Time and space efficient quantum algorithms for detecting cycles and testing bipartiteness

Cade, Chris; Montanaro, Ashley; Belovs, Aleksandrs

doi:10.26421/qic18.1-2-2

Cited by 8 publications

(15 citation statements)

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“…All of our algorithms are in the adjacency matrix model (see Section 2), in which one can query elements of the adjacency matrix of the input graph. This contrasts with work such as [7] which study similar problems in the adjacency list model.…”

Section: Contributions and Comparison To Previous Workmentioning

confidence: 77%

“…There is an optimal reduction from graph connectivity to st-connectivity [10]. Cade et al use an st-connectivity subroutine to create nearly query-optimal algorithms for cycle detection and bipartiteness [7] 1 . Finally, the st-connectivity span program algorithm underlies the learning graph framework [3], one of the most successful heuristics for span program algorithm design.…”

Section: Introductionmentioning

confidence: 99%

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Applications of the quantum algorithm for st-connectivity

DeLorenzo,

Kimmel,

Witter

2019

Preprint

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We present quantum algorithms for various problems related to graph connectivity. We give simple and query-optimal algorithms for cycle detection and odd-length cycle detection (bipartiteness) using a reduction to st-connectivity. Furthermore, we show that our algorithm for cycle detection has improved performance under the promise of large circuit rank or a small number of edges. We also provide algorithms for detecting even-length cycles and for estimating the circuit rank of a graph. All of our algorithms have logarithmic space complexity.

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Section: Contributions and Comparison To Previous Workmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Applications of the quantum algorithm for st-connectivity

DeLorenzo,

Kimmel,

Witter

2019

Preprint

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show abstract

“…The definitions in Refs. [3,10] only apply to non-binary inputs (q = 2). We say that |w is an optimal witness for x if it minimizes the right hand side of Eq.…”

Section: And We Denote By π H(x) the Orthogonal Projection Onto The S...mentioning

confidence: 99%

“…To prove Lemma 19, we use an analysis that mirrors the Boolean function decision algorithm of Belovs and Reichardt [3, Section 5.2] and Cade et al [10,Section C.2] and the dual adversary algorithm of Reichardt [20, Algorithm 1]. Our approach differs from these previous algorithms in the addition of a parameter that controls the precision of our phase estimation; we do not always run phase estimation with a precision that is as high as those in previous works, which is what causes our false negatives.…”

Section: Function Evaluationmentioning

confidence: 99%

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Leveraging Unknown Structure in Quantum Query Algorithms

Anderson,

Chung,

Kimmel

2020

Preprint

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Quantum span program algorithms for function evaluation commonly have reduced query complexity when promised that the input has a certain structure. We design a modified span program algorithm to show these speed-ups persist even without having a promise ahead of time, and we extend this approach to the more general problem of state conversion. For example, there is a span program algorithm that decides whether two vertices are connected in an n-vertex graph with O(n 3/2 ) queries in general, but with O( √ kn) queries if promised that, if there is a path, there is one with at most k edges. Our algorithm uses Õ( √ kn) queries to solve this problem if there is a path with at most k edges, without knowing k ahead of time.

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Feedback‐Assisted Quantum Search by Continuous‐Time Quantum Walks

et al. 2022

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The quantum search of a target node on a cycle graph by means of a quantum walk assisted by continuous measurement and feedback are addressed. Unlike previous spatial search approaches, where the oracle is described as a projector on the target state, a dynamical oracle implemented through a feedback Hamiltonian is considered. The idea is based on continuously monitoring the position of the quantum walker on the graph and then applying a unitary feedback operation based on the information obtained from measurement. The feedback changes the couplings between the nodes and it is optimized at each time via a numerical procedure. The stochastic trajectories describing the evolution for graphs of dimensions up to N = 15 are numerically simulated, and the performance of the protocol is quantified via the average fidelity between the state of the walker and the target node. Different constraints on the control strategy are discussed. For unbounded controls, the protocol is able to quickly localize the walker on the target node. How the performance is lowered by posing an upper bound on the control couplings is then discussed. Finally, it is shown how a digital feedback protocol seems in general as efficient as the continuous bounded one.

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Time and space efficient quantum algorithms for detecting cycles and testing bipartiteness

Cited by 8 publications

References 0 publications

Applications of the quantum algorithm for st-connectivity

Applications of the quantum algorithm for st-connectivity

Leveraging Unknown Structure in Quantum Query Algorithms

Feedback‐Assisted Quantum Search by Continuous‐Time Quantum Walks

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