Mathematical Methods for Practical Quantum Computing

我的研究计划的目标是为被称为量子计算机的计算设备设计实际应用程序。量子计算机可以利用量子力学现象。这使他们能够有效地解决某些问题，而这些问题尚无有效的经典方法。1994年，Peter Shor证明了量子计算机可以在多项式时间内分解整数，从而提供了这种量子加速的最著名的例子。这一运行时间与最著名的经典算法的指数运行时间形成了鲜明对比。自从Shor发现以来，许多利用量子计算机能力的算法已经被引入，应用范围从密码学到材料科学。这种承诺的效率提高为解决与建造量子计算机相关的挑战提供了巨大的动力，由此产生的研究努力最近最终导致了小型但完全可编程的量子计算机的开发。*尽管取得了巨大的实验进展，但量子计算机的应用仍然很遥远。根据目前的估计，运行量子算法的成本超过了迄今为止对硬件增长最乐观的预测。因此，量子计算机不太可能使用现有技术解决实际感兴趣的问题。量子计算机实际应用的主要障碍之一是将量子算法表示为逻辑量子电路时产生的开销。这一过程将算法的抽象描述映射到量子电路的显式描述，通常使用十多年前开发的技术进行。当时，这些方法被认为是足够的，因为与建造可靠的量子计算机相关的挑战还远远没有得到满足。然而，根据最近的实验进展，这些方法似乎不够充分。*我的研究计划旨在减少与构建逻辑电路及其分解为基本操作相关的开销。我计划开发构建量子电路的新方法，并为它们的优化开发可靠的工具。我预计，我的计划所产生的贡献将对量子计算领域产生重大影响，并将有助于将量子计算机转变为科学发现的工具。

The goal of my research program is to devise practical applications for computing devices known as quantum computers.****** Quantum computers can harness quantum mechanical phenomena. This allows them to efficiently solve certain problems for which no efficient classical methods are known. In 1994, Peter Shor provided the most famous example of such a quantum speedup by proving that quantum computers can factor integers in polynomial time. This running time is in striking contrast with the exponential running time of the best known classical algorithms. In the time since Shor's discovery, many algorithms leveraging the power of quantum computers have been introduced with applications ranging from cryptography to materials science. This promised increase in efficiency has provided great incentive to solve the challenges associated with building quantum computers and the resulting research efforts recently culminated in the development of small but fully programmable quantum computers.****** Despite this great experimental progress, applications of quantum computers remain distant. According to the current estimates, the cost of running quantum algorithms exceeds by far the most optimistic previsions for hardware growth. As a result, quantum computers are unlikely to solve problems of practical interests using available techniques. One of the main obstacles to the practical application of quantum computers is the overhead incurred when expressing a quantum algorithm as a logical quantum circuit. This process, which maps the abstract description of an algorithm to the explicit description of a quantum circuit, is often carried out using techniques developed more than a decade ago. At the time, these methods were considered sufficient because the challenges associated with building reliable quantum computers were so far from being met. In light of recent experimental progress, however, these methods appear inadequate.****** My research program aims at reducing the overhead associated with the construction of logical circuits and their decomposition into basic operations. I plan to develop new methods for the construction of quantum circuits and reliable tools for their optimization. I expect that the contributions stemming from my program will have significant effects on the field of quantum computation and will assist in turning quantum computers into instruments of scientific discovery.