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1. Efficient Quantum Algorithms for Testing Symmetries of Open Quantum Systems

   https://doi.org/10.1142/S1230161223500178  

   Bandyopadhyay, Rahul; Rubin, Alex H.; Radulaski, Marina; Wilde, Mark M. (November 2023, Open Systems & Information Dynamics)

   Symmetry is an important and unifying notion in many areas of physics. In quantum mechanics, it is possible to eliminate degrees of freedom from a system by leveraging symmetry to identify the possible physical transitions. This allows us to simplify calculations and characterize potentially complicated dynamics of the system with relative ease. Previous works have focused on devising quantum algorithms to ascertain symmetries by means of fidelity-based symmetry measures. In our present work, we develop alternative symmetry testing quantum algorithms that are efficiently implementable on quantum computers. Our approach estimates asymmetry measures based on the Hilbert–Schmidt distance, which is significantly easier, in a computational sense, than using fidelity as a metric. The method is derived to measure symmetries of states, channels, Lindbladians, and measurements. We apply this method to a number of scenarios involving open quantum systems, including the amplitude damping channel and a spin chain, and we test for symmetries within and outside the finite symmetry group of the Hamiltonian and Lindblad operators. 

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2. The orbit-polynomial: a novel measure of symmetry in networks

   https://doi.org/10.1109/ACCESS.2020.2970059  

   Dehmer, M.; Chen, Z.; Emmert-Streib, F.; Mowshowitz, A.; Varmuza, K.; Feng, L.; Jodlbauer, H.; Shi, Y.; and Tao, J. (January 2020, IEEE access)

   Research on the structural complexity of networks has produced many useful results in graph theory and applied disciplines such as engineering and data analysis. This paper is intended as a further contribution to this area of research. Here we focus on measures designed to compare graphs with respect to symmetry. We do this by means of a novel characteristic of a graph G, namely an ``orbit polynomial.'' A typical term of this univariate polynomial is of the form czn, where c is the number of orbits of size n of the automorphism group of G. Subtracting the orbit polynomial from 1 results in another polynomial that has a unique positive root, which can serve as a relative measure of the symmetry of a graph. The magnitude of this root is indicative of symmetry and can thus be used to compare graphs with respect to that property. In what follows, we will prove several inequalities on the unique positive roots of orbit polynomials corresponding to different graphs, thus showing differences in symmetry. In addition, we present numerical results relating to several classes of graphs for the purpose of comparing the new symmetry measure with existing ones. Finally, it is applied to a set of isomers of the chemical compound adamantane C10H16. We believe that the measure can be quite useful for tackling applications in chemistry, bioinformatics, and structure-oriented drug design. 

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3. Theory of quantum circuits with Abelian symmetries

   https://doi.org/10.1103/PhysRevResearch.6.043292  

   Marvian, Iman (December 2024, Physical Review Research)

   Quantum circuits with gates (local unitaries) respecting a global symmetry have broad applications in quantum information science and related fields, such as condensed-matter theory and quantum thermodynamics. However, despite their widespread use, fundamental properties of such circuits are not well understood. Recently, it was found that generic unitaries respecting a global symmetry cannot be realized, even approximately, using gates that respect the same symmetry. This observation raises important open questions: What unitary transformations can be realized with $k$-local gates that respect a global symmetry? In other words, in the presence of a global symmetry, how does the locality of interactions constrain the possible time evolution of a composite system? In this work, we address these questions for the case of Abelian (commutative) symmetries and develop constructive methods for synthesizing circuits with such symmetries. Remarkably, as a corollary, we find that, while the locality of interactions still imposes additional constraints on realizable unitaries, certain restrictions observed in the case of non-Abelian symmetries do not apply to circuits with Abelian symmetries. For instance, in circuits with a general non-Abelian symmetry such as $\mathrm{SU}\left(d\right)$, the unitary realized in a subspace with one irreducible representation (charge) of the symmetry dictates the realized unitaries in multiple other sectors with inequivalent representations of the symmetry. Furthermore, in certain sectors, rather than all unitaries respecting the symmetry, the realizable unitaries are the symplectic or orthogonal subgroups of this group. We prove that none of these restrictions appears in the case of Abelian symmetries. This result suggests that global non-Abelian symmetries may affect the thermalization of quantum systems in ways not possible under Abelian symmetries. Published by the American Physical Society2024 

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4. Fractionalization of coset non-invertible symmetry and exotic Hall conductance

   https://doi.org/10.21468/SciPostPhys.17.3.095  

   Hsin, Po-Shen; Kobayashi, Ryohei; Zhang, Carolyn (January 2024, SciPost Physics)

   We investigate fractionalization of non-invertible symmetry in (2+1)D topological orders. We focus on coset non-invertible symmetries obtained by gauging non-normal subgroups of invertible0 $0$-form symmetries. These symmetries can arise as global symmetries in quantum spin liquids, given by the quotient of the projective symmetry group by a non-normal subgroup as invariant gauge group. We point out that such coset non-invertible symmetries in topological orders can exhibit symmetry fractionalization: each anyon can carry a “fractional charge” under the coset non-invertible symmetry given by a gauge invariant superposition of fractional quantum numbers. We present various examples using field theories and quantum double lattice models, such as fractional quantum Hall systems with charge conjugation symmetry gauged and finite group gauge theory from gauging a non-normal subgroup. They include symmetry enrichedS_3 $S_3$andO(2) $O\left(2\right)$gauge theories. We show that such systems have a fractionalized continuous non-invertible coset symmetry and a well-defined electric Hall conductance. The coset symmetry enforces a gapless edge state if the boundary preserves the continuous non-invertible symmetry. We propose a general approach for constructing coset symmetry defects using a “sandwich” construction: non-invertible symmetry defects can generally be constructed from an invertible defect sandwiched by condensation defects. The anomaly free condition for finite coset symmetry is also identified. 

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5. Chern-Weil global symmetries and how quantum gravity avoids them

   https://doi.org/10.1007/JHEP11(2021)053  

   Heidenreich, Ben; McNamara, Jacob; Montero, Miguel; Reece, Matthew; Rudelius, Tom; Valenzuela, Irene (November 2021, Journal of High Energy Physics)

   A bstract We draw attention to a class of generalized global symmetries, which we call “Chern-Weil global symmetries,” that arise ubiquitously in gauge theories. The Noether currents of these Chern-Weil global symmetries are given by wedge products of gauge field strengths, such as F 2 ∧ H 3 and tr( $$ {F}_2^2 $$ F 2 2 ), and their conservation follows from Bianchi identities. As a result, they are not easy to break. However, it is widely believed that exact global symmetries are not allowed in a consistent theory of quantum gravity. As a result, any Chern-Weil global symmetry in a low-energy effective field theory must be either broken or gauged when the theory is coupled to gravity. In this paper, we explore the processes by which Chern-Weil symmetries may be broken or gauged in effective field theory and string theory. We will see that many familiar phenomena in string theory, such as axions, Chern-Simons terms, worldvolume degrees of freedom, and branes ending on or dissolving in other branes, can be interpreted as consequences of the absence of Chern-Weil symmetries in quantum gravity, suggesting that they might be general features of quantum gravity. We further discuss implications of breaking and gauging Chern-Weil symmetries for particle phenomenology and for boundary CFTs of AdS bulk theories. Chern-Weil global symmetries thus offer a unified framework for understanding many familiar aspects of quantum field theory and quantum gravity. 

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