Combining Viewpoints in Quantum Theory

数学、计算机科学和物理学之间存在着一种美丽而奇特的共生关系。纯粹的数学推理可能会给物理世界带来不可思议的后果。反之亦然，物理直觉和实验可以发现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，它也是物理和数学的特例。毕竟，计算机是物理对象，因此受物理定律的支配。然而，推动计算机科学的经典问题可以独立于计算机的物理构造方式来回答。例如，抽象推理本身就可以决定计算机在原则上能否被编程来解决给定的问题，如果是的话，效率如何。从我第一次了解起，我就一直对物理科学中纯粹思维的这种不合理的有效性感到惊讶。这吸引了我对量子计算机科学的兴趣，它位于数学、计算机科学和物理学的交界处。量子计算机本质上是我们可以控制的小型量子力学系统，用于让自然比任何经典计算机更有效地解决某些问题。了解足够详细的量子计算，使其能够大规模部署，显然将改变我们的社会。高级量子编程有几个障碍。最基本的问题直指量子力学反直觉的核心。问题是，量子力学的机制削弱了通常如此有效的逻辑思维和直觉的力量。例如，如果我给你一块饼干，可以选择茶或咖啡，你会得到茶和饼干，或者咖啡和饼干。但根据量子力学定律，这一最基本的逻辑真理不再成立。这是因为人们一次只能从一个经典观点中提取量子系统的数据。为了更多地了解该系统，我们需要结合来自多个经典观点的测量结果。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之间工作和切换。然而，由于某种原因，经典观点之间的转换未能得到系统的研究，最可能的原因是量子系统通常被孤立地研究。幸运的是，计算机科学、物理学和数学之间的联系比上面勾勒的更深。理论计算机科学擅长处理整个系统社区，包括复合系统，这些系统都是并行存在的。因此，计算机科学技术的转移实际上影响了物理和数学，而这些概念并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类别中，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，让非专业程序员更容易获得量子协议和算法的设计，这将带来实际好处；我的目标是两全其美。

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.