Parthasarathy, K. R. A remark on the unitary group of a tensor product of \(n\) finite-dimensional Hilbert spaces. (English) Zbl 1025.47047 Proc. Indian Acad. Sci., Math. Sci. 113, No. 1, 3-13 (2003). Introduction: From the theory of quantum computing and quantum circuits (as outlined, for example, in [M. A. Nielsen and I. L. Chuang, “Quantum computation and quantum information” (Cambridge Univ. Pr.) (2000; Zbl 1049.81015)]) it is now well-known that every unitary operator on the \(n\)-fold tensor product \((\mathbb{C}^2)^{\otimes^n}\) of copies of the two-dimensional Hilbert space \(\mathbb{C}^2\) can be expressed as a composition of a finite number of unitary operators living on pair products \(H_i\otimes H_j\) where \(H_i\) and \(H_j\) denote the \(i\)th and \(j\)th copies of \(\mathbb{C}^2\). The proof outlined in [loc. cit.] also yields an upper bound on the number of such ‘pair product’ operators as a function of \(n\). Following more or less their lines of proof and using a key lemma suggested to me by R. Jaikumar, we present a generalization when copies of \(\mathbb{C}^2\) are replaced by arbitrary finite-dimensional complex Hilbert spaces. Thus the present note is of a pedagogical and expository nature. MSC: 47N50 Applications of operator theory in the physical sciences 81P68 Quantum computation 46M05 Tensor products in functional analysis Keywords:\(n\)-qubit quantum computer; qubits; gates; controlled gates Citations:Zbl 1049.81015 PDFBibTeX XMLCite \textit{K. R. Parthasarathy}, Proc. Indian Acad. Sci., Math. Sci. 113, No. 1, 3--13 (2003; Zbl 1025.47047) Full Text: DOI References: [1] Jaikumar Radhakrishnan: Private communication, April 2001 [2] Nielsen, M. A.; Chuang, I. L., Quantum Computation and Quantum Information (2000), Cambridge: Cambridge University Press, Cambridge · Zbl 1049.81015 [3] Parthasarathy, K. R., Lectures on quantum computation and quantum error correcting codes, Notes by Amitava Bhattacharya (2001), Mumbai: Tata Institute of Fundamental Research, Mumbai This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.