Quantum Hamlets: Distributed Compilation of Large Algorithmic Graph States

TL;DR

This work investigates the problem of compiling the generation of graph states to arbitrarily many distributed homogeneous quantum processing units (QPUs), providing a scalable partitioning algorithm and graph state generation protocol to minimize the number of Bell pairs required, and designs a heuristic algorithm, BURY, that partitions graph states to require fewer Bell pairs for generation.

Abstract

We investigate the problem of compiling the generation of graph states to arbitrarily many distributed homogeneous quantum processing units (QPUs), providing a scalable partitioning algorithm and graph state generation protocol to minimize the number of Bell pairs required. To this goal, we consider the problem of balanced k graph partitioning with the objective of minimizing the sizes of the maximum matchings between partitions, a more natural measure of entanglement compared to the naive but common metric of cut edges. We show that our heuristic algorithm, BURY, partitions graph states to require fewer Bell pairs for generation than state-of-the-art k partition algorithms. Furthermore, we show that BURY reduces the cut-rank of the partitions, demonstrating that the partitioning found by our algorithm is likely to minimize the Bell pair utilization of any future improved distributed graph state generation protocol. Additionally, we discuss how one could straightforwardly apply our methods to the dynamic case where the graph state generation and measurement are performed concurrently. Our study of the balanced minimum maximum matching k partition problem and the heuristic algorithm we design provides a scalable foundation for reducing quantum network overhead for distributed measurement-based quantum computation (MBQC), as well as any scheme where distributed graph state generation is desired.

Figures (11)

• Figure 1: Level of abstraction at which this work operates. Each gray box in the figure correspond to a different hamlet (QPU), in this example, each of which has three villagers (logical qubits), and one mayor (communication qubit). The mayor qubits are endnodes in some quantum network and therefore, for a cost, we can place two mayors into entangled state. We assume operations between two villagers within the same hamlet to be free, while an operation that involves an interaction between villagers in separate villages requires their mayors to be entangled.
• Figure 2: Grafting edges to a node of the vertex cover during VCG. When generating the edges between a vertex of the vertex cover $v$ and its neighbors $n_1,n_2,n_{...}$ on a separate hamlet, $v$ is coupled to its mayor $M_v$, and its neighbors are coupled to theirs, $M_n$. Long-range entanglement is used to entangle the two mayors. This is shown in a). b) shows that Y measurement on $M_n$ creates a clique in the subgraph formed by $M_v$ and $n_1,n_2,n_{...}$. c) shows that Y measurement on $M_v$ yields our desired result of generating edges between $v$ and its remote neighbors. Notice that this procedure does not affect any preexisting edges that have been generated adjacent to $v,n_1,n_2,n_{...}$
• Figure 3: Example first step of the bury heuristic algorithm on a $4 \times4$ grid graph. On the left, the weights of the nodes are initialized to be equal to their degree plus one. The right shows the state of the algorithm after the first iteration on this graph. The top-left vertex has been chosen due to it's minimum value, and has been "buried". That is, it and its neighbors have been decided to be included in the same partition, or colored black as shown here. The costs to bury other nodes has been updated as well. Note that although the neighbors of the chosen node have been colored, they have not been buried yet, and therefore can be chosen in further iterations of the algorithm.
• Figure 4: Example of performance of METIS and BURY on a $6\times6$ square grid graph, partitioned into three hamlets. a) METIS partitions in a way that causing a maximum matching of size 4 between blue and green, 3 between red and green, and 3 between red and blue. For a total of 10 Bell pairs when generated by VCG. b) Our BURY heuristics partitions such that 4 Bell pairs are required between blue and green, 2 between red and blue, and 3 between red and green. For a total of 9.
• Figure 5: Performance of METIS and BURY on compiled QAOA graph states The $y$-axis corresponds to the number of Bell pairs required to generate the distributed graph state using the VCG protocol. This value is also exactly equal to the sum of the maximum matching sizes between all partitions. The $x$-axis corresponds to the size of the QAOA circuit the MBQC graph state was compiled from. The colors correspond to varying the number of hamlets while the keeping the number of total logical qubits across all hamlets constant for a given $x$-value. The marker type distinguishes our BURY heuristic from the METIS algorithm.