Circuit complexity across a topological phase transition

Abstract

where k n = 2π L (n + 1/2) with n = 0, 1, . . . , L/2 − 1.

Conformal field theory complexity from Euler-Arnold equations

“…Secondly, given the importance of differential regularisation (section 3.3), metric degeneracy (section 3.4) and gauge invariance (section 4.3.3) for the absence of cost assignment for complex phases, we recognise that it is the HS equation, α = 0, β = 1 in (6.2), which should yield the most plausible model of complexity. 12 When it comes to the ease of solving, the HS equation lies somewhere in between the KdV equation studied in [36,42], which however assigns non-zero complexity to complex phases, and our Fubini-Study result (4.4), which are considerably more complicated due to their partial integro-differential nature. One interesting physical quantity by which such equations and their solutions can be qualitatively characterized and compared are sectional curvatures.…”

“…13 However, we should remind the reader that our equation (4.4) is completely universal and negative sectional curvature arise for free CFTs 1+1 as well as for holographic CFTs 1+1 . It is also known that the metrics associated with the KdV and CH equations give rise to sectional curvatures of non-definite sign [82,83], while the geometry of the HS equation corresponding to an open subset of a sphere has 12 On a formal level, the degeneracy of the metric on the Virasoro-algebra means that it might be as well defined as a non-degenerate metric on the homogeneous space V ir/S 1 [75]. This operation of taking the group manifold modulo rotations corresponds to fixing the U(1)-gauge symmetry of section 4.3.3.…”

“…The approach elucidated above was originally introduced in [1][2][3] as a way of bounding qubit circuits preparing unitary transformations of interest in the context of quantum computing and currently serves as our best definition of complexity in the QFT setting. In the course of the past three years starting with [4,5] many results were delivered on complexity of quantum fields by focusing in the vast majority of cases on free QFTs soluble via Gaussian techniques [6], see [7][8][9][10][11][12][13][14][15][16][17][18] for sample developments.…”

Aspects of entanglement, chaos and complexity: from many-body to high energy systems

“…As part of this effort, [97,98,99] proposed to characterize QPTs, including topological ones, using a geometric notion of circuit complexity introduced by Nielsen [90,91]. Inspired by its computer science analogue, this object quantifies how difficult it is to construct a particular unitary operator that maps between a pair of given reference and target states, i.e., the minimum number of basic operations needed to implement this task.…”

“…Perhaps the simplest and one of the most studied among these non-equilibrium protocols is that of a quantum quench, where a parameter of the hamiltonian is suddenly changed and the system is let evolve under the new hamiltonian [105,106]. This also includes the study of the quench dynamics of circuit complexity [107,108,109,97,98,110]. Here we propose to go a step further in the endeavour of using circuit complexity as a novel tool to understand the dynamics of quantum many-body systems and explore a different non-equilibrium protocol corresponding to the periodic driving of many-body systems.…”