Search

Fault-tolerant quantum computation using two-dimensional topological quantum error correcting codes can benefit from multi-qubit long-range operations. By using simple commutation rules, a quantum circuit can be transpiled into a sequence of solely non-Clifford multi-qubit gates. Prior work on fault-tolerant compilation avoids optimal scheduling of such gates since they reduce the parallelizability of the circuit. We observe that the reduced parallelization potential is outweighed by the significant reduction in the number of gates. We therefore devise a method for scheduling multi-qubit lattice surgery using an earliest-available-first policy, solving the associated forest packing problem using a representation of the multi-qubit gates as Steiner trees. Our extensive testing on random and application-inspired circuits demonstrates the method's scalability and performance. We show that the transpilation significantly reduces the circuit length on the set of circuits tested, and that the resulting circuit of multi-qubit gates has a further reduction in the expected circuit execution time compared to serial execution., Comment: 23 pages, 7 figures, 4 tables

We establish a H\"{o}rmander type theorem for the multilinear pseudo-differential operators, which is also a generalization of the results in \cite{MR4322619} to symbols depending on the spatial variable. Most known results for multilinear pseudo-differential operators were obtained by assuming their symbols satisfy pointwise derivative estimates(Mihlin-type condition), that is, their symbols belong to some symbol classes $n$-$\mathcal{S}^m_{\rho, \delta}(\mathbb{R}^d)$, $0 \le \delta \le \rho \le1$, $0 \le \delta<1$ for some $m \le 0$. In this paper, we shall consider multilinear pseudo-differential operators whose symbols have limited smoothness described in terms of function space and not in a pointwise form(H\"ormander type condition). Our conditions for symbols are weaker than the Mihlin-type conditions in two senses: the one is that we only assume the first-order derivative conditions in the spatial variable and lower-order derivative conditions in the frequency variable, and the other is that we make use of $L^2$-average condition rather than pointwise derivative conditions for the symbols. As an application, we obtain some mapping properties for the multilinear pseudo-differential operators associated with symbols belonging to the classes $n$-$\mathcal{S}^{m}_{\rho,\delta}(\mathbb{R}^{d})$, $0 \le \rho \le 1$, $0 \le \delta<1$, $m \le 0$. Moreover, it can be pointed out that our results can be applied to wider classes of symbols which do not belong to the traditional symbol classes $n$-$\mathcal{S}^{m}_{\rho,\delta}(\mathbb{R}^{d})$.

This paper has been withdrawn since it contains some discrepancy with othe authers's recent result. We will not post this until this discrepancy is resolved., Comment: This paper has been withdrawn