arXiv subjects

Samuel S. Mendelson

Publications and source records attributed to Samuel S. Mendelson.

6 recordsLinked to original sources

arXiv2026

High-Performance Exact Synthesis of Two-Qubit Quantum Circuits

Exact synthesis provides unconditional optimality and canonical structure, but is often limited to small, carefully scoped regimes. We present an exact synthesis framework for two-qubit circuits over the Clifford+$T$ gate set that optimizes $T$-count exactly. Our approach exhausts a bounded search space, exploits algebraic canonicalization to avoid redundancy, and constructs a lookup table of optimal implementations that turns synthesis into a query. Algorithmically, we combine meet-in-the-middle ideas with provable pruning rules and problem-specific arithmetic designed for modern hardware. The result is an exact, reusable synthesis engine with substantially improved practical performance.

quant-ph↗

arXiv2025

Navigating Quantum Missteps in Agent-Based Modeling: A Schelling Model Case Study

Quantum computing promises transformative advances, but remains constrained by recurring misconceptions and methodological pitfalls. This paper demonstrates a fundamental incompatibility between traditional agent-based modeling (ABM) implementations and quantum optimization frameworks like Quadratic Unconstrained Binary Optimization (QUBO). Using Schelling's segregation model as a case study, we show that the standard practice of directly translating ABM state observations into QUBO formulations not only fails to deliver quantum advantage, but actively undermines computational efficiency. The fundamental issue is architectural. Traditional ABM implementations entail observing the state of the system at each iteration, systematically destroying the quantum superposition required for computational advantage. Through analysis of Schelling's segregation dynamics on lollipop networks, we demonstrate how abandoning the QUBO reduction paradigm and instead reconceptualizing the research question, from "simulate agent dynamics iteratively until convergence" to "compute minimum of agent moves required for global satisfaction", enables a faster classical solution. This structural reconceptualization yields an algorithm that exploits network symmetries obscured in traditional ABM simulations and QUBO formulations. It establishes a new lower bound which quantum approaches must outperform to achieve advantage. Our work emphasizes that progress in quantum agent-based modeling does not require forcing classical ABM implementations into quantum frameworks. Instead, it should focus on clarifying when quantum advantage is structurally possible, developing best-in-class classical baselines through problem analysis, and fundamentally reformulating research questions rather than preserving classical iterative state change observation paradigms.

quant-ph↗

arXiv2024

Exact Synthesis of Multiqubit Clifford-Cyclotomic Circuits

Let $n\geq 8$ be divisible by 4. The Clifford-cyclotomic gate set $\mathcal{G}_n$ is the universal gate set obtained by extending the Clifford gates with the $z$-rotation $T_n = \mathrm{diag}(1,ζ_n)$, where $ζ_n$ is a primitive $n$-th root of unity. In this note, we show that, when $n$ is a power of 2, a multiqubit unitary matrix $U$ can be exactly represented by a circuit over $\mathcal{G}_n$ if and only if the entries of $U$ belong to the ring $\mathbb{Z}[1/2,ζ_n]$. We moreover show that $\log(n)-2$ ancillas are always sufficient to construct a circuit for $U$. Our results generalize prior work to an infinite family of gate sets and show that the limitations that apply to single-qubit unitaries, for which the correspondence between Clifford-cyclotomic operators and matrices over $\mathbb{Z}[1/2,ζ_n]$ fails for all but finitely many values of $n$, can be overcome through the use of ancillas.

quant-ph↗

arXiv2023

Spectral properties of random graphs with fixed equitable partition

We define a graph to be $S$-regular if it contains an equitable partition given by a matrix $S$. These graphs are generalizations of both regular and bipartite, biregular graphs. An $S$-regular matrix is defined then as a matrix on an $S$-regular graph consistent with the graph's equitable partition. In this paper we derive the limiting spectral density for large, random $S$-regular matrices as well as limiting functions of certain statistics for their eigenvector coordinates as a function of eigenvalue. These limiting functions are defined in terms of spectral measures on $S$-regular trees. In general, these spectral measures do not have a closed-form expression; however, we provide a defining system of polynomials for them. Finally, we explore eigenvalue bounds of $S$-regular graph, proving an expander mixing lemma, Alon-Bopana bound, and other eigenvalue inequalities in terms of the eigenvalues of the matrix $S$.

math.CO↗

arXiv2019

On a special presentation of matrix algebras

Recognizing when a ring is a complete matrix ring is of significant importance in algebra. It is well-known folklore that a ring $R$ is a complete $n\times n$ matrix ring, so $R\cong M_{n}(S)$ for some ring $S$, if and only if it contains a set of $n\times n$ matrix units $\{e_{ij}\}_{i,j=1}^n$. A more recent and less known result states that a ring $R$ is a complete $(m+n)\times(m+n)$ matrix ring if and only if, $R$ contains three elements, $a$, $b$, and $f$, satisfying the two relations $af^m+f^nb=1$ and $f^{m+n}=0$. In many instances the two elements $a$ and $b$ can be replaced by appropriate powers $a^i$ and $a^j$ of a single element $a$ respectively. In general very little is known about the structure of the ring $S$. In this article we study in depth the case $m=n=1$ when $R\cong M_2(S)$. More specifically we study the universal algebra over a commutative ring $A$ with elements $x$ and $y$ that satisfy the relations $x^iy+yx^j=1$ and $y^2=0$. We describe completely the structure of these $A$-algebras and their underlying rings when $\gcd(i,j)=1$. Finally we obtain results that fully determine when there are surjections onto $M_2({\mathbb F})$ when ${\mathbb F}$ is a base field ${\mathbb Q}$ or ${\mathbb Z}_p$ for a prime number $p$.

math.RA↗

arXiv2019

Quantum-Assisted Clustering Algorithms for NISQ-Era Devices

In the NISQ-era of quantum computing, we should not expect to see quantum devices that provide an exponential improvement in runtime for practical problems, due to the lack of error correction and small number of qubits available. Nevertheless, these devices should be able to provide other performance improvements, particularly when combined with existing classical machines. In this article, we develop several hybrid quantum-classical clustering algorithms that can be employed as subroutines on small, NISQ-era devices. These new hybrid algorithms require a number of qubits that is at most logarithmic in the size of the data, provide performance improvement and/or runtime improvement over their classical counterparts, and do not require a black-box oracle. Consequently, we are able to provide a promising near-term application of NISQ-era devices.

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