Preskill評點最新跨界之作：量子信息遇到量子物質

凝聚態物理是研究各種各樣材料性質的古老理論，而量子信息是為了發展量子計算所形成的新理論。近十幾年來，越來越多的人意識到，為了理解低溫下量子材料的性質，我們需要深刻理解量子信息，特別是它們的量子糾纏。而量子信息理論的發展也越來越多地受到量子材料方面問題的激勵。這樣量子物質和量子信息相互推動，使這一交叉領域成為目前研究發展的一個熱點。新概念、新觀點、新理論，層出不窮。甚至量子信息有可能是量子物質的起源，因為所有基本粒子可能來源於糾纏的量子信息。但目前關於凝聚態物理的教科書很少從量子信息的角度看問題，關於量子信息的教科書也很少從凝聚態物理的角度看問題。所以目前急需一本跨界的教科書，介紹這一交叉領域的新眼光和新進展。在曾蓓的發動下，陳諧、周端陸和我一起寫了這本跨界的新書。希望能幫助學生和研究人員進入這一日新月異的交叉領域。我們很榮幸請到由高能物理轉做量子信息的跨界領軍人物——Preskill教授給我們的新書撰寫前言，介紹這一交叉領域的背景和發展。我們將其譯成中文，以饗《返樸》讀者。

——文小剛

撰文 | Preskill

翻譯 | 曹凝萍

In 1989 I attended a workshop at the University of Minnesota. The organizers had hoped the workshop would spawn new ideas about the origin of high- temperature superconductivity, which had recently been discovered. But I was especially impressed by a talk about the fractional quantum Hall effect by a young physicist named Xiao-Gang Wen.

1989年，我參加了一個在明尼蘇達大學（University of Minnesota）舉行的專題研討會。當時，高溫超導剛被發現，組織者希望此次研討會能催生一些關於高溫超導原理的新想法。但是此次研討會給我留下深刻印象的卻是一名年輕物理學家關於分數量子霍爾效應的報告，報告人的名字是文小剛。

From Wen I heard for the first time about a concept called topological order. He explained that for some quantum phases of two-dimensional matter the ground state becomes degenerate when the system resides on a surface of nontrivial topology such as a torus, and that the degree of degeneracy provides a useful signature for distinguishing different phases. I was fascinated.

從文小剛那裡，我第一次聽說了拓撲序（Topological Order）的概念：當把二維的量子物態放到有非平凡拓撲的表面（例如環面）上時，其能量最低的基態會出現好幾個，而基態的不同數目可以用來區分不同的量子相。這使我眼睛一亮。

同一個量子物態被放到有不同拓撲性質的表面上時，如果它最低能量的基態的數目會不一樣，那麽這個量子物態就被稱為有拓撲序。

Up until then, studies of phases of matter and the transitions between them usually built on principles annunciated decades earlier by Lev Landau. Landau had emphasized the crucial role of symmetry, and of local order parameters that distinguish different symmetry realizations. Though much of what Wen said went over my head, I did manage to glean that he was proposing a way to distinguish quantum phases founded on much different principles that Landau’s. As a particle physicist I deeply appreciated the power of Landau theory, but I was also keenly aware that the interface of topology and physics had already yielded many novel and fruitful insights.

在那之前，對物質的相及相變的研究通常建立在朗道（Lev Landau）數十年前所發展的思想之上。朗道強調了對稱性和局域序參量的關鍵作用，其中序參量可以用來區分不同的量子相。雖然文小剛講過的大部分內容從我左耳進右耳出，我還是看出了他所提出的這種區分量子相的方法是建立在與朗道完全不同的原理之上的。作為一名粒子物理學家，我深知朗道理論的強大和普適性，可我也清楚地知道拓撲學和物理的交融已然產生出了很多新穎和富有成果的見解。

Mulling over these ideas on the plane ride home, I scribbled a few lines of verse:

在回程的飛機上，細細回味這些想法，我潦草的寫下幾行詩：

Now we are allowed

To disavow Landau.

Wow...

如今我們知曉

允許否定朗道

哇……

（吾輩今獲 / 弗朗道之權 / 籲！）

Without knowing where it might lead, one could sense the opening of a new chapter.

雖不知這將把我們帶向何方，但我仍然能感覺到新篇章即將開啟。

At around that same time, another new research direction was beginning to gather steam, the study of quantum information. Richard Feynman and Yuri Manin had suggested that a computer processing quantum information might perform tasks beyond the reach of ordinary digital computers. David Deutsch formalized the idea, which attracted the attention of computer scientists, and eventually led to Peter Shor’s discovery that a quantum computer can factor large numbers in polynomial time. Meanwhile, Alexander Holevo, Charles Bennett and others seized the opportunity to unify Claude Shannon’s information theory with quantum physics, erecting new schemes for quantifying quantum entanglement and characterizing processes in which quantum information is acquired, transmitted, and processed.

也就是差不多這個時候，另一個新的研究方向——量子信息——開始醞釀。理查德·費曼（Richard Feynman）和尤裡·馬寧（Yuri Manin）提出，一台能夠處理量子信息的計算機可能可以執行普通數字計算機無法企及的任務。大衛·多伊奇（David Deutsch）將這一想法變成了具體的算法，從而吸引了計算機科學家的注意，這也導致彼得·肖爾（Peter Shor）最終發現量子計算機可以在多項式時間內完成大數的因數分解。與此同時， 亞歷山大·霍勒夫（Alexander Holevo）、查理·貝內特（Charles Bennett）和其他研究者抓住了時機，將香農的信息理論和量子物理相結合，建立了度量量子糾纏和刻畫量子信息的獲取、傳遞及處理過程的新體系。

The discovery of Shor’s algorithm caused a burst of excitement and activity, but quantum information science remained outside the mainstream of physics, and few scientists at that time glimpsed the rich connections between quantum information and the study of quantum matter. One notable exception was Alexei Kitaev, who had two remarkable insights in the 1990s. He pointed out that finding the ground state energy of a quantum system defined by a local Hamilto年n, when suitably formalized, is as hard as any problem whose solution can be verified with a quantum computer. This idea launched the study of Hamilto年n complexity. Kitaev also discerned the relationship between Wen’s concept of topological order and the quantum error-correcting codes that can protect delicate quantum superpositions from the ravages of environmental decoherence. Kitaev’s notion of a topological quantum computer, a mere theorist’s fantasy when proposed in 1997, is by now pursued in experimental laboratories around the world (though the technology still has far to go before truly scalable quantum computers will be capable of addressing hard problems).

雖然肖爾算法（Shor’s algorithm）的發現引爆了一輪熱烈的研究，但是量子信息科學依舊處在物理學的邊緣。當時，幾乎沒有科學家預見到量子信息和量子物質研究之間的豐富關聯，但阿列克謝·基塔耶夫（Alexei Kitaev）是其中一個例外。他在1990年代提出了兩個非凡的見解：他指出，嚴格表述後，尋找一個量子系統局域哈密頓量（Local Hamilto年n）的基態能量，與能由量子計算機驗證的最困難問題同樣困難。這一想法引發了對哈密頓量複雜度的研究。基塔耶夫還察覺到文小剛的拓撲序概念和量子糾錯碼之間的關聯，而後者可以用來保護脆弱的量子疊加免受環境退相乾的破壞。基塔耶夫在1997年提出關於拓撲量子計算機的概念時，這還僅僅是理論家的幻想。而如今全世界有很多實驗室都在嘗試這一思路（儘管距離大型量子計算機能夠真正解決困難問題還有很長的路要走）。

Thereafter progress accelerated, led by a burgeoning community of scientists working at the interface of quantum information and quantum matter. Guifre Vidal realized that many-particle quantum systems that are only slightly entangled can be succinctly described using tensor networks. This new method extended the reach of mean-field theory and provided an illuminating new perspective on the successes of the Density Matrix Renormalization Group (DMRG). By proving that the ground state of a local Hamilto年n with an energy gap has limited entanglement (the area law), Matthew Hastings showed that tensor network tools are widely applicable. These tools eventually led to a complete understanding of gapped quantum phases in one spatial dimension.

此後，越來越多的科學家開始在量子信息和量子物質的交叉領域開展研究，使得這方面的進展大大加速。基弗爾·維道（Guifre Vidal）發現，用張量網絡能夠簡潔地描述弱糾纏的多體量子系統。這種新方法擴展了平均場理論（Mean-Field Theory）的應用範圍，並提供了一個新視角來理解為什麽密度矩陣重整化群（Density Matrix Renormalization Group）如此成功。通過證明局域哈密頓量的有能隙的基態具有有限糾纏，馬修·黑斯廷斯（Matthew Hastings）證明，張量網絡理論有廣泛的應用範圍。這些工具最終使我們對一維有能隙量子態有了全面的理解。

The experimental discovery of topological insulators focused attention on the interplay of symmetry and topology. The more general notion of a symmetry-protected topological (SPT) phase arose, in which a quantum system has an energy gap in the bulk but supports gapless excitations confined to its boundary which are protected by specified symmetries. (For topological insulators the symmetries are particle-number conservation and time- reversal invariance.) Again, tensor network methods proved to be well suited for establishing a complete classification of one-dimensional SPT phases, and guided progress toward understanding higher dimensions, though many open questions remain.

拓撲絕緣體的實驗發現，使人們開始專注於對稱性和拓撲學的相互聯繫。一種更普適的觀點，對稱保護拓撲（Symmetry-Protected Topological，SPT）序誕生了。這些SPT系統的內部有能隙，但其在邊界上會有無能隙激發，而這一無能隙的特性會被相應的對稱性保護（對於拓撲絕緣體來說，對稱性意味著粒子數守恆和時間反演不變性）。同樣，張量網絡方法被證明非常適用於建立一維SPT相的完全分類，並指導人們研究和理解高維系統的SPT相（雖然還有很多問題懸而未決）。

We now have a much deeper understanding of topological order than when I first heard about it from Wen nearly 30 years ago. A central new insight is that topologically ordered systems have long-range entanglement, and that the entanglement has universal properties, like topological entanglement entropy, which are insensitive to the microscopic details of the Hamilto年n. Indeed, topological order is an intrinsic property of a quantum state and can be identified without reference to any particular Hamilto年n at all. To understand the meaning of long-range entanglement, imagine a quantum computer which applies a sequence of geometrically local operations to an input quantum state, producing an output product state which is completely disentangled. If the time required to complete this disentangling computation is independent of the size of the system, then we say the input state is short-ranged entangled; otherwise it is long-range entangled. More generally (loosely speaking), two states are in different quantum phases if no constant-time quantum computation can convert one state to the other. This fundamental connection between quantum computation and quantum order has many ramifications which are explored in this book.

與我30年前第一次聽文小剛提到拓撲序時相比，現在我們已經對拓撲序有了更深的認識。一個核心的新認知是拓撲序系統具有長程糾纏，這種糾纏具有一些拓撲不變性，例如拓撲糾纏熵，其數值在系統哈密頓量的微觀細節發生小的變化時保持不變。事實上，拓撲序是一種量子態的固有性質，可以在完全不參考任何具體哈密頓量的情況下被識別出來。為理解長程糾纏的意義，想象在一台量子計算機上，輸入一個量子糾纏態，進行一系列局部操作，最後輸出完全無糾纏的直積態。如果完全解除糾纏所需要的時間不隨著系統變大而增加，則稱輸入態有短程糾纏；反之，則為長程糾纏。更一般地來說（不是很嚴格），如果沒有耗時恆定的量子計算可以讓一個態轉化為另一個態，則這兩個態將屬於不同的量子相。本書深入探討了量子計算和量子序的本質關聯及其導致的眾多結果。

撲序的長程量子糾纏長什麽樣非常難以想象。通過中國結或者凱爾特結，也許讀者可以體會一下各種構型的長程糾纏的樣子。

When is the right time for a book that summarizes the status of an ongoing research area? It’s a subtle question. The subject should be sufficiently mature that enduring concepts and results can be identified and clearly explained. If the pace of progress is sufficiently rapid, and the topics emphasized are not well chosen, then an ill-timed book might become obsolete quickly. On the other hand, the subject ought not to be too mature; only if there are many exciting open questions to attack will the book be likely to attract a sizable audience eager to master the material.

對於一個蓬勃發展的研究領域來說，什麽時候寫一本書來總結是個非常微妙的問題。書的主題應該足夠成熟，相關概念和結果可以經得起考驗，且可以被識別和清晰地解釋。如果研究進展太過迅速，而且重點話題選擇不夠恰當，那麽一本不合時宜的書將很快被淘汰。而另一方面，題材不應太過成熟。只有存在許多激動人心且懸而未決的問題時，這本書才能吸引大量渴望掌握其知識的讀者。

I feel confident that Quantum Information Meets Quantum Matter is appearing at an opportune time, and that the authors have made wise choices about what to include. They are world-class experts, and are themselves responsible for many of the scientific advances explained here. The student or senior scientist who studies this book closely will be well grounded in the tools and ideas at the forefront of current research at the confluence of quantum information science and quantum condensed matter physics.

我有信心認為《量子信息遇見量子物質》這本書出現在一個非常恰當的時機。本書的作者都是世界級的專家，他們參與發展了書中所描述的許多科學成果，並對書的內容作出了明智的取捨。在量子信息科學和量子凝聚態物理這一交叉前沿領域，認真研究這本書的學生或科研人員都將在該領域的研究工具和觀點方面得到很大的收獲。

Indeed, I expect that in the years ahead a steadily expanding community of scientists, including computer scientists, chemists, and high-energy physicists, will want to be well acquainted with the ideas at the heart of Quantum Information Meets Quantum Matter. In particular, growing evidence suggests that the quantum physics of spacetime itself is an emergent manifestation of long-range quantum entanglement in an underlying more fundamental quantum theory. More broadly, as quantum technology grows ever more sophisticated, I believe that the theoretical and experimental study of highly complex many-particle systems will be an increasingly central theme of 21st century physical science. It that’s true, Quantum Information Meets Quantum Matter is bound to hold an honored place on the bookshelves of many scientists for years to come.

事實上，我預期在未來的歲月裡，會有越來越多的科學家（包括計算機科學家、化學家和高能物理學家）渴望理解《量子信息遇見量子物質》一書的核心觀點。特別是，越來越多的證據表明，時空本身的量子物理內涵，是長程量子糾纏在更基本的量子理論中的演生現象。更廣泛地說，隨著量子技術發展得越來越複雜，我相信關於高度複雜的多體系統的理論和實驗研究將會逐漸成為二十一世紀物理學的中心課題。在未來幾年裡，《量子信息遇見量子計算》勢必在許多科學家的書架上佔有一席之地。

作者簡介

約翰·普雷斯基爾（John Preskill），加州理工學院理論物理學教授。在研究生階段，他曾發表論文闡述了大統一理論中超重磁單極子在早期宇宙中的產生，這個問題導致了後來的宇宙暴漲理論的提出。

他從2000年開始擔任加州理工學院量子信息中心主任，近年來主要研究與量子計算和量子信息理論相關的數學問題。

普雷斯基爾因為和另外兩位理論物理學家霍金、基普·索恩（ Kip Thorne）關於黑洞信息佯謬所做的物理學賭局而為大眾所熟知。2004年，霍金宣布讓步，並贈送一本棒球百科全書給普雷斯基爾，但物理學界並未就這一問題達成最後共識。

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