Quantum algorithms for graph problems with cut queries

Abstract

Let G be an n-vertex graph with m edges. When asked a subset S of vertices, a cut query on G returns the number of edges of G that have exactly one endpoint in S. We show that there is a bounded-error quantum algorithm that determines all connected components of G after making O(log(n) 6 ) many cut queries. In contrast, it follows from results in communication complexity that any randomized algorithm even just to decide whether the graph is connected or not must make at least Ω(n/ log(n)) many cut queries. We further show that with O(log(n) 8 ) many cut queries a quantum algorithm can with high probability output a spanning forest for G.En route to proving these results, we design quantum algorithms for learning a graph using cut queries. We show that a quantum algorithm can learn a graph with maximum degree d after O(d log(n) 2 ) many cut queries, and can learn a general graph with O( √ m log(n) 3/2 ) many cut queries. These two upper bounds are tight up to the poly-logarithmic factors, and compare to Ω(dn) and Ω(m/ log(n)) lower bounds on the number of cut queries needed by a randomized algorithm for the same problems, respectively.The key ingredients in our results are the Bernstein-Vazirani algorithm, approximate counting with "OR queries", and learning sparse vectors from inner products as in compressed sensing.

On the query complexity of connectivity with global queries

“…We do this by showing that the aforementioned O(log 4 (n)) adjacency-matrixvector multiplication query algorithm to compute a spanning forest also works with the more restrictive Ax • (1 − x) queries. In addition to quantitatively improving the result of [LSZ21], this gives a much shorter proof and nicely separates the algorithm into a quantum part, simulating Ax • (1 − x) queries, and a classical randomized algorithm using Ax • (1 − x) queries.…”

“…We look at applications of matrix-vector multiplication algorithms to quantum algorithms using cut and BIS queries. Recent work of Lee, Santha, and Zhang showed that a quantum algorithm can output a spanning forest of a graph with high probability after O(log 8 (n)) cut queries [LSZ21]. This is in contrast to the randomized case where Ω(n/ log n) cut queries can be required to determine if a graph is connected [BFS86].…”

“…This is in contrast to the randomized case where Ω(n/ log n) cut queries can be required to determine if a graph is connected [BFS86]. A key observation made in [LSZ21] is that a quantum algorithm with cut queries can efficiently simulate a restricted version of a matrix-vector multiplication query to the adjacency matrix. Namely, if A is the adjacency matrix of an n-vertex simple graph, a quantum algorithm can compute Ax • (1 − x) for x ∈ {0, 1} n with O(log n) cut queries.…”

“…Lee, Santha, and Zhang [LSZ21] give a quantum algorithm to find a spanning forest of a simple n-vertex graph after O(log 8 (n)) cut queries. A key observation they make is that a quantum algorithm can efficiently simulate a restricted version of a matrix-vector multiplication query to the adjacency matrix.…”

“…Corollary 11 (cf. [LSZ21,Lemma 11]). Let G = (V, w) be an n-vertex weighted graph with integer weights in {0, 1, .…”