networked computers were classical! Unfortunately, as yetthese problems are not especially important in a practical setting, and suffer from someundesirable technical restrictions. A major challenge for the future of quantum computation and quantum information is to find problems of real-world importance for whichdistributed quantum computation offers a substantial advantage over distributed classicalcomputation.

更有趣的是分布式量子计算的结果。想象一下，你有两台联网的计算机，试图解决一个特定的问题。解决问题需要多少沟通?最近有研究表明，与传统的联网计算机相比，量子计算机解决某些问题所需的通信可以以指数方式减少!不幸的是，到目前为止，这些问题在实际设置中并不是特别重要，并且受到一些不受欢迎的技术限制。量子计算和量子信息的未来面临的一个主要挑战是找到分布式量子计算比分布式经典计算具有实质性优势的现实世界重要问题。

Let’s return to information theory proper. The study of information theory begins withthe properties of a single communications channel. In applications we often do not dealwith a single communications channel, but rather with networks of many channels. Thesubject of networked information theory deals with the information carrying propertiesof such networks of communications channels, and has been developed into a rich andintricate subject.

让我们回到信息论本身。信息论的研究始于单一通信信道的特性。在应用程序中，我们通常不处理单个通信通道，而是处理由多个通道组成的网络。网络信息论这门学科研究的是这种通信信道网络的信息承载特性，已经发展成为一门内容丰富而复杂的学科。

By contrast, the study of networked quantum information theory is very much in itsinfancy. Even for very basic questions we know little about the information carrying abilities of networks of quantum channels. Several rather striking preliminary results havebeen found in the past few years; however, no unifying theory of networked informationtheory exists for quantum channels. One example of networked quantum informationtheory should suffice to convince you of the value such a general theory would have.Imagine that we are attempting to send quantum information from Alice to Bob througha noisy quantum channel. If that channel has zero capacity for quantum information,then it is impossible to reliably send any information from Alice to Bob. Imagine insteadthat we consider two copies of the channel, operating in synchrony. Intuitively it is clear(and can be rigorously justified) that such a channel also has zero capacity to send quantum information. However, if we instead reverse the direction of one of the channels, asillustrated in Figure 1.1, it turns out that sometimes we can obtain a non-zero capacityfor the transmission of information from Alice to Bob! Counter-intuitive properties likethis illustrate the strange nature of quantum information. Better understanding the information carrying properties of networks of quantum channels is a major open problemof quantum computation and quantum information.

相比之下，网络量子信息理论的研究还处于起步阶段。即使对于非常基本的问题，我们对量子信道网络的信息承载能力也知之甚少。在过去的几年里，已经发现了几个相当惊人的初步结果;然而，对于量子信道，没有统一的网络信息理论。一个网络量子信息理论的例子应该足以让你相信这样一个通用理论的价值。想象一下，我们正试图通过一个有噪声的量子信道将量子信息从Alice发送给Bob。如果该信道的量子信息容量为零，那么就不可能可靠地从Alice向Bob发送任何信息。想象一下，我们考虑两个同步操作的通道副本。直觉上很清楚(并且可以严格证明)，这样的信道也没有能力发送量子信息。然而，如果我们将其中一个通道的方向相反，如图1.1所示，事实证明，有时我们可以获得从Alice到Bob传输信息的非零容量!像这样的反直觉性质说明了量子信息的奇怪本质。更好地理解量子信道网络的信息承载特性是量子计算和量子信息的一个重大开放问题。

Figure 1.1. Classically, if we have two very noisy channels of zero capacity running side by side, then the combinedchannel has zero capacity to send information. Not surprisingly, if we reverse the direction of one of the channels,we still have zero capacity to send information. Quantum mechanically, reversing one of the zero capacity channelscan actually allow us to send information!

图1.1。通常，如果我们有两个非常嘈杂的零容量信道并排运行，那么合并后的信道发送信息的容量为零。毫不奇怪，如果我们改变其中一个信道的方向，我们仍然没有能力发送信息。从量子力学的角度来看，逆转其中一个零容量通道实际上可以让我们发送信息!

私钥加密

Let’s switch fields one last time, moving to the venerable old art and science of cryptography. Broadly speaking, cryptography is the problem of doing communication or computation involving two or more parties who may not trust one another. The bestknown cryptographic problem is the transmission of secret messages. Suppose two partieswish to communicate in secret. For example, you may wish to give your credit card number to a merchant in exchange for goods, hopefully without any malevolent third partyintercepting your credit card number. The way this is done is to use a cryptographicprotocol. We’ll describe in detail how cryptographic protocols work later in the book, butfor now it will suffice to make a few simple distinctions. The most important distinctionis between private key cryptosystems and public key cryptosystems.

让我们最后一次切换领域，转向古老的密码学艺术和科学。从广义上讲，密码学是涉及两个或更多可能互不信任的各方进行通信或计算的问题。最著名的密码学问题是秘密消息的传输。假设双方希望秘密通信。例如，您可能希望将信用卡号码提供给商家以换取商品，希望没有任何恶意的第三方拦截您的信用卡号码。这样做的方法是使用加密协议。我们将在本书后面详细描述加密协议是如何工作的，但现在只需要做一些简单的区分就足够了。私钥密码系统和公钥密码系统之间最重要的区别。

The way a private key cryptosystem works is that two parties, ‘Alice’ and ‘Bob’, wishto communicate by sharing a private key, which only they know. The exact form of thekey doesn’t matter at this point – think of a string of zeroes and ones. The point is thatthis key is used by Alice to encrypt the information she wishes to send to Bob. AfterAlice encrypts she sends the encrypted information to Bob, who must now recover theoriginal information. Exactly how Alice encrypts the message depends upon the privatekey, so that to recover the original message Bob needs to know the private key, in orderto undo the transformation Alice applied.

私钥密码系统的工作方式是，双方“Alice”和“Bob”希望通过共享一个只有他们知道的私钥来进行通信。在这一点上，键的确切形式并不重要——想象一个由0和1组成的字符串。关键是Alice使用这个密钥来加密她希望发送给Bob的信息。在alice加密之后，她将加密的信息发送给Bob, Bob现在必须恢复原始信息。Alice加密消息的确切方式取决于私钥，因此为了恢复原始消息，Bob需要知道私钥，以便撤销Alice应用的转换。

Unfortunately, private key cryptosystems have some severe problems in many contexts.The most basic problem is how to distribute the keys? In many ways, the key distributionproblem is just as difficult as the original problem of communicating in private – amalevolent third party may be eavesdropping on the key distribution, and then use theintercepted key to decrypt some of the message transmission.

不幸的是，私钥密码系统在许多情况下存在一些严重的问题。最基本的问题是如何分配密钥?在许多方面，密钥分发问题就像最初的私下通信问题一样困难——恶意的第三方可能会窃听密钥分发，然后使用截获的密钥解密一些消息传输。

One of the earliest discoveries in quantum computation and quantum information wasthat quantum mechanics can be used to do key distribution in such a way that Alice andBob’s security can not be compromised. This procedure is known as quantum cryptography or quantum key distribution. The basic idea is to exploit the quantum mechanicalprinciple that observation in general disturbs the system being observed. Thus, if there isan eavesdropper listening in as Alice and Bob attempt to transmit their key, the presenceof the eavesdropper will be visible as a disturbance of the communications channel Aliceand Bob are using to establish the key. Alice and Bob can then throw out the key bitsestablished while the eavesdropper was listening in, and start over. The first quantumcryptographic ideas were proposed by Stephen Wiesner in the late 1960s, but unfortunately were not accepted for publication! In 1984 Charles Bennett and Gilles Brassard,building on Wiesner’s earlier work, proposed a protocol using quantum mechanics todistribute keys between Alice and Bob, without any possibility of a compromise. Sincethen numerous quantum cryptographic protocols have been proposed, and experimentalprototypes developed. At the time of this writing, the experimental prototypes are nearingthe stage where they may be useful in limited-scale real-world applications.

量子计算和量子信息的最早发现之一是，量子力学可以用来进行密钥分发，这样Alice和bob的安全性就不会被破坏。这个过程被称为量子密码学或量子密钥分发。其基本思想是利用量子力学原理，即观察一般会干扰被观察的系统。因此，如果窃听者在Alice和Bob试图传输他们的密钥时进行监听，窃听者的存在将作为对Alice和Bob用来建立密钥的通信信道的干扰而可见。然后，Alice和Bob可以扔掉窃听者窃听时建立的密钥，然后重新开始。第一个量子密码学的想法是由Stephen Wiesner在20世纪60年代末提出的，但不幸的是没有被接受发表!1984年，Charles Bennett和Gilles Brassard在Wiesner早期工作的基础上，提出了一个使用量子力学在Alice和Bob之间分发密钥的协议，没有任何妥协的可能性。从那时起，已经提出了许多量子加密协议，并开发了实验原型。在撰写本文时，实验原型已接近可用于有限规模的实际应用的阶段。

公钥加密

The second major type of cryptosystem is the public key cryptosystem. Public keycryptosystems don’t rely on Alice and Bob sharing a secret key in advance. Instead, Bobsimply publishes a ‘public key’, which is made available to the general public. Alicecan make use of this public key to encrypt a message which she sends to Bob. Whatis interesting is that a third party cannot use Bob’s public key to decrypt the message!Strictly speaking, we shouldn’t say cannot. Rather, the encryption transformation ischosen in a very clever and non-trivial way so that it is extremely difficult (though notimpossible) to invert, given only knowledge of the public key. To make inversion easy, Bobhas a secret key matched to his public key, which together enable him to easily performthe decryption. This secret key is not known to anybody other than Bob, who can thereforebe confident that only he can read the contents of Alice’s transmission, to the extent thatit is unlikely that anybody else has the computational power to invert the encryption,given only the public key. Public key cryptosystems solve the key distribution problemby making it unnecessary for Alice and Bob to share a private key before communicating.

第二种主要类型的密码系统是公钥密码系统。公钥密码系统不依赖于Alice和Bob事先共享密钥。相反，bob只是发布了一个“公钥”，公众可以使用它。alice可以使用这个公钥加密她发送给Bob的消息。有趣的是，第三方不能使用Bob的公钥来解密消息!严格地说，我们不应该说不能。相反，加密转换是以一种非常聪明且不平凡的方式选择的，因此在只知道公钥的情况下，很难(尽管不可能)进行反转。为了使反转变得容易，bob有一个与他的公钥匹配的密钥，这使他能够轻松地执行解密。除了鲍勃之外，任何人都不知道这个密钥，因此，鲍勃可以确信只有他才能读取爱丽丝传输的内容，以至于任何人都不太可能具有仅给定公钥的计算能力来反转加密。公钥密码系统通过使Alice和Bob在通信之前不必共享私钥来解决密钥分发问题。

Rather remarkably, public key cryptography did not achieve widespread use until themid-1970s, when it was proposed independently by Whitfield Diffie and Martin Hellman,and by Ralph Merkle, revolutionizing the field of cryptography. A little later, RonaldRivest, Adi Shamir, and Leonard Adleman developed the RSA cryptosystem, whichat the time of writing is the most widely deployed public key cryptosystem, believed tooffer a fine balance of security and practical usability. In 1997 it was disclosed that theseideas – public key cryptography, the Diffie–Hellman and RSA cryptosystems – wereactually invented in the late 1960s and early 1970s by researchers working at the Britishintelligence agency GCHQ.

值得注意的是，公钥密码学直到20世纪70年代中期才得到广泛使用，当时它由Whitfield Diffie和Martin Hellman以及Ralph Merkle独立提出，彻底改变了密码学领域。不久之后，RonaldRivest、Adi Shamir和Leonard Adleman开发了RSA密码系统，在撰写本文时，它是部署最广泛的公钥密码系统，被认为提供了安全性和实用性的良好平衡。1997年，有消息披露，公钥密码学、迪菲-赫尔曼和RSA密码系统实际上是由英国情报机构GCHQ的研究人员在20世纪60年代末和70年代初发明的。

公钥加密的安全性

The key to the security of public key cryptosystems is that it should be difficult toinvert the encryption stage if only the public key is available. For example, it turns outthat inverting the encryption stage of RSA is a problem closely related to factoring.Much of the presumed security of RSA comes from the belief that factoring is a problemhard to solve on a classical computer. However, Shor’s fast algorithm for factoring ona quantum computer could be used to break RSA! Similarly, there are other public keycryptosystems which can be broken if a fast algorithm for solving the discrete logarithmproblem – like Shor’s quantum algorithm for discrete logarithm – were known. Thispractical application of quantum computers to the breaking of cryptographic codes hasexcited much of the interest in quantum computation and quantum information.

公钥密码系统安全性的关键在于，如果只有公钥可用，则很难反转加密阶段。例如，RSA加密阶段的反转是一个与因式分解密切相关的问题。RSA假定的安全性很大程度上来自于这样一种信念，即分解是一个难以在传统计算机上解决的问题。然而，肖尔在量子计算机上的快速分解算法可以用来破解RSA!类似地，如果已知解决离散对数问题的快速算法(如Shor的离散对数量子算法)，也可以破解其他公钥密码系统。量子计算机在密码破解中的实际应用激发了人们对量子计算和量子信息的极大兴趣。

We have been looking at the historical antecedents for quantum computation andquantum information. Of course, as the field has grown and matured, it has sproutedits own subfields of research, whose antecedents lie mainly within quantum computationand quantum information.

我们一直在研究量子计算和量子信息的历史先例。当然，随着该领域的发展和成熟，它已经萌芽了自己的研究子领域，其前身主要是量子计算和量子信息。

Perhaps the most striking of these is the study of quantum entanglement. Entanglement is a uniquely quantum mechanical resource that plays a key role in many of themost interesting applications of quantum computation and quantum information; entanglement is iron to the classical world’s bronze age. In recent years there has been a tremendous effort trying to better understand the properties of entanglement consideredas a fundamental resource of Nature, of comparable importance to energy, information,entropy, or any other fundamental resource. Although there is as yet no complete theoryof entanglement, some progress has been made in understanding this strange property ofquantum mechanics. It is hoped by many researchers that further study of the propertiesof entanglement will yield insights that facilitate the development of new applications inquantum computation and quantum information.

也许其中最引人注目的是对量子纠缠的研究。纠缠是一种独特的量子力学资源，在量子计算和量子信息的许多最有趣的应用中起着关键作用;纠缠是铁到古典世界的青铜器时代。近年来，人们做出了巨大的努力，试图更好地理解纠缠的性质，纠缠被认为是自然的基本资源，与能量、信息、熵或任何其他基本资源同等重要。虽然至今还没有完整的纠缠理论，但是在理解量子力学的这个奇怪特性方面已经取得了一些进展。许多研究人员希望，对纠缠特性的进一步研究将产生有助于开发量子计算和量子信息新应用的见解。