Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Lower Bounds on Quantum Query and Learning Graph Complexities

Ansis Rosmanis

Published Jul 22, 2014• 2 citations
OpenAlex

Abstract

I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii In this thesis we study the power of quantum query algorithms and learning graphs; the latter essentially being very specialized quantum query algorithms themselves. We almost exclusively focus on proving lower bounds for these computational models. First, we study lower bounds on learning graph complexity. We consider two types of learning graphs: adaptive and, more restricted, non-adaptive learning graphs. We express both adaptive and non-adaptive learning graph complexities of Boolean-valued functions (i.e., decision problems) as semidefinite minimization problems, and derive their dual problems. For various functions, we construct feasible solutions to these dual problems, thereby obtaining lower bounds on the learn-ing graph complexity of the functions. Most notably, we prove an almost optimal Ω(n9/7/ log n)

Related papers

Applications of the Adversary Method in Quantum Query Algorithms

Aleksandrs Belovs

In the thesis, we use a recently developed tight characterisation of quantum query complexity, the adversary bound, to develop new quantum algorithms and lower bounds. Our results are as follows: * We develop a new technique for the construction of quantum algorithms: learning graphs. * We use learning graphs to improve quantum query complexity of the triangle detection and the $k$-distinctness problems. * We prove tight lower bounds for the $k$-sum and the triangle sum problems. * We construct quantum algorithms for some subgraph-finding problems that are optimal in terms of query, time and space complexities. * We develop a generalisation of quantum walks that connects electrical properties of a graph and its quantum hitting time. We use it to construct a time-efficient quantum algorithm for 3-distinctness.

arXiv:1402.3858•Feb 16, 2014•10 citations

On the Power of Non-adaptive Learning Graphs

Aleksandrs Belovs, Ansis Rosmanis

We introduce a notion of the quantum query complexity of a certificate structure. This is a formalisation of a well-known observation that many quantum query algorithms only require the knowledge of the disposition of possible certificates in the input string, not the precise values therein. Next, we derive a dual formulation of the complexity of a non-adaptive learning graph, and use it to show that non-adaptive learning graphs are tight for all certificate structures. By this, we mean that there exists a function possessing the certificate structure and such that a learning graph gives an optimal quantum query algorithm for it. For a special case of certificate structures generated by certificates of bounded size, we construct a relatively general class of functions having this property. The construction is based on orthogonal arrays, and generalizes the quantum query lower bound for the k-sum problem derived recently. Finally, we use these results to show that the best known learning graph for the triangle problem is almost optimal in these settings. This also gives a quantum query lower bound for the triangle-sum problem.

IEEE Conference on Computational Complexity (CCC)•Jun 1, 2013•19 citations

On the Power of Non-Adaptive Learning Graphs

Aleksandrs Belovs, Ansis Rosmanis

We introduce a notion of the quantum query complexity of a certificate structure. This is a formalisation of a well-known observation that many quantum query algorithms only require the knowledge of the disposition of possible certificates in the input string, not the precise values therein. Next, we derive a dual formulation of the complexity of a non-adaptive learning graph, and use it to show that non-adaptive learning graphs are tight for all certificate structures. By this, we mean that there exists a function possessing the certificate structure and such that a learning graph gives an optimal quantum query algorithm for it. For a special case of certificate structures generated by certificates of bounded size, we construct a relatively general class of functions having this property. The construction is based on orthogonal arrays, and generalizes the quantum query lower bound for the $k$-sum problem derived recently in arXiv:1206.6528. Finally, we use these results to show that the learning graph for the triangle problem from arXiv:1210.1014 is almost optimal in these settings. This also gives a quantum query lower bound for the triangle-sum problem.

arXiv:1210.3279•Oct 11, 2012•1 citation

Characterizing Algorithmic Efficiencies through Concentration

Hung Pham

Understanding inherent bottlenecks to efficient algorithm design lies at the heart of computer science. This question is significant both in the classical computing domain as well as in the emerging context of quantum computing. In this thesis, my goal is to characterize bottlenecks in designing efficient algorithms through the lens of a parameter called concentration of functions, starting with the domain of quantum information. My primary focus is probabilistically approximately correct (PAC) learning. I chose this domain since it allows us to approach the important subject of supervised learning in a rigorous and principled manner. For PAC learning, I propose a quantum algorithm to learn the class of concentrated Boolean functions with complexity $ O(\frac{M}{\epsilon^2})$ which offers an advantage over the best known classical PAC algorithms with complexity $\tilde{O}(n^2M)$, where M denotes the number of the concentration terms. I also show a lower bound $\Omega(M)$ to PAC learn this class of functions in distribution-independent settings. All this work is done in the context of the standard query model for PAC learning where the complexity measure is the number of queries, dubbed query complexity. I extend this work to include the learning model where functions are learned without any error, which is often called exact learning, and prove a query complexity lower bound of $\Omega(\frac{\epsilon \log M}{n} 2^n)$ in exact learning the class of concentrated Boolean functions. In the next part of the thesis, I focus my work on classical algorithms and explore a combinatorial counterpart of concentration called degree of symmetry. In this arena, graph isomorphism is my problem of choice. Once again, my goal is to characterize the efficiency of algorithms, in particular parallel algorithms, for graph isomorphism based on a concentration-related parameter. In particular, I propose a parallel algorithm in polynomial time using a quasi-polynomial number of processors for the Graph Isomorphism problem. My work builds on Babai’s celebrated quasi-polynomial algorithm and is work-preserving. The idea behind the parallelization explores the symmetry of the input structure for easier parallelization.

•Aug 1, 2021•0 citations

Efficient algorithms in quantum query complexity

Robin Kothari

In this thesis we provide new upper and lower bounds on the quantum query complexity of a diverse set of problems. Specifically, we study quantum algorithms for Hamiltonian simulation, matrix multiplication, oracle identification, and graph-property recognition. \n \nFor the Hamiltonian simulation problem, we provide a quantum algorithm with query complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Our algorithm is based on a new quantum algorithm for implementing unitary matrices that can be written as linear combinations of efficiently implementable unitary gates. This algorithm uses a new form of ``oblivious amplitude amplification'' that can be applied even though the reflection about the input state is unavailable. \n \nIn the oracle identification problem, we are given oracle access to an unknown N-bit string x promised to belong to a known set of size M, and our task is to identify x. We present the first quantum algorithm for the problem that is optimal in its dependence on N and M. Our algorithm is based on ideas from classical learning theory and a new composition theorem for solutions of the filtered gamma_2-norm semidefinite program. \n \nWe then study the quantum query complexity of matrix multiplication and related problems over rings, semirings, and the Boolean semiring in particular. Our main result is an output-sensitive algorithm for Boolean matrix multiplication that multiplies two n x n Boolean matrices with query complexity O(n sqrt{l}), where l is the sparsity of the output matrix. The algorithm is based on a reduction to the graph collision problem and a new algorithm for graph collision. \n \nFinally, we study the quantum query complexity of minor-closed graph properties and show that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity Theta(n^{3/2}) and those that do have such a characterization can be solved strictly faster, with o(n^{3/2}) queries. Our lower bound is based on a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.

•Aug 13, 2014•20 citations

Models of Query Complexity for Boolean Functions

Sourav Chakraborty

In this thesis we study various models of query complexity. A query algorithm computes a function under the restriction that the input can be accessed only by making probes to the the bits of the input. The query complexity of a function f is the minimum number of probes made by any query algorithm that computes f. In this thesis, we consider three different models of query complexity, (1) deterministic decision tree complexity (query complexity when the underlying algorithm is deterministic), (2) approximate decision tree complexity aka. property testing (query complexity when the underlying algorithm is probabilistic and only expected to ”approximately ” compute f) and quantum query complexity (query complexity when the underlying algorithm is allowed to make quantum queries). The main results in this thesis are: • We study the relation between deterministic decision tree complexity and other combinatorial measures of complexity measures like sensitivity and block sensitivity. We prove that for minterm-transitive functions the sensitivity is quadratically related to block sensitivity which is polynomially

•Aug 6, 2010•0 citations
I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii In this thesis we study the power of quantum query algorithms and learning graphs; the latter essentially being very specialized quantum query algorithms themselves. We almost exclusively focus on proving lower bounds for these computational models. First, we study lower bounds on learning graph complexity. We consider two types of learning graphs: adaptive and, more restricted, non-adaptive learning graphs. We express both adaptive and non-adaptive learning graph complexities of Boolean-valued functions (i.e., decision problems) as semidefinite minimization problems, and derive their dual problems. For various functions, we construct feasible solutions to these dual problems, thereby obtaining lower bounds on the learn-ing graph complexity of the functions. Most notably, we prove an almost optimal Ω(n9/7/ log n)
In the thesis, we use a recently developed tight characterisation of quantum query complexity, the adversary bound, to develop new quantum algorithms and lower bounds. Our results are as follows: * We develop a new technique for the construction of quantum algorithms: learning graphs. * We use learning graphs to improve quantum query complexity of the triangle detection and the kkk-distinctness problems. * We prove tight lower bounds for the kkk-sum and the triangle sum problems. * We construct quantum algorithms for some subgraph-finding problems that are optimal in terms of query, time and space complexities. * We develop a generalisation of quantum walks that connects electrical properties of a graph and its quantum hitting time. We use it to construct a time-efficient quantum algorithm for 3-distinctness.
We introduce a notion of the quantum query complexity of a certificate structure. This is a formalisation of a well-known observation that many quantum query algorithms only require the knowledge of the disposition of possible certificates in the input string, not the precise values therein. Next, we derive a dual formulation of the complexity of a non-adaptive learning graph, and use it to show that non-adaptive learning graphs are tight for all certificate structures. By this, we mean that there exists a function possessing the certificate structure and such that a learning graph gives an optimal quantum query algorithm for it. For a special case of certificate structures generated by certificates of bounded size, we construct a relatively general class of functions having this property. The construction is based on orthogonal arrays, and generalizes the quantum query lower bound for the k-sum problem derived recently. Finally, we use these results to show that the best known learning graph for the triangle problem is almost optimal in these settings. This also gives a quantum query lower bound for the triangle-sum problem.
We introduce a notion of the quantum query complexity of a certificate structure. This is a formalisation of a well-known observation that many quantum query algorithms only require the knowledge of the disposition of possible certificates in the input string, not the precise values therein. Next, we derive a dual formulation of the complexity of a non-adaptive learning graph, and use it to show that non-adaptive learning graphs are tight for all certificate structures. By this, we mean that there exists a function possessing the certificate structure and such that a learning graph gives an optimal quantum query algorithm for it. For a special case of certificate structures generated by certificates of bounded size, we construct a relatively general class of functions having this property. The construction is based on orthogonal arrays, and generalizes the quantum query lower bound for the kkk-sum problem derived recently in arXiv:1206.6528. Finally, we use these results to show that the learning graph for the triangle problem from arXiv:1210.1014 is almost optimal in these settings. This also gives a quantum query lower bound for the triangle-sum problem.
Understanding inherent bottlenecks to efficient algorithm design lies at the heart of computer science. This question is significant both in the classical computing domain as well as in the emerging context of quantum computing. In this thesis, my goal is to characterize bottlenecks in designing efficient algorithms through the lens of a parameter called concentration of functions, starting with the domain of quantum information. My primary focus is probabilistically approximately correct (PAC) learning. I chose this domain since it allows us to approach the important subject of supervised learning in a rigorous and principled manner. For PAC learning, I propose a quantum algorithm to learn the class of concentrated Boolean functions with complexity $ O(\frac{M}{\epsilon^2})$ which offers an advantage over the best known classical PAC algorithms with complexity O~(n2M)\tilde{O}(n^2M)O~(n2M), where M denotes the number of the concentration terms. I also show a lower bound Ω(M)\Omega(M)Ω(M) to PAC learn this class of functions in distribution-independent settings. All this work is done in the context of the standard query model for PAC learning where the complexity measure is the number of queries, dubbed query complexity. I extend this work to include the learning model where functions are learned without any error, which is often called exact learning, and prove a query complexity lower bound of Ω(ϵlog⁡Mn2n)\Omega(\frac{\epsilon \log M}{n} 2^n)Ω(nϵlogM​2n) in exact learning the class of concentrated Boolean functions. In the next part of the thesis, I focus my work on classical algorithms and explore a combinatorial counterpart of concentration called degree of symmetry. In this arena, graph isomorphism is my problem of choice. Once again, my goal is to characterize the efficiency of algorithms, in particular parallel algorithms, for graph isomorphism based on a concentration-related parameter. In particular, I propose a parallel algorithm in polynomial time using a quasi-polynomial number of processors for the Graph Isomorphism problem. My work builds on Babai’s celebrated quasi-polynomial algorithm and is work-preserving. The idea behind the parallelization explores the symmetry of the input structure for easier parallelization.
In this thesis we provide new upper and lower bounds on the quantum query complexity of a diverse set of problems. Specifically, we study quantum algorithms for Hamiltonian simulation, matrix multiplication, oracle identification, and graph-property recognition. \n \nFor the Hamiltonian simulation problem, we provide a quantum algorithm with query complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Our algorithm is based on a new quantum algorithm for implementing unitary matrices that can be written as linear combinations of efficiently implementable unitary gates. This algorithm uses a new form of ``oblivious amplitude amplification'' that can be applied even though the reflection about the input state is unavailable. \n \nIn the oracle identification problem, we are given oracle access to an unknown N-bit string x promised to belong to a known set of size M, and our task is to identify x. We present the first quantum algorithm for the problem that is optimal in its dependence on N and M. Our algorithm is based on ideas from classical learning theory and a new composition theorem for solutions of the filtered gamma_2-norm semidefinite program. \n \nWe then study the quantum query complexity of matrix multiplication and related problems over rings, semirings, and the Boolean semiring in particular. Our main result is an output-sensitive algorithm for Boolean matrix multiplication that multiplies two n x n Boolean matrices with query complexity O(n sqrt{l}), where l is the sparsity of the output matrix. The algorithm is based on a reduction to the graph collision problem and a new algorithm for graph collision. \n \nFinally, we study the quantum query complexity of minor-closed graph properties and show that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity Theta(n^{3/2}) and those that do have such a characterization can be solved strictly faster, with o(n^{3/2}) queries. Our lower bound is based on a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.
In this thesis we study various models of query complexity. A query algorithm computes a function under the restriction that the input can be accessed only by making probes to the the bits of the input. The query complexity of a function f is the minimum number of probes made by any query algorithm that computes f. In this thesis, we consider three different models of query complexity, (1) deterministic decision tree complexity (query complexity when the underlying algorithm is deterministic), (2) approximate decision tree complexity aka. property testing (query complexity when the underlying algorithm is probabilistic and only expected to ”approximately ” compute f) and quantum query complexity (query complexity when the underlying algorithm is allowed to make quantum queries). The main results in this thesis are: • We study the relation between deterministic decision tree complexity and other combinatorial measures of complexity measures like sensitivity and block sensitivity. We prove that for minterm-transitive functions the sensitivity is quadratically related to block sensitivity which is polynomially