A Multilevel Framework for Partitioning Quantum Circuits

TL;DR

This paper formalises and extends existing constructions for graphical quantum circuit partitioning and designs a new objective function that captures further possibilities for non-local operations via nested state teleportation and compares the entanglement requirements and runtimes with state-of-the-art methods, finding that it achieves the lowest entanglement costs in most cases.

Abstract

Executing quantum algorithms over distributed quantum systems requires quantum circuits to be divided into sub-circuits which communicate via entanglement-based teleportation. Naively mapping circuits to qubits over multiple quantum processing units (QPUs) results in large communication overhead, increasing both execution time and noise. This can be minimised by optimising the assignment of qubits to QPUs and the methods used for covering non-local operations. Formulations that are general enough to capture the spectrum of teleportation possibilities lead to complex problem instances which can be difficult to solve effectively. This highlights a need to exploit the wide range of heuristic techniques used in the graph partitioning literature. This paper formalises and extends existing constructions for graphical quantum circuit partitioning and designs a new objective function that captures further possibilities for non-local operations via nested state teleportation. We adapt the well-known Fiduccia-Mattheyses heuristic to the constraints and problem objective and explore multilevel techniques that coarsen hypergraphs and partition at multiple levels of granularity. We find that this reduces runtime and improves solution quality of standard partitioning. We place these techniques within a larger framework, through which we can extract full distributed quantum circuits including teleportation instructions. We compare the entanglement requirements and runtimes with state-of-the-art methods, finding that we achieve the lowest entanglement costs in most cases. Averaging over a wide range of circuits, we reduce the entanglement requirements by 35% compared with the next best-performing method. We also find that our techniques can scale to much larger circuit sizes than competing methods, provided the number of partitions is not too large.

Figures (33)

• Figure 1: The starting and ending processes. The starting process $S_{q,e}$ is a linear map which maps the state of the root qubit $q$ onto a joint state between $q$ and an auxiliary qubit $e$ in another QPU. The ending process $E_{q,e}$ applies the inverse map, disentangling the auxiliary qubit from the root qubit, returning the original state to $q$. Diagrammatic notation from Ref. wuEntanglementefficientBipartitedistributedQuantum2023a
• Figure 2: Methods for covering a non-local operation. On the left is the entanglement-assisted starting and ending processes $S_{q,e}$ and $E_{q,e}$, used for a gate teleportation. On the right, we flip the direction of the ending process to $E_{e,q}$, which allows us to teleport the state of $q$ onto $e$. The two circuits are equivalent up to relabelling of qubits after the ending process.
• Figure 3: The nested state teleportation procedure. A $2$-fold starting process is applied to entangle $q_1$, located in QPU $B$, with auxiliary qubits in $A$ and $C$. This fans the state out, allowing $CP(\theta)$ gates with $q_0$ and $q_3$ to be executed, as well as the local gate with $q_2$. The $H$ gate on $q_1$ requires the gate teleportation to be ended. Since there is another gate between $q_1$ and $q_3$, all ending processes should be re-routed towards $C$, collapsing the final state onto the auxiliary qubit. In fact, for each ending process, we may route the ending process to any QPU that still has an active link, provided the final ending process ends up at our chosen destination. The circuit on the right is equivalent to the one on the left up to a relabelling of qubits, as shown at the end of the wires.
• Figure 4: Correspondence between single and two-qubit gates and temporal hypergraphs. We represent any single-qubit gate as a grey node, storing the parameters $\theta, \phi, \lambda$ as node attributes. Two-qubit $CP(\theta)$ gates are represented as an edge between two black nodes, storing the phase parameter $\theta$ as an edge attribute.
• Figure 5: Base graph for a random, 8-qubit circuit. Grey nodes correspond to single-qubit gates, while black nodes correspond to qubits involved in $CP(\theta)$ gates, as illustrated in \ref{['fig:U_CP']}. State edges connect nodes representing the same qubit at different time steps. In \ref{['fig:CP5_8_FM']}, an optimised partitioning of the nodes is illustrated by dragging nodes vertically into different QPU regions. Cut state edges correspond to state teleportation, while cut gate edges correspond to gate teleportation.