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Pawel Wocjan

Publications and source records attributed to Pawel Wocjan.

At least 19 recordsLinked to original sources
arXiv2026

Quantum Algorithm for Identifying Hidden Graphs: Spectral Theory and Numerical Evidence

We give a quantum algorithm for a novel type of black-box problem: identifying a hidden $d$-regular base graph $G$ on $n$ vertices from oracle access to an obfuscated version of it, rather than traversing it. From $G$ we build the spired graph $G_{\rm spire}$ in three steps: each vertex is lifted into an exponentially large cluster, with adjacent clusters joined by a random bipartite graph; each cluster is then crowned with a balanced spire; finally, all vertices are randomly relabelled. Specializing to $G=K_2$ recovers the welded-trees graph. Our algorithm is conceptually simple: a continuous-time quantum walk on $G_{\rm spire}$, followed by a single Hadamard test at a classically precomputed time $t^*$; the algorithm returns the candidate whose predicted amplitude is closest to the measurement. The design rests on a rigorous spectral theory: from the apex of any spire, the walk is confined to a polynomial-dimensional invariant subspace evolving under the adjacency matrix of a simpler towered graph $G_{\rm tower}$; that matrix block-diagonalizes into $n$ independent tridiagonal systems of size $n$, each solved in closed form by a Chebyshev secular equation. Efficient numerics enabled by this decomposition supply $t^*$ and the predicted amplitudes. On the prism graphs $Y_m$ versus the Möbius ladders $M_m$ (each on $n=2m$ vertices), the numerical study supports a precise conjecture that $\widetilde O(n^2/\log n)$ measurements at evolution time of order $m^2$ suffice to distinguish the two families; we have tested $4 \le m \le 5121$ ($n$ up to $10242$). By analogy with the welded-trees lower bounds, we further conjecture that any classical algorithm requires queries exponential in $n$. Together these conjectures point to an exponential quantum speedup for the identification of an obfuscated base graph.

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arXiv2026

A Factorization Identity for Twisted Multinomial Coefficients with Application to Pilot States in Hamiltonian Decoded Quantum Interferometry

The $q$-multinomial coefficient, a classical object in enumerative combinatorics, counts permutations of multisets weighted by the number of inversions, with a single deformation parameter $q$. We introduce the twisted multinomial coefficient, in which each inversion between letters $i$ and $j$ carries a pair-dependent weight $ω_{ij}$ determined by a skew-symmetric matrix $Ω$. In general, no closed-form evaluation is known. Our main result is that under a natural structural condition on $Ω$ - predecessor-uniformity ($ω_{ij} = q_j$ for all $i<j$) - the twisted multinomial factorizes as a product of Gaussian ($q$-deformed) binomials with site-dependent parameters: $\binom{k}{k_1,\ldots,k_m}_Ω= \prod_j\binom{\ell_j}{k_j}_{q_j}$ where $\ell_j = k_1+\cdots+k_j$. This extends the standard product formula for the $q$-multinomial from a single parameter $q$ to $m-1$ independent parameters. The identity is purely combinatorial: it holds for arbitrary $q_j \in \mathbb{C}\setminus\{0\}$ without any algebraic constraints. We were led to this identity by studying pilot state preparation in Hamiltonian Decoded Quantum Interferometry (HDQI), a recently proposed quantum algorithm for preparing Gibbs and ground states. As an application, we show that the factorization yields an exact matrix product state (MPS) of bond dimension $k+1$ for the expansion coefficients of $h^k$ in a twisted algebra. We further show that the same site matrices deliver an exact MPS of bond dimension $\mathrm{deg}(\mathcal{P})+1$ for the expansion coefficients of $\mathcal{P}(h)$, for any polynomial $\mathcal{P}$, via a polynomial-dependent right boundary vector.

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arXiv2025

Quantized Markov Chain Couplings that Prepare Qsamples

We present a novel approach to quantizing Markov chains. The approach is based on the Markov chain coupling method, which is frequently used to prove fast mixing. Given a particular coupling, e.g., a grand coupling, we construct a completely positive and trace preserving map. This quantum map has a unique fixed point, which corresponds to the quantum sample (qsample) of the classical Markov chain's stationary distribution. We show that the convergence time of the quantum map is directly related to the coupling time of the Markov chain coupling.

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arXiv2025

Controlization Schemes Based on Orthogonal Arrays

Realizing controlled operations is fundamental to the design and execution of quantum algorithms. In quantum simulation and learning of quantum many-body systems, an important subroutine consists of implementing a controlled Hamiltonian time-evolution. Given only black-box access to the uncontrolled evolution $e^{-iHt}$, controlizing it, i.e., implementing $\mathrm{ctrl}(e^{-iHt}) = |0\rangle\langle 0|\otimes I + |1\rangle\langle 1 |\otimes e^{-iHt}$ is non-trivial. Controlization has been recently used in quantum algorithms for transforming unknown Hamiltonian dynamics [OKTM24] leveraging a scheme introduced in Refs. [NSM15, DNSM21]. The main idea behind the scheme is to intersperse the uncontrolled evolution with suitable operations such that the overall dynamics approximates the desired controlled evolution. Although efficient, this scheme uses operations randomly sampled from an exponentially large set. In the present work, we show that more efficient controlization schemes can be constructed with the help of orthogonal arrays for unknown 2-local Hamiltonians. We conduct a detailed analysis of their performance and demonstrate the resulting improvements through numerical experiments. This construction can also be generalized to $k$-local Hamiltonians. Moreover, our controlization schemes based on orthogonal arrays can take advantage of the interaction graph's structure and be made more efficient.

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arXiv2024

Techniques for learning sparse Pauli-Lindblad noise models

Error-mitigation techniques such as probabilistic error cancellation and zero-noise extrapolation benefit from accurate noise models. The sparse Pauli-Lindblad noise model is one of the most successful models for those applications. In existing implementations, the model decomposes into a series of simple Pauli channels with one- and two-local terms that follow the qubit topology. While the model has been shown to accurately capture the noise in contemporary superconducting quantum processors for error mitigation, it is important to consider higher-weight terms and effects beyond nearest-neighbor interactions. For such extended models to remain practical, however, we need to ensure that they can be learned efficiently. In this work we present new techniques that accomplish exactly this. We introduce twirling based on Pauli rotations, which enables us to automatically generate single-qubit learning correction sequences and reduce the number of unique fidelities that need to be learned. In addition, we propose a basis-selection strategy that leverages graph coloring and uniform covering arrays to minimize the number of learning bases. Taken together, these techniques ensure that the learning of the extended noise models remains efficient, despite their increased complexity.

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arXiv2023

Two conjectured strengthenings of Turán's theorem

We investigate two conjectured spectral graph theoretic strengthenings of Turán's theorem. Let $μ_1 \ge \ldots \ge μ_n$ denote the eigenvalues of a graph $G$ with $n$ vertices, $m$ edges and clique number $ω(G)$. The concise version of Turán's theorem is that $n/(n - d)$ is a lower bound for the clique number $ω(G)$, where $d$ is the average degree. Our first conjecture is that $d$ can be replaced in this bound with $\sqrt{s^+}$, where $s^+$ is the sum of the squares of the positive eigenvalues. We prove this conjecture for triangle-free, weakly perfect and Kneser graphs and for almost all graphs. We have also used various software tools to search for a counter-example. Nikiforov proved a spectral version of Turán's theorem that \[ μ_1^2 \le \frac{2m(ω(G) - 1)}{ω(G)}, \] and Bollobás and Nikiforov conjectured that for $G \not = K_n$ \[ μ_1^2 + μ_2^2 \le \frac{2m(ω(G) - 1)}{ω(G)}. \] For our second conjecture, we propose that for all graphs $(μ_1^2 + μ_2^2)$ in this inequality can be replaced by the sum of the squares of the $ω(G)$ largest eigenvalues, provided they are positive. We prove the conjecture for weakly perfect, Kneser, and classes of strongly regular graphs. We also provide experimental evidence and describe how the bound can be applied. Liu and Ning published a wide-ranging paper entitled ``Unsolved Problems in spectral graph theory'', and these two conjectures were placed second and fourth in their list of such problems.

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arXiv2023

Thermal State Preparation via Rounding Promises

A promising avenue for the preparation of Gibbs states on a quantum computer is to simulate the physical thermalization process. The Davies generator describes the dynamics of an open quantum system that is in contact with a heat bath. Crucially, it does not require simulation of the heat bath itself, only the system we hope to thermalize. Using the state-of-the-art techniques for quantum simulation of the Lindblad equation, we devise a technique for the preparation of Gibbs states via thermalization as specified by the Davies generator. In doing so, we encounter a severe technical challenge: implementation of the Davies generator demands the ability to estimate the energy of the system unambiguously. That is, each energy of the system must be deterministically mapped to a unique estimate. Previous work showed that this is only possible if the system satisfies an unphysical 'rounding promise' assumption. We solve this problem by engineering a random ensemble of rounding promises that simultaneously solves three problems: First, each rounding promise admits preparation of a 'promised' thermal state via a Davies generator. Second, these Davies generators have a similar mixing time as the ideal Davies generator. Third, the average of these promised thermal states approximates the ideal thermal state.

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arXiv2023

On the complexity of quantum partition functions

The partition function and free energy of a quantum many-body system determine its physical properties in thermal equilibrium. Here we study the computational complexity of approximating these quantities for $n$-qubit local Hamiltonians. First, we report a classical algorithm with $\mathrm{poly}(n)$ runtime which approximates the free energy of a given $2$-local Hamiltonian provided that it satisfies a certain denseness condition. Our algorithm combines the variational characterization of the free energy and convex relaxation methods. It contributes to a body of work on efficient approximation algorithms for dense instances of optimization problems which are hard in the general case, and can be viewed as simultaneously extending existing algorithms for (a) the ground energy of dense $2$-local Hamiltonians, and (b) the free energy of dense classical Ising models. Secondly, we establish polynomial-time equivalence between the problem of approximating the free energy of local Hamiltonians and three other natural quantum approximate counting problems, including the problem of approximating the number of witness states accepted by a QMA verifier. These results suggest that simulation of quantum many-body systems in thermal equilibrium may precisely capture the complexity of a broad family of computational problems that has yet to be defined or characterized in terms of known complexity classes. Finally, we summarize state-of-the-art classical and quantum algorithms for approximating the free energy and show how to improve their runtime and memory footprint.

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arXiv2022

Simpler (classical) and faster (quantum) algorithms for Gibbs partition functions

We present classical and quantum algorithms for approximating partition functions of classical Hamiltonians at a given temperature. Our work has two main contributions: first, we modify the classical algorithm of Štefankovič, Vempala and Vigoda (\emph{J.~ACM}, 56(3), 2009) to improve its sample complexity; second, we quantize this new algorithm, improving upon the previously fastest quantum algorithm for this problem, due to Harrow and Wei (SODA 2020). The conventional approach to estimating partition functions requires approximating the means of Gibbs distributions at a set of inverse temperatures that form the so-called cooling schedule. The length of the cooling schedule directly affects the complexity of the algorithm. Combining our improved version of the algorithm of Štefankovič, Vempala and Vigoda with the paired-product estimator of Huber (\emph{Ann.\ Appl.\ Probab.}, 25(2),~2015), our new quantum algorithm uses a shorter cooling schedule than previously known. This length matches the optimal length conjectured by Štefankovič, Vempala and Vigoda. The quantum algorithm also achieves a quadratic advantage in the number of required quantum samples compared to the number of random samples drawn by the best classical algorithm, and its computational complexity has quadratically better dependence on the spectral gap of the Markov chains used to produce the quantum samples.

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arXiv2022

Space-efficient Quantization Method for Reversible Markov Chains

In a seminal paper, Szegedy showed how to construct a quantum walk $W(P)$ for any reversible Markov chain $P$ such that its eigenvector with eigenphase $0$ is a quantum sample of the limiting distribution of the random walk and its eigenphase gap is quadratically larger than the spectral gap of $P$. The standard construction of Szegedy's quantum walk requires an ancilla register of Hilbert-space dimension equal to the size of the state space of the Markov chain. We show that it is possible to avoid this doubling of state space for certain Markov chains that employ a symmetric proposal probability and a subsequent accept/reject probability to sample from the Gibbs distribution. For such Markov chains, we give a quantization method which requires an ancilla register of dimension equal to only the number of different energy values, which is often significantly smaller than the size of the state space. To accomplish this, we develop a technique for block encoding Hadamard products of matrices which may be of wider interest.

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arXiv2022

Quantum-enhanced Markov chain Monte Carlo

Sampling from complicated probability distributions is a hard computational problem arising in many fields, including statistical physics, optimization, and machine learning. Quantum computers have recently been used to sample from complicated distributions that are hard to sample from classically, but which seldom arise in applications. Here we introduce a quantum algorithm to sample from distributions that pose a bottleneck in several applications, which we implement on a superconducting quantum processor. The algorithm performs Markov chain Monte Carlo (MCMC), a popular iterative sampling technique, to sample from the Boltzmann distribution of classical Ising models. In each step, the quantum processor explores the model in superposition to propose a random move, which is then accepted or rejected by a classical computer and returned to the quantum processor, ensuring convergence to the desired Boltzmann distribution. We find that this quantum algorithm converges in fewer iterations than common classical MCMC alternatives on relevant problem instances, both in simulations and experiments. It therefore opens a new path for quantum computers to solve useful--not merely difficult--problems in the near term.

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arXiv2021

Spectral upper bound on the quantum k-independence number of a graph

A well known upper bound for the independence number $α(G)$ of a graph $G$, due to Cvetković, is that \begin{equation*} α(G) \le n^0 + \min\{n^+ , n^-\} \end{equation*} where $(n^+, n^0, n^-)$ is the inertia of $G$. We prove that this bound is also an upper bound for the quantum independence number $α_q$(G), where $α_q(G) \ge α(G)$ and for some graphs $α_q(G) \gg α(G)$. We identify numerous graphs for which $α(G) = α_q(G)$, thus increasing the number of graphs for which $α_q$ is known. We also demonstrate that there are graphs for which the above bound is not exact with any Hermitian weight matrix, for $α(G)$ and $α_q(G)$. Finally, we show this result in the more general context of spectral bounds for the quantum $k$-independence number, where the $k$-independence number is the maximum size of a set of vertices at pairwise distance greater than $k$.

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arXiv2021

Szegedy Walk Unitaries for Quantum Maps

Szegedy developed a generic method for quantizing classical algorithms based on random walks [Proceedings of FOCS, 2004, pp. 32-41]. A major contribution of his work was the construction of a walk unitary for any reversible random walk. Such unitary posses two crucial properties: its eigenvector with eigenphase $0$ is a quantum sample of the limiting distribution of the random walk and its eigenphase gap is quadratically larger than the spectral gap of the random walk. It was an open question if it is possible to generalize Szegedy's quantization method for stochastic maps to quantum maps. We answer this in the affirmative by presenting an explicit construction of a Szegedy walk unitary for detailed balanced Lindbladians -- generators of quantum Markov semigroups -- and detailed balanced quantum channels. We prove that our Szegedy walk unitary has a purification of the fixed point of the Lindbladian as eigenvector with eigenphase $0$ and that its eigenphase gap is quadratically larger than the spectral gap of the Lindbladian. To construct the walk unitary we leverage a canonical form for detailed balanced Lindbladians showing that they are structurally related to Davies generators. We also explain how the quantization method for Lindbladians can be applied to quantum channels. We give an efficient quantum algorithm for quantizing Davies generators that describe many important open-system dynamics, for instance, the relaxation of a quantum system coupled to a bath. Our algorithm extends known techniques for simulating quantum systems on a quantum computer.

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arXiv2020

Spectral Lower Bounds for the Quantum Chromatic Number of a Graph -- Part II

Hoffman proved that a graph $G$ with eigenvalues $μ_1 \ge \ldots \ge μ_n$ and chromatic number $χ(G)$ satisfies: \[ χ\ge 1 + κ\] where $κ$ is the smallest integer such that \[ μ_1 + \sum_{i=1}^κ μ_{n+1-i} \le 0. \] We strengthen this well known result by proving that $χ(G)$ can be replaced by the quantum chromatic number, $χ_q(G)$, where for all graphs $χ_q(G) \le χ(G)$ and for some graphs $χ_q(G)$ is significantly smaller than $χ(G)$. We also prove a similar result, and investigate implications of these inequalities for the quantum chromatic number of various classes of graphs, which improves many known results. For example, we demonstrate that the Kneser graph $KG_{p,2}$ has $χ_q = χ= p - 2$.

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arXiv2020

More Tales of Hoffman: bounds for the vector chromatic number of a graph

Let $χ(G)$ denote the chromatic number of a graph and $χ_v(G)$ denote the vector chromatic number. For all graphs $χ_v(G) \le χ(G)$ and for some graphs $χ_v(G) \ll χ(G)$. Galtman proved that Hoffman's well-known lower bound for $χ(G)$ is in fact a lower bound for $χ_v(G)$. We prove that two more spectral lower bounds for $χ(G)$ are also lower bounds for $χ_v(G)$. We then use one of these bounds to derive a new characterization of $χ_v(G)$.

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arXiv2019

Spectral lower bounds for the orthogonal and projective ranks of a graph

The orthogonal rank of a graph $G=(V,E)$ is the smallest dimension $ξ$ such that there exist non-zero column vectors $x_v\in\mathbb{C}^ξ$ for $v\in V$ satisfying the orthogonality condition $x_v^\dagger x_w=0$ for all $vw\in E$. We prove that many spectral lower bounds for the chromatic number, $χ$, are also lower bounds for $ξ$. This result complements a previous result by the authors, in which they showed that spectral lower bounds for $χ$ are also lower bounds for the quantum chromatic number $χ_q$. It is known that the quantum chromatic number and the orthogonal rank are incomparable. We conclude by proving an inertial lower bound for the projective rank $ξ_f$, and conjecture that a stronger inertial lower bound for $ξ$ is also a lower bound for $ξ_f$.

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arXiv2019

Conjectured bound for the distribution of eigenvalues of a graph

Let $(n^+, n^0, n^-)$ denote the inertia of a graph $G$ with $n$ vertices. Nordhaus-Gaddum bounds are known for inertia, except for an upper bound for $n^-$. We conjecture that for any graph \[ n^-(G) + n^-(\bar{G}) \le 1.5(n - 1), \] and prove this bound for various classes of graphs and for almost all graphs. We consider the relationship between this bound and the number of eigenvalues that lie within the interval $-1$ to $0$, which we denote $n_{(-1,0)}(G)$. We conjecture that for any graph \[ n_{(-1,0)}(G) \le 0.5(n - 1). \] and prove this bound for almost all graphs. We also investigate extremal graphs for both bounds and show that both bounds are equivalent for regular graphs.

math.CO↗
arXiv2018

An inertial upper bound for the quantum independence number of a graph

A well known upper bound for the independence number $α(G)$ of a graph $G$, is that \[ α(G) \le n^0 + \min\{n^+ , n^-\}, \] where $(n^+, n^0, n^-)$ is the inertia of $G$. We prove that this bound is also an upper bound for the quantum independence number $α_q$(G), where $α_q(G) \ge α(G)$. We identify numerous graphs for which $α(G) = α_q(G)$ and demonstrate that there are graphs for which the above bound is not exact with any Hermitian weight matrix, for $α(G)$ and $α_q(G)$. This result complements results by the authors that many spectral lower bounds for the chromatic number are also lower bounds for the quantum chromatic number.

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