Symmetry is a unifying concept in physics. In quantum information and beyond, it is known that quantum states possessing symmetry are not useful for certain information-processing tasks. For example, states that commute with a Hamiltonian realizing a time evolution are not useful for timekeeping during that evolution, and bipartite states that are highly extendible are not strongly entangled and thus not useful for basic tasks like teleportation. Motivated by this perspective, this paper details several quantum algorithms that test the symmetry of quantum states and channels. For the case of testing Bose symmetry of a state, we show that there is a simple and efficient quantum algorithm, while the tests for other kinds of symmetry rely on the aid of a quantum prover. We prove that the acceptance probability of each algorithm is equal to the maximum symmetric fidelity of the state being tested, thus giving a firm operational meaning to these latter resource quantifiers. Special cases of the algorithms test for incoherence or separability of quantum states. We evaluate the performance of these algorithms on choice examples by using the variational approach to quantum algorithms, replacing the quantum prover with a parameterized circuit. We demonstrate this approach for numerous examples using the IBM quantum noiseless and noisy simulators, and we observe that the algorithms perform well in the noiseless case and exhibit noise resilience in the noisy case. We also show that the maximum symmetric fidelities can be calculated by semi-definite programs, which is useful for benchmarking the performance of these algorithms for sufficiently small examples. Finally, we establish various generalizations of the resource theory of asymmetry, with the upshot being that the acceptance probabilities of the algorithms are resource monotones and thus well motivated from the resource-theoretic perspective.

翻译：对称性是物理学中的一个统一概念。在量子信息及其他领域，已知具有对称性的量子态对某些信息处理任务无用。例如，与实现时间演化的哈密顿量对易的态在此演化过程中对时间计量无用，而高度可扩展的双粒子态并非强纠缠态，因此对诸如量子隐形传态等基本任务无效。基于这一观点，本文详细阐述了若干用于测试量子态与量子通道对称性的量子算法。对于测试态中的玻色对称性，我们展示了一种简单高效的量子算法，而对其他类型对称性的测试则依赖于量子证明者的辅助。我们证明每个算法的接受概率等于被测试态的最大对称保真度，从而为后者的资源度量赋予明确的操作意义。这些算法的特例可用于测试量子态的非相干性或可分离性。我们通过变分量子算法方法评估这些算法在典型实例上的性能，即用参数化电路替代量子证明者。我们使用IBM量子无噪声及有噪声模拟器对多个实例进行演示，观察到算法在无噪声情况下表现良好，且在噪声情况下展现出噪声鲁棒性。我们还证明了最大对称保真度可通过半定规划计算，这对在足够小规模的实例上基准测试这些算法性能十分有用。最后，我们建立了不对称资源理论的多种推广形式，其关键结论是算法的接受概率是资源单调量，因此从资源论视角来看具备充分的动机。