603 research outputs found
An Algorithmic Argument for Nonadaptive Query Complexity Lower Bounds on Advised Quantum Computation
This paper employs a powerful argument, called an algorithmic argument, to
prove lower bounds of the quantum query complexity of a multiple-block ordered
search problem in which, given a block number i, we are to find a location of a
target keyword in an ordered list of the i-th block. Apart from much studied
polynomial and adversary methods for quantum query complexity lower bounds, our
argument shows that the multiple-block ordered search needs a large number of
nonadaptive oracle queries on a black-box model of quantum computation that is
also supplemented with advice. Our argument is also applied to the notions of
computational complexity theory: quantum truth-table reducibility and quantum
truth-table autoreducibility.Comment: 16 pages. An extended abstract will appear in the Proceedings of the
29th International Symposium on Mathematical Foundations of Computer Science,
Lecture Notes in Computer Science, Springer-Verlag, Prague, August 22-27,
200
Quantum lower bound for inverting a permutation with advice
Given a random permutation as a black box and ,
we want to output . Supplementary to our input, we are given
classical advice in the form of a pre-computed data structure; this advice can
depend on the permutation but \emph{not} on the input . Classically, there
is a data structure of size and an algorithm that with the help
of the data structure, given , can invert in time , for
every choice of parameters , , such that . We prove a
quantum lower bound of for quantum
algorithms that invert a random permutation on an fraction of
inputs, where is the number of queries to and is the amount of
advice. This answers an open question of De et al.
We also give a quantum lower bound for the simpler but
related Yao's box problem, which is the problem of recovering a bit ,
given the ability to query an -bit string at any index except the
-th, and also given bits of advice that depend on but not on .Comment: To appear in Quantum Information & Computation. Revised version based
on referee comment
A quantum view on convex optimization
In this dissertation we consider quantum algorithms for convex optimization. We start by considering a black-box setting of convex optimization. In this setting we show that quantum computers require exponentially fewer queries to a membership oracle for a convex set in order to implement a separation oracle for that set. We do so by proving that Jordan's quantum gradient algorithm can also be applied to find sub-gradients of convex Lipschitz functions, even though these functions might not even be differentiable. As a corollary we get a quadraticly faster algorithm for convex optimization using membership queries. As a second set of results we give sub-linear time quantum algorithms for semidefinite optimization by speeding up the iterations of the Arora-Kale algorithm. For the problem of finding approximate Nash equilibria for zero-sum games we then give specific algorithms that improve the error-dependence and only depend on the sparsity of the game, not it's size. These last results yield improved algorithms for linear programming as a corollary. We also show several lower bounds in these settings, matching the upper bounds in most or all parameters
Transition role of entangled data in quantum machine learning
Entanglement serves as the resource to empower quantum computing. Recent
progress has highlighted its positive impact on learning quantum dynamics,
wherein the integration of entanglement into quantum operations or measurements
of quantum machine learning (QML) models leads to substantial reductions in
training data size, surpassing a specified prediction error threshold. However,
an analytical understanding of how the entanglement degree in data affects
model performance remains elusive. In this study, we address this knowledge gap
by establishing a quantum no-free-lunch (NFL) theorem for learning quantum
dynamics using entangled data. Contrary to previous findings, we prove that the
impact of entangled data on prediction error exhibits a dual effect, depending
on the number of permitted measurements. With a sufficient number of
measurements, increasing the entanglement of training data consistently reduces
the prediction error or decreases the required size of the training data to
achieve the same prediction error. Conversely, when few measurements are
allowed, employing highly entangled data could lead to an increased prediction
error. The achieved results provide critical guidance for designing advanced
QML protocols, especially for those tailored for execution on early-stage
quantum computers with limited access to quantum resources
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
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