555 research outputs found
Quantum Adversary (Upper) Bound
We describe a method to upper bound the quantum query complexity of Boolean
formula evaluation problems, using fundamental theorems about the general
adversary bound. This nonconstructive method can give an upper bound on query
complexity without producing an algorithm. For example, we describe an oracle
problem which we prove (non-constructively) can be solved in queries,
where the previous best quantum algorithm uses a polylogarithmic number of
queries. We then give an explicit -query algorithm for this problem based
on span programs.Comment: Journal version. Edited for clarity and conciseness. Haar transform
algorithm remove
Applications of the Adversary Method in Quantum Query Algorithms
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 -distinctness problems.
* We prove tight lower bounds for the -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.Comment: PhD Thesis, 169 page
Quantum Algorithms for Attacking Hardness Assumptions in Classical and Post‐Quantum Cryptography
In this survey, the authors review the main quantum algorithms for solving the computational problems that serve as hardness assumptions for cryptosystem. To this end, the authors consider both the currently most widely used classically secure cryptosystems, and the most promising candidates for post-quantum secure cryptosystems. The authors provide details on the cost of the quantum algorithms presented in this survey. The authors furthermore discuss ongoing research directions that can impact quantum cryptanalysis in the future
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