# Understanding Quantum Algorithms via Query Complexity

@article{Ambainis2017UnderstandingQA, title={Understanding Quantum Algorithms via Query Complexity}, author={Andris Ambainis}, journal={ArXiv}, year={2017}, volume={abs/1712.06349} }

Query complexity is a model of computation in which we have to compute a function $f(x_1, \ldots, x_N)$ of variables $x_i$ which can be accessed via queries. The complexity of an algorithm is measured by the number of queries that it makes. Query complexity is widely used for studying quantum algorithms, for two reasons. First, it includes many of the known quantum algorithms (including Grover's quantum search and a key subroutine of Shor's factoring algorithm). Second, one can prove lower…

## 53 Citations

### Variational learning algorithms for quantum query complexity

- Education
- 2022

Quantum query complexity plays an important role in studying quantum algorithms, which captures the most known quantum algorithms, such as search and period ﬁnding. A query algorithm applies U t O x…

### From the sum-of-squares representation of a Boolean function to an optimal exact quantum query algorithm

- Computer ScienceQuantum Information Processing
- 2021

A primary algorithm framework is proposed with three basic steps that can be used to investigate the quantum query model with low complexity, such as Deutsch’s problem, a five-bit symmetric Boolean function and the characterization of Boolean functions with exact quantum 2-query complexity.

### Fast Classical and Quantum Algorithms for Online k-server Problem on Trees

- Computer ScienceICTCS
- 2021

A quantum algorithm to find the first marked element in a collection of $m$ objects, that works even in the presence of two-sided bounded errors on the input oracle, that has worst-case complexity $O(\sqrt{m})$.

### A Note on the Quantum Query Complexity of Permutation Symmetric Functions

- Mathematics, Computer ScienceITCS
- 2019

This paper improves the result of [AA14] and shows that for any permutation symmetric function f, the quantum query complexity is at most polynomially smaller than the classical randomized query complexity.

### Unitary property testing lower bounds by polynomials

- Computer ScienceArXiv
- 2022

A generalized polynomial method for unitary property testing problems, leveraging connections with invariant theory, is applied to obtain lower bounds on problems such as determining recurrence times of unitaries, approximating the dimension of a marked subspace, and approximates the entanglement entropy of a marking state.

### A Quantum Query Complexity Trichotomy for Regular Languages

- Computer Science2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS)
- 2019

The algorithm for star-free languages is viewed as a nontrivial generalization of Grover's algorithm which extends the quantum quadratic speedup to a much wider range of string-processing algorithms than was previously known.

### A tight lower bound for non-coherent index erasure

- Computer Science, MathematicsQuantum Inf. Comput.
- 2022

A tight $\Omega(\sqrt{n})$ lower bound is proved on the quantum query complexity of the non-coherent case of the index erasure problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary function-dependent state.

### Quantum Algorithms for Some Strings Problems Based on Quantum String Comparator

- Computer Science, MathematicsMathematics
- 2022

We study algorithms for solving three problems on strings. These are sorting of n strings of length k, “the Most Frequent String Search Problem”, and “searching intersection of two sequences of…

### Extended Learning Graphs for Triangle Finding

- Computer ScienceAlgorithmica
- 2019

New quantum algorithms for Triangle Finding improving its best previously known quantum query complexities for both dense and sparse instances are presented and a framework is presented in order to easily combine and analyze them.

### A Tight Lower Bound for Index Erasure

- Computer ScienceITCS
- 2020

A tight $\Omega(\sqrt{n})$ lower bound is proved on the quantum query complexity of the non-coherent case of the Index Erasure problem, where, in addition to preparing the required superposition, the algorithm is allowed to leave the ancillary memory in an arbitrary function-dependent state.

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