# Span-program-based quantum algorithm for the rank problem

@article{Belovs2011SpanprogrambasedQA, title={Span-program-based quantum algorithm for the rank problem}, author={Aleksandrs Belovs}, journal={ArXiv}, year={2011}, volume={abs/1103.0842} }

Recently, span programs have been shown to be equivalent to quantum query algorithms. It is an open problem whether this equivalence can be utilized in order to come up with new quantum algorithms. We address this problem by providing span programs for some linear algebra problems.
We develop a notion of a high level span program, that abstracts from loading input vectors into a span program. Then we give a high level span program for the rank problem. The last section of the paper deals with…

## 14 Citations

### Span Programs are Equivalent to Quantum Query Algorithms

- Computer ScienceSIAM J. Comput.
- 2014

It is proved that for any boolean function, the optimal “witness size” of a span program equals the general adversary bound, therefore, span program witness size and quantum query complexity are equivalent measures.

### Span-Program-Based Quantum Algorithms for Graph Bipartiteness and Connectivity

- Computer Science
- 2015

Given a query access to the adjacency matrix of a simple graph G with n vertices, this work provides two new span-program-based quantum algorithms that provide an algorithm for testing if the graph is bipartite that uses $$On\sqrt{n}$$ quantum queries.

### Span-Program-Based Quantum Algorithms for Graph Bipartiteness and Connectivity

- Computer ScienceMEMICS
- 2015

Given a query access to the adjacency matrix of a simple graph with vertices, this work provides two new span-program-based quantum algorithms: an algorithm for testing if the graph is bipartite that uses $O(n\sqrt{n})$ quantum queries; an algorithmFor testing ifThe graph is connected that uses O(n(n)(n)} quantum queries.

### Span-program-based quantum algorithm for tree detection

- Computer ScienceArXiv
- 2013

This paper presents a time-efficient span-program-based quantum algorithm for the subgraph/not-a-minor problem for $T, and shows that this problem can be solved by a bounded-error quantum algorithm with $O(n)$ query complexity and time complexity.

### Span programs for functions with constant-sized 1-certificates: extended abstract

- Computer ScienceSTOC '12
- 2012

The power of the approach is proved by designing a quantum algorithm for the triangle problem with query complexity O(n35/27) that is better than O( n13/10) of the best previously known algorithm by Magniez et al.

### Quantum Algorithm for k-distinctness with Prior Knowledge on the Input

- Computer Science
- 2011

The learning graph technique from arXiv:1105.4024 is used to give a quantum algorithm for k-distinctness problem that runs in o(n^{3/4})$ queries, for a fixed $k$, given some prior knowledge on the structure of the input.

### Span program for non-binary functions

- Computer ScienceQuantum Inf. Comput.
- 2019

It is concluded that non-binary span programs characterize the quantum query complexity of f up to a factor of order at most $\sqrt{\ell-1}$ and this factor cannot be improved.

### Succinct Non-Interactive Zero Knowledge Arguments from Span Programs and Linear Error-Correcting Codes

- Computer ScienceIACR Cryptol. ePrint Arch.
- 2013

A new linear PCP is proposed for the Circuit-SAT, based on a combination of standard span programs that verify the correctness of every individual gate and high-distance linear error-correcting codes that check the consistency of wire assignments.

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

- Computer ScienceQuantum Inf. Comput.
- 2018

Space and time efficient quantum algorithms for deciding the two properties in the adjacency array model are described, which run in time $\tilde{O}(n\sqrt{d_m})$ and also require $O(\log n)$ space, where d_m$ is the maximum degree of any vertex in the input graph.

### Span Programs, Electrical Flows, and Beyond: New Approaches to Quantum Algorithms

- Computer Science
- 2014

This dissertation explores several new approaches to developing quantum algorithms, including span programs, electrical flows and nonsparse Hamiltonian simulation, and gives two quantum algorithms for approximating the effective resistance between two given vertices in an electrical network.

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