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arXiv:quant-ph/0507171

An Introduction to Cartan's KAK Decomposition for QC Programmers

Robert R. Tucci

Published Jul 18, 2005• 42 citations
arXivHTMLPDF

Abstract

This paper presents no new results; its goals are purely pedagogical. A special case of the Cartan Decomposition has found much utility in the field of quantum computing, especially in its sub-field of quantum compiling. This special case allows one to factor a general 2-qubit operation (i.e., an element of U(4)) into local operations applied before and after a three parameter, non-local operation. In this paper, we give a complete and rigorous proof of this special case of Cartan's Decomposition. From the point of view of QC programmers who might not be familiar with the subtleties of Lie Group Theory, the proof given here has the virtues, that it is constructive in nature, and that it uses only Linear Algebra. The constructive proof presented in this paper is implemented in some Octave/Matlab m-files that are included with the paper. Thus, this paper serves as documentation for the attached m-files.

DOI
10.48550/arxiv.quant-ph/0507171

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This paper presents no new results; its goals are purely pedagogical. A special case of the Cartan Decomposition has found much utility in the field of quantum computing, especially in its sub-field of quantum compiling. This special case allows one to factor a general 2-qubit operation (i.e., an element of U(4)) into local operations applied before and after a three parameter, non-local operation. In this paper, we give a complete and rigorous proof of this special case of Cartan's Decomposition. From the point of view of QC programmers who might not be familiar with the subtleties of Lie Group Theory, the proof given here has the virtues, that it is constructive in nature, and that it uses only Linear Algebra. The constructive proof presented in this paper is implemented in some Octave/Matlab m-files that are included with the paper. Thus, this paper serves as documentation for the attached m-files.
We provide a new algorithm that translates a unitary matrix into a quantum circuit according to the G=KAK theorem in Lie group theory. With our algorithm, any matrix decomposition corresponding to type-AIII KAK decompositions can be derived according to the given Cartan involution. Our algorithm contains, as its special cases, Cosine-Sine decomposition (CSD) and Khaneja-Glaser decomposition (KGD) in the sense that it derives the same quantum circuits as the ones obtained by them if we select suitable Cartan involutions and square root matrices. The selections of Cartan involutions for computing CSD and KGD will be shown explicitly. As an example, we show explicitly that our method can automatically reproduce the well-known efficient quantum circuit for the nnn-qubit quantum Fourier transform.
The KAK decomposition is a fundamental tool in Lie theory and quantum computing. Despite its widespread use, the mathematical foundations remain incomplete, particularly regarding the precise conditions for the decomposition and the characterization of equivalence classes under multiplication by elements of KKK. Here, we present a mathematical theory of the KAK decomposition for connected compact semisimple Lie groups and derive the decomposition for SU(4)\mathrm{SU}(4)SU(4). In particular, we clarify the relationship between various definitions of a Cartan decomposition in the literature and give a complete proof of a general KAK decomposition theorem. We then distinguish two distinct notions of KAK equivalence classes, double coset equivalence and projective equivalence, thereby addressing mathematical inconsistencies regarding KAK classification in the literature. Specifically, for SU(4)\mathrm{SU}(4)SU(4), we show that local equivalence classes under multiplication by SU(2)⊗SU(2)\mathrm{SU}(2)\otimes \mathrm{SU}(2)SU(2)⊗SU(2) are geometrically represented not by the usual "Weyl chamber" as claimed in the existing literature. Instead, the "Weyl chamber" is only recovered by the projective-local equivalence which disregards global phases. We develop a systematic theory for determining equivalence and uniqueness for both notions of equivalence. Our work establishes a rigorous Lie-theoretic foundation for the theory of quantum gates and circuits.
We provide a new algorithm that translates a unitary matrix into a quantum circuit according to the G=KAK theorem in Lie group theory. With our algorithm, any matrix decomposition corresponding to type-AIII KAK decompositions can be derived according to the given Cartan involution. Our algorithm contains, as its special cases, Cosine-Sine decomposition (CSD) and Khaneja-Glaser decomposition (KGD) in the sense that it derives the same quantum circuits as the ones obtained by them if we select suitable Cartan involutions and square root matrices. The selections of Cartan involutions for computing CSD and KGD will be hown explicitly. As an example, we show explicitly that our method can automatically reproduce the well-known efficient quantum circuit for the n-qubit quantum Fourier transform.
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.
We show the applicability of the Cartan decomposition of Lie algebras to quantum circuits. This approach can be used to synthesize circuits that can efficiently implement any desired unitary operation. Our method finds explicit quantum circuit representations of the algebraic generators of the relevant Lie algebras allowing the direct implementation of a Cartan decomposition on a quantum computer. The construction is recursive and allows us to expand any circuit down to generators and rotation matrices on individual qubits, where through our recursive algorithm we find that the generators themselves can be expressed with controlled-not (cnot) and swap gates explicitly. Our approach is independent of the standard cnot implementation and can be easily adapted to other cross-qubit circuit elements. In addition to its versatility, we also achieve near-optimal counts when working with cnot gates, achieving an asymptotic cnot cost of 21164n\frac{21}{16}{4}^{n}1621​4n for nnn qubits.
A scheme to perform the Cartan decomposition for the Lie algebra su(N) of arbitrary finite dimensions is introduced. The schme is based on two algebraic structures, the conjugate partition and the quotient algebra, that are easily generated by a Cartan subalgebra and generally exist in su(N). In particular, the Lie algebras su(2^p) and every su(2^{p-1} < N < 2^p) share the isomorphic structure of the quotient algebra. This structure enables an efficient algorithm for the recursive and exhaustive construction of Cartan decompositions. Further with the scheme, a unitary transformation in SU(N) can be recursively decomposed into a product of certain designated operators, e.g., local and nonlocal gates. Such a recursive decomposition of a transformation implies an evolution path on the manifold of the group.