1.2 量子比特

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1.2 量子比特

The bit is the fundamental concept of classical computation and classical information. Quantum computation and quantum information are built upon an analogous concept, the quantum bit, or qubit for short. In this section we introduce the properties of single and multiple qubits, comparing and contrasting their properties to those of classical bits.

比特是经典计算和经典信息的基本概念。量子计算和量子信息是建立在一个类似的概念，量子比特，或简称量子位。在本节中，我们将介绍单量子位和多量子位的特性，并将其与经典比特的特性进行比较和对比。

What is a qubit? We’re going to describe qubits as mathematical objects with certain specific properties. ‘But hang on’, you say, ‘I thought qubits were physical objects.’ It’s true that qubits, like bits, are realized as actual physical systems, and in Section 1.5 and Chapter 7 we describe in detail how this connection between the abstract mathematical point of view and real systems is made. However, for the most part we treat qubits as abstract mathematical objects. The beauty of treating qubits as abstract entities is that it gives us the freedom to construct a general theory of quantum computation and quantum information which does not depend upon a specific system for its realization.

什么是量子位?我们将量子位描述为具有某些特定属性的数学对象。“等等，”你会说，“我以为量子位是物理对象。的确，像比特一样，量子位被实现为实际的物理系统，在第1.5节和第7章中，我们详细描述了抽象数学观点和实际系统之间的联系是如何建立的。然而，在大多数情况下，我们将量子位视为抽象的数学对象。将量子比特视为抽象实体的美妙之处在于，它使我们可以自由地构建量子计算和量子信息的一般理论，而不依赖于特定的系统来实现。

What then is a qubit? Just as a classical bit has a state – either 0 or 1 – a qubit also has a state. Two possible states for a qubit are the states $|0\rang$ and $|1\rang$ , which as you might guess correspond to the states 0 and 1 for a classical bit. Notation like ‘| ’ is called the Dirac notation, and we’ll be seeing it often, as it’s the standard notation for states in quantum mechanics. The difference between bits and qubits is that a qubit can be in a state other than $|0\rang$ or $|1\rang$ . It is also possible to form linear combinations of states, often called superpositions:

\tag{1.1} |ψ\rang = α |0\rang + β |1\rang .

The numbers $α$ and $β$ are complex numbers, although for many purposes not much is lost by thinking of them as real numbers. Put another way, the state of a qubit is a vector in a two-dimensional complex vector space. The special states $|0\rang$ and $|1\rang$ are known as computational basis states, and form an orthonormal basis for this vector space.

数字 $α$ 和 $β$ 是复数，尽管在许多情况下，将它们视为实数并没有多少损失。换句话说，量子比特的状态是二维复向量空间中的一个向量。特殊状态 $|0\rang$ 和 $|1\rang$ 被称为计算基状态，它们构成了这个向量空间的一个标准正交基。

We can examine a bit to determine whether it is in the state 0 or 1. For example, computers do this all the time when they retrieve the contents of their memory. Rather remarkably, we cannot examine a qubit to determine its quantum state, that is, the values of α and β. Instead, quantum mechanics tells us that we can only acquire much more restricted information about the quantum state. When we measure a qubit we get either the result 0, with probability $|α|^2$ , or the result 1, with probability $|β|^2$ . Naturally, $|α|^2 + |β|^2 = 1$ , since the probabilities must sum to one. Geometrically, we can interpret this as the condition that the qubit’s state be normalized to length 1. Thus, in general a qubit’s state is a unit vector in a two-dimensional complex vector space.

我们可以检查一个比特以确定它是处于状态0还是1。例如，计算机在检索内存内容时总是这样做。值得注意的是，我们不能检查一个量子位来确定它的量子态，即α和β的值。相反，量子力学告诉我们，我们只能获得更有限的关于量子态的信息。当我们测量一个量子位时，我们得到的结果要么是0，概率是 $|α|^2$ ，要么是1，概率是 $|β|^2$ 。自然地， $|α|^2 + |β|^2 = 1$ ，因为概率之和必须为1。几何上，我们可以将其解释为量子位的状态归一化为长度为1的条件。因此，一般来说，量子比特的状态是二维复向量空间中的单位向量。

This dichotomy between the unobservable state of a qubit and the observations we can make lies at the heart of quantum computation and quantum information. In most of our abstract models of the world, there is a direct correspondence between elements of the abstraction and the real world, just as an architect’s plans for a building are in correspondence with the final building. The lack of this direct correspondence in quantum mechanics makes it difficult to intuit the behavior of quantum systems; however, there is an indirect correspondence, for qubit states can be manipulated and transformed in ways which lead to measurement outcomes which depend distinctly on the different properties of the state. Thus, these quantum states have real, experimentally verifiable consequences, which we shall see are essential to the power of quantum computation and quantum information.

量子比特的不可观察状态和我们可以观察到的这种二分法是量子计算和量子信息的核心。在我们对世界的大多数抽象模型中，抽象元素与现实世界之间存在着直接的对应关系，就像建筑师对建筑物的规划与最终建筑的对应关系一样。由于量子力学中缺乏这种直接对应关系，因此很难直观地了解量子系统的行为;然而，有一种间接的对应关系，因为量子比特状态可以被操纵和转换，从而导致明显依赖于状态不同属性的测量结果。因此，这些量子态具有真实的、实验可验证的结果，我们将看到这对量子计算和量子信息的力量至关重要。

The ability of a qubit to be in a superposition state runs counter to our ‘common sense’ understanding of the physical world around us. A classical bit is like a coin: either heads or tails up. For imperfect coins, there may be intermediate states like having it balanced on an edge, but those can be disregarded in the ideal case. By contrast, a qubit can exist in a continuum of states between $|0\rang$ and $|1\rang$ – until it is observed. Let us emphasize again that when a qubit is measured, it only ever gives ‘0’ or ‘1’ as the measurement result – probabilistically. For example, a qubit can be in the state

\tag{1.2} \frac1{\sqrt{2}}|0\rang+\frac1{\sqrt{2}}|1\rang

量子比特处于叠加态的能力与我们对周围物理世界的“常识”理解背道而驰。经典比特就像硬币:要么正面朝上，要么反面朝上。对于不完美的硬币，可能存在中间状态，例如在边缘上保持平衡，但在理想情况下可以忽略这些状态。相比之下，量子位可以存在于 $|0\rang$ 和 $|1\rang$ 之间的连续状态中，直到它被观察到。让我们再次强调，当一个量子位被测量时，它只会给出“0”或“1”作为测量结果-概率。例如，一个量子位可以处于这样的状态：

\tag{1.2} \frac1{\sqrt{2}}|0\rang+\frac1{\sqrt{2}}|1\rang

which, when measured, gives the result 0 fifty percent $(|\frac1{\sqrt{2}}|^2)$ of the time, and the result 1 fifty percent of the time. We will return often to this state, which is sometimes denoted $|+\rang$ .

当测量时，结果为0的概率是50％ $(|\frac1{\sqrt{2}}|^2)$ ，结果为1的概率是50％。我们经常会回到这种状态，有时表示为 $|+\rang$ 。

Despite this strangeness, qubits are decidedly real, their existence and behavior extensively validated by experiments (discussed in Section 1.5 and Chapter 7), and many different physical systems can be used to realize qubits. To get a concrete feel for how a qubit can be realized it may be helpful to list some of the ways this realization may occur: as the two different polarizations of a photon; as the alignment of a nuclear spin in a uniform magnetic field; as two states of an electron orbiting a single atom such as shown in Figure 1.2. In the atom model, the electron can exist in either the so-called ‘ground’ or ‘excited’ states, which we’ll call $|0\rang$ and $|1\rang$ , respectively. By shining light on the atom, with appropriate energy and for an appropriate length of time, it is possible to move the electron from the $|0\rang$ state to the $|1\rang$ state and vice versa. But more interestingly, by reducing the time we shine the light, an electron initially in the state $|0\rang$ can be moved ‘halfway’ between $|0\rang$ and $|1\rang$ , into the $|+\rang$ state.

alt text
Figure 1.2. Qubit represented by two electronic levels in an atom.

图1.2。由原子中的两个电子能级表示的量子位。

Naturally, a great deal of attention has been given to the ‘meaning’ or ‘interpretation’ that might be attached to superposition states, and of the inherently probabilistic nature of observations on quantum systems. However, by and large, we shall not concern ourselves with such discussions in this book. Instead, our intent will be to develop mathematical and conceptual pictures which are predictive.

自然，人们对叠加态的“意义”或“解释”以及量子系统观测的固有概率性质给予了大量关注。然而，总的来说，我们不打算在本书中讨论这些问题。相反，我们的目的将是发展数学和概念性的图像预测。

One picture useful in thinking about qubits is the following geometric representation.
下面的几何表示法对思考量子比特很有用。

Because $|α|^2 + |β|^2 = 1$ , we may rewrite Equation (1.1) as
因为 $|α|^2 + |β|^2 = 1$ ，我们可以将式(1.1)重写为

\tag{1.3} |\psi\rang=e^{i\gamma}(\cos\frac\theta2|0\rang+e^{i\varphi}\sin\frac\theta2|1\rang)

where $\theta$ , $\psi$ and $\gamma$ are real numbers. In Chapter 2 we will see that we can ignore the factor of $e^{i\gamma}$ out the front, because it has no observable effects, and for that reason we can effectively write
其中 $\theta$ , $\psi$ 和 $\gamma$ 是实数。在第二章中，我们将看到我们可以忽略前面的 $e^{i\gamma}$ 因子，因为它没有可观察到的影响，因此我们可以有效地写

\tag{1.4} |\psi\rang=\cos\frac\theta2|0\rang+e^{i\varphi}\sin\frac\theta2|1\rang

The numbers $\theta$ and $\psi$ define a point on the unit three-dimensional sphere, as shown in Figure 1.3. This sphere is often called the Bloch sphere; it provides a useful means of visualizing the state of a single qubit, and often serves as an excellent testbed for ideas about quantum computation and quantum information. Many of the operations on single qubits which we describe later in this chapter are neatly described within the Bloch sphere picture. However, it must be kept in mind that this intuition is limited because there is no simple generalization of the Bloch sphere known for multiple qubits.
数字 $\theta$ 和 $\psi$ 在单位三维球面上定义了一个点，如图1.3所示。这个球通常被称为“布洛赫球”;它提供了一种可视化单个量子比特状态的有用方法，并且经常作为量子计算和量子信息思想的优秀测试平台。我们将在本章后面描述的对单个量子位的许多操作都在布洛赫球图中得到了简洁的描述。然而，必须记住，这种直觉是有限的，因为对于已知的多个量子位，布洛赫球没有简单的推广。

Figure 1.3. Bloch sphere representation of a qubit.

How much information is represented by a qubit? Paradoxically, there are an infinite number of points on the unit sphere, so that in principle one could store an entire text of Shakespeare in the infinite binary expansion of $\theta$ . However, this conclusion turns out to be misleading, because of the behavior of a qubit when observed. Recall that measurement of a qubit will give only either 0 or 1. Furthermore, measurement changes the state of a qubit, collapsing it from its superposition of $|0\rang$ and $|1\rang$ to the specific state consistent with the measurement result. For example, if measurement of $|+\rang$ gives 0, then the post-measurement state of the qubit will be $|0\rang$ . Why does this type of collapse occur? Nobody knows. As discussed in Chapter 2, this behavior is simply one of the fundamental postulates of quantum mechanics. What is relevant for our purposes is that from a single measurement one obtains only a single bit of information about the state of the qubit, thus resolving the apparent paradox. It turns out that only if infinitely many identically prepared qubits were measured would one be able to determine $α$ and $β$ for a qubit in the state given in Equation (1.1).

一个量子位代表了多少信息?矛盾的是，单位球面上有无限多的点，所以原则上，人们可以将莎士比亚的整部作品存储在 $\theta$ 的无限二进制展开中。然而，由于观察到量子位的行为，这个结论被证明是误导性的。回想一下，量子位的测量只会给出0或1。此外，测量改变了量子比特的状态，将其从 $|0\rang$ 和 $|1\rang$ 的叠加态坍缩到与测量结果一致的特定状态。例如，如果 $|+\rang$ 的测量结果为0，那么量子位元的后测量状态将为 $|0\rang$ 。为什么会发生这种类型的崩溃?没有人知道。正如第二章所讨论的，这种行为只是量子力学的基本假设之一。与我们的目的相关的是，从一次测量中，人们只能获得有关量子比特状态的一位信息，从而解决了表面上的悖论。事实证明，只有测量无限多个相同准备的量子位，人们才能确定方程(1.1)中给定状态下量子位的 $α$ 和 $β$ 。

But an even more interesting question to ask might be: how much information is represented by a qubit if we do not measure it? This is a trick question, because how can one quantify information if it cannot be measured? Nevertheless, there is something conceptually important here, because when Nature evolves a closed quantum system of qubits, not performing any ‘measurements’, she apparently does keep track of all the continuous variables describing the state, like $α$ and $β$ . In a sense, in the state of a qubit, Nature conceals a great deal of ‘hidden information’. And even more interestingly, we will see shortly that the potential amount of this extra ‘information’ grows exponentially with the number of qubits. Understanding this hidden quantum information is a question that we grapple with for much of this book, and which lies at the heart of what makes quantum mechanics a powerful tool for information processing.

但一个更有趣的问题可能是:如果我们不测量一个量子位，它能代表多少信息?这是一个棘手的问题，因为如果无法测量，人们如何量化信息?然而，这里有一些概念上重要的东西，因为当自然进化出一个封闭的量子比特系统，不进行任何“测量”时，她显然会跟踪描述状态的所有连续变量，比如 $α$ 和 $β$ 。从某种意义上说，在量子比特的状态下，大自然隐藏了大量的“隐藏信息”。更有趣的是，我们很快就会看到，这些额外“信息”的潜在数量随着量子比特的数量呈指数级增长。理解这种隐藏的量子信息是我们在本书中努力解决的一个问题，也是量子力学成为信息处理强大工具的核心所在。