Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Bipartitioning of Graph States for Distributed Measurement-Based Quantum Computing

Kjell Fredrik Pettersen, Matthias Heller, Giorgio Sartor, Raoul Heese

TL;DR

This work tackles distributing graph-state resources for measurement-based quantum computing across two QPUs by minimizing inter-node entanglement. It introduces a simulated-annealing framework augmented with an efficient incremental update for the cut rank, leveraging key matrices to achieve $O(n^2)$ per-swap updates of $\rho_{X,Y}(G)$. The main contributions are the incremental algorithm, the detailed case analysis for swap updates, and numerical validation on grid, sparse, and QAOA-inspired graphs showing reduced requirements for inter-node Bell pairs. The approach offers a practical pathway to scalable distributed MBQC by enabling efficient qubit assignment that minimizes cross-node entanglement, with potential extensions to more partitions and advanced optimization strategies.

Abstract

Measurement-Based Quantum Computing (MBQC) is inherently well-suited for Distributed Quantum Computing (DQC): once a resource state is prepared and distributed across a network of quantum nodes, computation proceeds through local measurements coordinated by classical communication. However, since non-local gates acting on different Quantum Processing Units (QPUs) are a bottleneck, it is crucial to optimize the qubit assignment to minimize inter-node entanglement of the shared resource. For graph state resources shared across two QPUs, this task reduces to finding bipartitions with minimal cut rank. We introduce a simulated annealing-based algorithm that efficiently updates the cut rank when two vertices swap sides across a bipartition, such that computing the new cut rank from scratch, which would be much more expensive, is not necessary. We show that the approach is highly effective for determining qubit assignments in distributed MBQC by testing it on grid graphs and the measurement-based Quantum Approximate Optimization Algorithm (QAOA).

Bipartitioning of Graph States for Distributed Measurement-Based Quantum Computing

TL;DR

This work tackles distributing graph-state resources for measurement-based quantum computing across two QPUs by minimizing inter-node entanglement. It introduces a simulated-annealing framework augmented with an efficient incremental update for the cut rank, leveraging key matrices to achieve per-swap updates of . The main contributions are the incremental algorithm, the detailed case analysis for swap updates, and numerical validation on grid, sparse, and QAOA-inspired graphs showing reduced requirements for inter-node Bell pairs. The approach offers a practical pathway to scalable distributed MBQC by enabling efficient qubit assignment that minimizes cross-node entanglement, with potential extensions to more partitions and advanced optimization strategies.

Abstract

Measurement-Based Quantum Computing (MBQC) is inherently well-suited for Distributed Quantum Computing (DQC): once a resource state is prepared and distributed across a network of quantum nodes, computation proceeds through local measurements coordinated by classical communication. However, since non-local gates acting on different Quantum Processing Units (QPUs) are a bottleneck, it is crucial to optimize the qubit assignment to minimize inter-node entanglement of the shared resource. For graph state resources shared across two QPUs, this task reduces to finding bipartitions with minimal cut rank. We introduce a simulated annealing-based algorithm that efficiently updates the cut rank when two vertices swap sides across a bipartition, such that computing the new cut rank from scratch, which would be much more expensive, is not necessary. We show that the approach is highly effective for determining qubit assignments in distributed MBQC by testing it on grid graphs and the measurement-based Quantum Approximate Optimization Algorithm (QAOA).
Paper Structure (28 sections, 23 equations, 7 figures, 1 algorithm)

This paper contains 28 sections, 23 equations, 7 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Distributing graph states for MBQC
  3. Distribution rule
  4. Proof
  5. Example
  6. An incremental algorithm for cut rank
  7. Key matrices in the cut rank calculations
  8. The cut rank calculation process
  9. First case: $i\in X^F$ and $j\in Y^F$
  10. Second case: $i\in X^B$ and $j\in Y^F$
  11. $P^X_1(i)$ is true
  12. $P^X_1(i)$ is false
  13. Third case: $i\in X^F$ and $j\in Y^B$
  14. $P^Y_1(j)$ is true
  15. $P^Y_1(j)$ is false
  16. ...and 13 more sections

Figures (7)

  • Figure 1: Example of how to embed a partitioned graph such that the cut rank of its partition corresponds to the number of edges between them. The transformation between the original graph $G$ and the transformed graph $G'$ can be achieved with local transformations and node removals as defined in \ref{['eqn:G:example']}. In the context of , both graphs are locally equivalent and demonstrate how the entanglement connections are shared across two . For $G'$, the qubits $6$ to $9$ correspond to two EPR pairs.
  • Figure 2: Execution times of the simulated annealing algorithm for a balanced bipartition of $n\times n$ grid graphs using naive Gauss-Jordan elimination for calculating the rank in each iteration, vs. our proposed algorithm using update rules.
  • Figure 3: Cut rank results from our proposed algorithm using update rules for $n\times n$ grid graphs.
  • Figure 4: Results from our proposed algorithm using update rules for $100$ sparse Erdős-Rényi random graphs $G(n,p)$ for each $n$, where $p=c/n$ and the first partition set has size $P_1n$.
  • Figure 5: Exemplary circuit and corresponding implementation for .
  • ...and 2 more figures