Developing fast and efficient quantum compiling tools is critical to translate theoretical gains in quantum computation power to real world performance. Finding robust techniques to compile a quantum algorithm into the shortest sequence of standard universal quantum gates has profound impact to quantum computation in both near and long term. In the near term with the quantum hardware available currently or in the near future, halving the circuit lengths can mean the difference between success and failure of a computation task. In the long term, these techniques make fault-tolerant computing possible, which requires optimization of vast, complex circuits. The most viable approach to achieve this goal is to break up a large circuit into few-qubit subcircuits where reduction of the circuit length even merely by a constant factor manifests as an exponential improvement in computation performance. While optimal algorithms for single-qubit circuits have been well-understood, current efforts focus on solving the optimal compiling problem of two-qubit circuits. Achievement of this important milestone leading to further applications to larger circuits relies on a convergent effort with cross-disciplinary expertise, engaging students at both undergraduate and graduate levels in research and education programs, and participation of industrial collaborators.
The general quantum compiling problem being addressed here is as follows: given some unitary matrix representing the operation of a computation algorithm and a universal set of quantum gates, find a sequence of gates that either is equivalent to the unitary matrix (called exact synthesis) or approximates the unitary matrix within a desired precision (called inexact synthesis). The most efficient compiler is a circuit synthesis algorithm optimized by minimization of a given cost-metrics, such as the gate-count, especially for the gates that are difficulty to implement in a fault-tolerance manner. Optimal or at least nearly-optimal algorithms of two-qubit circuits for exact and inexact synthesis are achievable utilizing techniques based on number theory and matrix decomposition. It is estimated that the gate-count can be reduced by a constant-factor ranging roughly 1~10. Algorithms for either case have stand along values and are complementary to each other. They can be explored simultaneously and completed independently. Furthermore, these algorithms can be the basis to build synthesis techniques for even larger qubit-number circuits.
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