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 quantummechanical 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 largescale deployment will clearly transform our society.
There are several obstructions to highlevel 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 quantummechanical 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 nonspecialist programmers; I aim to have my biscuit and eat it too.
