@phdthesis{17208,
  abstract     = {Can current quantum computers provide a speedup over their classical counterparts for some kinds of problems? In this thesis, with a focus on ground state search/preparation, we address some of the challenges that both quantum annealing and variational quantum algorithms suffer from, hindering any possible practical speedup in comparison to the best classical counterparts. 

In the first part of the thesis, we study the performance of quantum annealing for solving a particular combinatorial optimization problem called 3-XOR satisfability (3-XORSAT). The classical problem is mapped into a ground state search of a 3-local classical Hamiltonian $H_C$. We consider how modifying the initial problem, by adding more interaction terms to the corresponding Hamiltonian, leads to the emergence of a first-order phase transition during the annealing process. This phenomenon causes the total annealing duration, $T$, required to prepare the ground state of $H_C$ with a high probability to increase exponentially with the size of the problem. Our findings indicate that with the growing complexity of problem instances, the likelihood of encountering first-order phase transitions also increases, making quantum annealing an impractical solution for these types of combinatorial optimization problems.

In the second part, we focus on the problem of barren plateaus in generic variational quantum algorithms. Barren plateaus correspond to flat regions in the parameter space where the gradient of the cost function is zero in expectation, and with the variance decaying exponentially with the system size, thus obstructing an efficient parameter optimization.  We propose an algorithm to circumvent Barren Plateaus by monitoring the entanglement entropy of k-local reduced density matrices, alongside a method for estimating entanglement entropy via classical shadow tomography. We illustrate the approach with the paradigmatic example of the variational quantum eigensolver, and show that our algorithm effectively avoids barren plateaus in the initialization as well as during the optimization stage. 

Lastly, in the last two Chapters of this thesis, we focus on the quantum approximate optimization algorithm (QAOA), originally introduced as an algorithm for solving generic combinatorial optimization problems in near-term quantum devices. Specifically, we focus on how to develop rigorous initialization strategies with guarantee improvement. Our motivation for this study lies in that for random initialization, the optimization typically leads to local minima with poor performance. Our main result corresponds to the analytical construction of index-1 saddle points or transition states, stationary points with a single direction of descent, as a tool for systematically exploring the QAOA optimization landscape. This leads us to propose a novel greedy parameter initialization strategy that guarantees for the energy to decrease with an increasing number of circuit layers. Furthermore, with precise estimates for the negative Hessian eigenvalue and its eigenvector, we establish a lower bound for energy improvement following a QAOA iteration.},
  author       = {Medina Ramos, Raimel A},
  issn         = {2663-337X},
  keywords     = {Quantum computing, Variational Quantum Algorithms, Optimization},
  pages        = {133},
  publisher    = {Institute of Science and Technology Austria},
  title        = {{Exploring the optimization landscape of variational quantum algorithms}},
  doi          = {10.15479/at:ista:17208},
  year         = {2024},
}

@inproceedings{10004,
  abstract     = {Markov chains are the de facto finite-state model for stochastic dynamical systems, and Markov decision processes (MDPs) extend Markov chains by incorporating non-deterministic behaviors. Given an MDP and rewards on states, a classical optimization criterion is the maximal expected total reward where the MDP stops after T steps, which can be computed by a simple dynamic programming algorithm. We consider a natural generalization of the problem where the stopping times can be chosen according to a probability distribution, such that the expected stopping time is T, to optimize the expected total reward. Quite surprisingly we establish inter-reducibility of the expected stopping-time problem for Markov chains with the Positivity problem (which is related to the well-known Skolem problem), for which establishing either decidability or undecidability would be a major breakthrough. Given the hardness of the exact problem, we consider the approximate version of the problem: we show that it can be solved in exponential time for Markov chains and in exponential space for MDPs.},
  author       = {Chatterjee, Krishnendu and Doyen, Laurent},
  booktitle    = {Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science},
  isbn         = {978-1-6654-4896-3},
  issn         = {1043-6871},
  keywords     = {Computer science, Heuristic algorithms, Memory management, Automata, Markov processes, Probability distribution, Complexity theory},
  location     = {Rome, Italy},
  pages        = {1--13},
  publisher    = {Institute of Electrical and Electronics Engineers},
  title        = {{Stochastic processes with expected stopping time}},
  doi          = {10.1109/LICS52264.2021.9470595},
  year         = {2021},
}

@article{11675,
  abstract     = {We consider the problems of maintaining an approximate maximum matching and an approximate minimum vertex cover in a dynamic graph undergoing a sequence of edge insertions/deletions. Starting with the seminal work of Onak and Rubinfeld (in: Proceedings of the ACM symposium on theory of computing (STOC), 2010), this problem has received significant attention in recent years. Very recently, extending the framework of Baswana et al. (in: Proceedings of the IEEE symposium on foundations of computer science (FOCS), 2011) , Solomon (in: Proceedings of the IEEE symposium on foundations of computer science (FOCS), 2016) gave a randomized dynamic algorithm for this problem that has an approximation ratio of 2 and an amortized update time of O(1) with high probability. This algorithm requires the assumption of an oblivious adversary, meaning that the future sequence of edge insertions/deletions in the graph cannot depend in any way on the algorithm’s past output. A natural way to remove the assumption on oblivious adversary is to give a deterministic dynamic algorithm for the same problem in O(1) update time. In this paper, we resolve this question. We present a new deterministic fully dynamic algorithm that maintains a O(1)-approximate minimum vertex cover and maximum fractional matching, with an amortized update time of O(1). Previously, the best deterministic algorithm for this problem was due to Bhattacharya et al. (in: Proceedings of the ACM-SIAM symposium on discrete algorithms (SODA), 2015); it had an approximation ratio of (2+ε) and an amortized update time of O(logn/ε2). Our result can be generalized to give a fully dynamic O(f3)-approximate algorithm with O(f2) amortized update time for the hypergraph vertex cover and fractional hypergraph matching problem, where every hyperedge has at most f vertices.},
  author       = {Bhattacharya, Sayan and Chakrabarty, Deeparnab and Henzinger, Monika H},
  issn         = {1432-0541},
  journal      = {Algorithmica},
  keywords     = {Dynamic algorithms, Data structures, Graph algorithms, Matching, Vertex cover},
  number       = {4},
  pages        = {1057--1080},
  publisher    = {Springer Nature},
  title        = {{Deterministic dynamic matching in O(1) update time}},
  doi          = {10.1007/s00453-019-00630-4},
  volume       = {82},
  year         = {2020},
}

@article{11676,
  abstract     = {We study the problem of maximizing a monotone submodular function with viability constraints. This problem originates from computational biology, where we are given a phylogenetic tree over a set of species and a directed graph, the so-called food web, encoding viability constraints between these species. These food webs usually have constant depth. The goal is to select a subset of k species that satisfies the viability constraints and has maximal phylogenetic diversity. As this problem is known to be NP-hard, we investigate approximation algorithms. We present the first constant factor approximation algorithm if the depth is constant. Its approximation ratio is (1−1e√). This algorithm not only applies to phylogenetic trees with viability constraints but for arbitrary monotone submodular set functions with viability constraints. Second, we show that there is no (1−1/e+ϵ)-approximation algorithm for our problem setting (even for additive functions) and that there is no approximation algorithm for a slight extension of this setting.},
  author       = {Dvořák, Wolfgang and Henzinger, Monika H and Williamson, David P.},
  issn         = {1432-0541},
  journal      = {Algorithmica},
  keywords     = {Approximation algorithms, Submodular functions, Phylogenetic diversity, Viability constraints},
  number       = {1},
  pages        = {152--172},
  publisher    = {Springer Nature},
  title        = {{Maximizing a submodular function with viability constraints}},
  doi          = {10.1007/s00453-015-0066-y},
  volume       = {77},
  year         = {2017},
}

@article{11668,
  abstract     = {We study multiple keyword sponsored search auctions with budgets. Each keyword has multiple ad slots with a click-through rate. The bidders have additive valuations, which are linear in the click-through rates, and budgets, which are restricting their overall payments. Additionally, the number of slots per keyword assigned to a bidder is bounded.

We show the following results: (1) We give the first mechanism for multiple keywords, where click-through rates differ among slots. Our mechanism is incentive compatible in expectation, individually rational in expectation, and Pareto optimal. (2) We study the combinatorial setting, where each bidder is only interested in a subset of the keywords. We give an incentive compatible, individually rational, Pareto-optimal, and deterministic mechanism for identical click-through rates. (3) We give an impossibility result for incentive compatible, individually rational, Pareto-optimal, and deterministic mechanisms for bidders with diminishing marginal valuations.},
  author       = {Colini-Baldeschi, Riccardo and Leonardi, Stefano and Henzinger, Monika H and Starnberger, Martin},
  issn         = {2167-8383},
  journal      = {ACM Transactions on Economics and Computation},
  keywords     = {Algorithms, Economics, Clinching ascending auction, auctions with budgets, Sponsored search auctions},
  number       = {1},
  publisher    = {Association for Computing Machinery},
  title        = {{On multiple keyword sponsored search auctions with budgets}},
  doi          = {10.1145/2818357},
  volume       = {4},
  year         = {2015},
}

@article{11679,
  abstract     = {We are given a set T = {T 1 ,T 2 , . . .,T k } of rooted binary trees, each T i leaf-labeled by a subset L(Ti)⊂{1,2,...,n} . If T is a tree on {1,2, . . .,n }, we let T|L denote the minimal subtree of T induced by the nodes of L and all their ancestors. The consensus tree problem asks whether there exists a tree T * such that, for every i , T∗|L(Ti) is homeomorphic to T i .

We present algorithms which test if a given set of trees has a consensus tree and if so, construct one. The deterministic algorithm takes time min{O(N n 1/2 ), O(N+ n 2 log n )}, where N=∑i|Ti| , and uses linear space. The randomized algorithm takes time O(N log3 n) and uses linear space. The previous best for this problem was a 1981 O(Nn) algorithm by Aho et al. Our faster deterministic algorithm uses a new efficient algorithm for the following interesting dynamic graph problem: Given a graph G with n nodes and m edges and a sequence of b batches of one or more edge deletions, then, after each batch, either find a new component that has just been created or determine that there is no such component. For this problem, we have a simple algorithm with running time O(n 2 log n + b 0 min{n 2 , m log n }), where b 0 is the number of batches which do not result in a new component. For our particular application, b0≤1 . If all edges are deleted, then the best previously known deterministic algorithm requires time O(mn−−√) to solve this problem. We also present two applications of these consensus tree algorithms which solve other problems in computational evolutionary biology.},
  author       = {Henzinger, Monika H and King, V. and Warnow, T.},
  issn         = {1432-0541},
  journal      = {Algorithmica},
  keywords     = {Algorithms, Data structures, Evolutionary biology, Theory of databases},
  pages        = {1--13},
  publisher    = {Springer Nature},
  title        = {{Constructing a tree from homeomorphic subtrees, with applications to computational evolutionary biology}},
  doi          = {10.1007/pl00009268},
  volume       = {24},
  year         = {1999},
}