arXiv subjects

Shyam Dhamapurkar

Publications and source records attributed to Shyam Dhamapurkar.

6 recordsLinked to original sources

arXiv2026

Localization Without Disorder: Quantum Walks on Structured Graphs

Continuous-time quantum walks (CTQWs) exhibit localization phenomena that differ fundamentally from their classical counterparts, yet the precise relationship between network structure, spectral degeneracy, and confined dynamics remains incompletely understood. In this work, we present a complete analytical characterization of localization in CTQWs on two highly symmetric graph families: barbell graphs and star-of-cliques graphs. These networks combine pronounced spectral degeneracy with modular structure, enabling exact diagonalization and explicit computation of both eigenstate and dynamical inverse participation ratios (IPRs). Our analysis reveals that localization is governed by the interplay between degenerate subspaces, which generate families of confined modes, and hybridization between invariant subspaces, which redistributes spectral weight. Notably, the dynamical IPR can exceed expectations based solely on eigenstate IPRs, demonstrating that coherent superposition within degenerate eigenspaces enhances confinement. By connecting IPR values to the effective number of vertices visited, we provide a structural diagnostic for predicting quantum transport outcomes in modular networks, establishing that connectivity alone can determine where and how strongly a quantum walk localizes.

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arXiv2024

Quantum walks advantage on the dihedral group for uniform sampling problem

Random walk algorithms are crucial for sampling and approximation problems in statistical physics and theoretical computer science. The mixing property is necessary for Markov chains to approach stationary distributions and is facilitated by walks. Quantum walks show promise for faster mixing times than classical methods but lack universal proof, especially in finite group settings. Here, we investigate the continuous-time quantum walks on Cayley graphs of the dihedral group $D_{2n}$ for odd $n$, generated by the smallest inverse closed symmetric subset. We present a significant finding that, in contrast to the classical mixing time on these Cayley graphs, which typically takes at least order $Ω(n^2 \log(1/2ε))$, the continuous-time quantum walk mixing time on $D_{2n}$ is of order $O(n (\log n)^5 \log(1/ε))$, achieving a quadratic improvement over the classical case. Our paper advances the general understanding of quantum walk mixing on Cayley graphs, highlighting the improved mixing time achieved by continuous-time quantum walks on $D_{2n}$. This work has potential applications in algorithms for a class of sampling problems based on non-abelian groups.

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arXiv2024

Weaving Complex Graph on simple low-dimensional qubit lattices

In quantum computing, the connectivity of qubits placed on two-dimensional chips limits the scalability and functionality of solid-state quantum computers. This paper presents two approaches to constructing complex quantum networks from simple qubit arrays, specifically grid lattices. The first approach utilizes a subset of qubits as tunable couplers, effectively yielding a range of non-trivial graph-based Hamiltonians. The second approach employs dynamic graph engineering by periodically activating and deactivating couplers, enabling the creation of effective quantum walks with longer-range couplings. Numerical simulations verify the effective dynamics of these approaches. In terms of these two approaches, we explore implementing various graphs, including cubes and fullerenes, etc, on two-dimensional lattices. These techniques facilitate the realization of analog quantum simulation, particularly continuous-time quantum walks discussed in detail in this manuscript, for different computational tasks on superconducting quantum chips despite their inherent low dimensional simple architecture.

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arXiv2023

Quantum walks as thermalizations, with application to fullerene graphs

We consider to what extent quantum walks can constitute models of thermalization, analogously to how classical random walks can be models for classical thermalization. In a quantum walk over a graph, a walker moves in a superposition of node positions via a unitary time evolution. We show a quantum walk can be interpreted as an equilibration of a kind investigated in the literature on thermalization in unitarily evolving quantum systems. This connection implies that recent results concerning the equilibration of observables can be applied to analyse the node position statistics of quantum walks. We illustrate this in the case of a family of graphs known as fullerenes. We find that a bound from Short et al., implying that certain expectation values will at most times be close to their time-averaged value, applies tightly to the node position probabilities. Nevertheless, the node position statistics do not thermalize in the standard sense. In particular, quantum walks over fullerene graphs constitute a counter-example to the hypothesis that subsystems equilibrate to the Gibbs state. We also exploit the bridge created to show how quantum walks can be used to probe the universality of the eigenstate thermalisation hypothesis (ETH) relation. We find that whilst in C60 with a single walker, the ETH relation does not hold for node position projectors, it does hold for the average position, enforced by a symmetry of the Hamiltonian. The findings suggest a unified study of quantum walks and quantum self-thermalizations is natural and feasible.

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arXiv2023

Quantum walk mixing is faster than classical on periodic lattices

This work focuses on the quantum mixing time, which is crucial for efficient quantum sampling and algorithm performance. We extend Richter's previous analysis of continuous time quantum walks on the periodic lattice $\mathbb{Z}_{n_1}\times \mathbb{Z}_{n_2}\times \dots \times \mathbb{Z}_{n_d}$, allowing for non-identical dimensions $n_i$. We present two quantum walks that achieve faster mixing compared to classical random walks. The first is a coordinate-wise quantum walk with a mixing time of $O\left(\left(\sum{i=1}^{d} n_i \right) \log{(d/ε)}\right)$ and $O(d \log(d/ε))$ measurements. The second is a continuous-time quantum walk with $O(\log(1/ε))$ measurements, conjectured to have a mixing time of $O\left(\sum_{i=1}^d n_i(\log(n_1))^2 \log(1/ε)\right)$. Our results demonstrate a quadratic speedup over classical mixing times on the generalized periodic lattice. We provide analytical evidence and numerical simulations supporting the conjectured faster mixing time. The ultimate goal is to prove the general conjecture for quantum walks on regular graphs.

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arXiv2021

Set membership with two classical and quantum bit probes

We consider the following problem: Given a set S of at most n elements from a universe of size m, represent it in memory as a bit string so that membership queries of the form "Is x in S?" can be answered by making at most t probes into the bit string. Let s(m,n,t) be the minimum number of bits needed by any such scheme. We obtain new upper bounds for s(m,n,t=2), which match or improve all the previously known bounds. We also consider the quantum version of this problem and obtain improved upper bounds.

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