arXiv subjects

Parker Kuklinski

Publications and source records attributed to Parker Kuklinski.

13 recordsLinked to original sources

arXiv2026

Harmonic sequence state-preparation

We demonstrate an efficient circuit to prepare a quantum state with amplitudes proportional to a harmonic sequence. We do this by first preparing a large quantum state with linearly related amplitudes and then applying a quantum Fourier transform; this has a direct analogy to the fact that the Fourier coefficients of a sawtooth wave follow a harmonic sequence. We then consider an extension of this problem by block-encoding a matrix with a harmonic sequence along its diagonal. The cost of both circuits is dominated by the costs associated with the quantum Fourier transform.

quant-ph↗

arXiv2025

Detailed assessment of calculating drag force with quantum computers: Explicit time-evolution precludes exponential advantage for nonlinear differential equations

This study examines the potential for fault-tolerant quantum computers to provide utility in fluid dynamics simulations, with a focus on drag force calculations for ship hull design. We assess whether quantum algorithms can surpass classical computational limits by generating detailed quantum resource estimates (QREs) in terms of logical qubits and $T$-gate counts. Our analysis is based on a quantum algorithm leveraging Carleman linearization of the lattice Boltzmann method (LBM), which has been suggested to offer exponential speedup. We develop efficient block encodings for LBM matrices and a method for amplitude-encoding drag force. We apply the method to the simple case of fluid flow past a sphere across a range of Reynolds numbers ($\mathrm{Re}$). We estimate the required (logical qubits)$\times$($T$-gates), finding them to be prohibitively large, ranging from $10^{21}$ to $10^{39}$. While classical simulations scale as $O(\mathrm{Re}^3)$, our QREs exhibit a modest polynomial scaling of $O(\mathrm{Re}^{2.68})$, indicating no exponential quantum advantage. We attribute this limitation to an intrinsic power-law relationship between spatial grid resolution and time-stepping requirements that is a fundamental characteristic of explicit methods for evolving nonlinear differential equations. Thus, quantum computers are unlikely to provide utility in applications that require time-evolving fluids and other systems of nonlinear differential equations.

quant-ph↗

arXiv2025

Efficient block-encodings require structure

Block-encodings are ubiquitous in quantum computing as a way to represent data within a unitary operator. While several unstructured methods are applicable to arbitrary data, these techniques are burdened by hidden costs and poor accuracy. In this paper, we demonstrate that, even for a small 6-qubit encoding, these structure-agnostic techniques require wildly intractable resources. We compare these resources with an encoding method which respects a mathematical representation of the block, leading to the conclusion that unstructured encoding mehtods should only be used in the most extenuating circumstances. This finding runs contrary to existing literature on quantum algorithms which often employ structure-agnostic methods.

quant-ph↗

arXiv2025

A simpler Gaussian state-preparation

The ability to efficiently state-prepare Gaussian distributions is critical to the success of numerous quantum algorithms. The most popular algorithm for this subroutine (Kitaev-Webb) has favorable polynomial resource scaling, however it faces enormous resource overheads making it functionally impractical. In this paper, we present a new, more intuitive method which uses exactly $n-1$ rotations, $(n-1)(n-2)/2$ two-qubit controlled rotations, and $\lfloor(n-1)/2\rfloor$ ancilla to state-prepare an $n$-qubit Gaussian state. We then apply optimizations to the circuit to render it linear in T-depth. This method can be extended to state-preparations of complex functions with polynomial phase.

quant-ph↗

arXiv2024

S-FABLE and LS-FABLE: Fast approximate block-encoding algorithms for unstructured sparse matrices

The Fast Approximate BLock-Encoding algorithm (FABLE) is a technique to block-encode arbitrary $N\times N$ dense matrices into quantum circuits using at most $O(N^2)$ one and two-qubit gates and $\mathcal{O}(N^2\log{N})$ classical operations. The method nontrivially transforms a matrix $A$ into a collection of angles to be implemented in a sequence of $y$-rotation gates within the block-encoding circuit. If an angle falls below a threshold value, its corresponding rotation gate may be eliminated without significantly impacting the accuracy of the encoding. Ideally many of these rotation gates may be eliminated at little cost to the accuracy of the block-encoding such that quantum resources are minimized. In this paper we describe two modifications of FABLE to efficiently encode sparse matrices; in the first method termed Sparse-FABLE (S-FABLE), for a generic unstructured sparse matrix $A$ we use FABLE to block encode the Hadamard-conjugated matrix $H^{\otimes n}AH^{\otimes n}$ (computed with $\mathcal{O}(N^2\log N)$ classical operations) and conjugate the resulting circuit with $n$ extra Hadamard gates on each side to reclaim a block-approximation to $A$. We demonstrate that the FABLE circuits corresponding to block-encoding $H^{\otimes n}AH^{\otimes n}$ significantly compress and that overall scaling is empirically favorable (i.e. using S-FABLE to block-encode a sparse matrix with $\mathcal{O}(N)$ nonzero entries requires approximately $\mathcal{O}(N)$ rotation gates and $\mathcal{O}(N\log N)$ CNOT gates). In the second method called `Lazy' Sparse-FABLE (LS-FABLE), we eliminate the quadratic classical overhead altogether by directly implementing scaled entries of the sparse matrix $A$ in the rotation gates of the S-FABLE oracle. This leads to a slightly less accurate block-encoding than S-FABLE, while still demonstrating favorable scaling to FABLE similar to that found in S-FABLE.

quant-ph↗

arXiv2023

Blind Number Sequencing

A popular challenge on the social media app TikTok is to place the output of a random number generator in ascending order without arriving at a contradiction. Most players rely on intuition to construct their sequences and debate has surfaced in the comments about the optimal strategy to win the game. To this end, we determine an optimal strategy that offers a 21 percent improvement in winning odds over the intuitive equal-spacing strategy in the popular hashtag 20numberchallenge variant.

math.PR↗

arXiv2023

An uncountable union of line segments with null two-dimensional measure

In this paper we construct an uncountable union of line segments $T$ which has full intersection with the sets $(\{ 0\} \times [0, 1]) \cup (\{ 1\} \times [0, 1])\subset\mathbb{R}^2$ but has null two-dimensional measure. Further results are proved on the decay rate of $μ(T)$ if the line segments comprising $T$ are replaced with increasingly fine approximations by parallelograms.

math.CA↗

arXiv2021

A Generalized Lerche-Newberger Formula

The Lerche-Newberger formula simplifies harmonic sums of Bessel functions and has seen application in plasma physics and frequency modulated quantum systems. In this paper, we rigorously prove the formula and extend the classical result to a family of multi-dimensional extensions of the single variable Bessel functions called generalized Bessel functions. Since prevailing definitions of these functions do not accommodate arbitrary complex order, we use an auxiliary family of functions called generalized Anger functions and show that the single-variable result holds in multiple dimensions for a certain selection of parameters. We conclude by applying these results to physical systems.

math.CA↗

arXiv2021

Identities and Properties of Multi-Dimensional Generalized Bessel Functions

The Generalized Bessel Function (GBF) extends the single variable Bessel function to several dimensions and indices in a nontrivial manner. Two-dimensional GBFs have been studied extensively in the literature and have found application in laser physics, crystallography, and electromagnetics. In this article, we document several properties of $m$-dimensional GBFs including an underlying partial differential equation structure, asymptotics for simultaneously large order and argument, and analysis of generalized Neumann, Kapteyn, and Schlömilch series. We extend these results to mixed-type GBFs where appropriate.

math.GM↗

arXiv2020

Bifurcation Curves of Two-Dimensional Quantum Walks

The quantum walk differs fundamentally from the classical random walk in a number of ways, including its linear spreading and initial condition dependent asymmetries. Using stationary phase approximations, precise asymptotics have been derived for one-dimensional two-state quantum walks, one-dimensional three-state Grover walks, and two-dimensional four-state Grover walks. Other papers have investigated asymptotic behavior of a much larger set of two-dimensional quantum walks and it has been shown that in special cases the regions of polynomial decay can be parameterized. In this paper, we show that these regions of polynomial decay are bounded by algebraic curves which can be explicitly computed. We give examples of these bifurcation curves for a number of two-dimensional quantum walks.

quant-ph↗

arXiv2020

Conditional Probability Distributions of Finite Absorbing Quantum Walks

Quantum walks are known to have nontrivial interactions with absorbing boundaries. In particular it has been shown that an absorbing boundary in the one dimensional quantum walk partially reflects information, as observed by absorption probability computations. In this paper, we shift our sights from the local phenomena of absorption probabilities to the global behavior of finite absorbing quantum walks in one dimension. We conduct our analysis by approximating the eigenbasis of the associated absorbing quantum walk operator matrix Q_n where n is the lattice size. The conditional probability distributions of these finite absorbing quantum walks exhibit distinct behavior at various timescales, namely wavelike reflections for t = O(n), fractional quantum revivals for t = O(n^2), and stability for t = O(n^3). At the end of this paper, we demonstrate the existence of these quantum revivals in other sufficiently regular quantum walk systems.

quant-ph↗

arXiv2019

Absorption Probabilities of Quantum Walks

Quantum walks are known to have nontrivial interaction with absorbing boundaries. In particular, Ambainis et.\ al.\ \cite{ambainis01} showed that in the $(\Z ,C_1,H)$ quantum walk (one-dimensional Hadamard walk) an absorbing boundary partially reflects information. These authors also conjectured that the left absorption probabilities $P_n^{(1)}(1,0)$ related to the finite absorbing Hadamard walks $(\Z ,C_1,H,\{ 0,n\} )$ satisfy a linear fractional recurrence in $n$ (here $P_n(1,0)$ is the probability that a Hadamard walk particle initialized in $|1\rangle |R\rangle$ is eventually absorbed at $|0\rangle$ and not at $|n\rangle$). This result, as well as a third order linear recurrence in initial position $m$ of $P_n^{(m)}(1,0)$, was later proved by Bach and Borisov \cite{bach09} using techniques from complex analysis. In this paper we extend these results to general two state quantum walks and three-state Grover walks, while providing a partial calculation for absorption in $d$-dimensional Grover walks by a $d-1$-dimensional wall. In the one-dimensional cases, we prove partial reflection of information, a linear fractional recurrence in lattice size, and a linear recurrence in initial position.

quant-ph↗

arXiv2016

Grover Walks on a Line with Absorbing Boundaries

In this paper, we study Grover walks on a line with one and two absorbing boundaries. In particular, we present some results for the absorbing probabilities both in a semi-finite and finite line. Analytical expressions for these absorbing probabilities are presented by using the combinatorial approach. These results are perfectly matched with numerical simulations. We show that the behavior of Grover walks on a line with absorbing boundaries is strikingly different from that of classical walks and that of Hadamard walks.

quant-ph↗