Generalized swap networks for near-term quantum computing

The practical use of many types of near-term quantum computers requires accounting for their limited connectivity. One way of overcoming limited connectivity is to insert swaps in the circuit so that logical operations can be performed on physically adjacent qubits, which we refer to as solving the `routing via matchings' problem. We address the routing problem for families of quantum circuits defined by a hypergraph wherein each hyperedge corresponds to a potential gate. Our main result is that any unordered set of $k$-qubit gates on distinct $k$-qubit subsets of $n$ logical qubits can be ordered and parallelized in $O(n^{k-1})$ depth using a linear arrangement of $n$ physical qubits; the construction is completely general and achieves optimal scaling in the case where gates acting on all $\binom{n}{k}$ sets of $k$ qubits are desired. We highlight two classes of problems for which our method is particularly useful. First, it applies to sets of mutually commuting gates, as in the (diagonal) phase separators of Quantum Alternating Operator Ansatz (Quantum Approximate Optimization Algorithm) circuits. For example, a single level of a QAOA circuit for Maximum Cut can be implemented in linear depth, and a single level for $3$-SAT in quadratic depth. Second, it applies to sets of gates that do not commute but for which compilation efficiency is the dominant criterion in their ordering. In particular, it can be adapted to Trotterized time-evolution of fermionic Hamiltonians under the Jordan-Wigner transformation, and also to non-standard mixers in QAOA. Using our method, a single Trotter step of the electronic structure Hamiltonian in an arbitrary basis of $n$ orbitals can be done in $O(n^3)$ depth while a Trotter step of the unitary coupled cluster singles and doubles method can be implemented in $O(n² \eta)$ depth, where $\eta$ is the number of electrons.