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A related open question concerns the geometry of the complexity metric itself: while uniform penalty factors yield a positively curved metric whose geodesic flow is not ergodic, non-uniform penalty factors can render some sectional curvatures negative, though not all of them.
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On the geometry of quantum complexity ↗
Another desirable property of complexity metric is the ergodicity of geodesics, which is important to apply thermodynamical arguments to complexity evolution [ 47 , 48 , 49 ] . Ergodicity in this context refers to the general idea that the trajectory of a generic state along a geodesic will eventual…
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Another desirable property of complexity metric is the ergodicity of geodesics, which is important to apply thermodynamical arguments to complexity evolution [ 47 , 48 , 49 ] . Ergodicity in this context refers to the general idea that the trajectory of a generic state along a geodesic will eventually visit all the allowed portions of the unitary space. There are classical mathematical results (see e.g. [ 51 ] ) showing that the geodesic flow on a manifold with all negative sectional curvatures is ergodic. The complexity metric with uniform penalty factors is positively curved in all the directions and does not have an ergodic geodesic flow. The introduction of non-uniform penalty factors can make some of the sectional curvatures negative [ 5 ] , but not all of them.