Right-invariant Metrics Research Articles

Polynomial Equivalence of Complexity Geometries

This paper proves the polynomial equivalence of a broad class of definitions of quantum computational complexity. We study right-invariant metrics on the unitary group—often called `complexity geometries' following the definition of quantum complexity proposed by Nielsen—and delineate the equivalence class of metrics that have the same computational power as quantum circuits. Within this universality class, any unitary that can be reached in one metric can be approximated in any other metric in the class with a slowdown that is at-worst polynomial in the length and number of qubits and inverse-polynomial in the permitted error. We describe the equivalence classes for two different kinds of error we might tolerate: Killing-distance error, and operator-norm error. All metrics in both equivalence classes are shown to have exponential diameter; all metrics in the operator-norm equivalence class are also shown to give an alternative definition of the quantum complexity class BQP. My results extend those of Nielsen et al., who in 2006 proved that one particular metric is polynomially equivalent to quantum circuits. The Nielsen et al. metric is incredibly highly curved. I show that the greatly enlarged equivalence class established in this paper also includes metrics that have modest curvature. I argue that the modest curvature makes these metrics more amenable to the tools of differential geometry, and therefore makes them more promising starting points for Nielsen's program of using differential geometry to prove complexity lowerbounds. In a previous paper my collaborators and I—inspired by the UV/IR decoupling that happens in the phenomenon of renormalization—conjectured that high- dimensional metrics that look very different at short scales will often nevertheless give rise at long scales to the same emergent effective geometry. The results of this paper provide evidence for those conjectures, since many complexity metrics that have radically different penalty factors and therefore radically different short- distance properties are shown to belong to the same long-distance equivalence class.

Circuit complexity in U(1) gauge theory

In this paper, we study circuit complexity for a free vector field of a [Formula: see text] gauge theory in Coulomb gauge, and Gaussian states. We introduce a quantum circuit model with Gaussian states, including reference and target states. Using Nielsen’s geometric approach, the complexity then can be found as the shortest geodesic in the space of states. This geodesic is based on the notion of geodesic distance on the Lie group of Bogoliubov transformations equipped with a right-invariant metric. We use the framework of the covariance matrix to compute circuit complexity between Gaussian states. We apply this framework to the free vector field in general dimensions where we compute the circuit complexity of the ground state of the Hamiltonian.

Six-dimensional Lie–Einstein metrics

A metric is said to be Lie–Einstein if it is both a one-sided invariant metric on a Lie group and also an Einstein space. In this article right-invariant metrics are used thoughout. Several Lie–Einstein metrics on six-dimension Lie groups are constructed. The associated Lie algebras are necessarily solvable and are assumed to be indecomposable. They belong to two classes studied by Mubarakzyanov and Turkowski, respectively. Since the general problem of finding Lie–Einstein metrics in dimension six appears to be intractable, a certain ansatz for the form of the metric and class of Lie algebra is adopted. Nineteen Mubarakzyanov and five Turkowski Lie algebras are shown to be Lie–Einstein.

Geodesic distance for right-invariant metrics on diffeomorphism groups: critical Sobolev exponents

We study the geodesic distance induced by right-invariant metrics on the group $\operatorname{Diff}_c(M)$ of compactly supported diffeomorphisms of a manifold $M$, and show that it vanishes for the critical Sobolev norms $W^{s,n/s}$, where $n$ is the dimension of $M$ and $s\in(0,1)$. This completes the proof that the geodesic distance induced by $W^{s,p}$ vanishes if $sp\le n$ and $s<1$, and is positive otherwise. The proof is achieved by combining the techniques of two recent papers --- [JM19] by the authors, which treated the sub-critical case, and [BHP18] of Bauer, Harms and Preston, which treated the critical 1-dimensional case.

Principles and symmetries of complexity in quantum field theory

Based on general and minimal properties of the discrete circuit complexity, we define the complexity in continuous systems in a geometrical way. We first show that the Finsler metric naturally emerges in the geometry of the complexity in continuous systems. Due to fundamental symmetries of quantum field theories, the Finsler metric is more constrained and consequently, the complexity of SU(n) operators is uniquely determined as a length of a geodesic in the Finsler geometry. Our Finsler metric is bi-invariant contrary to the right-invariance of discrete qubit systems. We clarify why the bi-invariance is relevant in quantum field theoretic systems. After comparing our results with discrete qubit systems we show most results in k-local right-invariant metric can also appear in our framework. Based on the bi-invariance of our formalism, we propose a new interpretation for the Schrödinger’s equation in isolated systems – the quantum state evolves by the process of minimizing “computational cost”.

Vanishing geodesic distance for right-invariant Sobolev metrics on diffeomorphism groups

We study the geodesic distance induced by right-invariant metrics on the group $\operatorname{Diff}_\text{c}(M)$ of compactly supported diffeomorphisms, for various Sobolev norms $W^{s,p}$. Our main result is that the geodesic distance vanishes identically on every connected component whenever $s<\min\{n/p,1\}$, where $n$ is the dimension of $M$. We also show that previous results imply that whenever $s > n/p$ or $s \ge 1$, the geodesic distance is always positive. In particular, when $n\ge 2$, the geodesic distance vanishes if and only if $s<1$ in the Riemannian case $p=2$, contrary to a conjecture made in Bauer et al. [BBHM13].

Circuit complexity for free fermions

We study circuit complexity for free fermionic field theories and Gaussian states. Our definition of circuit complexity is based on the notion of geodesic distance on the Lie group of special orthogonal transformations equipped with a right-invariant metric. After analyzing the differences and similarities to bosonic circuit complexity, we develop a comprehensive mathematical framework to compute circuit complexity between arbitrary fermionic Gaussian states. We apply this framework to the free Dirac field in four dimensions where we compute the circuit complexity of the Dirac ground state with respect to several classes of spatially unentangled reference states. Moreover, we show that our methods can also be applied to compute the complexity of excited energy eigenstates of the free Dirac field. Finally, we discuss the relation of our results to alternative approaches based on the Fubini-Study metric, the relevance to holography and possible extensions.

The Camassa–Holm equation as an incompressible Euler equation: A geometric point of view

The group of diffeomorphisms of a compact manifold endowed with the L2 metric acting on the space of probability densities gives a unifying framework for the incompressible Euler equation and the theory of optimal mass transport. Recently, several authors have extended optimal transport to the space of positive Radon measures where the Wasserstein–Fisher–Rao distance is a natural extension of the classical L2-Wasserstein distance. In this paper, we show a similar relation between this unbalanced optimal transport problem and the Hdiv right-invariant metric on the group of diffeomorphisms, which corresponds to the Camassa–Holm (CH) equation in one dimension. Geometrically, we present an isometric embedding of the group of diffeomorphisms endowed with this right-invariant metric in the automorphisms group of the fiber bundle of half densities endowed with an L2 type of cone metric. This leads to a new formulation of the (generalized) CH equation as a geodesic equation on an isotropy subgroup of this automorphisms group; On S1, solutions to the standard CH thus give radially 1-homogeneous solutions of the incompressible Euler equation on R2 which preserves a radial density that has a singularity at 0. An other application consists in proving that smooth solutions of the Euler–Arnold equation for the Hdiv right-invariant metric are length minimizing geodesics for sufficiently short times.

Metrization Theorem for Uniform Loops with the Invertibility Property

In this paper, we have proved a metrization theorem that gives the sufficient conditions for a uniform IP-loop X to be metrizable by a left-invariant metric. It is shown that by consideration of topological IP-loop dual to X we obtain an analogical theorem for the case of the right-invariant metric.

Noncommutative Integration and Symmetry Algebra of the Dirac Equation on the Lie Groups

The algebra of first-order symmetry operators of the Dirac equation on four-dimensional Lie groups with right-invariant metric is investigated. It is shown that the algebra of symmetry operators is in general not a Lie algebra. Noncommutative reduction mediated by spin symmetry operators is investigated. For the Dirac equation on the Lie group SO(2,1) a parametric family of particular solutions obtained by the method of noncommutative integration over a subalgebra containing a spin symmetry operator is constructed.

Anisotropically Weighted and Nonholonomically Constrained Evolutions on Manifolds

We present evolution equations for a family of paths that results from anisotropically weighting curve energies in non-linear statistics of manifold valued data. This situation arises when performing inference on data that have non-trivial covariance and are anisotropic distributed. The family can be interpreted as most probable paths for a driving semi-martingale that through stochastic development is mapped to the manifold. We discuss how the paths are projections of geodesics for a sub-Riemannian metric on the frame bundle of the manifold, and how the curvature of the underlying connection appears in the sub-Riemannian Hamilton–Jacobi equations. Evolution equations for both metric and cometric formulations of the sub-Riemannian metric are derived. We furthermore show how rank-deficient metrics can be mixed with an underlying Riemannian metric, and we relate the paths to geodesics and polynomials in Riemannian geometry. Examples from the family of paths are visualized on embedded surfaces, and we explore computational representations on finite dimensional landmark manifolds with geometry induced from right-invariant metrics on diffeomorphism groups.

Diffeomorphic Density Matching by Optimal Information Transport

We address the following problem: given two smooth densities on a manifold, find an optimal diffeomorphism that transforms one density into the other. Our framework builds on connections between the Fisher-Rao information metric on the space of probability densities and right-invariant metrics on the infinite-dimensional manifold of diffeomorphisms. This optimal information transport, and modifications thereof, allows us to construct numerical algorithms for density matching. The algorithms are inherently more efficient than those based on optimal mass transport or diffeomorphic registration. Our methods have applications in medical image registration, texture mapping, image morphing, non-uniform random sampling, and mesh adaptivity. Some of these applications are illustrated in examples.

Riemannian Geometry of the Contactomorphism Group

Given an odd-dimensional compact manifold and a contact form, we consider the group of contact transformations of the manifold (contactomorphisms) and the subgroup of those transformations that precisely preserve the contact form (quantomorphisms). If the manifold also has a Riemannian metric, we can consider the \(L^2\) inner product of vector fields on it, which by restriction gives an inner product on the tangent space at the identity of each of the groups that we consider. We then obtain right-invariant metrics on both the contactomorphism and quantomorphism groups. We show that the contactomorphism group has geodesics at least for short time and that the quantomorphism group is a totally geodesic subgroup of it. Furthermore we show that the geodesics in this smaller group exist globally. Our methodology is to use the right invariance to derive an “Euler–Arnold” equation from the geodesic equation and to show using ODE methods that it has solutions which depend smoothly on the initial conditions. For global existence we then derive a “quasi-Lipschitz” estimate on the stream function, which leads to a Beale–Kato–Majda criterion which is automatically satisfied for quantomorphisms. Special cases of these Euler–Arnold equations are the Camassa–Holm equation (when the manifold is one-dimensional) and the quasi-geostrophic equation in geophysics.

Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle

In this paper, we study the geodesic flow of a right-invariant metric induced by a general Fourier multiplier on the diffeomorphism group of the circle and on some of its homogeneous spaces. This study covers in particular right-invariant metrics induced by Sobolev norms of fractional order. We show that, under a certain condition on the symbol of the inertia operator (which is satisfied for the fractional Sobolev norm $H^{s}$ for $s \ge 1/2$), the corresponding initial value problem is well-posed in the smooth category and that the Riemannian exponential map is a smooth local diffeomorphism. Paradigmatic examples of our general setting cover, besides all traditional Euler equations induced by a local inertia operator, the Constantin-Lax-Majda equation, and the Euler-Weil-Petersson equation.

Constrained Diffeomorphic Shape Evolution

We design optimal control strategies in spaces of diffeomorphisms and shape spaces in which the Eulerian velocities of the evolving deformations are constrained to belong to a suitably chosen finite-dimensional space, which is also following the motion. This results in a setting that provides a great flexibility in the definition of Riemannian metrics, extending previous approaches in which shape spaces are built as homogeneous spaces under the action of the diffeomorphism group equipped with a right-invariant metric. We provide specific instances of this general setting, and describe in detail the resulting numerical algorithms, with experimental illustrations in the case of plane curves.

The geometry of a vorticity model equation

We provide rigorous evidence of the fact that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics describes the geodesic flow on the subgroup of orientation-preserving diffeomorphisms fixing one point, with respect to right-invariant metric induced by the homogeneous Sobolev norm $H^{1/2}$ and show the local existence of the geodesics in the extended group of diffeomorphisms of Sobolev class $H^{k}$ with $k\ge 2$.

ON THE GEODESIC FLOW ON THE GROUP OF DIFFEOMORPHISMS OF THE CIRCLE WITH A FRACTIONAL SOBOLEV RIGHT-INVARIANT METRIC

We show that the geodesic flow on the infinite-dimensional group of diffeomorphisms of the circle, endowed with a fractional Sobolev metric at the identity, is described by the generalized Constantin–Lax–Majda equation with parameter a = -½.

Control of mechanical systems on Lie groups and ideal hydrodynamics

In contrast to the Euler–Poincare reduction of geodesic flows of left- or right-invariant metrics on Lie groups to the corresponding Lie algebra (or its dual), one can consider the reduction of the geodesic flows to the group itself. The reduced vector field has a remarkable hydrodynamic interpretation: it is the velocity field for a stationary flow of an ideal fluid. Right- or left-invariant symmetry fields of the reduced field define vortex manifolds for such flows.

A Bi-Hamiltonian Supersymmetric Geodesic Equation

A supersymmetric extension of the Hunter–Saxton equation is constructed. We present its bi-Hamiltonian structure and show that it arises geometrically as a geodesic equation on the space of superdiffeomorphisms of the circle that leave a point fixed endowed with a right-invariant metric.

The Hunter–Saxton Equation: A Geometric Approach

We provide a rigorous foundation for the geometric interpretation of the Hunter–Saxton equation as the equation describing the geodesic flow of the $\dot{H}^1$ right-invariant metric on the quotient space $Rot(\mathbb{S})\backslash\mathcal{D}^k(\mathbb{S})$ of the infinite-dimensional Banach manifold $\mathcal{D}^k(\mathbb{S})$ of orientation-preserving $H^k$-diffeomorphisms of the unit circle $\mathbb{S}$ modulo the subgroup of rotations $Rot(\mathbb{S})$. Once the underlying Riemannian structure has been established, the method of characteristics is used to derive explicit formulas for the geodesics corresponding to the $\dot{H}^1$ right-invariant metric, yielding, in particular, new explicit expressions for the spatially periodic solutions of the initial-value problem for the Hunter–Saxton equation.