藍色情懷

編結量子計算

Computing with Quantum Knots

《科學人》雜誌五月號　
撰文╱柯林斯（Graham P. Collins）翻譯／高湧泉

量子位元的特殊性質，令科學家對量子計算興致勃勃。如果能再結合數學上拓撲的概念，或可創造出新的量子計算方式。
A machine based on bizarre particles called anyons that represents a calculation as a set of braids in spacetime might be a shortcut to practical quantum computation.

我們可以利用特殊粒子世界線（軌跡）的絞辮來執行量子計算，這種計算對於普通（古典）電腦來說是無能為力的。我們所用的特殊粒子存在於一種稱為二維電子氣的流體中。

鼓吹量子電腦的人向我們保證，這種電腦可以執行一般電腦無能為力的計算。在這一類只能依賴量子電腦的 計算任務當中，有些具有很重要的應用價值。舉例而言，只要有電腦能夠在合理的時間之內，將很大的數字分解成其組成因數，就可以破解某些廣為使用的密碼系 統。幾乎所有用於保護高度敏感資料的密碼系統，都會被某個量子算則所破解。

Quantum computers promise to perform calculations believed to be impossible for ordinary computers. Some of those calculations are of great real-world importance. For example, certain widely used encryption methods could be cracked given a computer capable of breaking a large number into its component factors within a reasonable length of time. Virtually all encryption methods used for highly sensitive data are vulnerable to one quantum algorithm or another.

量子電腦為什麼有更強的計算能力？答案在於量子電腦所處理的資訊是以量子位元代表，而非普通位元。一 個普通的古典位元只能是0或1；標準的微晶片架構很嚴謹地在執行這種古典二分法。但是相對而言，一個量子位元則可以處於一種所謂的「疊加狀態」，這是一種 可以讓0的一部份與1的一部份共存的狀態。我們可以把可能的量子位元狀態看成是球面上的一點。北極代表古典狀態1，南極是0，所有介於兩者之間的點則代表 0或1的所有可能疊加狀態（見2003年1月號〈奇妙的量子棋步〉）。量子電腦之所以具有特殊能力，原因就在於量子位元能夠自由地在整個球面上漫遊。

The extra power of a quantum computer comes about because it operates on information represented as qubits, or quantum bits, instead of bits. An ordinary classical bit can be either a 0 or a 1, and standard microchip architectures enforce that dichotomy rigorously. A qubit, in contrast, can be in a so-called superposition state, which entails proportions of 0 and 1 coexisting together. One can think of the possible qubit states as points on a sphere. The north pole is a classical 1, the south pole a 0, and all the points in between are all the possible superpositions of 0 and 1 [see 「Rules for a Complex Quantum World,」 by Michael A. Nielsen; Scientific American, November 2002]. The freedom that qubits have to roam across the entire sphere helps to give quantum computers their unique capabilities.

可惜的是，量子電腦似乎非常難製造。一般而言，我們利用侷限於某個地方的粒子（例如單獨的原子離子或 電子）的某些量子性質，來代表量子位元。但是它們的疊加狀態極為脆弱，只要它們和週遭環境（包括所有組成電腦的材料）有一點點不期而來的交互作用，那麼疊 加狀態就會被破壞。如果量子位元不能和環境仔細地隔絕起來，這種干擾就會造成計算上的錯誤。

Unfortunately, quantum computers seem to be extremely diffi cult to build. The qubits are typically expressed as certain quantum properties of trapped particles, such as individual atomic ions or electrons. But their superposition states are exceedingly fragile and can be spoiled by the tiniest stray interactions with the ambient environment, which includes all the material making up the computer itself. If qubits are not carefully isolated from their surroundings, such disturbances will introduce errors into the computation.

因此在設計量子電腦時，絕大多數將焦點置於減少量子位元與環境的交互作用。研究人員知道，一旦錯誤率 可以降低到每一萬步計算約只會出現一次，修正錯誤的步驟就可以用於補償個別量子位元的衰落。可以運作的量子電腦需要含有大量的量子位元，而每個量子位元與 環境的隔離必須好到讓錯誤率如前述的那樣低，建造這樣的量子電腦是極困難的工作，物理學家距離成功還很遙遠。

Most schemes to design a quantum computer therefore focus on finding ways to minimize the interactions of the qubits with the environment. Researchers know that if the error rate can be reduced to around one error in every 10,000 steps, then error-correction procedures can be implemented to compensate for decay of individual qubits. Constructing a functional machine that has a large number of qubits isolated well enough to have such a low error rate is a daunting task that physicists are far from achieving.

有一些研究者試圖走另一條很不一樣的路來建造量子電腦。在這個新辦法裡，脆弱的量子狀態所依賴的是物理系統的拓撲性質。拓撲是一門數學，它研究的對像是物體在平滑變形（例如伸長、擠壓、彎曲、但不得切斷或連接起來）之下仍會保持不變的性質。拓撲涵蓋的項 目之一是扭結（knot）理論。微小的擾動並不會改變物體的拓撲性質。例如，一條弦綁成一個扭結的封閉迴圈，和沒有扭結的封閉迴圈相比，在拓撲上兩者是不 同的（見65頁〈拓撲與扭結〉）。將沒有扭結的封閉迴圈變成一個封閉迴圈加上扭結的唯一辦法是切斷弦，綁出扭結，再將弦的兩端封起來。同樣的，要把一個拓撲量子位元轉變成另一種狀態，也非得利用類似的激烈方式不可，來自環境的一點點推擠是改變不了拓撲量子位元的。

A few researchers are pursuing a very different way to build a quantum computer. In their approach the delicate quantum states depend on what are known as topological properties of a physical system. Topology is the mathematical study of properties that are unchanged when an object is smoothly deformed, by actions such as stretching, squashing and bending but not by cutting or joining. It embraces such subjects as knot theory. Small perturbations do not change a topological property. For example, a closed loop of string with a knot tied in it is topologically different from a closed loop with no knot [see box on opposite page]. The only way to change the closed loop into a closed loop plus knot is to cut the string, tie the knot and then reseal the ends of the string together. Similarly, the only way to convert a topological qubit to a different state is to subject it to some such violence. Small nudges from the environment will not do the trick.

乍看之下，拓撲量子電腦根本不像是個電腦。它用來計算的是結成絞辮的弦，而不是傳統意義上的實體弦。 這種用於計算的弦是物理學家所稱的世界線，它所代表的是穿過時間與空間的粒子。（你可以這麼想像：這樣一條弦的長度代表粒子在時間軸上的運動，其厚度則代 表粒子的實體大小。）此外，這種計算所牽涉到的粒子並非你最初可能想像的電子或質子。其實這種量子電腦所牽涉到的粒子是準粒子 （quasiparticle），它是二維電子系統的激發態，它們的行為和高能物理中的粒子與反粒子很像。這些粒子還有個麻煩之處：它們是一種特別型態的 準粒子，稱為任意子，具有建構量子電腦所需要的數學性質。

At fi rst sight, a topological quantum computer does not seem much like a computer at all. It works its calculations on braided strings—but not physical strings in the conventional sense. Rather, they are what physicists refer to as world lines, representations of particles as they move through time and space. (Imagine that the length of one of these strings represents a particle's movement through time and that its thickness represents the particle's physical dimensions.) Moreover, even the particles involved are unlike the electrons and protons that one might fi rst imagine. They are instead quasiparticles—excitations in a two-dimensional electronic system that behave a lot like the particles and antiparticles of high-energy physics.And as a further complication, the quasiparticles are of a special type called anyons, which have the desired mathematical properties.

執行一次這種量子計算的過程大約是這樣子的：首先，創造許多對任意子，將它們沿著一條線排列。（見66頁〈拓撲量子計算的原理〉）每一對任意子就如同一個粒子與其反粒子，是純粹由能量所創造出來的。

Here is a what a computation might look like: fi rst, create pairs of anyons and place them along a line [see box on page 60]. Each anyon pair is rather like a particle and its corresponding antiparticle, created out of pure energy.

其次，以明確的順序讓一對對相鄰的任意子彼此環繞。每一個任意子的世界線基本上就構成一條線，任意子 這種對調的運動便製造出了一串這些世界線的絞辮。量子計算就藏在如此形成的特定絞辮裡。任意子的最終狀態存放了計算的結果，這狀態的性質取決於絞辮，而非 任何偶然的電磁交互作用。同時絞辮是拓撲性的（把線搖動一下並不會改變絞辮），所以它在本質上就不受外界的影響。目前在微軟工作的基塔耶夫（Alexei Y. Kitaev）首先於1997年，提出以這種方式來利用任意子執行計算。

Next, move pairs of adjacent anyons around one another in a carefully determined sequence. Each anyon's world line forms a thread, and the movements of the anyons as they are swapped this way and that produce a braiding of all the threads. The quantum computation is encapsulated in the particular braid so formed. The fi nal states of the anyons, which embody the result of the computation, are determined by the braid and not by any stray electric or magnetic interaction. And because the braid is topological—nudging the threads a little bit this way and that does not change the braiding—it is inherently protected from outside disturbances. The idea of using anyons to carry out computations in this fashion was proposed in 1997 by Alexei Y. Kitaev, now at Microsoft.

目前也在微軟從事研究的傅利曼（Michael H. Freedman）於1988年秋天在哈佛大學演講，主題就是利用量子拓撲進行計算的可能性。他在1998年發表了一篇研究論文，闡述了他的想法。傅利曼 的想法奠基於一項數學發現：某些屬於「結不變量」的數學量，和二維曲面隨著時間而演變的量子物理有關。如果我們可以創造物理系統的某個狀況，同時對它做適 當的測量，就可以約略自動計算出結不變量，不然我們就得透過傳統電腦執行冗長又不方便的計算。我們也可以利用類似的捷徑來執行同樣困難、但有實際應用價值 的計算。

Michael H. Freedman, now at Microsoft, lectured at Harvard University in the fall of 1988 on the possibility of using quantum topology for computation. These ideas, published in a research paper in 1998, built on the discovery that certain mathematical quantities known as knot invariants were associated with the quantum physics of a twodimensional surface evolving in time. If an instance of the physical system could be created and an appropriate measurement carried out, the knot invariant would be approximately computed automatically instead of via an inconveniently long calculation on a conventional computer. Equally diffi cult problems of more real-world importance would have similar shortcuts.

雖然這一切聽起來只不過是和現實無關的理論玄想而已，但是最近對於分數量子霍爾效應的實驗，已經讓任意子的想法比較紮實一些，研究者已經設想出更多的實驗以便執行初步的拓撲量子計算。

Although it all sounds like wild theorizing quite removed from reality, recent experiments in a field known as fractional quantum Hall physics have put the anyon scheme on firmer footing. Further experiments have been proposed to carry out the rudiments of a topological quantum computation.

拓撲與扭結一個封閉迴圈（a）的拓撲不會因為扭曲成另一種形狀（b）而改變，
但是會不同於帶有扭結的封閉迴圈（c）的拓撲。如果只是把迴圈扭來扭去，並不會
造出扭結，我們必須切斷迴圈，綁個結，然後再把兩端接起來才能得到扭結（c）。
由此可知，迴圈的拓撲不會受到微擾（如果只是扭來扭去）的影響。

Topology of a closed loop (a) is unaltered if the string is pushed around to form another
 shape (b) but is different from that of a closed loop with a knot tied in it (c). The knot 
cannot be formed just by moving around the string. Instead one must cut the string, tie 
the knot and rejoin the ends. Consequently, the topology of the loop is insensitive to 
perturbations that only push the string around.

任意子
Anyons

前面提過，拓撲量子電腦藉由交換粒子的位置，來把粒子的世界線纏成絞辮。量子物理與古典物理的基本差異之一就在於粒子在對調之後，它們的狀態究竟為何。在古典物理中，如果你在位置A和B各放置一個電子，然後再對調這兩個電子，那麼兩個電子最後的狀態和初 始的狀態並沒有什麼不同，原因是電子是不可區分的粒子，所以我們也無法區分最終狀態與初始狀態。然而在量子力學中，情況就不是這麼簡單了。

as previously mentioned, a topological quantum computer braids world lines by swapping the positions of particles. How particles behave when swapped is one of the many ways that quantum physics differs fundamentally from classical physics. In classical physics, if you have two electrons at locations A and B and you interchange their positions, the fi nal state is the same as the initial state. Because the electrons are indistinguishable, so, too, are the initial and fi nal states. Quantum mechanics is not so simple.

為什麼？因為量子力學是用波函數來描述粒子的狀態。這個函數涵蓋了粒子所有的性質，包括在各處找到粒子的機率、測量到粒子具有各種速度的機率等。譬如說，如果波函數在某個區域有很大的量值，則我們就比較可能在那裡發現粒子。

The difference arises because quantum mechanics describes the state of a particle with a quantity called the wave function, a wave in space that encapsulates all the properties of the particle—the probability of fi nding it at various locations, the probability of measuring it at various velocities, and so on. For example, a particle is most likely to be found in a region where the wave function has a large amplitude.

我們用一個共同的波函數來描述一對電子。當兩個電子交換之後，波函數會和原來的波函數相差了一個負號。這個改變將波峰變成波谷、波谷變成波峰。但波動的振幅大小不會受到影響。

A pair of electrons is described by a joint wave function, and when the two electrons are exchanged, the resulting joint wave function is minus one times the original. That changes peaks of the wave into troughs, and vice versa, but it has no effect on the amplitude of the oscillations. In fact, it does not change any measurable quantity of the two electrons considered by themselves.

事實上，兩個電子互換並不會影響兩個電子本身可以測量的量，真正受到影響的，是如何與其他的電子干涉。當我們將兩個波疊加起來，就會有干涉現象。兩個波相互干涉，若波峰和波峰落在一起，則波幅就會增高（此即「建設性干涉」）；如果波峰和波谷落在一起， 則疊加的波幅就會降低（此即「破壞性干涉」）。如果彼此干涉的兩個波之一改變了本身正負號（即此波函數多乘了-1這個因子），則此波的波峰與波谷就會對調，而將建設性干涉之處（一個亮點）變成破壞性干涉（一個暗點）。

What it does change is how the electrons might interfere with other electrons. Interference occurs when two waves are added together. When two waves interfere, the combination has a high amplitude where peaks of one align with peaks of the other (「constructive interference」) and has a low amplitude where peaks align with troughs (「destructive interference」). Multiplying one of the waves by a phase of minus one interchanges its peaks and troughs and thus changes constructive interference, a bright spot, to destructive interference, a dark spot.

電子並不是唯一會因為交換位置而改變波函數正負號的粒子，質子、中子、以及任何一個所謂的「費米子」也會如此。與費米子不同的另一大類粒子是「玻色子」，兩個玻色子對調時，它們的波函數維持不變，可以說此波函數乘上了+1這個因子。

It is not just electrons that pick up a factor of minus one in this way but also protons, neutrons and in general any particle of a class called fermions. Bosons, the other chief class of particles, have wave functions that are unchanged when two particles are swapped. You might say that their wave functions are multiplied by a factor of plus one.

數學上我們可以證明，三維空間中的粒子只可能是費米子或是玻色子。但是在二維空間，粒子卻不必然是費米子或是玻色子，它們還可能是「任意子」：當兩個這種粒子對調時，波函數會乘上一個絕對值等於1的複數（相位）因子。我們可以用角度來代表此複數因子：0 度所對應的因子為1，180度對應的因子為-1，0度與180度之間的角度對應到某個複數。例如，90度對應到虛數i，即-1的平方根。就如同把波函數乘 以-1不會影響個別粒子的可測量性質，將波函數乘上絕對值等於1的複數也不會影響個別粒子的可測量性質，因為那些可測量的性質只和波動的振幅大小有關。不 過這項額外的複數因子卻可以改變兩個複數波相互干涉的情形。

Deep mathematical reasons require that quantum particles in three dimensions must be either fermions or bosons. In two dimensions, another possibility arises: the factor might be a complex phase. A complex phase can be thought of as an angle. Zero degrees corresponds to the number one; 180 degrees is minus one. Angles in-between are complex numbers. For example, 90 degrees corresponds to i, the square root of minus one. As with a factor of minus one, multiplying a wave function by a phase has absolutely no effect on the measured properties of the individual particle, because all that matters for those properties are the amplitudes of the oscillations of the wave. Nevertheless, the phase can change how two complex waves interfere.

任意子之所以稱為任意子，是因為任意一個複數相位因子都可能出現，而不是像玻色子或費米子那樣，只能多出+1或-1的因子而已。

Particles that pick up a complex phase on being swapped are called anyons because any complex phase might appear, not just a phase of plus or minus one. Particles of a given species, however, always pick up the same phase.

平面上的電子
Electrons in Flatland

任意子只能存在於二維的世界；這麼一來，我們如何能夠在真實的三維世界中，製造出可用來執行拓撲計算 的一對對任意子？答案在於量子粒子的平面世界。我們可以小心地製作兩片砷化鉀半導體，以使得一層電子「氣」能夠存在於界面上。這些電子可以在二維界面上自 由運動，但是無法在垂直於界面的第三維空間上運動。物理學家已經仔細研究過這種電子系統（稱為二維電子氣），尤其是當系統在極低溫下並處於強磁場中的行 為，因為在這些條件之下，電子氣會展現不尋常的量子性質。

anyons exist only in a two-dimensional world. How can we produce
pairs of them for topological computing when we live in three dimensions? The answer lies in the fl atland realm of quasiparticles. Two slabs of gallium arsenide semiconductor can be carefully engineered to accommodate a 「gas」 of electrons at their interface. The electrons move freely in the two dimensions of the interface but are constrained from moving in the third dimension, which would take them off the interface. Physicists have intensely studied such systems of electrons, called two-dimensional electron gases, particularly when the systems are immersed in high transverse magnetic fi elds at extremely low temperatures, because of the unusual quantum properties exhibited under these conditions.

例如，在分數量子霍爾效應中，電子氣的激發態就像是帶有不到一個電子電荷的粒子。其他的激發態則可以 將磁通量完整帶在身邊，就好像這些磁通量是粒子的一部份。2005年，紐約州立大學石溪分校的高德曼（Vladimir J. Goldman）、卡密諾（Fernando E. Camino）與周威宣稱，他們已用實驗直接證明了出現於分數量子霍爾效應中的準粒子是任意子，對於用拓撲方式來從事量子計算而言，這是關鍵的一步。然而 就這些準粒子是否真的具有任意子性質來說，仍有一些研究者還在尋找其他證據，因為某些非量子效應可能可以造成高德曼等人所看到的結果。

For example, in the fractional quantum Hall effect, excitations in the electron gas behave like particles having a fraction of the charge of the electron. Other excitations carry units of the magnetic flux around with them as though the flux were an integral part of the particle. In 2005 Vladimir J. Goldman, Fernando E. Camino and Wei Zhou of Stony Brook University claimed to have direct experimental confi rmation that quasiparticles occurring in the fractional quantum Hall state are anyons, a crucial fi rst step in the topological approach to quantum computation. Some researchers, however, still seek independent lines of evidence for the quasiparticles' anyonic nature because certain nonquantum effects could conceivably produce the results seen by Goldman and his colleagues.

　　在二維世界中，當我們對調兩個粒子時，還得注意一個重要的新問題：粒子交換時，是循著順時針的路徑或是 逆時針的路徑？波函數所獲得的額外複數相位因子取決於所取的路徑；這兩種路徑在拓撲上是不同的，因為我們無法在不讓兩條路徑交錯、不讓粒子相撞的情況下， 連續地將順時針的路徑轉成逆時針的路徑。

　　In two dimensions, an important new issue arises in the swapping of two particles: Do the particles follow clockwise tracks or counterclockwise tracks as they are interchanged? The phase picked up by the wave function depends on that property. The two alternative paths are topologically distinct, because the experimenter cannot continuously deform the clockwise paths into counterclockwise paths without crossing the paths and having the particles collide somewhere.

　　建造拓撲電腦還有另一項困難：任意子必須是所謂「非阿貝爾式」（nonabelian），也就是說粒 子對調時的先後順序是非常重要的。想像你有三個相同的任意子排在一起，分別位於A、B、C的位置上。首先，對調位於A和B的任意子；其次，對調現在位於B 和C的粒子；最後的波函數會等於原始波函數乘上某個複數因子。假設我們反過來先對調位於B和C的任意子，然後才對調位於A和B的任意子；在這樣的順序之 下，如果最後的波函數所乘上複數因子和前面第一種對調的方式一樣，則我們說任意子是「阿貝爾式」。如果波函數所乘上的複數因子取決於對調的順序，則那種任 意子就是「非阿貝爾式」。（非阿貝爾式的性質之所以會出現，原因就在於對於這些任意子來說，波函數所乘的複數因子其實是個矩陣，而兩個矩陣相乘的結果本來 就會取決於相乘的順序。）

　　To build a topological quantum computer requires one additional complication: the anyons must be what is called nonabelian. This property means that the order in which particles are swapped is important. Imagine that you have three identical anyons in a row, at positions A, B and C. First swap the anyons at positions A and B. Next swap the anyons now located at B and C. The result will be the original wave function modified by some factor. Suppose instead that the anyons at B and C are swapped first, followed by swapping those at A and B. If the result is the wave function multiplied by the same factor as before, the anyons are called abelian. If the factors differ depending on the order of the swapping, they are nonabelian anyons. (The nonabelian property arises because for these anyons, the factor that multiplies the wave function is a matrix of numbers, and the result of multiplying two matrices depends on the order in which they are multiplied.)

　　高德曼等人的實驗所牽涉的是阿貝爾任意子。不過理論學家有很好的理由相信某些分數量子霍爾效應的準粒子是非阿貝爾任 意子。科學家為了找出這個問題的答案，已經設計了一些實驗。其中一項實驗是由傅利曼與馬里蘭大學的達斯沙爾馬（Sankar Das Sarma）以及微軟的納亞克（Chetan Nayak）所提出的，並由以色列魏玆曼科學學院的史登（Ady Stern）與哈佛大學的郝柏林（Bertrand Halperin）提出了重要的改進。加州理工學院的基塔耶夫與邦德森（Parsa Bonderson）與現在任教於加州大學河濱分校的須騰格（Kirill Shtengel）也提出了另一項實驗。

　　The experiment by Goldman's team involved abelian anyons. Nevertheless, theorists have strong reason to believe that certain fractional quantum Hall quasiparticles are indeed nonabelian. Experiments have been proposed to settle that question. One was suggested by Freedman, along with Sankar Das Sarma of the University of Maryland at College Park and Chetan Nayak of Microsoft, with important refinements proposed by Ady Stern of the Weizmann Institute in Israel and Bertrand Halperin of Harvard University; the second was presented by Kitaev, Parsa Bonderson of the California Institute of Technology and Kirill Shtengel, now at the University of California, Riverside.

絞辮與閘

Braids and Gates

　　一旦有了非阿貝爾任意子，就可以具體製造出所謂辮群（braid group）的表現。這種群的數學結構描述了將一排線編織成絞辮的所有可能方式。任何絞辮都是由一序列的基本運作所建造出來的，這個基本運作只牽涉到相鄰 兩條線的順時鐘或逆時鐘運動。每一串可能的基本任意子運作都會對應到一條絞辮，反之亦然。此外，每條絞辮還會對應到一個非常複雜的矩陣，這個矩陣是每次任 意子交換所對應的矩陣的總和。

　　Once you have nonabelian anyons, you can generate a physical representation of what is called the braid group. This mathematical structure describes all the ways that a given row of threads can be braided together. Any braid can be built out of a series of elementary operations in which two adjacent threads are moved, by either a clockwise or a counterclockwise motion. Every possible sequence of anyon manipulations corresponds to a braid, and vice versa. Also corresponding to each braid is a very complicated matrix, the result of combining all the individual matrices of every anyon exchange.

　　現在我們就有了一切所需的要素，來瞭解這些絞辮究竟如何對應到量子計算。在傳統的電腦中，電腦的狀態是由所有位 元的狀態（電腦暫存器中某一組特定的0和1序列）來代表。同樣的，量子電腦是由所有量子位元的狀態來代表。在拓撲量子電腦中，代表量子位元的是一群群的任意子。

　　Now we have all the elements in place to see how these braids correspond to a quantum computation. In a conventional computer, the state of the computer is represented by the combined state of all its bits—the particular sequence of 0s and 1s in its register. Similarly, a quantum computer is represented by the combined state of all its qubits. In a topological quantum computer, the qubits may be represented by groups of anyons.

　　在量子電腦中，我們用一個矩陣來描述從所有量子位元的初始狀態到最終狀態的過程──最終狀態即是這個矩陣乘上所有量子位元的總初始波函 數。很明顯地，類似的事情也發生於拓撲量子電腦中：這時的矩陣是對應到某特定絞辮的矩陣，而絞辮來自於某一程序的任意子運作。所以，我們已經驗證了用任意 子所執行的運作會導致量子計算。

　　In a quantum computer, the process of going from the initial state of all the qubits to the final state is described by a matrix that multiplies the joint wave function of all the qubits. The similarity to what happens in a topological quantum computer is obvious: in that case, the matrix is the one associated with the particular braid corresponding to the sequence of anyon manipulations. Thus, we have verified that the operations carried out on the anyons result in a quantum computation.

　　我們還必須確認另一項重要的特徵：拓撲量子電腦是否能執行任何傳統電腦所能做的計算？傅利曼與印第安納大學的 拉森（Michael Larsen），以及現在任職於微軟的王正漢三人，於2002年證明，拓撲量子電腦的確能夠模擬任何標準電腦的任何計算，只不過還有個小弱點：這種模擬是 近似的而已。但是對於任何想達到的精確度（例如萬分之一）來說，都可以找到一條絞辮，能模擬所需要做的計算到那個精確度。所要求的精確度越高，絞辮擰轉的 次數就越多。幸運的是，所需擰轉次數增加的非常慢，所以要達到非常高的精確度其實並不困難。可是傅利曼等人的證明，並沒有講清楚如何決定實際上哪個絞辮對 應到計算，這要靠拓撲量子電腦的特定設計，尤其是所使用的任意子類別，以及它們和基本量子位元的關係。

　　Another important feature must be confirmed: Can our topological quantum computer perform any computation that a conventional quantum computer can? Freedman, working with Michael Larsen of Indiana University and Zheng-han Wang, now at Microsoft, proved in 2002 that a topological quantum computer can indeed simulate any computation of a standard quantum computer, but with one catch: the simulation is approximate. Yet given any desired accuracy, such as one part in 10⁴, a braid can be found that will simulate the required computation to that accuracy. The finer the accuracy required, the greater the number of twists in the braid. Fortunately, the number of twists required increases very slowly, so it is not too difficult to achieve very high accuracy. The proof does not, however, indicate how to determine which actual braid corresponds to a computation—that depends on the specific design of topological quantum computer, in particular the species of anyons employed and their relation to elementary qubits.

　　2005年，弗羅里達州立大學的波奈斯蒂（Nicholas E. Bonesteel）與同校的同仁，以及在朗訊科技貝爾實驗室的合作者開始設法尋找從事特定計算的絞辮。這一研究小組明確說明了如何建構一個所謂的「控制 反閘」（controlled NOT gate，簡稱CNOT gate），只要用上六個任意子，精確度即可達千分之二。一個CNOT閘有兩個輸入：控制位元和目標位元。如果控制位元是1，則CNOT閘會將目標位元從 0變成1，或從1變成0；如果控制位元是0，則目標位元就不會改變。CNOT閘的網絡可以作用於量子位元上，我們只要用CNOT閘網絡以及另一項運算（將 個別量子位元乘上複數相位因子）就能完成任何計算。這項結果也可以用來證實拓撲量子電腦能夠執行任何量子計算。

　　The problem of finding braids for doing specific computations was tackled in 2005 by Nicholas E. Bonesteel of Florida State University, along with colleagues there and at Lucent Technologies's Bell Laboratories. The team showed explicitly how to construct a so-called controlled NOT (or CNOT) gate to an accuracy of two parts in 10³ by braiding six anyons. A CNOT gate takes two inputs: a control bit and a target bit. If the control bit is 1, it changes the target bit from 0 to 1, or vice versa. Otherwise the bits are unaltered. Acting on qubits, any computation can be built from a network of CNOT gates and one other operation—the multiplication of individual qubits by a complex phase. This result serves as another confirmation that topological quantum computers can perform any quantum computation.

　　一般相信量子電腦可以執行古典 電腦不可能從事的計算。可是拓撲量子電腦的計算能力可能比一般量子電腦更好嗎？傅利曼、基塔耶夫與王正漢三人證明了這是不可能的。他們證明傳統量子電腦可 以有效率地模擬拓撲量子電腦的運算，同時有無限的精確度。也就是說，傳統量子電腦也可以執行任何拓撲量子電腦可以執行的計算。這個結果暗示了一項一般性定 理：任何使用量子資源的計算系統，只要足夠高明，就具有完全相同的計算能力。（邱契與涂林在1930年代，曾提出了一個適用於古典電腦的類似命題。）

　　Quantum computers can perform feats believed to be impossible for classical computers. Is it possible that a topological computer is more powerful than a conventional quantum computer? Another theorem, proved by Freedman, Kitaev and Wang, shows that is not the case. They demonstrated that the operation of a topological quantum computer can be simulated efficiently to arbitrary accuracy on a conventional quantum computer, meaning that anything that a topological quantum computer can compute a conventional quantum computer can also compute. This result suggests a general theorem: any sufficiently advanced computation system that makes use of quantum resources has exactly the same computational abilities. (An analogous thesis for classical computing was proposed by Alonzo Church and Alan Turing in the 1930s.)

粒子進，答案出

Particles In, Answers Out

　　我前面略過了建造實用拓撲量子電腦所需的兩個關鍵過程：在計算開始之前量子位元的初始化，以及最後結果的讀取。

　　I have glossed over two processes that are crucial to building a practical topological quantum computer: the initialization of the qubits before the start of the computation and the readout of the answer at the end.

　 　初始化的步驟牽涉到造出一對對的準粒子，關鍵的問題是弄清楚所創造的準粒子的類別。基本的步驟是讓「測試任意子」繞過造出的準粒子對，然後測量這項步驟 如何改變了「測試任意子」；它們的變化取決於它們所通過的任意子的種類。（如果測試任意子改變了，它就不能和其夥伴徹底地相互消滅。）不屬於我們需要類型 的任意子對就會被拋棄。

　　The initialization step involves generating quasiparticle pairs, and the problem is knowing what species of quasiparticle has been created. The basic procedure is to pass test anyons around the generated pairs and then measure how the test anyons have been altered by that process, which depends on the species of the anyons that they have passed. (If a test anyon is altered, it will no longer be cleanly annihilated with its partner.) Anyon pairs not of the required type would be discarded.

　　讀取的步驟也牽涉到如何測量任意子的狀態。如果任意子相隔很遠，就不可能測量：任意子必須成對的拉近在一起才可以測 量。約略地講，這個步驟就像是檢查任意子對是否可以消滅得很乾淨（就如同真正的粒子和反粒子對），還是消滅後仍會殘留下電荷與磁通量；任意子的狀態是從它 們誕生之時很精確的反粒子關係編織而來的，檢驗的結果會揭露任意子的狀態在編織後究竟有何變化。

　　The readout step also involves measuring anyon states. While the anyons are widely separated, that measurement is impossible: the anyons must be brought together in pairs to be measured. Roughly speaking, it is like checking to see if the pairs annihilate cleanly, like true antiparticles, or if they leave behind residues of charge and flux, which reveals how their states have been altered by braiding from the exact antiparticle relation in which they began their lives.

　　其實拓撲量子電腦也並非能夠完全免於錯誤。錯 誤的主要來源是基底材料中的熱擾動，這種擾動可以產生多餘的一對任意子。這兩個任意子會和執行計算的絞辮交纏在一起，最後才相互消滅（見67頁〈防止隨機 的錯誤〉）。幸運的是，拓撲量子電腦運作所需的溫度極低，在此低溫下，熱擾動受到抑制。此外，當多餘的任意子必須運動更遠的距離，才足以造成干擾時，這會 引起錯誤的不良過程出現的機率會以指數函數的形式降低。所以，我們可以如此達成任何所需的精確度：建造一個夠大的電腦，同時在編織任意子的時候，讓這些工 作的任意子相距夠遠。

　　Also, it is not true that a topological computer is totally immune to errors. The main source of error is thermal fluctuations in the substrate material, which can generate an extra pair of anyons. Both the anyons then intertwine themselves with the braid of the computation, and finally the pair annihilates again [see box on page 61]. Fortunately, the thermal generation process is suppressed at the low temperature at which a topological computer would operate. Furthermore, the probability of the entire bad process occurring decreases exponentially as the distance traveled by the interlopers increases. Thus, one can achieve any required degree of accuracy by building a sufficiently large computer and keeping the working anyons far enough apart as they are braided.

　　拓撲量子電腦仍還在其嬰兒階段：基本的運作要素，如非阿貝爾任意子，尚待證實，最簡單的邏輯閘也還沒做出來。前面提過的 傅利曼、達斯沙爾馬以及納亞克的實驗應可以達成這兩項目標──只要所涉的任意子的確如預期般的是非阿貝爾任意子，那麼他們的裝置就可以在量子位元狀態上執 行邏輯上的反（NOT）的運算。傅利曼等三人估計此一過程的錯誤率會是10^-30或更小。錯誤率為什麼能夠這麼小？因為當溫度下降而且尺度增加時，錯誤的機率會以指數函數的形式快速下降。是拓撲的結構成就了如此快速降低的錯誤率，在較傳統的量子計算辦法中，不能如法炮製。

　　Topological quantum computing remains in its infancy. The basic working elements, nonabelian anyons, have not yet been demonstrated to exist, and the simplest of logic gates has yet to be built. The previously mentioned experiment of Freedman, Das Sarma and Nayak would achieve both those goals—if the anyons involved do turn out to be nonabelian, as expected, the device would carry out the logical NOT operation on the qubit state. The trio estimated that the error rate for the process would be 10^-30 or less. Such a tiny error rate occurs because the probability of errors is exponentially suppressed as the temperature is lowered and the length scale increased. That exponential factor is the essential contribution of topology, and it has no analogue in the more traditional approaches to quantum computing.

　 　由於拓撲量子計算有這種低得不得了的錯誤率（比當今任何其他量子計算方法所能達到的，還要低好幾個數量級），所以它變得非常引人矚目。此外，製造分數量 子霍爾裝置的技術已經成熟，因為這些技術正是微晶片工業的技術。唯一的問題是這些裝置必須在極低溫（以絕對溫度單位而論，約為幾毫K）才能運作，因為只有 這樣，神奇的準粒子才會穩定。

　　The promise of extraordinarily low error rates—many orders of magnitude lower than those achieved by any other quantum computation scheme to date—is what makes topological quantum computing so attractive. Also, the technologies involved in making fractional quantum Hall devices are mature, being precisely those of the microchip industry; the only catch is that the devices have to operate at extremely low temperatures—on the order of millikelvins—for the magical quasiparticles to be stable.

　　如果非阿貝爾任意子真的存在，那麼拓撲量子電腦就可能從後超越過傳統量子電腦，搶先將個別的量子位元與邏輯閘放 大成為紮紮實實的機器，像是個名副其實的「電腦」。用量子結與量子辮來計算，最初只是另一種奇妙的方案而已，但是對於實現實際可行又沒有錯誤的量子計算而 言，這種辦法卻可能成為標準方式。

　　If nonabelian anyons actually exist, topological quantum computers could well leapfrog conventional quantum computer designs in the race to scale up from individual qubits and logic gates to fully fledged machines more deserving of the name “computer.” Carrying out calculations with quantum knots and braids, a scheme that began as an esoteric alternative, could become the standard way to implement practical, error-free quantum computation.

用量子絞辮來計算

Overview/Quantum Braids

　　■量子電腦保證可以遠比古典電腦更具威力。但是量子電腦如要能運作，它們的錯誤率必須極低。如果要從傳統的量子電腦設計來達成這麼低的錯誤率，目前的技術還沒有辦法做到。

　 　■Quantum computers promise to greatly exceed the abilities of classical computers, but to function at all, they must have very low error rates. Achieving the required low error rates with conventional designs is far beyond current technological capabilities.

　　■另外一種設計是所謂的拓撲量子電腦，它用另一類全然不同的物理系統來實現量子計算。由於拓撲性質不會受到微擾的影響，所以等於有內建的防錯能力，例如不受由系統與環境的偶發交互作用影響而引發錯誤。

　　■An alternative design is the so-called topological quantum computer, which would use a radically different physical system to implement quantum computation. Topological properties are unchanged by small perturbations, leading to a built-in resistance to errors such as those caused by stray interactions with the surrounding environment.

　　■拓撲量子計算所利用的是稱為任意子的激發態，這是一種理論假設，具有奇異的粒子般結構，只出現於二維世界。最近的實驗顯示，任意子存在於特別的平面半導體結構中，這些結構必須冷卻至接近絕對零度並放置在強磁場內。

　　 ■Topological quantum computing would make use of theoretically postulated excitations called anyons, bizarre particlelike structures that are possible in a two-dimensional world. Experiments have recently indicated that anyons exist in special planar semiconductor structures cooled to near absolute zero and immersed in strong magnetic fields.

1.Topologically Protected Qubits from a Possible Non-Abelian Fractional
Quantum Hall State. Sankar Das Sarma, Michael Freedman and Chetan Nayak
in Physical Review Letters, Vol. 94, pages 166802-1–168802-4; April 29,
2005.

2.Devices Based on the Fractional Quantum Hall Effect May Fulfill the
Promise of Quantum Computing. Charles Day in Physics Today, Vol. 58,
pages 21-24; October 2005.

3.Anyon There? David Lindley in Physical Review Focus, Vol. 16, Story 14; 
November 2, 2005. http://focus.aps.org/story/v16/st14

4.Topological Quantum Computation. John Preskill. Lecture notes
available at www.theory.caltech.edu/~preskill/ph219/topological.pdf