量子位 评奖_什么是量子位？

Welcome to Introduction to Quantum Computing. I am your guide, Associate Professor Chris Ferrie, a researcher in the UTS Centre for Quantum Software and Information. This is Lecture 2. It would probably be a good idea to have read the previous Lectures before continuing.

欢迎来到《量子计算入门》。 我是您的指南， UTS量子软件和信息中心的研究员Chris Ferrie副教授 。 这是第二讲 。 在继续之前先阅读以前的讲座可能是一个好主意。

What did you learn last week?Last week you reviewed the necessary maths to get started on quantum computing. You also had a high-level introduction to the field. Most importantly, you had your first taste of quantum programming by creating “hello quantum world” in several quantum programming languages.What will you learn this week?This week you will meet the qubit — the basic unit of quantum information. You will perform computations using Dirac notation, the secret weapon of the quantum ninja. By applying gates and measurements to your qubit, you will know how to perform a single qubit quantum computation.What will you be able to do at the end of this week 2?At the end of this module, you should be able to answer these questions:What is quantum information and how is it represented?
What is a quantum state?
How are quantum logic operations represented?
What is a quantum circuit and how is it analysed?
How do you “read” quantum information?

信息不是物理的 (Information is not physical)

Information is an overloaded word with many meanings depending on context. Here we will take a computational view. An abstract model of a computation looks like this: Input → Compute → Output. The input is one from a finite set of possibilities. We can always uniquely identify which it is with bits, answers to yes/no questions. A sequence of n bits can represent 2ⁿ different possibilities. So, no matter what the input to a computation is, we can always think about it in terms of bits without losing anything. The same goes for the output. There are many concrete representations of bits, the most common being sequences of 0’s and 1’s.

信息是一个重载的单词，具有取决于上下文的多种含义。 在这里，我们将进行计算。 计算的抽象模型如下所示：输入→计算→输出。 输入来自一组有限的可能性。 我们始终可以用位唯一地标识它是什么，回答是/否的问题。 n位序列可以表示2种不同的可能性。 因此，无论计算的输入是什么，我们始终可以按位考虑它，而不会丢失任何内容。 输出也是如此。 有很多具体的位表示形式，最常见的是0和1的序列。

We can phrase computational questions within this model. For example, we can ask what sorts of operations preserve bits. This would tell us all the “allowed” computations that turn sequences of input bits into output bits. But we might also ask which, if any, algorithms can be implemented efficiently. If every algorithm required a number of steps that grew exponentially as the input size grew, computer science would be a non-starter. But even if there were algorithms that can be implemented efficiently (there are of course), it would probably still be the case that performing the computation by hand would be impractical. No one wants to add two 1 million digit numbers, even if “addition” is efficient. So we might then ask if a physical machine can be built to perform very large computations for us. And indeed they can — that’s digital electronics.

我们可以在此模型中表达计算问题。 例如，我们可以询问哪种操作保留位。 这将告诉我们所有将输入位序列转换为输出位的“允许”计算。 但我们也可能会问，哪些算法(如果有)可以有效实施。 如果每种算法都需要随着输入大小的增长而呈指数增长的许多步骤，那么计算机科学将是一个起步。 但是，即使有可以有效实施的算法(当然也有)， 手动执行计算仍然是不切实际的情况。 即使“加法”是有效的，也没有人愿意添加两个一百万个数字。 因此，我们可能会问是否可以构建一台物理机器来为我们执行非常大的计算。 而且确实可以，那就是数字电子产品。

Now let’s ask a similar set of questions, but not of bits. Let’s imagine an abstract model of computation where the inputs are sequences of complex vectors. Let’s take the simplest non-trivial case of 2-dimensional vectors.

现在，让我们问类似的问题集，而不是问题。 让我们想象一个抽象的计算模型，其中输入是复数向量的序列。 让我们以二维向量的最简单的平凡情况为例。

The notation |𝜓⟩ anticipates qubits, but for now they are just abstract vectors, like our abstract bits from before. So we ask first, what sorts of operations preserve these vectors — that is, take input vectors and produce output vectors. Let’s constrain the search to those things that are linear and preserve the length of the vector as well. Now we ask if any of these operations can be implemented efficiently in the sense that the number of elementary steps doesn’t grow exponentially as the number of input vectors increases. Of course we can create our own efficient algorithms by design (they might not do anything interesting, but we can make them). Even then, though, the fact remains that as the input size increases, our capacity to calculate things “by hand” becomes limited. In this case, those same digital machines could do the computation for us, but it turns out that they cannot do so efficiently. So, the question becomes, can a physical machine be built to perform the computation for us efficiently?

| 𝜓⟩符号表示可以预见的量子位，但目前它们只是抽象矢量，就像我们以前的抽象位一样。 因此，我们首先要问，哪些类型的操作会保留这些向量-即，采用输入向量并产生输出向量。 让我们将搜索限制在那些线性的东西上，并保留向量的长度。 现在，我们要问这些操作中的任何一个是否可以有效地实现，即基本步数不会随着输入向量数量的增加而呈指数增长。 当然，我们可以通过设计来创建自己的高效算法(它们可能没有做任何有趣的事情，但是我们可以使它们成为现实)。 即使这样，事实仍然是，随着输入大小的增加，我们“手动”计算事物的能力受到限制。 在这种情况下，那些相同的数字机器可以为我们做计算，但是事实证明它们不能高效地做。 因此，问题就变成了，可以建造一台物理机器来为我们高效地执行计算吗？

And, indeed, it seems the answer is yes! That machine would be a quantum computer. The 2-dimensional vectors can be physically represented by quantum mechanical systems, such as atomic energy levels, spin, polarisation, and many more. But that is in some sense beside the point. The point is, whereas we think of bits as “information”, this new model of computation with vectors has a new type of computational information. Since it can be represented with 2-dimensional quantum systems, we call them “quantum bits”, or qubits.

而且，的确，答案似乎是肯定的！ 那台机器将是一台量子计算机。 二维向量可以用量子力学系统来物理表示，例如原子能级，自旋，极化等。 但这在某种意义上是不重要的。 关键是，尽管我们将位视为“信息”，但是这种带有向量的新计算模型具有一种新型的计算信息。 因为它可与2维量子系统来表示，我们称之为“量子位”，或量子位 。

Wow. That was a long way to go just to get to qubits, which I’m sure you’ve already heard of. The reason I want you to think about this perspective is that it avoids the usual unnecessary attachment of qubits to physical things, which very quickly leads to confusion since we cannot really think concretely about “quantum things” having definite and objective properties. In any case, if you have been following, “quantum information” is specified in qubits — 2-dimensional complex vectors with unit norm.

哇。 仅仅为了获得量子比特，还有很长的路要走，我相信您已经听说过。 我想让您考虑这种观点的原因是，它避免了量子位通常与物理事物的不必要连接，这很快就会导致混乱，因为我们无法真正具体地考虑具有确定性和客观属性的“量子事物”。 无论如何，如果您一直在遵循，则在量子位(具有单位范数的二维复数向量)中指定“量子信息”。

Next time you hear bits are 0 or 1 and qubits are 0 and 1 at the same, think about how illogical that statement is and how utterly useless it will be if you tried to use it in your task to program a quantum computer.

下次当您听到比特是0或1，而量子比特是0 和 1时 ，请考虑一下该语句多么不合逻辑，以及如果您尝试在任务中使用它来对量子计算机进行编程，那么它将完全无用。

Bits are variables that can take one of two values. The state of bit is what particular value it has. Similarly qubits are 2-dimensional complex vectors with unit norm. The state of a qubit is the particular vector value the qubit has. Sometimes we are sloppy with the language and use these things interchangeably. Sorry about that..

位是可以采用两个值之一的变量。 位的状态是它具有的特定值。 类似地，量子位是具有单位范数的二维复数向量。 量子位的状态是该量子位具有的特定矢量值。 有时我们对语言草率，并且可以交替使用这些东西。 对于那个很抱歉..

狄拉克的遗产：请真正的量子比特站起来吗 (Dirac’s legacy: would the real quantum bits please stand up)

When you ask a quantum information theorist what powers quantum computation you’ll get a myriad of answers — superposition, parallelism, entanglement, coherence, yadda yadda yadda. I will say, what power quantum computation — at least as far as it can get now — is Dirac notation.

当您问量子信息理论家什么能为量子计算提供动力时，您将得到无数的答案-叠加，并行，纠缠，相干，yadda yadda yadda。 我要说的是，至少到现在为止，功率量子计算是狄拉克符号。

Dirac notation, also called bra–ket notation, is a way to write objects in linear algebra. We already used it above when we introduced qubits with |𝜓⟩. This is called a “ket”. The vertical bar “|” and the right angle bracket “⟩” come together to form the ket. The thing on the inside is just a label. The ket | ⟩ represents a vector in a particular complex linear space. Other notation used for the same thing uses underlines, arrows, or bold notation.

狄拉克(Dirac)表示法，也称为Braket表示法，是一种在线性代数中写入对象的方法。 在上面用| 𝜓⟩引入qubit时，我们已经在使用它。 这被称为“ ket”。 竖线“ |” 和直角括号“⟩”一起形成ket。 里面的东西只是一个标签。 地毯| ⟩表示特定复数线性空间中的向量。 用于同一事物的其他符号使用下划线，箭头或粗体符号。

However you denote your vectors, you need to be able to distinguish them from the coefficients. Kets are perfect for this. They also make it convenient to enumerate bases in binary. We choose binary labeling to associate with the standard one when taking the concrete vector space of column vectors with complex valued entries. This is called the computational basis in quantum computing. When we expand vectors in this basis and find more than one non-zero component, we call that linear combination of basis vectors a superposition. In addition to |0⟩ and |1⟩, we have labels for some other special vectors, we call them the “plus state” |+⟩, “minus state” |−⟩ and some others.

无论您表示矢量，都需要能够将它们与系数区分开。 Kets非常适合此。 它们还使枚举二进制数的基数变得方便。 当采用具有复杂值条目的列向量的具体向量空间时，我们选择二进制标记与标准标记关联。 这称为量子计算中的计算基础 。 当我们在此基础上展开向量并找到多个非零分量时，我们将基础向量的线性组合称为叠加。 除了|0⟩和|1⟩之外，我们还有一些其他特殊矢量的标签，我们称它们为“正状态” | +⟩，“负状态” | −⟩等。

In this canonical representation, used extensively in introductions to linear algebra, row vectors are created with the conjugate transpose, which swaps rows for columns and complex conjugates each entry of the vector. These are not vectors, but can be placed in one-to-one correspondence with them. In Dirac notation, they are called “bras”, and are written as ⟨ |, with an appropriate label as needed.

在此规范表示形式(广泛用于线性代数入门)中，使用共轭转置来创建行向量， 共轭转置将行交换为列，并将复数共轭转换为向量的每个项。 这些不是向量，但可以与它们一一对应。 在狄拉克(Dirac)表示法中，它们称为“ bras”，并用⟨|表示，并根据需要添加适当的标签。

Abstractly, ⟨𝜙| maps any vector |𝜓⟩ to the inner product between |𝜙⟩ and |𝜓⟩, which in Dirac notation is ⟨𝜙|𝜓⟩. The computational basis is an orthonormal basis, which facilitates quick calculations in Dirac notation. For example, the norm of the vector |𝜓⟩ can be calculated as follows.

，| 将任何矢量| 𝜓⟩映射到| 𝜙⟩和| 𝜓⟩之间的内积，用Dirac表示法是⟨𝜙 | 𝜓⟩。 计算基础是正交基础，这有助于Dirac表示法的快速计算。 例如，向量||的范数可以如下计算。

更改量子位 (To change a qubit)

Those pictures we drew above are called circuits, both in classical computation and quantum computation. Each line, or wire, represents a qubit which can be labeled at various points to identify what the state of the qubit is. Circuits represent how the qubit state changes, one step at time, from left to right. The “steps” are unitary matrices, those that preserve the norm of the vector. In the context of computation, these are called gates.

我们上面绘制的那些图片在经典计算和量子计算中都称为电路 。 每一行中，或导线 ，表示可以在不同的点被标记以识别该量子位的状态是什么的一个量子位。 电路表示量子位状态如何从左到右一步变化。 “步骤”是unit矩阵，那些矩阵保持向量的范数。 在计算的上下文中，这些称为门。

There are lots of important examples and you will get to know them quite intimately. Some quantum gates act just as logical operations would act on the binary digits representing the computational basis states. In some sense, these don’t add much beyond classical computation — though, they are important parts of more complicated quantum circuits.

有很多重要的例子，您将非常熟悉它们。 一些量子门的作用就像逻辑运算将作用在代表计算基础状态的二进制数字上一样。 从某种意义上讲，它们并没有增加经典计算之外的内容，尽管它们是更复杂的量子电路的重要组成部分。

Other gates, like the relative phase gate, modify superpositions. By adding minus signs, components of vectors can start to cancel. This phenomenon is called inference and without it, the computation could be easily simulated with a classical algorithm. Sometimes it is said that quantum computation is just classical probabilistic computation with negative numbers.

其他门(例如相对相位门)会修改叠加。 通过添加减号，向量的成分可以开始消除。 这种现象称为推论 ，如果没有推论 ，则可以使用经典算法轻松模拟计算。 有时据说量子计算只是带有负数的经典概率计算。

None of the above gates generate superpositions starting from basis states. Probably the most important gate is the Hadamard gate, which generates the states |+⟩ and |−⟩ from |0⟩ and |1⟩. A great deal of quantum algorithms, and all that you will meet, start by changing the canonical |0⟩ state to a “uniform superposition”. We will see next week how this generalises to multiple qubit states.

以上所有门都不从基态生成叠加。 可能最重要的门是Hadamard门 ，它从|0⟩和|1⟩生成状态| +⟩和| −⟩。 大量的量子算法以及您将要遇到的所有一切，都始于将规范的|0⟩状态更改为“均匀叠加”。 下周我们将看到这如何推广到多个量子位状态。

生于统治 (Born to rule)

The last topic we need to discuss is the infamous quantum measurement. Regardless of what the state of the qubit |𝜓⟩ = 𝛼|0⟩ + 𝛽|1⟩ is in, “looking” at the qubit results in only ever |0⟩ or |1⟩. In physics, “looking” is called measurement, and is the result of fundamental probabilistic nature of quantum mechanics, which continues to cause much consternation to philosophers. For us, this is the model we must use, and so we will use it without much fuss. Of course, the result isn’t completely random and depends on the current state of the qubit we intend to measure. If the qubit state is |𝜓⟩ = 𝛼|0⟩ + 𝛽|1⟩, then the probability that |0⟩ will be observed is Pr(0|𝜓) = |𝛼|² and the similarly Pr(1|𝜓) = |𝛽|². Of course one of the alternatives must occur and so |𝛼|² + |𝛽|² = 1, which is where the unit norm and its preservation come from. When analysing a general quantum circuit, the numbers you need to know at the end of calculation are these probabilities.

我们需要讨论的最后一个主题是臭名昭著的量子测量。 不管量子位| 𝜓⟩ = 𝛼 |0⟩+ 𝛽 |1⟩处于什么状态，“看”量子位只会得到|0⟩或|1⟩。 在物理学中，“看”被称为测量，它是量子力学的基本概率性质的结果，这继续引起哲学家的极大震惊。 对于我们来说，这是我们必须使用的模型，因此我们将在使用它时不必大惊小怪。 当然，结果不是完全随机的，取决于我们要测量的量子位的当前状态。 如果量子位状态为| 𝜓⟩ = 𝛼 |0⟩+ 𝛽 |1⟩，那么|0⟩将被观测到的概率为Pr(0 | 𝜓)= | 𝛼 |²，类似地，Pr(1 | 𝜓) = | 𝛽 |²。 当然，必须出现一种替代方法，因此| 𝛼 |²+ | ​​𝛽 |²= 1，这是单位范数及其保存的来源。 分析一般的量子电路时，在计算结束时您需要知道的数字就是这些概率。

The very last point is that the state of the qubit after measurement is definitely in the outcome observed. For example, if a qubit in state |𝜓⟩ was measured and |0⟩ observed, then the state after the measurement is |0⟩. This is independent of the initial state |𝜓⟩ and thus “looking” at quantum information is not a reversible process. This abrupt change is sometimes called collapse. And that is what I’m going to do if I write one more wor…

最后一点是，测量后的量子位状态肯定存在于观察到的结果中。 例如，如果测量了状态| 𝜓⟩的量子位并观察到|0⟩，则测量后的状态为|0⟩。 这与初始状态| 𝜓⟩无关，因此“看”量子信息不是可逆的过程。 这种突然的变化有时称为崩溃。 如果我再写一份工作，那将是我要做的……

资源资源 (Resources)

A good reference for all the standard quantum gates used in quantum information is the Wikipedia entry on Quantum logic gates.

有关量子信息中使用的所有标准量子门的一个很好的参考，是有关量子逻辑门的Wikipedia条目。

We have played a little fast and loose with “linear algebra”. The topic is much deeper than matrices and vectors. Research in quantum information science often requires a more abstract understanding of linear algebra. If you are up for a challenge, John Watrous’s text The Theory of Quantum Information is an excellent resource for learning the tools of the quantum information scientist.

我们使用“线性代数”玩得有点快。 这个话题比矩阵和向量要深得多。 量子信息科学的研究通常需要对线性代数有更抽象的理解。 如果您要挑战，John Watrous的著作《量子信息论》是学习量子信息科学家工具的绝佳资源。