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Konrad Deka

Publications and source records attributed to Konrad Deka.

9 recordsLinked to original sources
arXiv2025

Determination of the fifth Busy Beaver value

The Busy Beaver value $S(n)$ is the maximum number of steps that an $n$-state 2-symbol Turing machine can perform from the all-zero tape before halting. $S$ was historically introduced by Tibor Rad\'o in 1962 as one of the simplest examples of an uncomputable function. We prove that $S(5) = 47,176,870$ using the Coq proof assistant. The proof enumerates $181,385,789$ Turing machines with 5 states and, for each machine, decides whether it halts or not. Our result marks the first determination of a new Busy Beaver value in over 40 years and the first Busy Beaver value ever to be formally verified, attesting to the effectiveness of massively collaborative online research (bbchallenge$.$org).

cs.LO↗
arXiv2025

Turing machines deciders, part I

The Busy Beaver Challenge (or bbchallenge) aims at collaboratively solving the following conjecture: "$S(5) = 47{,}176{,}870$" [Rad\'o, 1962], [Marxen and Buntrock, 1990], [Aaronson, 2020]. This conjecture says that if a 5-state Turing machine runs for more than 47,176,870 steps without halting, then it will never halt -- starting from the all-0 tape. Proving this conjecture amounts to deciding whether 181,385,789 Turing machines with 5 states halt or not -- starting from the all-0 tape [bbchallenge, 2025]. To do so, we write $\textit{deciders}$: programs that take as input a Turing machine and output either HALT, NONHALT, or UNKNOWN. Each decider is specialised in recognising a particular type of non-halting behavior. After two years of work, the Busy Beaver Challenge achieved its goal in July 2024 by delivering a proof of "$S(5) = 47{,}176{,}870$" formalised in Coq [bbchallenge, 2025]. In this document, we present deciders that were developed before the Coq proof and which were mainly not used in the proof; nonetheless, they are relevant techniques for analysing Turing machines. Part II of this work is the decider section of our paper showing "$S(5) = 47{,}176{,}870$" [bbchallenge, 2025], presenting the deciders that were used in the Coq proof.

cs.LO↗
arXiv2025

Bowen's Problem 32 and the conjugacy problem for systems with specification

We show that Rufus Bowen's Problem 32 on the classification of symbolic systems with the specification property does not admit a solution that would use concrete invariants. To this end, we construct a class of symbolic systems with the specification property and show that the conjugacy relation on this class is too complicated to admit such a classification. More generally, we gauge the complexity of the classification problem for symbolic systems with the specification property. Along the way, we also provide answers to two questions related to the classification of pointed systems with the specification property: to a question of Ding and Gu related to the complexity of the classification of pointed Cantor systems with the specification property and to a question of Bruin and Vejnar related to the complexity of the classification of pointed Hilbert cube systems with the specification property.

math.LO↗
arXiv2024

Simultaneously optimizing symmetry shifts and tensor factorizations for cost-efficient Fault-Tolerant Quantum Simulations of electronic Hamiltonians

In fault-tolerant quantum computing, the cost of calculating Hamiltonian eigenvalues using the quantum phase estimation algorithm is proportional to the constant scaling the Hamiltonian matrix block-encoded in a unitary circuit. We present a method to reduce this scaling constant for the electronic Hamiltonians represented as a linear combination of unitaries. Our approach combines the double tensor-factorization method of Burg et al. with the the block-invariant symmetry shift method of Loaiza and Izmaylov. By extending the electronic Hamiltonian with appropriately parametrized symmetry operators and optimizing the tensor-factorization parameters, our method achieves a 25% reduction in the block-encoding scaling constant compared to previous techniques. The resulting savings in the number of non-Clifford T-gates, which are an essential resource for fault-tolerant quantum computation, are expected to accelerate the feasiblity of practical Hamiltonian simulations. We demonstrate the effectiveness of our technique on Hamiltonians of industrial and biological relevance, including the nitrogenase cofactor (FeMoCo) and cytochrome P450.

quant-ph↗
arXiv2024

Diagonal operator decomposition on restricted topologies via enumeration of quantum state subsets

Various quantum algorithms require usage of arbitrary diagonal operators as subroutines. For their execution on a physical hardware, those operators must be first decomposed into target device's native gateset and its qubit connectivity for entangling gates. Here, we assume that the allowed gates are exactly the CX gate and the parameterized phase gate. We introduce a framework for the analysis of CX-only circuits and through its lens provide solution constructions for several different device topologies (fully-connected, linear and circular). We also introduce two additional variants of the problem. Those variants can be used in place of exact decomposition of the diagonal operator when the circuit following it satisfies a set of prerequisites, enabling further reduction in the CX cost of implementation. Finally, we discuss how to exploit the framework for the decomposition of a particular, rather than general, diagonal operator.

quant-ph↗
arXiv2024

Clifford circuits over non-cyclic abelian groups

We present a discussion of the generalized Clifford group over non-cyclic finite abelian groups. These Clifford groups appear naturally in the theory of topological error correction and abelian anyon models. We demonstrate a generalized Gottesman-Knill theorem, stating that every Clifford circuit can be efficiently classically simulated. We additionally provide circuits for a universal quantum computing scheme based on local two-qudit Clifford gates and magic states.

quant-ph↗
arXiv2023

Anyons in a highly-entangled toric xy model

While ostensibly coined in 1989 by Xiao-Gang Wen, the term "topological order" has been in use since 1972 to describe the behavior of the classical xy model. It has been noted that the xy model does not have Wen's topological order since it is also subject a non-topological U(1) gauge action. We show in a sense this is the only obstruction. That is, if gauge invariance is enforced energetically then the $xy$ model becomes purely topologically ordered. In fact, we show that the quantum $xy$ topological order is an infinite lattice limit of Kitaev's quantum double model applied to the group G=Z.

cond-mat.str-el↗
arXiv2022

Borel complexity of sets of points with prescribed Birkhoff averages in Polish dynamical systems with a specification property

We study the descriptive complexity of sets of points defined by placing restrictions on statistical behaviour of their orbits in dynamical systems on Polish spaces. A particular examples of such sets are the set of generic points of a $T$-invariant Borel probability measure, but we also consider much more general sets (for example, $\alpha$-Birkhoff regular sets and the irregular set appearing in multifractal analysis of ergodic averages of a continuous real-valued function). We show that many of these sets are Borel. In fact, all these sets are Borel when we assume that our space is compact. We provide examples of these sets being non-Borel, properly placed at the first level of the projective hierarchy (they are complete analytic or co-analytic). This proves that the compactness assumption is in some cases necessary to obtain Borelness. When these sets are Borel, we use the Borel hierarchy to measure their descriptive complexity. We show that the sets of interest are located at most at the third level of the hierarchy. We also use a modified version of the specification property to show that for many dynamical systems these sets are properly located at the third level. To demonstrate that the specification property is a sufficient, but not necessary condition for maximal descriptive complexity of a set of generic points, we provide an example of a compact minimal system with an invariant measure whose set of generic points is $\Pi^0_3$-complete.

math.DS↗