arXiv subjects

Daniel Gibney

Publications and source records attributed to Daniel Gibney.

14 recordsLinked to original sources

arXiv2026

Average-Case and Smoothed Near-Optimality for Color-Code Decoding

Minimum-weight decoding for two-dimensional color codes is NP-hard (Walters and Turner 2026), motivating the search for approximation guarantees beyond worst-case exact decoding. We study a block-based decoder for triangular color-code lattices. The decoder satisfies the deterministic additive guarantee $\lvert E_{\mathrm{alg}}\rvert \leq \operatorname{OPT}(S)+O(n/\tau)$, where $n$ is the number of vertices and $\tau$ is the wall spacing. We show that this additive guarantee becomes a near-optimal multiplicative guarantee under natural noise models. For constant-rate i.i.d. face noise and constant local degree, choosing $\tau=\Theta(\epsilon^{-1})$ gives a $(1+\epsilon)$-approximation with probability $1-\exp(-\Omega(n))$, in time $n2^{O(\epsilon^{-1})}$. We also prove a smoothed analogue: the same near-optimality guarantee holds when an arbitrary adversarial error pattern is perturbed by independent constant-rate noise. Finally, in the low-probability regime $p=o(1/\log^2 n)$, the syndrome decomposes into small active regions with high probability, allowing independent component-wise decoding and yielding an exact minimum-weight correction in time $n2^{O((\log n)^{3/2})}$. These results show that, despite worst-case hardness, color-code decoding admits strong average-case, smoothed, and sparse-regime guarantees.

cs.DS↗

arXiv2026

Open-source implementation of the anti-Hermitian contracted Schr\"odinger equation for electronic ground and excited states

Efficient simulation of strongly correlated electrons has become a routine tool in molecular electronic structure theory due to recent advances in approximate configuration interaction (CI) techniques. Nonetheless, the quantitative and predictive description of molecular electronic states remains a significant challenge due to the difficulty of computing all-electron correlation beyond CI. Here, we describe a new open-source implementation of the anti-Hermitian contracted Schr\"odinger equation (ACSE) for use in accurate simulation of all-electron correlation in molecules. In contrast to standard approaches via multireference perturbation theory, the scaling of the ACSE does not depend on the complexity of the strongly correlated reference wavefunction. Furthermore, the ACSE employs the exact electronic Hamiltonian, rather than an approximate perturbative Hamiltonian. Our benchmark results demonstrate good accuracy for main group and transition metal systems, in weakly and strongly correlated regimes, with various basis sets, and for ground and excited states. The results suggest that the ACSE has potential as a scalable and robust technique for simulating all-electron correlation in molecular ground and excited states.

quant-ph↗

arXiv2025

Contextual Pattern Mining and Counting

Given a string $P$ of length $m$, a longer string $T$ of length $n>m$, and two integers $l\geq 0$ and $r\geq 0$, the context of $P$ in $T$ is the set of all string pairs $(L,R)$, with $|L|=l$ and $|R|=r$, such that the string $LPR$ occurs in $T$. We introduce two problems related to the notion of context: (1) the Contextual Pattern Mining (CPM) problem, which given $T$, $(m,l,r)$, and an integer $\tau>0$, asks for outputting the context of each substring $P$ of length $m$ of $T$, provided that the size of the context of $P$ is at least $\tau$; and (2) the Contextual Pattern Counting (CPC) problem, which asks for preprocessing $T$ so that the size of the context of a given query string $P$ of length $m$ can be found efficiently. For CPM, we propose a linear-work algorithm that either uses only internal memory, or a bounded amount of internal memory and external memory, which allows much larger datasets to be handled. For CPC, we propose an $\widetilde{\mathcal{O}}(n)$-space index that can be constructed in $\widetilde{\mathcal{O}}n)$ time and answers queries in $\mathcal{O}(m)+\widetilde{\mathcal{O}}(1)$ time. We further improve the practical performance of the CPC index by optimizations that exploit the LZ77 factorization of $T$ and an upper bound on the query length. Using billion-letter datasets from different domains, we show that the external memory version of our CPM algorithm can deal with very large datasets using a small amount of internal memory while its runtime is comparable to that of the internal memory version. Interestingly, we also show that our optimized index for CPC outperforms an approach based on the state of the art for the reporting version of CPC [Navarro, SPIRE 2020] in terms of query time, index size, construction time, and construction space, often by more than an order of magnitude.

cs.DS↗

arXiv2025

Benchmarking and contrasting exchange-correlation functional differences in response to static correlation in unrestricted Kohn-Sham and a hybrid 1-electron reduced density matrix functional theory

A hybrid Kohn-Sham Density Functional Theory (KS-DFT) and 1-electron Reduced Density Matrix Functional Theory (1-RDMFT) has recently been developed to describe strongly correlated systems at mean-field computational cost. This approach relies on combining a Reduced Density Matrix Functional to capture strong correlation effects with existing exchange correlation (XC) functionals to capture the remaining dynamical correlation effects. In this work, we systematically benchmark the performance of nearly 200 different XC functionals available within LibXC in this DFA 1-RDMFT framework, contrasting it with their performance in unrestricted KS-DFT. We identify optimal XC functionals for use within DFA 1-RDMFT and elucidate fundamental trends in the response of different XC functionals to strong correlation in both DFA 1-RDMFT and UKS-DFT.

physics.comp-ph↗

arXiv2024

Enhancing Density Functional Theory for Static Correlation in Large Molecules

A critical challenge for density functional theory (DFT) in practice is its limited ability to treat static electron correlation, leading to errors in its prediction of charges, multiradicals, and reaction barriers. Recently, we combined one- and two-electron reduced density matrix theories with DFT to obtain a universal $O(N^3)$ generalization of DFT for static correlation. In this Letter, we enhance the theory's treatment of large molecules by renormalizing the trace of the two-electron identity matrix in the correction using Cauchy-Schwarz inequalities of the electron-electron repulsion matrix. We apply the resulting functional theory to linear hydrogen chains as well as the prediction of the singlet-triplet gap and equilibrium geometries of a series of acenes. This renormalization of the generalized DFT retains the $O(N^{3})$ computational scaling of DFT while enabling the accurate treatment of static correlation for a broad range of molecules and materials.

physics.chem-ph↗

arXiv2023

Near-Optimal Quantum Algorithms for Bounded Edit Distance and Lempel-Ziv Factorization

Classically, the edit distance of two length-$n$ strings can be computed in $O(n^2)$ time, whereas an $O(n^{2-\epsilon})$-time procedure would falsify the Orthogonal Vectors Hypothesis. If the edit distance does not exceed $k$, the running time can be improved to $O(n+k^2)$, which is near-optimal (conditioned on OVH) as a function of $n$ and $k$. Our first main contribution is a quantum $\tilde{O}(\sqrt{nk}+k^2)$-time algorithm that uses $\tilde{O}(\sqrt{nk})$ queries, where $\tilde{O}(\cdot)$ hides polylogarithmic factors. This query complexity is unconditionally optimal, and any significant improvement in the time complexity would resolve a long-standing open question of whether edit distance admits an $O(n^{2-\epsilon})$-time quantum algorithm. Our divide-and-conquer quantum algorithm reduces the edit distance problem to a case where the strings have small Lempel-Ziv factorizations. Then, it combines a quantum LZ compression algorithm with a classical edit-distance subroutine for compressed strings. The LZ factorization problem can be classically solved in $O(n)$ time, which is unconditionally optimal in the quantum setting. We can, however, hope for a quantum speedup if we parameterize the complexity in terms of the factorization size $z$. Already a generic oracle identification algorithm yields the optimal query complexity of $\tilde{O}(\sqrt{nz})$ at the price of exponential running time. Our second main contribution is a quantum algorithm that achieves the optimal time complexity of $\tilde{O}(\sqrt{nz})$. The key tool is a novel LZ-like factorization of size $O(z\log^2n)$ whose subsequent factors can be efficiently computed through a combination of classical and quantum techniques. We can then obtain the string's run-length encoded Burrows-Wheeler Transform (BWT), construct the $r$-index, and solve many fundamental string processing problems in time $\tilde{O}(\sqrt{nz})$.

cs.DS↗

arXiv2023

Compressibility-Aware Quantum Algorithms on Strings

Sublinear time quantum algorithms have been established for many fundamental problems on strings. This work demonstrates that new, faster quantum algorithms can be designed when the string is highly compressible. We focus on two popular and theoretically significant compression algorithms -- the Lempel-Ziv77 algorithm (LZ77) and the Run-length-encoded Burrows-Wheeler Transform (RL-BWT), and obtain the results below. We first provide a quantum algorithm running in $\tilde{O}(\sqrt{zn})$ time for finding the LZ77 factorization of an input string $T[1..n]$ with $z$ factors. Combined with multiple existing results, this yields an $\tilde{O}(\sqrt{rn})$ time quantum algorithm for finding the RL-BWT encoding with $r$ BWT runs. Note that $r = \tilde{\Theta}(z)$. We complement these results with lower bounds proving that our algorithms are optimal (up to polylog factors). Next, we study the problem of compressed indexing, where we provide a $\tilde{O}(\sqrt{rn})$ time quantum algorithm for constructing a recently designed $\tilde{O}(r)$ space structure with equivalent capabilities as the suffix tree. This data structure is then applied to numerous problems to obtain sublinear time quantum algorithms when the input is highly compressible. For example, we show that the longest common substring of two strings of total length $n$ can be computed in $\tilde{O}(\sqrt{zn})$ time, where $z$ is the number of factors in the LZ77 factorization of their concatenation. This beats the best known $\tilde{O}(n^\frac{2}{3})$ time quantum algorithm when $z$ is sufficiently small.

cs.DS↗

arXiv2022

Comparison of Density-Matrix Corrections to Density Functional Theory

Density functional theory (DFT), one of the most widely utilized methods available to computational chemistry, fails to describe systems with statically correlated electrons. To address this shortcoming, in previous work we transformed DFT into a one-electron reduced density matrix theory (1-RDMFT) via the inclusion of a quadratic one-electron reduced density matrix (1-RDM) correction. Here, we combine our 1-RDMFT approach with different DFT functionals as well as Hartree-Fock to elucidate the method's dependence on the underlying functional selection. Furthermore, we generalize the information density matrix functional theory (iDMFT), recently developed as a correction to the Hartree-Fock method, by incorporating density functionals in place of the Hartree-Fock functional. We relate iDMFT mathematically to our approach and benchmark the two with a common set of functionals and systems.

physics.chem-ph↗

arXiv2022

The Complexity of Approximate Pattern Matching on De Bruijn Graphs

Aligning a sequence to a walk in a labeled graph is a problem of fundamental importance to Computational Biology. For finding a walk in an arbitrary graph with $|E|$ edges that exactly matches a pattern of length $m$, a lower bound based on the Strong Exponential Time Hypothesis (SETH) implies an algorithm significantly faster than $O(|E|m)$ time is unlikely [Equi et al., ICALP 2019]. However, for many special graphs, such as de Bruijn graphs, the problem can be solved in linear time [Bowe et al., WABI 2012]. For approximate matching, the picture is more complex. When edits (substitutions, insertions, and deletions) are only allowed to the pattern, or when the graph is acyclic, the problem is again solvable in $O(|E|m)$ time. When edits are allowed to arbitrary cyclic graphs, the problem becomes NP-complete, even on binary alphabets [Jain et al., RECOMB 2019]. These results hold even when edits are restricted to only substitutions. The complexity of approximate pattern matching on de Bruijn graphs remained open. We investigate this problem and show that the properties that make de Bruijn graphs amenable to efficient exact pattern matching do not extend to approximate matching, even when restricted to the substitutions only case with alphabet size four. We prove that determining the existence of a matching walk in a de Bruijn graph is NP-complete when substitutions are allowed to the graph. In addition, we demonstrate that an algorithm significantly faster than $O(|E|m)$ is unlikely for de Bruijn graphs in the case where only substitutions are allowed to the pattern. This stands in contrast to pattern-to-text matching where exact matching is solvable in linear time, like on de Bruijn graphs, but approximate matching under substitutions is solvable in subquadratic $O(n\sqrt{m})$ time, where $n$ is the text's length [Abrahamson, SIAM J. Computing 1987].

cs.DS↗

arXiv2022

Density Functional Theory Transformed into a One-electron Reduced Density Matrix Functional Theory for the Capture of Static Correlation

Density functional theory (DFT), the most widely adopted method in modern computational chemistry, fails to describe accurately the electronic structure of strongly correlated systems. Here we show that DFT can be formally and practically transformed into a one-electron reduced-density-matrix (1-RDM) functional theory, which can address the limitations of DFT while retaining favorable computational scaling compared to wavefunction-based approaches. In addition to relaxing the idempotency restriction on the 1-RDM in the kinetic energy term, we add a quadratic 1-RDM-based term to DFT's density-based exchange-correlation functional. Our approach, which we implement by quadratic semidefinite programming at DFT's computational scaling of $O(r^{3})$, yields substantial improvements over traditional DFT in the description of static correlation in chemical structures and processes such as singlet biradicals and bond dissociations.

physics.chem-ph↗

arXiv2020

Quantifying Membership Inference Vulnerability via Generalization Gap and Other Model Metrics

We demonstrate how a target model's generalization gap leads directly to an effective deterministic black box membership inference attack (MIA). This provides an upper bound on how secure a model can be to MIA based on a simple metric. Moreover, this attack is shown to be optimal in the expected sense given access to only certain likely obtainable metrics regarding the network's training and performance. Experimentally, this attack is shown to be comparable in accuracy to state-of-art MIAs in many cases.

cs.LG↗

arXiv2020

Simple Reductions from Formula-SAT to Pattern Matching on Labeled Graphs and Subtree Isomorphism

The CNF formula satisfiability problem (CNF-SAT) has been reduced to many fundamental problems in P to prove tight lower bounds under the Strong Exponential Time Hypothesis (SETH). Recently, the works of Abboud, Hansen, Vassilevska W. and Williams (STOC 16), and later, Abboud and Bringmann (ICALP 18) have proposed basing lower bounds on the hardness of general boolean formula satisfiability (Formula-SAT). Reductions from Formula-SAT have two advantages over the usual reductions from CNF-SAT: (1) conjectures on the hardness of Formula-SAT are arguably much more plausible than those of CNF-SAT, and (2) these reductions give consequences even for logarithmic improvements in a problems upper bounds. Here we give tight reductions from Formula-SAT to two more problems: pattern matching on labeled graphs (PMLG) and subtree isomorphism. Previous reductions from Formula-SAT were to sequence alignment problems such as Edit Distance, LCS, and Frechet Distance and required some technical work. This paper uses ideas similar to those used previously, but in a decidedly simpler setting, helping to illustrate the most salient features of the underlying techniques.

cs.CC↗

arXiv2020

On the Complexity of BWT-runs Minimization via Alphabet Reordering

The Burrows-Wheeler Transform (BWT) has been an essential tool in text compression and indexing. First introduced in 1994, it went on to provide the backbone for the first encoding of the classic suffix tree data structure in space close to the entropy-based lower bound. Recently, there has been the development of compact suffix trees in space proportional to "$r$", the number of runs in the BWT, as well as the appearance of $r$ in the time complexity of new algorithms. Unlike other popular measures of compression, the parameter $r$ is sensitive to the lexicographic ordering given to the text's alphabet. Despite several past attempts to exploit this, a provably efficient algorithm for finding, or approximating, an alphabet ordering which minimizes $r$ has been open for years. We present the first set of results on the computational complexity of minimizing BWT-runs via alphabet reordering. We prove that the decision version of this problem is NP-complete and cannot be solved in time $2^{o(σ+ \sqrt{n})}$ unless the Exponential Time Hypothesis fails, where $σ$ is the size of the alphabet and $n$ is the length of the text. We also show that the optimization problem is APX-hard. In doing so, we relate two previously disparate topics: the optimal traveling salesperson path and the number of runs in the BWT of a text, providing a surprising connection between problems on graphs and text compression. Also, by relating recent results in the field of dictionary compression, we illustrate that an arbitrary alphabet ordering provides a $O(\log^2 n)$-approximation. We provide an optimal linear-time algorithm for the problem of finding a run minimizing ordering on a subset of symbols (occurring only once) under ordering constraints, and prove a generalization of this problem to a class of graphs with BWT like properties called Wheeler graphs is NP-complete.

cs.DS↗

arXiv2019

On the Hardness and Inapproximability of Recognizing Wheeler Graphs

In recent years several compressed indexes based on variants of the Burrows-Wheeler transformation have been introduced. Some of these index structures far more complex than a single string, as was originally done with the FM-index [Ferragina and Manzini, J. ACM 2005]. As such, there has been an effort to better understand under which conditions such an indexing scheme is possible. This led to the introduction of Wheeler graphs [Gagie it et al., Theor. Comput. Sci., 2017]. A Wheeler graph is a directed graph with edge labels which satisfies two simple axioms. Wheeler graphs can be indexed in a way which is space efficient and allows for fast traversal. Gagie et al. showed that de Bruijn graphs, generalized compressed suffix arrays, and several other BWT related structures can be represented as Wheeler graphs. Here we answer the open question of whether or not there exists an efficient algorithm for recognizing if a graph is a Wheeler graph. We demonstrate:(i) Recognizing if a graph is a Wheeler graph is NP-complete for any edge label alphabet of size $σ\geq 2$, even for DAGs. It can be solved in linear time for $σ=1$; (ii) An optimization variant called Wheeler Graph Violation (WGV) which aims to remove the minimum number of edges needed to obtain a Wheeler graph is APX-hard, even for DAGs. Hence, unless P = NP, there exists constant $C > 1$ such that there is no $C$-approximation algorithm. We show conditioned on the Unique Games Conjecture, for every constant $C \geq 1$, it is NP-hard to find a $C$-approximation to WGV; (iii) The Wheeler Subgraph problem (WS) which aims to find the largest Wheeler subgraph is in APX for $σ=O(1)$; (iv) For the above problems there exist efficient exponential time exact algorithms, relying on graph isomorphism being computed in strictly sub-exponential time; (v) A class of graphs where the recognition problem is polynomial time solvable.

cs.CC↗