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arXiv:quant-ph/0505128 · Journal of Mathematical Physics

A constructive algorithm for the Cartan decomposition of SU(2N)

Henrique N. Sá Earp, Jiannis K. Pachos

Published Aug 1, 2005• 29 citations
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Abstract

We present an explicit numerical method to obtain the Cartan-Khaneja-Glaser decomposition of a general element G∊SU(2N) in terms of its “Cartan” and “non-Cartan” components. This effectively factors G in terms of group elements that belong in SU(2n) with n<N, a procedure that can be iterated down to n=2. We show that every step reduces to solving the zeros of a matrix polynomial, obtained by truncation of the Baker-Campbell-Hausdorff formula, numerically. All computational tasks involved are straightforward and the overall truncation errors are well under control.

DOI
10.1063/1.2008210

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We present an explicit numerical method to obtain the Cartan-Khaneja-Glaser decomposition of a general element G∊SU(2N) in terms of its “Cartan” and “non-Cartan” components. This effectively factors G in terms of group elements that belong in SU(2n) with n<N, a procedure that can be iterated down to n=2. We show that every step reduces to solving the zeros of a matrix polynomial, obtained by truncation of the Baker-Campbell-Hausdorff formula, numerically. All computational tasks involved are straightforward and the overall truncation errors are well under control.
We present an explicit numerical method to obtain the Cartan-Khaneja-Glaser decomposition of a general element G of SU(2^N) in terms of its `Cartan' and `non-Cartan' components. This effectively factors G in terms of group elements that belong in SU(2^n) with n<N, a procedure that can be iterated down to n=2. We show that every step reduces to solving the zeros of a matrix polynomial, obtained by truncation of the Baker-Campbell-Hausdorff formula, numerically. All computational tasks involved are straightforward and the overall truncation errors are well under control.
Cartan-Khaneja-Glaser decomposition of SU(2n)SU(2^n)SU(2n) via involutive automorphisms
We present a novel algorithm for performing the Cartan-Khaneja-Glaser decomposition of unitary matrices in SU(2n)\mathrm{SU}\left(2^n\right)SU(2n), a critical task for efficient quantum circuit design. Building upon the approach introduced by Sá Earp and Pachos (2005), we overcome key limitations of their method, such as reliance on ill-defined matrix logarithms and the convergence issues of truncated Baker-Campbell-Hausdorff (BCH) series. Our reformulation leverages the algebraic structure of involutive automorphisms and symmetric Lie algebra decompositions to yield a stable and recursive factorization process. We provide a full Python implementation of the algorithm, available in an open-source repository, and validate its performance on matrices in SU(8)\mathrm{SU}(8)SU(8) and SU(16)\mathrm{SU}(16)SU(16) using random unitary benchmarks. The algorithm produces decompositions that are directly suited to practical quantum hardware, with factors that can be implemented near-optimally using standard gate sets.
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.
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.
Recently, Vatan and Williams utilize a matrix decomposition of SU(2n)SU(2^n)SU(2n) introduced by Khaneja and Glaser to produce {\tt CNOT}-efficient circuits for arbitrary three-qubit unitary evolutions. In this note, we place the Khaneja Glaser Decomposition ({\tt KGD}) in context as a SU(2n)=KAKSU(2^n)=KAKSU(2n)=KAK decomposition by proving that its Cartan involution is type {\bf AIII}, given n≥3n \geq 3n≥3. The standard type {\bf AIII} involution produces the Cosine-Sine Decomposition (CSD), a well-known decomposition in numerical linear algebra which may be computed using mature, stable algorithms. In the course of our proof that the new decomposition is type {\bf AIII}, we further establish the following. Khaneja and Glaser allow for a particular degree of freedom, namely the choice of a commutative algebra a\mathfrak{a}a, in their construction. Let χ1n\chi_1^nχ1n​ be a {\tt SWAP} gate applied on qubits 111, nnn. Then χ1nvχ1n=k1  a  k2\chi_1^n v \chi_1^n=k_1\; a \; k_2χ1n​vχ1n​=k1​ak2​ is a KGD for $\mathfrak{a}=\mbox{span}_{\mathbb{R}} \{ \chi_1^n ( \ket{j}\bra{N-j-1} -\ket{N-j-1}\bra{j}) \chi_1^n \}$ if and only if v=(χ1nk1χ1n)(χ1naχ1n)(χ1nk2χ1n)v=(\chi_1^n k_1 \chi_1^n) (\chi_1^n a \chi_1^n)(\chi_1^n k_2 \chi_1^n)v=(χ1n​k1​χ1n​)(χ1n​aχ1n​)(χ1n​k2​χ1n​) is a CSD.
In this paper we provide an explicit parameterization of arbitrary unitary transformation acting on n qubits, in terms of one and two qubit quantum gates. The construction is based on successive Cartan decompositions of the semi-simple Lie group, SU(2^n). The decomposition highlights the geometric aspects of building an arbitrary unitary transformation out of quantum gates and makes explicit the choice of pulse sequences for the implementation of arbitrary unitary transformation on $n coupled spins. Finally we make observations on the optimality of the design procedure.