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Quantum property testing in sparse directed graphs

Simon Apers, Frédéric Magniez, Sayantan Sen, Dániel Szabó

TL;DR

This work investigates quantum property testing for sparse directed graphs, focusing on the unidirectional model where only outgoing edges are queryable. It delivers a near-quadratic quantum speedup for testing $k$-source-subgraph-freeness with a bound of $O(N^{\frac{1}{2}(1-\frac{1}{2^k-1})})$, and proves near-tight lower bounds via a dual-polynomial approach on an intermediate $k$-collision problem, connecting subgraph-freeness to collision-finding techniques. It also shows that not all graph properties admit quantum speedups by proving a linear quantum lower bound for testing 3-colorability in the undirected bounded-degree model. The results illuminate when quantum advantages arise in graph property testing and establish a framework using dual polynomials to derive quantum lower bounds in property testing contexts, with potential implications for a broader class of sparse-graph problems.

Abstract

We initiate the study of quantum property testing in sparse directed graphs, and more particularly in the unidirectional model, where the algorithm is allowed to query only the outgoing edges of a vertex. In the classical unidirectional model, the problem of testing $k$-star-freeness, and more generally $k$-source-subgraph-freeness, is almost maximally hard for large $k$. We prove that this problem has almost quadratic advantage in the quantum setting. Moreover, we show that this advantage is nearly tight, by showing a quantum lower bound using the method of dual polynomials on an intermediate problem for a new, property testing version of the $k$-collision problem that was not studied before. To illustrate that not all problems in graph property testing admit such a quantum speedup, we consider the problem of $3$-colorability in the related undirected bounded-degree model, when graphs are now undirected. This problem is maximally hard to test classically, and we show that also quantumly it requires a linear number of queries.

Quantum property testing in sparse directed graphs

TL;DR

This work investigates quantum property testing for sparse directed graphs, focusing on the unidirectional model where only outgoing edges are queryable. It delivers a near-quadratic quantum speedup for testing -source-subgraph-freeness with a bound of , and proves near-tight lower bounds via a dual-polynomial approach on an intermediate -collision problem, connecting subgraph-freeness to collision-finding techniques. It also shows that not all graph properties admit quantum speedups by proving a linear quantum lower bound for testing 3-colorability in the undirected bounded-degree model. The results illuminate when quantum advantages arise in graph property testing and establish a framework using dual polynomials to derive quantum lower bounds in property testing contexts, with potential implications for a broader class of sparse-graph problems.

Abstract

We initiate the study of quantum property testing in sparse directed graphs, and more particularly in the unidirectional model, where the algorithm is allowed to query only the outgoing edges of a vertex. In the classical unidirectional model, the problem of testing -star-freeness, and more generally -source-subgraph-freeness, is almost maximally hard for large . We prove that this problem has almost quadratic advantage in the quantum setting. Moreover, we show that this advantage is nearly tight, by showing a quantum lower bound using the method of dual polynomials on an intermediate problem for a new, property testing version of the -collision problem that was not studied before. To illustrate that not all problems in graph property testing admit such a quantum speedup, we consider the problem of -colorability in the related undirected bounded-degree model, when graphs are now undirected. This problem is maximally hard to test classically, and we show that also quantumly it requires a linear number of queries.
Paper Structure (39 sections, 36 theorems, 33 equations)

This paper contains 39 sections, 36 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Context
  3. Property testing of directed bounded degree graphs
  4. Related works on collision finding
  5. Our results
  6. Technical overview
  7. Subgraph-finding algorithm
  8. Collision-freeness lower bound
  9. The (dual) polynomial method
  10. Collision function
  11. Binary encoding
  12. Composition
  13. High correlation: proof of \ref{['claim:second_point']}
  14. Comparison with polynomial_strikes_back
  15. 3-colorability lower bound
  16. ...and 24 more sections

Key Result

Theorem 1.1

The quantum query complexity of testing $k$-source-subgraph-freeness in the unidirectional model is $O(N^{\frac{1}{2}\left(1 - \frac{1}{2^k -1}\right)})$.

Theorems & Definitions (66)

  • Theorem 1.1: Restated in \ref{['theo:kstarfree_ub_main']}
  • Theorem 1.2: Corollary of \ref{['lowerbound-col_intro']}
  • Theorem 1.3: Restated in \ref{['lowerbound-col']}
  • Theorem 1.4: Restated in \ref{['theo:3colorlbmain']}
  • Definition 2.1: Property Testing
  • Theorem 2.2
  • Remark 2.3
  • Definition 2.4: Source component
  • Definition 2.5: $k$-star
  • Proposition 2.6
  • ...and 56 more