Abstract

To simulate quantum systems on classical or quantum computers, the continuous observables (e.g., coordinate and momentum or energy and time) must be reduced to discrete ones. In this paper, we consider the continuous observables represented in the positional systems as power series in the radix multiplied over the summands (“digits”), which turn out to be Hermitian operators with discrete spectrum. We investigate the obtained quantum mechanical operators of digits, the commutation relations between them, and the effects of the choice of a numeral system on lattices and representations. Renormalizations of diverging sums naturally occur in constructing the digital representation.

中文翻译：

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量子力学中连续可观测量的数字表示

摘要

为了在经典或量子计算机上模拟量子系统，连续可观测量（例如坐标和动量或能量和时间）必须简化为离散可观测量。在本文中，我们将位置系统中的连续可观测量视为基数乘以被加数（“数字”）的幂级数，结果是具有离散谱的埃尔米特算子。我们研究了所获得的数字量子力学算子、它们之间的交换关系，以及数字系统的选择对格和表示的影响。在构造数字表示时自然会发生发散和的重整化。