Article
Physics, Multidisciplinary
Ward L. Vleeshouwers, Vladimir Gritsev
Summary: Unitary matrix integrals over symmetric polynomials have important applications in random matrix theory, gauge theory, number theory, and enumerative combinatorics. Our study provides novel results on these integrals and applies them to correlation functions of long-range random walks involving hard-core bosons. We propose a generalized identity for computing these integrals, allowing us to derive expressions for unitary matrix integrals over Schur polynomials. We also present a particle-hole duality between different LRRW models and suggest using fermionic systems instead of bosonic systems for computing correlation functions.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2023)
Article
Computer Science, Theory & Methods
Eiichi Bannai, Manabu Oura, Da Zhao
Summary: This study demonstrates that similar results can be obtained for the invariants of the complex Clifford group under certain conditions, and confirms a conjecture proposed by Zhu, Kueng, Grassl, and Gross.
DESIGNS CODES AND CRYPTOGRAPHY
(2021)
Article
Quantum Science & Technology
Miguel A. Ruiz-Ortiz, Ehyter M. Martin-Gonzalez, Diego Santiago-Alarcon, Salvador E. Venegas-Andraca
Summary: We propose a new probabilistic definition for the hitting time of a continuous-time quantum walk into a marked set of nodes, using measurements with respect to the jump times of a Poisson process. We also derive a formula for the mean hitting time based on our definition, Wald's theorem, and a stochastic process that models our quantum measurement outcomes. This stochastic process results in a Markov chain that incorporates the expected values of the squared norm of random unitary matrix entries, providing a way to embed a Markov chain in a continuous-time quantum walk.
QUANTUM INFORMATION PROCESSING
(2023)
Article
Mathematics
Johannes Christensen, Klaus Thomsen
Summary: In this paper, KMS states for a transient random walk on a discrete group are studied, and a complete description is provided under specific conditions. Examples are also constructed to demonstrate that the structure of the KMS states can be more complicated beyond certain cases.
MONATSHEFTE FUR MATHEMATIK
(2021)
Article
Statistics & Probability
Matthieu Dussaule, Ilya Gekhtman
Summary: This paper focuses on the relationship between admissible symmetric finitely supported probability measures on relatively hyperbolic groups and Floyd-Ancona type inequalities, determining the precise homeomorphism type of the r-Martin boundary, introducing the concept of spectral degeneracy, and providing a criterion for the stability of the Martin boundary. It is shown that the criterion is always satisfied in small rank, ensuring the strong stability of the Martin boundary for admissible symmetric finitely supported probability measures on geometrically finite Kleinian groups of dimension at most 5.
PROBABILITY THEORY AND RELATED FIELDS
(2021)
Article
Computer Science, Information Systems
Michal Oszmaniec, Adam Sawicki, Michal Horodecki
Summary: In this work, quantitative connections between epsilon-nets and approximate unitary t-designs are studied, revealing their relationship in d-dimensional Hilbert space and their applications in quantum computing. The results show near optimality and the potential for new construction methods in quantum computing.
IEEE TRANSACTIONS ON INFORMATION THEORY
(2022)
Article
Quantum Science & Technology
Allan Wing-Bocanegra, Salvador E. E. Venegas-Andraca
Summary: This paper proposes a method to map evolution operators to quantum circuits and applies it to generate quantum walks on common topologies. Experimental executions on IBM Quantum Composer platform and Qiskit Aer simulator show results close to analytical distributions.
QUANTUM INFORMATION PROCESSING
(2023)
Article
Mathematics, Applied
Russell Lyons, Yuval Peres
Summary: The final configuration of lamps for simple random walk on the lamplighter group over Z(d) (d >= 3) is proven to be the Poisson boundary, extending the previous results to more general types of walks on more general groups.
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics
Erik Aas, Arvind Ayyer, Svante Linusson, Samu Potka
Summary: This article investigates a random walk method and its application in Weil groups. By computing correlations, important limiting directions are obtained.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2023)
Article
Mathematics
Jeremie Brieussel, Tianyi Zheng
Summary: The study addresses the inverse problem of finite generated groups involving speed, entropy, isoperimetric profile, return probability, and L-p-compression functions. Additionally, it proves a recent conjecture related to joint evaluation of speed and entropy exponents and provides a new proof of the existence of uncountably many pairwise non-quasi-isometric solvable groups. Furthermore, a formula linking the L-p-compression exponent of a group and its wreath product with the cyclic group for p in [1, 2] is obtained.
ANNALS OF MATHEMATICS
(2021)
Article
Astronomy & Astrophysics
Judah F. Unmuth-Yockey
Summary: The study proposes a quantum algorithm to calculate low-energy expectation values of a quantum Hamiltonian by sampling the partition function of the associated average energy. The sampling is performed using an accept/reject Metropolis-style algorithm on the quantum gates of the circuit itself. Observables calculated under the canonical ensemble are extrapolated from higher energies to the ground state using samples from these circuits.
Article
Mathematics
J. P. McCarthy
Summary: The necessary and sufficient conditions for a Markov chain to be ergodic are that the chain is irreducible and aperiodic. This is demonstrated in the case of random walks on finite groups by the distribution of the driving probability support. The study of ergodicity of random walks on finite quantum groups is extended by considering the support projection of the driving state.
COMMUNICATIONS IN ALGEBRA
(2021)
Article
Quantum Science & Technology
Francisco Orts, Ernestas Filatovas, Ester M. Garzon, Gloria Ortega
Summary: This paper proposes a customizable circuit design to generate random numbers that can be used by current quantum devices. It also presents a state-of-the-art comparator circuit, which is highly efficient in terms of qubits, T-count, and T-depth. Both circuits provide valuable tools for quantum applications and algorithms that require random number generation or comparison operations.
EPJ QUANTUM TECHNOLOGY
(2023)
Article
Geochemistry & Geophysics
Noel Perez, Pablo Venegas, Diego Benitez, Felipe Grijalva, Roman Lara, Mario Ruiz
Summary: This study systematically tested various feature groups in automatic event classifiers for the classification of long-period and volcano-tectonic seismic events at Cotopaxi volcano. The results showed that the shape and texture feature groups were the most appropriate for classifying these events among different classifiers.
IEEE GEOSCIENCE AND REMOTE SENSING LETTERS
(2022)
Article
Mathematics, Applied
Matthieu Dussaule
Summary: This is the first paper in a series of two papers discussing local limit theorems in relatively hyperbolic groups. The paper presents rough estimates for the Green function and introduces the concept of relative automaticity, which is useful in both papers. It also defines the notion of spectral positive recurrence for random walks on relatively hyperbolic groups. Finally, using the estimates for the Green function, the paper proves that the probability p(n) of returning to the origin at time n asymptotically behaves as R(-n)n(-3/2) for spectrally positive-recurrent random walks, where R is the inverse of the spectral radius of the random walk.
ERGODIC THEORY AND DYNAMICAL SYSTEMS
(2022)