Combining Viewpoints in Quantum Theory

数学、计算机科学和物理学有着美丽而奇妙的共生关系。纯粹的数学推理可以不可思议地对物理世界产生影响。反之亦然，物理直觉和实验可以揭示看似完全抽象和脱离现实的数学真理。一方面，计算机科学可以被视为模拟物理系统或辅助数学探索的工具。另一方面，它也是物理学和数学的特例。计算机毕竟是物理对象，因此受物理定律的支配。然而，驱动计算机科学的经典问题可以独立于计算机的物理构建方式来回答。例如，抽象推理本身就可以决定计算机在原则上是否可以被编程来解决给定的问题，如果可以的话，效率有多高。自从我第一次获得知识以来，我一直对物理科学中纯思想的这种不合理的有效性感到惊讶。这吸引了我到量子计算机科学，它位于数学，计算机科学和物理学的界面。量子计算机本质上是我们可以控制的小型量子力学系统，用于使自然界比任何经典计算机更有效地解决某些问题。深入了解量子计算的细节，使其能够大规模部署，显然将改变我们的社会。其中最基本的几点直接触及量子力学反直觉性的核心。问题是量子力学的体系削弱了逻辑思维和直觉的力量，而逻辑思维和直觉通常是如此有效。例如，如果我给你一块饼干，让你选择茶或咖啡，你会希望得到茶和饼干，或者咖啡和饼干。但在量子力学定律下，这一最基本的逻辑真理不再成立。这是由于人们一次只能从一个经典观点从量子系统中提取数据。要了解更多关于系统的信息，我们需要从多个经典观点将联合收割机测量组合起来。同样，量子计算机比经典计算机强大得多，正是因为量子程序员能够在不同的经典观点中工作并在不同的经典观点之间切换。然而，由于某种原因，经典观点之间的转换没有得到系统的研究，很可能是因为量子系统通常是孤立地研究的。幸运的是，计算机科学、物理学和数学之间的联系比上面所描述的还要深刻。理论计算机科学擅长处理整个系统社区，包括所有并行的复合系统。因此，计算机科学技术的转移实际上影响了物理学和数学，这些概念没有得到太多的关注。我将把量子系统和经典观点放在同一个范畴中，研究它们之间的动力学关系，并最终努力恢复抽象思维在这一领域的有效性。这将促进我们对自然的理论理解。与此同时，它将通过使量子协议和算法的设计更容易为非专业程序员所接受而带来实际好处;我的目标是得到我的饼干并吃掉它。

Mathematics, computer science, and physics enjoy a beautiful but curious symbiotic relationship. Pure mathematical reasoning can uncannily have consequences for the physical world. Vice versa, physical intuition and experiment can uncover mathematical truths that seemed entirely abstract and divorced from reality. Computer science, on the one hand, can be regarded as a tool to simulate physical systems or aid mathematical exploration. On the other hand, it is also a special case of both physics and mathematics. Computers are, after all, physical objects, and are therefore governed by the laws of physics. Nevertheless, the classic questions driving computer science can be answered independently from the physical way computers are built. For example, abstract reasoning alone can decide whether a computer can in principle be programmed to solve a given problem, and if so, how efficiently.Ever since my first acquantaince I have been amazed by this unreasonable effectiveness of pure thought in physical sciences. This has drawn me to quantum computer science, which lies at the interface of mathematics, computer science, and physics. Quantum computers are essentially small quantum-mechanical systems that we can control, used to make nature solve certain problems much more efficiently than any classical computer could. Understanding quantum computing in enough detail to allow its large-scale deployment will clearly transform our society.There are several obstructions to high-level quantum programming. The most fundamental ones run straight to the heart of the counterintuitiveness of quantum mechanics. The problem is that the regime of quantum mechanics diminishes the power of logical thought and intuition that is usually so effective. For example, if I were to offer you a biscuit and a choice of tea or coffee, you would expect to receive either tea and a biscuit, or coffee and a biscuit. But under quantum-mechanical laws, this most basic logical truth no longer holds. This is caused by the fact that one can only extract data from a quantum system from one classical viewpoint at a time. To learn more about the system, we need to combine measurements from multiple classical viewpoints. Similarly, quantum computers are so much more powerful than classical ones precisely because of the ability of a quantum programmer to work in, and switch between, different classical viewpoints. However, the switching between classical viewpoints has escaped systematic study for some reason, most probably because quantum systems are usually studied in isolation. Fortunately, the ties between computer science, physics, and mathematics run even deeper than sketched above. Theoretical computer science excels in handling entire communities of systems, including compound ones, that all live in parallel. Thus, transfer of computer science techniques in fact influences physics and mathematics, where such notions have not received much attention. I will place quantum systems and classical viewpoints on an equal footing in a single category, investigate the dynamical relationships between them, and eventually endeavour to restore the effectiveness of abstract thought in this realm. This will advance our theoretical understanding of nature. At the same time, it will have practical benefits by making the design of quantum protocols and algorithms more accessible to non-specialist programmers; I aim to have my biscuit and eat it too.