Optimal Quantum Circuits for General Multi-Qutrit Quantum Computation

Quantum circuits of a general quantum gate acting on multiple $d$-level quantum systems play a prominent role in multi-valued quantum computation. A recursive Cartan decomposition of semi-simple unitary Lie group $U\left(3^n\right)$ (arbitrary $n$-qutrit gate) is first proposed with a rigorous proof, which completely decomposes an $n$-qutrit gate into local and non-local operations. On this basis, an explicit quantum circuit is designed for implementing arbitrary two-qutrit gates, and the cost of the construction is 21 generalized controlled $X$ (GCX) and controlled increment (CINC) gates less than the earlier best result of 26 GGXs. Furthermore, the program is extended to the $n$-qutrit system, and the quantum circuit of generic $n$-qutrit gates contained $\frac{41}{96}·3^{2n}-4·3^{n-1}-(\frac{n^2}{2}+\frac{n}{4}-\frac{29}{32})$ GGXs and CINCs is presented. Such asymptotically optimal structure is the best known result so far and its strength becomes more remarkable as $n$ increases, for example, when $n=5$, the program saves 7146 GCXs compared to the previous best result. In addition, concrete recursive decomposition expressions is given for each non-local operation instead of only quantum circuit diagrams.