量子計算的樂趣：簡明導論 | 周末讀書

《量子計算的樂趣：簡明導論》

作者：杰德·布羅迪（Jed Brody）

普林斯頓大學出版社

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推薦語

想進入量子信息的世界嗎？從這本書開始吧！

相信很多想進入量子信息學，特別是量子計算領域的朋友，都有過這樣的經歷：一咬牙，一跺腳，把Nielson & Chuang的《Quantum Computing and Quantum Information》（《量子計算與量子信息》抱起來就準備一頓啃，不啃完不罷休。結果是讀完第一章氣焰就打消一大半，讀到第三章已然被各種“前景知識”狂轟濫炸得筋疲力盡，往下再一翻，發現一行字“Part II Quantum Computation”，然后就繃不住了：原來之前經歷的只是前菜！在托著腮幫子快速亂翻了一遍這個“Part II”的內容，看到那一大堆圈圈符號括號之后，再看看頁碼——這才哪到哪！然后掩卷，一聲長嘆，出去透透氣。再然后，這本書就成功加入了你那長長的“讀過，但從沒讀完過的”書單。

以上是我當年啃那本書的感受。如此繪聲繪色，正因印象深刻。我是學凝聚態物理出身，彼時正值我剛剛接觸這個新領域，我們需要解決的問題是：如何用超導電路來實現這些量子信息處理？該如何將這些量子信息學的東西與我們做的實驗（在極低溫下要怎么擼薛定諤的貓）建立起聯系？于是我又接觸到另一本書——Alex. Zagoskin的《Quantum Engineering：Theory and Design of Quantum Coherent structures》，這是一本介紹如何在超導或二維電子氣中構建人工量子體系、如何操控和測量這些人工量子體系的書。雖然同樣充斥著大量的公式，但顯然這本書更合我的胃口，以至于我下定決心要將這本書引入中國：在忍住惡心手擼了兩千多個公式之后，這本書的中文版《量子工程學》在中國科學技術大學出版社的幫助下已經出版。

而我今天要點評的其實是另一本書《The Joy of Quantum Computing》。兩個月前，朋友推薦給我，并郵寄了一本英文原版書給我。看到書名，我覺得這大概是老外新出的一本量子計算科普書吧，量子計算這么難講，倒想看看老外是怎么講清楚的。翻開一看，里面的公式推導還真不少！心中暗自嘟嚷：這么多的公式符號，哪來的Joy！接下來這本書就在我的案頭一趴好幾個星期。

年底很多的事情應付完之后，略有閑暇，我又想起了這本書。于是在一個晚上我翻開了它。坐在椅子上看完第一章，我轉成“臥趴”姿勢在床上又接連看到第五章，直到近一點，觸發了我給自己規定的晚睡極限，才終于放下。接下來幾天，我很快將這本書看完——畢竟它只有一百多頁，而Nielson & Chuang的書，這頁數才剛過第一部分！我居然這么快看完了一本英文原版的、量子計算專業書籍！簡直成就感拉滿好不好！更好玩的是，我近期在“199天手搓量子計算機”系列直播中講解了Shor算法和Grover算法，主要的參考源，正是這本書。

這本書成功避免進入我的“讀過，但從沒讀完過的”書單的原因，當然不至于這本書薄那么簡單。另一個更為重要的原因還是作者的貼心。盡管仍有很多的公式與符號，但書中所用到的數學，幾乎都是高中生就能掌握的知識，僅在倒數第二章簡述了一些關于矩陣的知識。這種對數學工具的精準把控，總算讓我能夠在幾乎不需要借助草稿紙就能跟上其中的數理邏輯。不過我還是要在這里提醒躍躍欲試的讀者：這并不意味著這本書的知識水準不足，不少內容，比如貝爾不等式檢驗、Shor算法質因數分解等不少的推導，仍需來回的推敲方能跟上其中的邏輯。作者Jed Brody能夠在不上特別高等的數學的情況下把這些量子信息處理的經典案例講明白，更得益于作者的耐心——用近乎手把手的方式在講解知識。作者是一位有多年教學經驗的教師，能夠站在讀者，或者說學生的角度來教知識，這真是一位良師應有的素養。

接下來，我要對書中的內容做一些“劇透”了，怕影響自己觀書體驗的讀者請謹慎！第一章，作者從一個信息學的經典問題——信息加密與傳輸入手。與傳統信息學中常用的角色——Alice、Bob不同，作者引入了兩位荷馬史詩中的英雄人物：奧德修斯和他的妻子佩內洛普，并假設這位美麗的妻子是一位高明的量子工程師。在介紹完傳統的加密方法之后，便開始介紹如何利用“量子比特”進行密碼的協商和傳輸：如何通過改變兩組正交的測量基進行信息的保留和擦除，這正是現代量子密碼學——量子密鑰分發的基礎。在這一章中，同時也將量子比特的基本概念介紹給了讀者。

接下來兩章，作者從一個最簡單的量子算法（正如編程語言中的“Hello World！”程序）——Deutsch算法及其多比特推廣Deutsch-Jozsa算法入手，講解了量子信息處理的基本邏輯操作：單比特門和兩比特門。在量子算法中，最重要、也是最難理解的邏輯操作，就是多比特門操作。多比特門操作可以將多個量子比特糾纏在一起，而正是這種量子糾纏的能力，讓量子計算有了超越經典計算的能力。Deutsch是平行宇宙的信徒，在他看來，量子計算就是在平行宇宙中進行計算，否則無法解釋為什么量子計算能夠帶來“指數級飛升”的強大能力。當然，這是他的觀點，而大多數人跟他持有不同觀點。無論如何，我們不應糾纏于這些解釋層面的問題，而應專注于行為層面。兩比特門，能夠讓其作用于的兩個量子比特的狀態不再獨立，必須當成一個整體來看。

既然講了糾纏，再接下來兩章，作者進一步介紹了另一個酷炫的操作——量子瞬態傳輸。這也是一個比較容易迷惑的行為，借助一對糾纏的量子比特，量子態可以瞬間傳遞，不依賴于時間和空間。這似乎是違反常理的，另一個影響極為深遠，且至今仍被無數實驗驗證為正確的理論——相對論告訴我們，信息、能量、物質，都不可能以超過光速傳遞。于是作者引入了另一個重要的定理：量子不可克隆定理。正是因為（未知）量子態不可克隆，決定了量子瞬態傳輸無法以超光速的方式傳遞信息，而我們更不可能回到過去。

接下來就是大名鼎鼎的貝爾不等式了。這是一個極為高明的檢驗：通過一個定量的值，就能夠判斷一個狀態是定域（經典）的還是非定域（量子）的。這一章是全書難度較高的部分，邏輯鏈較長，一不小心，就會迷失其中。不過只要讀者足夠認真，就會發現，其中所用到的數學仍然是大學生以下的，至少沒有矩陣和微積分！

有不少人在接觸到量子計算的時候會產生這樣的問題：既然量子計算號稱是“通用計算”，那它能做加減乘除嗎？答案是可以的。不過做加減乘除當然無法展現量子計算的神奇，這些操作是量子計算的一些平庸信息處理，且經典計算就能夠做得很好。不過，在下一章，作者還是展示了用量子計算機如何做“加法”，這里引入了一個著名的三比特門——Toffli門，它實際上是一個“控制-控制-非門”。盡管沒有經典計算的加法真值表那么直觀，但量子計算是能夠做加法的，其他減、乘、除等基本運算，都可以快速轉換成加法運算，這在經典計算的代數邏輯中就已經證明過了。

有了前面的鋪墊，接下來可以介紹兩個著名的算法：Grover算法和Shor算法了。實際上，通過這本書，我對Grover算法有了更深的理解：它不僅是一個無結構數據搜索這么簡單，事實上它是一個非常普適的算法，可以將各種NP問題映射成一個Grover搜索問題，你甚至可以用它來解方程！作者更是給出了一個如何在IBM的真實量子云平臺上求解“X+1=3”的案例。為了講清楚Shor算法，作者從QFT/IQFT（量子傅立葉變換）講起，再到量子相位估計算法，再到Shor算法。每個算法，作者都貼心地給出了計算案例，通過手把手帶著讀者推演計算過程，把這些精妙的算法核心邏輯展現給讀者。

再往后的三章，分別是對量子糾錯的快速預覽（真的是快速預覽！幾頁紙的介紹，讓我意猶未盡。我希望作者在未來的計劃中擴充這些內容）、量子操作的矩陣表示以及矩陣的基本操作、密度矩陣以及純態、混態的概念等。

既然作為一本入門書，作者貼心地在附錄中給出了擴展閱讀的推薦書目，并對它們一一做了點評。最后還留了習題！

總之，這是一本讓我能夠快速通讀的量子計算與量子信息學的入門書籍。作者不是泛泛地介紹各種量子信息處理中的酷炫概念，而是給出了嚴謹的推導。更難能可貴的是，對于嚴肅的讀者，只要具備高中數學知識，就能夠應對其中80%的內容了。讀完這本書有兩個好處，首先是我的信心大增，我居然啃完了一本嚴肅的量子信息書籍！其次當然就是，為進一步閱讀和學習更為深入和系統的量子信息學打下一個良好的基礎。我如果是先看的這本書，再去看Nielson & Chuang的書，沒準我能再多堅持幾章！

量子計算和量子信息處理是一門嚴肅的交叉科學。那些覺得這是在騙人的讀者，不妨先看完這本書再下定論，有可能會改變你的看法。那些有志于從事量子計算和量子信息領域科學和技術研究的年輕人，特別是大學生們，你們有福了，可以先看這本書，再去啃大部頭的《量子計算與量子信息》，不像我當年，上來就啃大部頭，結果弄得灰心喪氣鼻青臉腫。其他對量子科技感興趣的讀者，我也推薦閱讀這本書，從而建立起對一些量子信息處理核心概念的正確認知，從此再也不會受到“量子美容”、“量子算命”、“量子速讀”的忽悠。

——北京量子信息科學研究院研究員、相干科技創始人 金貽榮

章節試讀

Chapter 1

Forging the Quantum Key

There are a lot of reasons to keep data secret, accessible only to intended viewers. Examples include credit card numbers intended only for a seller, medical information intended only for health care providers, military intelligence intended only for allies, proprietary industrial processes intended only for collaborators, and photos from a meeting of the Nude Headstand Enthusiasts Club intended only for fellow club members (you said the site was password protected, Steve).

One way to keep data secure is to seal it in a bank vault, or in a safe wrapped with padlocked chains buried in a cobra-infested island in a sea swarming with sharks. The trouble with this kind of security is that data often needs to be shared. So we need a convenient way to share data remotely with intended recipients, and only with intended recipients.

All electronic data, whether text, images, videos, or anything else, is stored as combinations of 0’s and 1’s. 0 and 1 represent two different voltages in electronic circuits. The two voltages could be 0 volts and 1 volt, but that’s not the only choice. The two voltages could be 0 volts and 5 volts; we simply use 1 to represent 5 volts. The two voltages could be -4 volts and 3.5 volts; we arbitrarily pick one of these to call 0, and the other to call 1. The point is that we can analyze the 0’s and 1’s in data without paying any attention to the physical details of how they’re stored.

In fact, 0’s and 1’s can represent more than just voltages. The 0’s and 1’s in bar codes and QR codes are black and white stripes or squares. The 0’s and 1’s in CDs and DVDs are different thicknesses of a layer of plastic. As long as there are two, and only two, distinct physical conditions, we have 0’s and 1’s, and we can do classical computation.

Our electronic devices know how to convert 0’s and 1’s to videos, images, sounds, text, and so on. The details of this conversion are not our focus. We wish only to securely transmit 0’s and 1’s from a sender to a recipient, over a perilous distance fraught with eavesdroppers. In fact, we assume that eavesdroppers will be greedily poring over our data transmissions, combing through our 0’s and 1’s for valuable secrets.

So we have little choice, then, but to encrypt our data. We transmute our sequence of 0’s and 1’s into meaningless gibberish, a cipher, which only the intended recipient can decipher. There are many ways of achieving this. Near the end of our journey, we will meet the RSA cryptosystem, which is vulnerable to the quantum attack of Shor’s algorithm. For now, we will consider a simpler cryptosystem: the private, or secret, key.

It’s convenient to give names to the sender and receiver of data. The traditional names are Alice and Bob. But I think Alice and Bob deserve a vacation. So as Alice and Bob settle into their cozy rooms overlooking waves booming against a rocky shore silvered by moonlight, let’s meet our new heroes, Odysseus and Penelope. Odysseus is rightly regarded as the most cunning of warriors. Less well known is that his wife Penelope is the most cunning of quantum engineers.

A 0 or 1 is called a bit. For each bit of the message that Penelope wants to send to Odysseus, she needs a secret key bit. The message bit is combined with the key bit to form an encrypted bit, according to these rules:

0 combined with 0 is 0.

0 combined with 1 is 1.

1 combined with 1 is 0.

In other words, if the message bit and the key bit are the same, the encrypted bit is 0. If the message bit and the key bit are different, the key bit is 1. There’s a mathematical symbol, , called “exclusive OR,” that represents these rules:

Let’s represent the message bit by M, the key bit by K, and the encrypted bit by E. So E = M K. Penelope sends encrypted bit E to Odysseus. How can Odysseus recover the message bit M? Odysseus knows the key bit K; this is the secret information known only to Odysseus and Penelope. To recover the message bit M, all Odysseus has to do is combine the encrypted bit E with the key bit K according to the same rule: E K. Since E = M K, Odysseus is really computing E K = M K K. Now, K is either 0 or 1. Since 0 0 = 0 and 1 1 = 0,

whether K is 0 or 1. So Odysseus computes M K K = M 0. Because M is either 0 or 1, and because 0 0 = 0 and 1 0 = 1,

So Odysseus recovers the message bit, but only because he knows the key bit. A potential eavesdropper like Hector doesn’t know the key bit and cannot compute the message bit even if he glimpses the encrypted bit.

Let’s take an example. Suppose Penelope wants to send Odysseus the message 0010. Before Odysseus began his voyage, with masts creaking and 10-foot waves slapping the hull, he and Penelope agreed to use the secret key 1011. Penelope combines each bit of the message with the corresponding bit of the secret key to obtain the cipher, as shown in Table 1.1. The first encrypted bit is 0 1 = 1, the second is 0 0 = 0, the third is 1 1 = 0, and the fourth is 0 1 = 1. So the cipher is 1001, which Penelope sends to Odysseus. Hector spies on this message but can’t make heads or tails of it because he doesn’t know the secret key.

Table 1.1

Now, Odysseus receives the cipher 1001, and he combines each of its bits with the corresponding bit of the secret key, 1011, as shown in Table 1.2. The first bit becomes 1 1 = 0, the second bit becomes 0 0 = 0, the third bit becomes 0 1 = 1, and the fourth bit becomes 1 1 = 0. Thus, Odysseus has restored the (lurid and poignant) message, 0010.

Table 1.2

So far, there’s nothing quantum about this. Suppose, however, that Penelope and Odysseus decide they need to periodically change their secret key to keep Hector from guessing it. How can Penelope and Odysseus establish a secret key remotely? This is where Penelope’s quantum genius comes in.

Three thousand years ahead of her time, Penelope has perfected a single-atom version of an experiment that normally requires a beam of atoms. (The real experiment, with a beam of atoms, is called the Stern-Gerlach experiment.) Penelope launches silver atoms through a magnetic field and observes that each atom is deflected toward either the magnet’s north pole or south pole; no atom passes straight through. If the magnetic field is vertical, each atom is deflected either UP or DOWN. If the magnetic field is horizontal, each atom is deflected either RIGHT or LEFT.

Penelope observes that if an atom is deflected UP and then immediately enters another vertical magnetic field, it will again be deflected UP:

We could send the atom through a hundred vertical magnetic fields in a row, and it would get deflected UP every time. The atom apparently has an enduring property that determines its behavior in vertical magnetic fields.

Similarly, an atom deflected DOWN is again deflected DOWN when it immediately enters another vertical magnetic field. If an atom is deflected RIGHT in a horizontal magnetic field, it is again deflected RIGHT in another horizontal magnetic field; the same rule applies to an atom deflected LEFT.

Penelope further observes that if an atom is deflected UP, and then enters a horizontal magnetic field, it is equally likely to be deflected RIGHT or LEFT. If the atom then enters a vertical magnetic field, it is no longer certain to go UP; it is equally likely to go DOWN:

The horizontal magnetic field apparently erased the atom’s vertical-field property: The atom lost its reliable UP-ness and has become just as likely to deflect DOWN.

Similarly, an atom initially deflected DOWN is equally likely to be deflected RIGHT or LEFT in a horizontal magnetic field, after which it is equally likely to go UP and DOWN in a vertical magnetic field. An atom initially deflected either RIGHT or LEFT is equally likely to be deflected UP or DOWN in a vertical magnetic field, after which it is equally likely to go either direction in a horizontal field, regardless of its initial deflection.

This is 100% of the quantum physics we need to understand quantum key distribution. To summarize, a silver atom deflected in a magnetic field will be deflected the same way if it subsequently enters a magnetic field in the same direction—if it hasn’t been in any other magnetic fields. If the atom enters a magnetic field perpendicular to the field it initially passed through, it has a 50% chance of going either way, and if it later enters a magnetic field in the same direction as the original field it traversed, it has a 50% chance of going either way.

In effect, when a silver atom passes through a magnetic field, it is endowed with one bit of information about how it behaves in that field: UP or DOWN in a vertical field, and RIGHT or LEFT in a horizontal field. But when the atom passes through a field perpendicular to the original field, the original information is erased and replaced with information about how the atom behaves in the new field.

So, Penelope’s plan is this. She will represent a 0 by a silver atom initially deflected either UP or RIGHT. She will represent a 1 by a silver atom initially deflected either DOWN or LEFT. She launches the selected atom to Odysseus, across the azure tides of sea-roiling Poseidon. Odysseus randomly sets his magnetic field either vertical or horizontal, and he observes the deflection of the atom.

For example, suppose Penelope wants to transmit a 1 by sending Odysseus a DOWN atom. Suppose Odysseus chooses to set his magnetic field vertical. Then, he will observe the atom deflected DOWN. He knows that Penelope uses DOWN to represent 1, so he guesses that Penelope wanted to transmit a 1.

However, if Odysseus instead chooses a horizontal magnetic field for this atom, it equally likely deflects RIGHT or LEFT. If it deflects RIGHT, Odysseus guesses incorrectly that Penelope wanted to transmit a 0.

Suppose that the choices and results for the first four atoms are as shown in Table 1.3. After Odysseus measures all the atoms, he and Penelope reveal the directions of their magnetic fields in all cases. They don’t need to encode this announcement; eavesdroppers can do no harm now. Odysseus discards his guesses whenever he chose a different magnetic field direction than Penelope. So in the example in Table 1.3, he discards his guesses for the second and fourth atoms. He knows that his guesses for the first and third atoms were correct, so he and Penelope have now established two bits of their secret key: 11. They repeat with as many atoms as necessary to generate a sufficiently long key.

Table 1.3

Now, how do the laws of quantum physics guarantee that their key is secure? In other words, how can they be certain that no eavesdropper copied the data as it traveled from Penelope to Odysseus? If Hector tries to intercept the silver atom, he has to choose whether to set his magnetic field horizontal or vertical, just as Odysseus does. He observes the atom and passes it on to Odysseus, but his attempt at espionage is thwarted by quantum physics. Let’s see how.

Consider this sequence of choices and outcomes:

Penelope chooses a vertical magnetic field, and Hector chooses a horizontal magnetic field. The silver atom is equally likely to deflect RIGHT or LEFT in Hector’s magnetic field. Odysseus has chosen the same magnetic field as Penelope, but the silver atom, having been deflected RIGHT, is equally likely to deflect UP and DOWN. If it deflects UP, Odysseus’s guess, 0, differs from Penelope’s bit, even though they chose the same magnetic field direction.

To detect Hector’s meddling, Penelope and Odysseus sacrifice some of their key bits by revealing them to each other (and unavoidably to any eavesdropper monitoring their communication). If their key bits disagree, when they chose the same magnetic field direction, they must conclude that an eavesdropper meddled with their attempt to generate a secret key. So they have to abandon this attempt at a secret key, and maybe try again later.

Penelope and Odysseus have to compare a sufficiently large number of key bits, perhaps 10, to have a high probability of detecting an eavesdropper. This is because the eavesdropper corrupts only 25% of the key bits. Half of the time, the eavesdropper chooses the same magnetic field direction as Penelope. In this case, the eavesdropper observes the silver atom without changing it and passes it unaltered on to Odysseus. The other half of the time, the eavesdropper chooses a different magnetic field direction than Penelope. This effectively erases the information about deflection in the direction of Penelope’s magnetic field. So when Odysseus sets his magnetic field in the same direction as Penelope’s, he’s only 50% likely to re-create Penelope’s original deflection. In summary: Half of the time, Hector chooses a different magnetic field direction than Penelope, and when this occurs, the key bit is corrupted half of the time. Half of one half is 25%, the rate of key bit corruption.

If Penelope and Odysseus compare a subset of their key bits and find that they all agree, they conclude that no eavesdropper was present, and all their other key bits remain secret and secure. (They have to discard the bits they reveal because an eavesdropper could be eavesdropping on this communication, even if no eavesdropper intercepted the silver atoms.) This is a successful instance of quantum key distribution. Quantum key distribution can’t stop eavesdroppers from eavesdropping, but it reveals the presence of an eavesdropper if there is one.

Now, let’s rewrite UP, DOWN, RIGHT, and LEFT in the language of quantum computing. Let’s use the symbol to represent a silver atom deflected UP. This symbol, , is called a ket, which is the second syllable of bracket. is often pronounced “ket zero.” We’ll use to represent an atom deflected DOWN. and are two possible states of a quantum bit, or qubit.

Remember that classical bits, 0 and 1, can represent two voltages in a circuit, or black and white stripes in a bar code, or different thicknesses of a plastic layer in CDs and DVDs. Similarly, a qubit can be constructed of many different physical systems. A silver atom is only one possibility, and not a very feasible one; not all quantum engineers are as cunning as Penelope. A qubit can be made of a photon, such that and represent two different polarization directions. In IBM’s quantum processors that we’ll use throughout this book, and represent two different states of a superconducting circuit. In fact, we’d rather not specify how our qubits are constructed: We want to establish rules and algorithms that work for any qubits, however they are made.

I once asked Matthias Steffen, IBM’s chief quantum architect, how to think about the and states of a superconducting circuit. He told me that he’d given up on visualizing it. So let’s follow the lead of IBM’s chief quantum architect. We will establish rules that allow us to predict the results when qubits are measured. But we will not stumble far along the rocky path of wondering what qubits are doing when we’re not measuring them.

Whereas a classical bit is either 0 or 1, a qubit can be in some combination of and , written . α and β are called probability amplitudes, and they are related to the probabilities of different measurements. Now, there are different ways of measuring qubits, analogous to the different magnetic field directions for the silver atoms. If we do a measurement that results in either and , this is called a measurement in the computational basis. (The computational basis is sometimes called the z basis by association with the vertical, or z, direction.) The probability of measuring is , and the probability of measuring is . The total probability of measuring something is 1, which means

This condition is called normalization. If α and β are real numbers, then and . However, α and β are allowed to be complex numbers. In this case, , where is the comlex conjugate of . We will work exclusively with real numbers for most of our journey.

We assigned UP = and DOWN = . What about RIGHT and LEFT? Atoms deflected RIGHT and LEFT are equally likely to subsequently deflect UP or DOWN in a vertical magnetic field. This means and should both be 1/2. We’ll choose

and

When we write
), the probability amplitude of is , and the probability amplitude of is .

It’s convenient to define

and

In the language of qubits, we can now say that deflection in a horizontal magnetic field is a case of a measurement that yields either or . This is called a measurement in the x basis by association with the horizontal, or x, direction.

We can combine Eqs. (1.4a) and (1.4b) to write and in terms of and . The ket symbols can be manipulated exactly like algebraic symbols such as x and y. We can add Eqs. (1.4a) and (1.4b) together, to find . Solving for , we obtain

using
. Similarly, subtracting Eq. (1.4b) from Eq. (1.4a) yields. Solving for ,

Whereas Eq. (1.4) gives probability amplitudes of and , Eq. (1.5) gives probability amplitudes of and : probability amplitudes for measurements in the x basis. Remembering to square probability amplitudes to find probabilities, we see that a qubit in state or is equally likely to be found in or when measured in the x basis. This is a generalization of the fact that a silver atom deflected UP or DOWN is equally likely to deflect RIGHT or LEFT when entering a horizontal magnetic field.

When a qubit is measured, the state becomes whatever was measured. For example, if a qubit, initially in state , is measured in the x basis, it is equally likely to become or . Effectively, its original state is erased and replaced by the new one. This is a generalization of the rule we saw for the silver atoms: If an atom is initially deflected UP or DOWN, and then traverses a horizontal magnetic field, it will deflect RIGHT or LEFT without retaining any information about whether it had been deflected UP or DOWN. This is sometimes called the collapse of the state due to measurement.

Actually, this effect of measurement is not significant in most of the later chapters. Measurements will occur only at the end of our quantum circuits. And we will almost always measure in the computational basis, so the result of measuring a qubit will be either or . In fact, the result of the measurement will be recorded as a classical bit, 0 or 1. All we have to remember going forward is that if a qubit in state is measured, then the probability of measuring 0 is , and the probability of measuring 1 is .

To review, let’s repeat our example with Penelope, Hector, and Odysseus, but now using ket notation:

Penelope's initial state is , which equals , given by Eq. (1.5b). Hector measures this qubit in the x basis, so the result will be or . The probability amplitude of is , and the probability amplitude of is. We square these amplitudes to determine probabilities, and we find that the probability of measuring is 1/2, and so is the probability of measuring . Hector's measurement happens to yield .

Next, Odysseus measures this qubit in the computational basis, so we have to write in terms of computational basis states:
, as given in Eq. (1.4a). The probability amplitude isfor both and , sois the probability of obtaining either result. Odysseus happens to find , which is different from the state that Penelope sent him. If they share these facts with each other, they will know that Hector has meddled with their qubit.