Apel Harriet, Cubitt Toby
Department of Computer Science, University College London, London, UK.
Ann Henri Poincare. 2025;26(1):317-364. doi: 10.1007/s00023-024-01432-3. Epub 2024 Apr 10.
Analogue Hamiltonian simulation is a promising near-term application of quantum computing and has recently been put on a theoretical footing alongside experiencing wide-ranging experimental success. These ideas are closely related to the notion of duality in physics, whereby two superficially different theories are mathematically equivalent in some precise sense. However, existing characterisations of Hamiltonian simulations are not sufficiently general to extend to all dualities in physics. We give a generalised duality definition encompassing dualities transforming a strongly interacting system into a weak one and vice versa. We characterise the dual map on operators and states and prove equivalence ofduality formulated in terms of observables, partition functions and entropies. A building block is a strengthening of earlier results on entropy preserving maps-extensions of Wigner's celebrated theorem- -to maps that are entropy preserving up to an additive constant. We show such maps decompose as a direct sum of unitary and antiunitary components conjugated by a further unitary, a result that may be of independent mathematical interest.
模拟哈密顿量模拟是量子计算一个很有前景的近期应用，最近它在获得广泛实验成功的同时也有了理论基础。这些想法与物理学中的对偶性概念密切相关，即两个表面上不同的理论在某种精确意义上在数学上是等价的。然而，现有的哈密顿量模拟的特征描述不够通用，无法扩展到物理学中的所有对偶性。我们给出了一个广义对偶性定义，涵盖了将强相互作用系统转变为弱相互作用系统以及反之亦然的对偶性。我们刻画了算子和态上的对偶映射，并证明了在可观测量、配分函数和熵方面表述的对偶性的等价性。一个构建模块是对早期关于熵保持映射（维格纳著名定理的扩展）的结果进行强化，以得到直至一个加性常数都保持熵的映射。我们表明这样的映射分解为一个酉分量和一个反酉分量的直和，这两个分量由另一个酉算子共轭，这一结果可能具有独立的数学意义。
