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An explicit family of unitaries with exponentially minimal length Pauli geodesics

Wei Huang
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Abstract

Recently, Nielsen et al have proposed a geometric approach to quantum computation. They've shown that the size of the minimum quantum circuits implementing a unitary U, up to polynomial factors, equals to the length of minimal geodesic from identity I through U. They've investigated a large class of solutions to the geodesic equation, called Pauli geodesics. They've raised a natural question whether we can explicitly construct a family of unitaries U that have exponentially long minimal length Pauli geodesics? We give a positive answer to this question.

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Wei Huang. 2007-01-28. An explicit family of unitaries with exponentially minimal length Pauli geodesics. https://arxiv.org/abs/quant-ph/0701202

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