Double Bracket Flows Research Articles

Double-bracket quantum algorithms for diagonalization

This work proposes double-bracket iterations as a framework for obtaining diagonalizing quantum circuits. Their implementation on a quantum computer consists of interlacing evolutions generated by the input Hamiltonian with diagonal evolutions which can be chosen variationally. No qubit overheads or controlled-unitary operations are needed but the method is recursive which makes the circuit depth grow exponentially with the number of recursion steps. To make near-term implementations viable, the proposal includes optimization of diagonal evolution generators and of recursion step durations. Indeed, thanks to this numerical examples show that the expressive power of double-bracket iterations suffices to approximate eigenstates of relevant quantum models with few recursion steps. Compared to brute-force optimization of unstructured circuits double-bracket iterations do not suffer from the same trainability limitations. Moreover, with an implementation cost lower than required for quantum phase estimation they are more suitable for near-term quantum computing experiments. More broadly, this work opens a pathway for constructing purposeful quantum algorithms based on so-called double-bracket flows also for tasks different from diagonalization and thus enlarges the quantum computing toolkit geared towards practical physics problems.

The Toda Flow as a Porous Medium Equation

We describe the geometry of the incompressible porous medium (IPM) equation: we prove that it is a gradient dynamical system on the group of area-preserving diffeomorphisms and has a special double-bracket form. Furthermore, we show its similarities and differences with the dispersionless Toda system. The Toda flow describes an integrable interaction of several particles on a line with an exponential potential between neighbours, while its continuous version is an integrable PDE, whose physical meaning was obscure. Here we show that this continuous Toda flow can be naturally regarded as a special IPM equation, while the key double-bracket property of Toda is shared by all equations of the IPM type, thus manifesting their gradient and non-autonomous Hamiltonian origin. Finally, we comment on Toda and IPM modifications of the QR diagonalization algorithm, as well as describe double-bracket flows in an invariant setting of general Lie groups with arbitrary inertia operators.

Convergence Properties of an Isospectral Saddle-Point Dynamics

We introduce a saddle-point dynamics, evolving isospectrally on the product of two homogeneous space of symmetric matrices, each orthogonally equivalent to a positive diagonal matrix. This dynamics is a saddle-point version of the so-called double-bracket flow and corresponds to a minimax version of the Frobenius norm minimization problem considered in (Brockett, 1989, 1991). We study the set of equilibria of this dynamics and prove its asymptotic convergence properties under suitable conditions. We also demonstrate how, under appropriate conditions, this dynamics provides a learning strategy for a two-player zero-sum game, where the players strategy set is the set of permutations.

Coordinated Motion Design on Lie Groups

The present paper proposes a unified geometric framework for coordinated motion on Lie groups. It first gives a general problem formulation and analyzes ensuing conditions for coordinated motion. Then, it introduces a precise method to design control laws in fully actuated and underactuated settings with simple integrator dynamics. It thereby shows that coordination can be studied in a systematic way once the Lie group geometry of the configuration space is well characterized. Applying the proposed general methodology to particular examples allows to retrieve control laws that have been proposed in the literature on intuitive grounds. A link with Brockett's double bracket flows is also made. The concepts are illustrated on SO(3) , SE(2) and SE(3) .

A MATHEMATICAL REVISION OF THE LANDAU-LIFSHITZ EQUATION

For the Landau-Lifshitz equation a Lie type linear system representation in the Minkowski space M(superscript 3+1) has been derived previously [25]. The internal symmetry group is a proper orthochronous Lorentz group SO0(3,1), and the numerical method based on the internal symmetry was developed in [29]. This paper derives another four new representations of the Landau-Lifshitz equation. We prove that this equation admits two generators: one conservative and one dissipative, as well as two brackets: Poisson bracket and dissipative bracket. Upon embedding the Landau-Lifshitz equation into a skew-symmetric matrix space, we can develop a double-bracket flow representation. The conserved magnetization' magnitude is just the result of the isospectrality for an iso-spectral flow equation. Finally, on the cotangent bundle of an invariant manifold of the constant magnetization magnitude, we introduce the Lie-Poisson bracket to construct an evolutional differential equations system. The magnetization trajectory traces a coadjoint orbit in the Poisson manifold under a coadjoint action of the rotation group SO(3). The six different representations including the one by Bloch et al. [3] are compared.

Observation of Generalized Euler Equations

AbstractWe investigate conditons for generalized double bracket flows to be Morse–Smale.The results yield necessary and sufficient conditions for observability in finite time. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Accessibility of a Class of Generalized Double-Bracket Flows

We investigate a generalization of Brockett's celebrated double bracket flow that is closely related to matrix Riccati differential equations. Using known results on the classification of transitive Lie group actions on homogeneous spaces, necessary and sufficient conditions for acces- sibility of the generalized double bracket flow on Grassmann manifolds are derived. This leads to sufficient Lie-algebraic conditions for controllability ofthe generalized double bracket flow. Accessi- bility on the Lagrangian Grassmann manifold is studied as well, with applications to matrix Riccati differential equations from optimal control.

Accessibility of the matrix Riccati differential equation

AbstractA known result on the classification of transitive Lie group actions on complex Grassmann manifolds is exploited to derive a necessary and sufficient accessibility criterion for the complex matrix differential Riccati equation. We treat both cases, the symmetric as well as the non‐symmetric Riccati equation. Corresponding accessibility results for the real Riccati equation are also available, but not stated here. An application to the accessibility of generalized double bracket flows is given. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical integration methods for the double-bracket flow

In this paper new methods up to order four based on the Magnus expansion are proposed for the numerical integration of the double-bracket equation. The Magnus series is constructed term-by-term by means of recurrences and a bound on the convergence domain is also provided. The new integrators preserve the most salient qualitative features of the flow and are computationally more efficient than other standard Lie-group solvers, such as the Runge–Kutta–Munthe-Kaas class of algorithms.

On the Discretization of Double-Bracket Flows

This paper extends the method of Magnus series to Lie-algebraic equations originating in double-bracket flows. We show that the solution of the isospectral flow Y'=[[Y,N],Y] , Y(0)=Y 0 ∈\Sym(n) , can be represented in the form Y(t)=e Ω(t) Y 0 e -Ω(t) , where the Taylor expansion of Ω can be constructed explicitly, term-by-term, identifying individual expansion terms with certain rooted trees with bicolor leaves. This approach is extended to other Lie-algebraic equations that can be appropriately expressed in terms of a finite ``alphabet.''

Scaled Toda-like flows

This paper discusses the class of isospectral flows X ̇ = [X, A ∘ X] , where ∘ denotes the Hadamard product and [., .] is the Lie bracket. The presence of A allows arbitrary and independent scaling for each element in the matrix X. The time-1 mapping of the scaled Toda-like flow still enjoys a QR-like iteration. The scaled structure includes the classical Toda flow, Brockett's double bracket flow, and other interesting flows as special cases. Convergence proof is thus unified and simplified. The effect of scaling on a variety of applications is demonstrated by examples.