A Remark on the Unitary Group of a Tensor Product of 𝑛 Finite-Dimensional Hilbert Spaces

[摘要] Let $H_i, 1 ≤ i ≤ n$ be complex finite-dimensional Hilbert spaces of dimension $d_i, 1 ≤ i ≤ n$ respectively with $d_i ≥ 2$ for every 𝑖. By using the method of quantum circuits in the theory of quantum computing as outlined in Nielsen and Chuang [2] and using a key lemma of Jaikumar [1] we show that every unitary operator on the tensor product $H = H_1 otimes H_2 otimesldots otimes H_n$ can be expressed as a composition of a finite number of unitary operators living on pair products $H_i otimes H_j, 1 ≤ i, j ≤ n$. An estimate of the number of operators appearing in such a composition is obtained.

[学科分类] 数学（综合）

[关键词] ð‘›-qubit quantum computer;qubits;gates;controlled gates.