Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal Scheduling of Graph States via Path Decompositions

Samuel J. Elman, Jason Gavriel, Ryan L. Mann

TL;DR

This paper addresses how to optimally schedule measurements in measurement-based quantum computation on graph states by linking scheduling to graph path decompositions. The authors prove that the spatial cost of a graph-state computation equals the pathwidth plus one, $sc(\ket{G}) = pw(G) + 1$, and show that approximating this cost is NP-hard in general, while a fixed-parameter tractable algorithm exists for graphs with bounded spatial cost. The key contribution is the exact equivalence between measurement schedules and path decompositions, enabling efficient optimal scheduling for bounded resources and informing graph-state design for fault-tolerant architectures. The work has practical implications for resource-efficient MBQC, highlighting low-degree graphs such as square lattices to minimize active qubits and bus channels, and it provides insights into classical simulability bounds for MBQC with bounded pathwidth graphs.

Abstract

We study the optimal scheduling of graph states in measurement-based quantum computation, establishing an equivalence between measurement schedules and path decompositions of graphs. We define the spatial cost of a measurement schedule based on the number of simultaneously active qubits and prove that an optimal measurement schedule corresponds to a path decomposition of minimal width. Our analysis shows that approximating the spatial cost of a graph is $\textsf{NP}$-hard, while for graphs with bounded spatial cost, we establish an efficient algorithm for computing an optimal measurement schedule.

Optimal Scheduling of Graph States via Path Decompositions

TL;DR

This paper addresses how to optimally schedule measurements in measurement-based quantum computation on graph states by linking scheduling to graph path decompositions. The authors prove that the spatial cost of a graph-state computation equals the pathwidth plus one, , and show that approximating this cost is NP-hard in general, while a fixed-parameter tractable algorithm exists for graphs with bounded spatial cost. The key contribution is the exact equivalence between measurement schedules and path decompositions, enabling efficient optimal scheduling for bounded resources and informing graph-state design for fault-tolerant architectures. The work has practical implications for resource-efficient MBQC, highlighting low-degree graphs such as square lattices to minimize active qubits and bus channels, and it provides insights into classical simulability bounds for MBQC with bounded pathwidth graphs.

Abstract

We study the optimal scheduling of graph states in measurement-based quantum computation, establishing an equivalence between measurement schedules and path decompositions of graphs. We define the spatial cost of a measurement schedule based on the number of simultaneously active qubits and prove that an optimal measurement schedule corresponds to a path decomposition of minimal width. Our analysis shows that approximating the spatial cost of a graph is -hard, while for graphs with bounded spatial cost, we establish an efficient algorithm for computing an optimal measurement schedule.
Paper Structure (6 sections, 5 theorems, 1 equation, 2 figures)

This paper contains 6 sections, 5 theorems, 1 equation, 2 figures.

Table of Contents

  1. Introduction
  2. Framework
  3. Optimal Scheduling
  4. Computational Complexity
  5. Implementations
  6. Conclusion & Outlook

Key Result

Theorem 1

Let $G=(V, E)$ be a graph and let $\mathcal{X}=(X_i)_{i=1}^n$ be a sequence of subsets of $V$. The following statements are equivalent:

Figures (2)

  • Figure 1: A graph that has a path decomposition given by $(\{a,b,c,d\}, \{c,d,e,f\})$. The width of this decomposition is three.
  • Figure 2: An implementation of the square lattice graph state. The white squares represent logical qubits and the shaded squares represent magic-state factories.

Theorems & Definitions (11)

  • Definition 1: Measurement schedule
  • Definition 2: Path decomposition
  • Theorem 1
  • proof
  • Corollary 2
  • Corollary 3
  • proof
  • Corollary 4
  • proof
  • Corollary 5
  • ...and 1 more