arXiv subjects

David A. Meyer

Publications and source records attributed to David A. Meyer.

At least 19 recordsLinked to original sources

arXiv2026

Universal entropic occupation statistics in disordered bosonic resonators

Programmable microcavities support grand-canonical photon gases. We show that bosonic state counting creates an entropic staircase of most-probable total occupations. For detuning density continuous and nonzero at the chemical-potential threshold, the active-cell fraction is asymptotically linear at low temperature and the conditional law has a universal limit; for uniform disorder the law is exact over a finite temperature interval. For two modes it is the Gauss--Kuzmin distribution, linking photonic thermodynamics and metric number theory. We outline finite-array and dye-microcavity tests.

quant-ph↗

arXiv2026

Spatial Search by Nonlinear Quantum Walk

Many-body quantum systems with effective nonlinearities have been shown to speed up quantum search on the complete graph, \textit{i.e.}, the combinatorial version of Grover's algorithm, at the expense of the number of particles needed for the effective nonlinearity to hold. Physically, however, data may not be arranged in an all-to-all network, and the task of searching incomplete graphs is the spatial search problem. We explore spatial search using a continuous-time nonlinear quantum walk on a variety of graphs. First, we consider incomplete graphs that are ``sufficiently complete'' so as to asymptotically search like the complete graph under a continuous-time (linear) quantum walk, which includes strongly regular graphs such as Paley graphs, regular graphs such as hypercubes, and irregular graphs such as complete bipartite graphs. For these sufficiently complete graphs, we analytically prove nonlinear speedups for Paley graphs and for complete bipartite graphs whose two partite sets both have size $Θ(N)$, for suitable cubic and cubic-quintic nonlinearities, and we give numerical evidence for stronger nonlinearities and for hypercubes. Second, we explore arbitrary-dimensional cubic lattices, and we numerically show that certain nonlinearities speed up search on sufficiently high dimensional lattices. Thus, nonlinear quantum search can remain viable even when the underlying graph is incomplete.

quant-ph↗

arXiv2026

Network-Level Measures of Mobility from Aggregated Origin-Destination Data

We introduce a framework for defining and interpreting collective mobility measures from spatially and temporally aggregated origin--destination (OD) data. Rather than characterizing individual behavior, these measures describe properties of the mobility system itself: how network organization, spatial structure, and routing constraints shape and channel population movement. In this view, aggregate mobility flows reveal aspects of connectivity, functional organization, and large-scale daily activity patterns encoded in the underlying transport and spatial network. To support interpretation and provide a controlled reference for the proposed time-elapsed calculations, we first employ an independent, network-driven synthetic data generator in which trajectories arise from prescribed system structure rather than observed data. This controlled setting provides a concrete reference for understanding how the proposed measures reflect network organization and flow constraints. We then apply the measures to fully anonymized data from the NetMob 2024 Data Challenge, examining their behavior under realistic limitations of spatial and temporal aggregation. While such data constraints restrict dynamical resolution, the resulting metrics still exhibit interpretable large-scale structure and temporal variation at the city scale.

stat.AP↗

arXiv2026

Mobility Trajectories from Network-Driven Markov Dynamics

We present a generative model of human mobility in which trajectories arise as realizations of a prescribed, time-dependent Markov dynamics defined on a spatial interaction network. The model constructs a hierarchical routing structure with hubs, corridors, feeder paths, and metro links, and specifies transition matrices using gravity-type distance decay combined with externally imposed temporal schedules and directional biases. Population mass evolves as indistinguishable, memoryless movers performing a single transition per time step. When aggregated, the resulting trajectories reproduce structured origin-destination flows that reflect network geometry, temporal modulation, and connectivity constraints. By applying the Perron-Frobenius theorem to the daily evolution operator, we identify a unique periodic invariant population distribution that serves as a natural non-transient reference state. We verify consistency between trajectory-level realizations and multi-step Markov dynamics, showing that discrepancies are entirely attributable to finite-population sampling. The framework provides a network-centric, privacy-preserving approach to generating mobility trajectories and studying time-elapsed flow structure without invoking individual-level behavioral assumptions.

cs.SI↗

arXiv2025

Reconstructing Minkowski geometry from causal separations

Aleksandrov, and then Zeeman, showed that the causal relations among the set of points in a Minkowski space of dimension greater than 2 determine the Minkowski space structure of the set up to a global conformal factor. We show that in any dimension the distances between causally related pairs of points determine the distances between spatially related pairs of points, and thus completely determine the Minkowski space structure of the set. This is a step in the direction of proving that causal sets arising from a Poisson process in a Lorentzian manifold determine that manifold up to the degree of approximation inherent in the intensity of the Poisson process -- the Hauptvermutung of causal set theory.

gr-qc↗

arXiv2025

Conserved Quantities in Linear and Nonlinear Quantum Search

In this tutorial, which contains some original results, we bridge the fields of quantum computing algorithms, conservation laws, and many-body quantum systems by examining three algorithms for searching an unordered database of size $N$ using a continuous-time quantum walk, which is the quantum analogue of a continuous-time random walk. The first algorithm uses a linear quantum walk, and we apply elementary calculus to show that the success probability of the algorithm reaches 1 when the jumping rate of the walk takes some critical value. We show that the expected value of its Hamiltonian $H_0$ is conserved. The second algorithm uses a nonlinear quantum walk with effective Hamiltonian $H(t) = H_0 + λ|ψ|^2$, which arises in the Gross-Pitaevskii equation describing Bose-Einstein condensates. When the interactions between the bosons are repulsive, $λ> 0$, and there exists a range of fixed jumping rates such that the success probability reaches 1 with the same asymptotic runtime of the linear algorithm, but with a larger multiplicative constant. Rather than the effective Hamiltonian, we show that the expected value of $H_0 + \frac{1}{2} λ|ψ|^2$ is conserved. The third algorithm utilizes attractive interactions, corresponding to $λ< 0$. In this case there is a time-varying critical function for the jumping rate $γ_c(t)$ that causes the success probability to reach 1 more quickly than in the other two algorithms, and we show that the expected value of $H(t)/[γ_c(t) N]$ is conserved.

quant-ph↗

arXiv2024

Constant-Time Quantum Search with a Many-Body Quantum System

The optimal runtime of a quantum computer searching a database is typically cited as the square root of the number of items in the database, which is famously achieved by Grover's algorithm. With parallel oracles, however, it is possible to search faster than this. We consider a many-body quantum system that naturally effects parallel queries, and we show that its parameters can be tuned to search a database in constant time, assuming a sufficient number of interacting particles. In particular, we consider Bose-Einstein condensates with pairwise and three-body interactions in the mean-field limit, which effectively evolve by a nonlinear Schrödinger equation with cubic and quintic nonlinearities. We solve the unstructured search problem formulated as a continuous-time quantum walk searching the complete graph in constant time. Depending on the number of marked vertices, however, the success probability can peak sharply, necessitating high precision time measurement to observe the system at this peak. Overcoming this, we prove that the relative coefficients of the cubic and quintic terms can be tuned to eliminate the need for high time-measurement precision by widening the peak in success probability or having it plateau. Finally, we derive a lower bound on the number of atoms needed for the many-body system to evolve by the effective nonlinearity.

quant-ph↗

arXiv2019

Mean Field Approximation for Identical Bosons on the Complete Graph

Non-linear dynamics in the quantum random walk setting have been shown to enable conditional speedup of Grover's algorithm. We examine the mean field approximation required for the use of the Gross-Pitaevskii equation on identical bosons evolving on the complete graph. We show that the states of such systems are parameterized by the basis of Young diagrams and determine their one- and two-party marginals. We find that isolated particles are required for good agreement with the mean field approximation, proving that without isolated particles the matrix fidelity agreement is bounded from above by $1/2$.

quant-ph↗

arXiv2019

Pairwise Concurrence in Cyclically Symmetric Quantum States

We provide an initial characterization of pairwise concurrence in quantum states which are invariant under cyclic permutations of party labeling. We prove that maximal entanglement can be entirely described by adjacent pairs, then give explicit descriptions of those states in specific subsets of 4 and 5 qubit states - X states. We also construct a monogamy bound on shared concurrences in the same subsets in 4 and 5 qubits, finding that above non-maximal entanglement thresholds, no other entanglements are possible.

quant-ph↗

arXiv2017

Symmetric 3 Qubit State Invariants

For pure symmetric 3-qubit states there are only three algebraically independent entanglement measures; one choice is the pairwise concurrence $\mathcal C$, the 3-tangle $τ$, and the Kempe invariant $κ$. Using a canonical form for symmetric $N$-qubit states derived from their Majorana representation, we derive the explicit achievable region of triples $(\mathcal C,τ,κ)$.

quant-ph↗

arXiv2017

Entanglement Constraints on States Locally Connected to the Greenberger-Horne-Zeilinger State

The multi-qubit GHZ state possesses tangles with elegant transformation properties under stochastic local operations and classical communication. Since almost all pure 3-qubit states are connected to the GHZ state via SLOCC, we derive a necessary and sufficient achievability inequality on arbitrary 3-qubit tangles, which is a strictly stronger constraint than both the monogamy inequality and the marginal eigenvalue inequality. We then show that entanglement shared with any single party in the n-qubit GHZ SLOCC equivalence class is precisely accounted for by the sum of its k-tangles, recently coined the strong monogamy equality, acknowledging competing but agreeing definitions of the k-tangle on this class, one of which is then computable for arbitrary mixed states. Strong monogamy is known to not hold arbitrarily, and so we introduce a unifying outlook on entanglement constraints in light of basic real algebraic geometry.

quant-ph↗

arXiv2017

Polynomial Monogamy Relations for Entanglement Negativity

The notion of non-classical correlations is a powerful contrivance for explaining phenomena exhibited in quantum systems. It is well known, however, that quantum systems are not free to explore arbitrary correlations---the church of the smaller Hilbert space only accepts monogamous congregants. We demonstrate how to characterize the limits of what is quantum mechanically possible with a computable measure, entanglement negativity. We show that negativity only saturates the standard linear monogamy inequality in trivial cases implied by its monotonicity under LOCC, and derive a necessary and sufficient inequality which, for the first time, is a non-linear higher degree polynomial. For very large quantum systems, we prove that the negativity can be distributed at least linearly for the tightest constraint and conjecture that it is at most linear.

quant-ph↗

arXiv2016

Irreconcilable Difference Between Quantum Walks and Adiabatic Quantum Computing

Continuous-time quantum walks and adiabatic quantum evolution are two general techniques for quantum computing, both of which are described by Hamiltonians that govern their evolutions by Schrödinger's equation. In the former, the Hamiltonian is fixed, while in the latter, the Hamiltonian varies with time. As a result, their formulations of Grover's algorithm evolve differently through Hilbert space. We show that this difference is fundamental; they cannot be made to evolve along each other's path without introducing structure more powerful than the standard oracle for unstructured search. For an adiabatic quantum evolution to evolve like the quantum walk search algorithm, it must interpolate between three fixed Hamiltonians, one of which is complex and introduces structure that is stronger than the oracle for unstructured search. Conversely, for a quantum walk to evolve along the path of the adiabatic search algorithm, it must be a chiral quantum walk on a weighted, directed star graph with structure that is also stronger than the oracle for unstructured search. Thus the two techniques, although similar in being described by Hamiltonians that govern their evolution, compute by fundamentally irreconcilable means.

quant-ph↗

arXiv2016

Quantum cellular automata without particles

Quantum cellular automata (QCA) constitute space and time homogeneous discrete models for quantum field theories (QFTs). Although QFTs are defined without reference to particles, computations are done in terms of Feynman diagrams, which are explicitly interpreted in terms of interacting particles. Similarly, the easiest QCA to construct are quantum lattice gas automata (QLGA). A natural question then is, which QCA are not QLGA? Here we construct a non-trivial example of such a QCA; it provides a simple model in $1+1$ dimensions with no particle interpretation at the scale where the QCA dynamics are homogeneous.

quant-ph↗

arXiv2015

Estimating an Activity Driven Hidden Markov Model

We define a Hidden Markov Model (HMM) in which each hidden state has time-dependent $\textit{activity levels}$ that drive transitions and emissions, and show how to estimate its parameters. Our construction is motivated by the problem of inferring human mobility on sub-daily time scales from, for example, mobile phone records.

stat.ML↗

arXiv2015

Expansive maps are isometries

We show that expansive maps from a dense subset of a compact metric space into the metric space itself are isometries

math.MG↗

arXiv2015

A quantum algorithm for Viterbi decoding of classical convolutional codes

We present a quantum Viterbi algorithm (QVA) with better than classical performance under certain conditions. In this paper the proposed algorithm is applied to decoding classical convolutional codes, for instance; large constraint length $Q$ and short decode frames $N$. Other applications of the classical Viterbi algorithm where $Q$ is large (e.g. speech processing) could experience significant speedup with the QVA. The QVA exploits the fact that the decoding trellis is similar to the butterfly diagram of the fast Fourier transform, with its corresponding fast quantum algorithm. The tensor-product structure of the butterfly diagram corresponds to a quantum superposition that we show can be efficiently prepared. The quantum speedup is possible because the performance of the QVA depends on the fanout (number of possible transitions from any given state in the hidden Markov model) which is in general much less than $Q$. The QVA constructs a superposition of states which correspond to all legal paths through the decoding lattice, with phase a function of the probability of the path being taken given received data. A specialized amplitude amplification procedure is applied one or more times to recover a superposition where the most probable path has a high probability of being measured.

quant-ph↗

arXiv2015

Distinguishing symmetric quantum oracles and quantum group multiplication

Given a unitary representation of a finite group on a finite-dimensional Hilbert space, we show how to find a state whose translates under the group are distinguishable with the highest probability. We apply this to several quantum oracle problems, including the GROUP MULTIPLICATION problem, in which the product of an ordered $n$-tuple of group elements is to be determined by querying elements of the tuple. For any finite group $G$, we give an algorithm to find the product of two elements of $G$ with a single quantum query with probability $2/|G|$. This generalizes Deutsch's Algorithm from $Z_2$ to an arbitrary finite group. We further prove that this algorithm is optimal. We also introduce the HIDDEN CONJUGATING ELEMENT PROBLEM, in which the oracle acts by conjugating by an unknown element of the group. We show that for many groups, including dihedral and symmetric groups, the unknown element can be determined with probability $1$ using a single quantum query.

quant-ph↗