Preskill评点最新跨界之作:量子信息遇到量子物质

凝聚态物理是研究各种各样材料性质的古老理论,而量子信息是为了发展量子计算所形成的新理论。近十几年来,越来越多的人意识到,为了理解低温下量子材料的性质,我们需要深刻理解量子信息,特别是它们的量子纠缠。而量子信息理论的发展也越来越多地受到量子材料方面问题的激励。这样量子物质和量子信息相互推动,使这一交叉领域成为目前研究发展的一个热点。新概念、新观点、新理论,层出不穷。甚至量子信息有可能是量子物质的起源,因为所有基本粒子可能来源于纠缠的量子信息。但目前关于凝聚态物理的教科书很少从量子信息的角度看问题,关于量子信息的教科书也很少从凝聚态物理的角度看问题。所以目前急需一本跨界的教科书,介绍这一交叉领域的新眼光和新进展。在曾蓓的发动下,陈谐、周端陆和我一起写了这本跨界的新书。希望能帮助学生和研究人员进入这一日新月异的交叉领域。我们很荣幸请到由高能物理转做量子信息的跨界领军人物——Preskill教授给我们的新书撰写前言,介绍这一交叉领域的背景和发展。我们将其译成中文,以飨《返朴》读者。

——文小刚

撰文 | Preskill

翻译 | 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,我参加了一个在明尼苏达大学(University of Minnesota)举行的专题研讨会。当时,高温超导刚被发现,组织者希望此次研讨会能催生一些关于高温超导原理的新想法。但是此次研讨会给我留下深刻印象的却是一名年轻物理学家关于分数量子霍尔效应的报告,报告人的名字是文小刚。

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.

从文小刚那里,我第一次听说了拓扑序(Topological Order)的概念:当把二维的量子物态放到有非平凡拓扑的表面(例如环面)上时,其能量最低的基态会出现好几个,而基态的不同数目可以用来区分不同的量子相。这使我眼睛一亮。

同一个量子物态被放到有不同拓扑性质的表面上时,如果它最低能量的基态的数目会不一样,那么这个量子物态就被称为有拓扑序。

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,对物质的相及相变的研究通常建立在朗道(Lev Landau)数十年前所发展的思想之上。朗道强调了对称性和局域序参量的关键作用,其中序参量可以用来区分不同的量子相。虽然文小刚讲过的大部分内容从我左耳进右耳出,我还是看出了他所提出的这种区分量子相的方法是建立在与朗道完全不同的原理之上的。作为一名粒子物理学家,我深知朗道理论的强大和普适性,可我也清楚地知道拓扑学和物理的交融已然产生出了很多新颖和富有成果的见解。

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

在回程的飞机上,细细回味这些想法,我潦草的写下几行诗:

Now we are allowed

To disavow Landau.

Wow...

如今我们知晓

允许否定朗道

哇……

(吾辈今获 / 弗朗道之权 / 吁!)

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

虽不知这将把我们带向何方,但我仍然能感觉到新篇章即将开启。

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.

也就是差不多这个时候,另一个新的研究方向——量子信息——开始酝酿。理查德·费曼(Richard Feynman)和尤里·马宁(Yuri Manin)提出,一台能够处理量子信息的计算机可能可以执行普通数字计算机无法企及的任务。大卫·多伊奇(David Deutsch)将这一想法变成了具体的算法,从而吸引了计算机科学家的注意,这也导致彼得·肖尔(Peter Shor)最终发现量子计算机可以在多项式时间内完成大数的因数分解。与此同时, 亚历山大·霍勒夫(Alexander Holevo)、查理·贝内特(Charles Bennett)和其他研究者抓住了时机,将香农的信息理论和量子物理相结合,建立了度量量子纠缠和刻画量子信息的获取、传递及处理过程的新体系。

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 Hamiltonian, when suitably formalized, is as hard as any problem whose solution can be verified with a quantum computer. This idea launched the study of Hamiltonian 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).

虽然肖尔算法(Shor’s algorithm)的发现引爆了一轮热烈的研究,但是量子信息科学依旧处在物理学的边缘。当时,几乎没有科学家预见到量子信息和量子物质研究之间的丰富关联,但阿列克谢·基塔耶夫(Alexei Kitaev)是其中一个例外。他在1990年代提出了两个非凡的见解:他指出,严格表述后,寻找一个量子系统局域哈密顿量(Local Hamiltonian)的基态能量,与能由量子计算机验证的最困难问题同样困难。这一想法引发了对哈密顿量复杂度的研究。基塔耶夫还察觉到文小刚的拓扑序概念和量子纠错码之间的关联,而后者可以用来保护脆弱的量子叠加免受环境退相干的破坏。基塔耶夫在1997年提出关于拓扑量子计算机的概念时,这还仅仅是理论家的幻想。而如今全世界有很多实验室都在尝试这一思路(尽管距离大型量子计算机能够真正解决困难问题还有很长的路要走)。

对拓扑序中的非阿贝尔粒子进行一系列的相互交换就可以进行量子计算。这些交换的不同编织相当于不同的计算程序。

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 Hamiltonian 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.

此后,越来越多的科学家开始在量子信息和量子物质的交叉领域开展研究,使得这方面的进展大大加速。基弗尔·维道(Guifre Vidal)发现,用张量网络能够简洁地描述弱纠缠的多体量子系统。这种新方法扩展了平均场理论(Mean-Field Theory)的应用范围,并提供了一个新视角来理解为什么密度矩阵重整化群(Density Matrix Renormalization Group)如此成功。通过证明局域哈密顿量的有能隙的基态具有有限纠缠,马修·黑斯廷斯(Matthew Hastings)证明,张量网络理论有广泛的应用范围。这些工具最终使我们对一维有能隙量子态有了全面的理解。

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.

拓扑绝缘体的实验发现,使人们开始专注于对称性和拓扑学的相互联系。一种更普适的观点,对称保护拓扑(Symmetry-Protected Topological,SPT)序诞生了。这些SPT系统的内部有能隙,但其在边界上会有无能隙激发,而这一无能隙的特性会被相应的对称性保护(对于拓扑绝缘体来说,对称性意味着粒子数守恒和时间反演不变性)。同样,张量网络方法被证明非常适用于建立一维SPT相的完全分类,并指导人们研究和理解高维系统的SPT相(虽然还有很多问题悬而未决)。

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 Hamiltonian. Indeed, topological order is an intrinsic property of a quantum state and can be identified without reference to any particular Hamiltonian 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.

与我30年前第一次听文小刚提到拓扑序时相比,现在我们已经对拓扑序有了更深的认识。一个核心的新认知是拓扑序系统具有长程纠缠,这种纠缠具有一些拓扑不变性,例如拓扑纠缠熵,其数值在系统哈密顿量的微观细节发生小的变化时保持不变。事实上,拓扑序是一种量子态的固有性质,可以在完全不参考任何具体哈密顿量的情况下被识别出来。为理解长程纠缠的意义,想象在一台量子计算机上,输入一个量子纠缠态,进行一系列局部操作,最后输出完全无纠缠的直积态。如果完全解除纠缠所需要的时间不随着系统变大而增加,则称输入态有短程纠缠;反之,则为长程纠缠。更一般地来说(不是很严格),如果没有耗时恒定的量子计算可以让一个态转化为另一个态,则这两个态将属于不同的量子相。本书深入探讨了量子计算和量子序的本质关联及其导致的众多结果。

扑序的长程量子纠缠长什么样非常难以想象。通过中国结或者凯尔特结,也许读者可以体会一下各种构型的长程纠缠的样子。

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.

对于一个蓬勃发展的研究领域来说,什么时候写一本书来总结是个非常微妙的问题。书的主题应该足够成熟,相关概念和结果可以经得起考验,且可以被识别和清晰地解释。如果研究进展太过迅速,而且重点话题选择不够恰当,那么一本不合时宜的书将很快被淘汰。而另一方面,题材不应太过成熟。只有存在许多激动人心且悬而未决的问题时,这本书才能吸引大量渴望掌握其知识的读者。

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.

我有信心认为《量子信息遇见量子物质》这本书出现在一个非常恰当的时机。本书的作者都是世界级的专家,他们参与发展了书中所描述的许多科学成果,并对书的内容作出了明智的取舍。在量子信息科学和量子凝聚态物理这一交叉前沿领域,认真研究这本书的学生或科研人员都将在该领域的研究工具和观点方面得到很大的收获。

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.

事实上,我预期在未来的岁月里,会有越来越多的科学家(包括计算机科学家、化学家和高能物理学家)渴望理解《量子信息遇见量子物质》一书的核心观点。特别是,越来越多的证据表明,时空本身的量子物理内涵,是长程量子纠缠在更基本的量子理论中的演生现象。更广泛地说,随着量子技术发展得越来越复杂,我相信关于高度复杂的多体系统的理论和实验研究将会逐渐成为二十一世纪物理学的中心课题。在未来几年里,《量子信息遇见量子计算》势必在许多科学家的书架上占有一席之地。

作者简介

约翰·普雷斯基尔(John Preskill),加州理工学院理论物理学教授。在研究生阶段,他曾发表论文阐述了大统一理论中超重磁单极子在早期宇宙中的产生,这个问题导致了后来的宇宙暴涨理论的提出。

他从2000年开始担任加州理工学院量子信息中心主任,近年来主要研究与量子计算和量子信息理论相关的数学问题。

普雷斯基尔因为和另外两位理论物理学家霍金、基普·索恩( Kip Thorne)关于黑洞信息佯谬所做的物理学赌局而为大众所熟知。2004年,霍金宣布让步,并赠送一本棒球百科全书给普雷斯基尔,但物理学界并未就这一问题达成最后共识。

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