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arXiv:2404.02358

Solving the $KP$ problem with the Global Cartan Decomposition

Elija Perrier, Christopher S. Jackson

Published Apr 2, 2024• 0 citations
arXivHTMLPDF

Abstract

Geometric methods have useful application for solving problems in a range of quantum information disciplines, including the synthesis of time-optimal unitaries in quantum control. In particular, the use of Cartan decompositions to solve problems in optimal control, especially lambda systems, has given rise to a range of techniques for solving the so-called $KP$-problem, where target unitaries belong to a semi-simple Lie group manifold $G$ whose Lie algebra admits a $\mathfrak{g}=\mathfrak{k} \oplus \mathfrak{p}$ decomposition and time-optimal solutions are represented by subRiemannian geodesics synthesised via a distribution of generators in $\mathfrak{p}$. In this paper, we propose a new method utilising global Cartan decompositions $G=KAK$ of symmetric spaces $G/K$ for generating time-optimal unitaries for targets $-iX \in [\frak{p},\frak{p}] \subset \frak{k}$ with controls $-iH(t) \in \frak{p}$. Target unitaries are parametrised as $U=kac$ where $k,c \in K$ and $a = e^{iΘ}$ with $Θ\in \frak{a}$. We show that the assumption of $dΘ=0$ equates to the corresponding time-optimal unitary control problem being able to be solved analytically using variational techniques. We identify how such control problems correspond to the holonomies of a compact globally Riemannian symmetric space, where local translations are generated by $\mathfrak{p}$ and local rotations are generated by $[\mathfrak{p},\mathfrak{p}]$.

DOI
10.48550/arxiv.2404.02358

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Solving the KPKPKP problem with the Global Cartan Decomposition
Geometric methods have useful application for solving problems in a range of quantum information disciplines, including the synthesis of time-optimal unitaries in quantum control. In particular, the use of Cartan decompositions to solve problems in optimal control, especially lambda systems, has given rise to a range of techniques for solving the so-called KPKPKP-problem, where target unitaries belong to a semi-simple Lie group manifold GGG whose Lie algebra admits a g=k⊕p\mathfrak{g}=\mathfrak{k} \oplus \mathfrak{p}g=k⊕p decomposition and time-optimal solutions are represented by subRiemannian geodesics synthesised via a distribution of generators in p\mathfrak{p}p. In this paper, we propose a new method utilising global Cartan decompositions G=KAKG=KAKG=KAK of symmetric spaces G/KG/KG/K for generating time-optimal unitaries for targets −iX∈[p,p]⊂k-iX \in [\frak{p},\frak{p}] \subset \frak{k}−iX∈[p,p]⊂k with controls −iH(t)∈p-iH(t) \in \frak{p}−iH(t)∈p. Target unitaries are parametrised as U=kacU=kacU=kac where k,c∈Kk,c \in Kk,c∈K and a=eiΘa = e^{iΘ}a=eiΘ with Θ∈aΘ\in \frak{a}Θ∈a. We show that the assumption of dΘ=0dΘ=0dΘ=0 equates to the corresponding time-optimal unitary control problem being able to be solved analytically using variational techniques. We identify how such control problems correspond to the holonomies of a compact globally Riemannian symmetric space, where local translations are generated by p\mathfrak{p}p and local rotations are generated by [p,p][\mathfrak{p},\mathfrak{p}][p,p].
The control of spin chains represents a very interesting problem from the point of view of quantum computation. The problem consists in defining a procedure to obtain any possible evolution operator of the spin chain by means of in external magnetic field. The set of possible evolution operators of the system corresponds to the unitary group SU(2/sup N/) (where N is the number of atoms in the chain) and the interactions involved can he set to correspond to elements in the corresponding Lie algebra. As a consequence, the whole problem can be formulated in Lie algebraic terms and the design issues be reduced to a suitable decomposition of the group elements. The goal of this paper is to introduce a Cartan decomposition for the unitary group based on a minimal switching decomposition of the special orthogonal group. We analyze its implications from the point of view of time optimality in the construction of a program as a sequence of quantum gates.
This chapter discusses in detail the case of two-level systems and the associated Lie group. It gives an example of a general decomposition of SU(n), namely the decomposition into planar rotation, which is quite simple and can be obtained by straightforward computations. This decomposition gives a direct way to parametrize the special unitary group. The chapter presents the basics of Cartan decompositions of semisimple Lie groups. It gives some examples of applications of Lie group decompositions and introduces some notions of Lie group and Lie algebra theory. The chapter discusses Cartan decompositions of Lie groups. Cartan decompositions appear to be ubiquitous in quantum control and dynamics and in quantum information theory.
Decompositions of the Lie group of unitary matrices are very useful tools in the control and analysis of quantum dynamics. In this paper, we survey recent results on decompositions, concerning their significance in terms of symmetries, entanglement, and their applications to control and dynamics. Several decompositions can be obtained by recursively applying the Cartan classification of the symmetric spaces of the classical Lie groups. The emphasis is on a novel recursive procedure to decompose the unitary evolution of bipartite quantum systems of arbitrary dimensions, in simpler factors. This procedure makes transparent the contributions of the entangling and non entangling transformations
The use of geometric and symmetry techniques in quantum and classical information processing has a long tradition across the physical sciences as a means of theoretical discovery and applied problem solving. In the modern era, the emergent combination of such geometric and symmetry-based methods with quantum machine learning (QML) has provided a rich opportunity to contribute to solving a number of persistent challenges in fields such as QML parametrisation, quantum control, quantum unitary synthesis and quantum proof generation. In this thesis, we combine state-of-the-art machine learning methods with techniques from differential geometry and topology to address these challenges. We present a large-scale simulated dataset of open quantum systems to facilitate the development of quantum machine learning as a field. We demonstrate the use of deep learning greybox machine learning techniques for estimating approximate time-optimal unitary sequences as geodesics on subRiemannian symmetric space manifolds. Finally, we present novel techniques utilising Cartan decompositions and variational methods for analytically solving quantum control problems for certain classes of Riemannian symmetric space. Owing to its multidisciplinary nature, this work contains extensive supplementary background information in the form of Appendices. Each supplementary Appendix is tailored to provide additional background material in a relatively contained way for readers whom may be familiar with some, but not all, of these diverse scientific disciplines. The Appendices reproduce or paraphrase standard results in the literature with source material identified at the beginning of each Appendix. Proofs are omitted for brevity but can be found in the cited sources and other standard texts.
Abstract Simulating the time evolution of a Hamiltonian system on a classical computer is hard—The computational power required to even describe a quantum system scales exponentially with the number of its constituents, let alone integrate its equations of motion. Hamiltonian simulation on a quantum machine is a possible solution to this challenge—Assuming that a quantum system composing of spin-½ particles can be manipulated at will, then it is tenable to engineer the interaction between those particles according to the one that is to be simulated, and thus predict the value of physical quantities by simply performing the appropriate measurements on the system. Establishing a linkage between the unitary operators described mathematically as a logic solution and the unitary operators recognizable as quantum circuits for execution, is therefore essential for algorithm design and circuit implementation. Most current techniques are fallible because of truncation errors or the stagnation at local solutions. This work offers an innovative avenue by tackling the Cartan decomposition with the notion of Lax dynamics. Within the integration errors that is controllable, this approach gives rise to a genuine unitary synthesis that not only is numerically feasible, but also can be utilized to gauge the quality of results produced by other means, and extend the knowledge to a wide range of applications. This paper aims at establishing the theoretic and algorithmic foundations by exploiting the geometric properties of Hamiltonian subalgebras and describing a common mechanism for deriving the Lax dynamics.
In this paper, we study some control problems that derive from time optimal control of coupled spin dynamics in NMR spectroscopy and quantum information and computation. Time optimal control helps to minimize relaxation losses. The ability to synthesize, local unitaries, much more rapidly than evolution of couplings, gives a natural time scale separation in these problems. The generators of evolution, $\g$, are decomposed into fast generators $\k$ (local Hamiltonians) and slow generators $\p$ (couplings) as a Cartan decomposition $\g = \p \oplus \k$. Using this decomposition, we exploit some convexity ideas to completely characterize the reachable set and time optimal control for these problems. In this paper, we carry out a second order analysis of time optimality.