Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0702173

Krawtchouk matrices from classical and quantum random walks

Philip Feinsilver, Jerzy Kocik

Published Feb 16, 2007• 0 citations
arXivHTMLPDF

Abstract

Krawtchouk's polynomials occur classically as orthogonal polynomials with respect to the binomial distribution. They may be also expressed in the form of matrices, that emerge as arrays of the values that the polynomials take. The algebraic properties of these matrices provide a very interesting and accessible example in the approach to probability theory known as quantum probability. First it is noted how the Krawtchouk matrices are connected to the classical symmetric Bernoulli random walk. And we show how to derive Krawtchouk matrices in the quantum probability context via tensor powers of the elementary Hadamard matrix. Then connections with the classical situation are shown by calculating expectation values in the quantum case.

DOI
10.48550/arxiv.quant-ph/0702173

Related papers

Krawtchouk-Griffiths systems I: matrix approach

Philip Feinsilver

We call Krawtchouk-Griffiths systems, or KG-systems, systems of multivariate polynomials orthogonal with respect to corresponding multinomial distributions.The original Krawtchouk polynomials are orthogonal with respect to a binomial distribution.Our approach is to work directly with matrices comprising the values of the polynomials at points of a discrete grid based on the possible counting values of the underlying multinomial distribution.The starting point for the construction of a KG-system is a generating matrix satisfying the K-condition, orthogonality with respect to the basic probability distribution associated to an individual step of the multinomial process.The variables of the polynomials corresponding to matrices may be interpreted as quantum observables in the real case, or quantum variables in the complex case.The structure of the recurrence relations for the orthogonal polynomials is presented with multiplication operators as the matrices corresponding to the quantum variables.An interesting feature is that the associated random walks correspond to the Lie algebra of the representation of symmetric tensor powers of matrices.

arXiv:1611.06991•Communications on Stochastic Analysis•Sep 1, 2016•4 citations

Matrix valued Szego polynomials and quantum random walks

Maŕıa José Cantero, F. Alberto Grünbaum, Moral, L. +1 more

We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation. We show how the theory of CMV matrices gives a natural tool to study these processes and to give results that are analogous to those that Karlin and McGregor developed to study (classical) birth-and-death processes using orthogonal polynomials on the real line. In perfect analogy with the classical case the study of QRWs on the set of non-negative integers can be handled using scalar valued (Laurent) polynomials and a scalar valued measure on the circle. In the case of classical or quantum random walks on the integers one needs to allow for matrix valued versions of these notions. We show how our tools yield results in the well known case of the Hadamard walk, but we go beyond this translation invariant model to analyze examples that are hard to analyze using other methods. More precisely we consider QRWs on the set of non- negative integers. The analysis of these cases leads to phenomena that are absent in the case of QRWs on the integers even if one restricts oneself to a constant coin. This is illustrated here by studying recurrence properties of the walk, but the same method can be used for other purposes. The presentation here aims at being selfcontained, but we re- frain from trying to give an introduction to quantum random walks, a subject well surveyed in the literature we quote. For two excellent reviews, see [1, 18]. See also the recent notes [19].

arXiv:0901.2244•Jan 15, 2009•1 citation

A Difference Operator Approach to Quantum Random Walks: Parseval Identity, Krawtchouk Matrices, and Hermite Limits

Chien-Wen Hwang

We introduce a discrete difference operator D_k to study the one-dimensional quantum random walk (QRW) with the Hadamard coin. Explicit combinatorial expressions are obtained for the probability amplitudes a(n,k) and b(n,k), which encode the final step direction and carry alternating signs that reflect the merging of leftward steps. Removing these signs and the coin-state distinction recovers the classical binomial distribution. The symmetric and antisymmetric combinations $a\pm b$ are shown to coincide with diagonal and sub-diagonal entries of the Krawtchouk matrix. Using cross identities among Krawtchouk matrix elements, we prove by induction that the amplitudes satisfy a Parseval identity sum (a^2+b^2)=2^n-1, establishing probability conservation in the Krawtchouk formulation. The operator D_k acts as a discrete Hermite polynomial generator: the ratios h_m = (n choose k)^{-1} D_k^m (n choose k) admit explicit closed forms and converge to Hermite polynomials in the continuous limit. At the discrete level, D_k connects successive Krawtchouk matrices and acts as a coherence generator and a raising operator. The h_m quantify the position-dependent degree of quantum interference on both halves of the distribution, either individually (for x >= 0) or through an inverse-Pascal combination of several h_m (for x < 0), and the ballistic O(n^2) scaling of the variance emerges from the superposition of all excited states. Numerical illustrations for n=6 and n=10 corroborate the analytical results.

quant-pharXiv:2607.24837•Jul 27, 2026•0 citations

Classical and quantum walks on paths associated with exceptional Krawtchouk polynomials

Hiroshi Miki, Satoshi Tsujimoto, Luc Vinet

Classical and quantum walks on some finite paths are introduced. It is shown that these walks have explicit solutions given in terms of exceptional Krawtchouk polynomials, and their properties are explored. In particular, fractional revival is shown to take place in the corresponding quantum walks.

arXiv:2201.02337•Journal of Mathematical Physics•Oct 1, 2022•4 citations

Matrix‐valued Szegő polynomials and quantum random walks

Maŕıa José Cantero, L. Moral, F. Alberto Grünbaum +1 more

Abstract We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation. We show how the theory of CMV matrices gives a natural tool to study these processes and to give results that are analogous to those that Karlin and McGregor developed to study (classical) birth‐and‐death processes using orthogonal polynomials on the real line. In perfect analogy with the classical case, the study of QRWs on the set of nonnegative integers can be handled using scalar‐valued (Laurent) polynomials and a scalar‐valued measure on the circle. In the case of classical or quantum random walks on the integers, one needs to allow for matrix‐valued versions of these notions. We show how our tools yield results in the well‐known case of the Hadamard walk, but we go beyond this translation‐invariant model to analyze examples that are hard to analyze using other methods. More precisely, we consider QRWs on the set of nonnegative integers. The analysis of these cases leads to phenomena that are absent in the case of QRWs on the integers even if one restricts oneself to a constant coin. This is illustrated here by studying recurrence properties of the walk, but the same method can be used for other purposes. The presentation here aims at being self‐contained, but we refrain from trying to give an introduction to quantum random walks, a subject well surveyed in the literature we quote. For two excellent reviews, see [1, 9]. See also the recent notes [20]. © 2009 Wiley Periodicals, Inc.

Communications on Pure and Applied Mathematics•Dec 29, 2009•105 citations

Spectral analysis of quantum Markov chains on the line and their statistics.

Manuel D. de la Iglesia, Carlos F. Lardizabal, Newton Loebens

Inspired by the classical spectral analysis of birth-death chains using orthogonal polynomials, we study an analogous set of constructions in the context of open quantum dynamics and related walks. In such setting, block tridiagonal matrices and matrix-valued orthogonal polynomials are the natural objects to be considered. We recall the problems of the existence of a matrix of measures or weight matrix together with concrete calculations of basic statistics of the walk, such as site recurrence and first passage time probabilities, with these notions being defined in terms of a quantum trajectories formalism. The discussion concentrates on the models of quantum Markov chains, due to S. Gudder, and on the particular class of open quantum walks, due to S. Attal et al. The folding trick for birth-death chains on the integers is revisited in this setting together with applications of the matrix-valued Stieltjes transform associated with the measures, thus extending recent results on the subject. Finally, we consider the case of non-symmetric weight matrices and explore some examples.

•Jul 21, 2021•0 citations
Krawtchouk's polynomials occur classically as orthogonal polynomials with respect to the binomial distribution. They may be also expressed in the form of matrices, that emerge as arrays of the values that the polynomials take. The algebraic properties of these matrices provide a very interesting and accessible example in the approach to probability theory known as quantum probability. First it is noted how the Krawtchouk matrices are connected to the classical symmetric Bernoulli random walk. And we show how to derive Krawtchouk matrices in the quantum probability context via tensor powers of the elementary Hadamard matrix. Then connections with the classical situation are shown by calculating expectation values in the quantum case.
We call Krawtchouk-Griffiths systems, or KG-systems, systems of multivariate polynomials orthogonal with respect to corresponding multinomial distributions.The original Krawtchouk polynomials are orthogonal with respect to a binomial distribution.Our approach is to work directly with matrices comprising the values of the polynomials at points of a discrete grid based on the possible counting values of the underlying multinomial distribution.The starting point for the construction of a KG-system is a generating matrix satisfying the K-condition, orthogonality with respect to the basic probability distribution associated to an individual step of the multinomial process.The variables of the polynomials corresponding to matrices may be interpreted as quantum observables in the real case, or quantum variables in the complex case.The structure of the recurrence relations for the orthogonal polynomials is presented with multiplication operators as the matrices corresponding to the quantum variables.An interesting feature is that the associated random walks correspond to the Lie algebra of the representation of symmetric tensor powers of matrices.
We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation. We show how the theory of CMV matrices gives a natural tool to study these processes and to give results that are analogous to those that Karlin and McGregor developed to study (classical) birth-and-death processes using orthogonal polynomials on the real line. In perfect analogy with the classical case the study of QRWs on the set of non-negative integers can be handled using scalar valued (Laurent) polynomials and a scalar valued measure on the circle. In the case of classical or quantum random walks on the integers one needs to allow for matrix valued versions of these notions. We show how our tools yield results in the well known case of the Hadamard walk, but we go beyond this translation invariant model to analyze examples that are hard to analyze using other methods. More precisely we consider QRWs on the set of non- negative integers. The analysis of these cases leads to phenomena that are absent in the case of QRWs on the integers even if one restricts oneself to a constant coin. This is illustrated here by studying recurrence properties of the walk, but the same method can be used for other purposes. The presentation here aims at being selfcontained, but we re- frain from trying to give an introduction to quantum random walks, a subject well surveyed in the literature we quote. For two excellent reviews, see [1, 18]. See also the recent notes [19].
We introduce a discrete difference operator D_k to study the one-dimensional quantum random walk (QRW) with the Hadamard coin. Explicit combinatorial expressions are obtained for the probability amplitudes a(n,k) and b(n,k), which encode the final step direction and carry alternating signs that reflect the merging of leftward steps. Removing these signs and the coin-state distinction recovers the classical binomial distribution. The symmetric and antisymmetric combinations a±ba\pm ba±b are shown to coincide with diagonal and sub-diagonal entries of the Krawtchouk matrix. Using cross identities among Krawtchouk matrix elements, we prove by induction that the amplitudes satisfy a Parseval identity sum (a^2+b^2)=2^n-1, establishing probability conservation in the Krawtchouk formulation. The operator D_k acts as a discrete Hermite polynomial generator: the ratios h_m = (n choose k)^{-1} D_k^m (n choose k) admit explicit closed forms and converge to Hermite polynomials in the continuous limit. At the discrete level, D_k connects successive Krawtchouk matrices and acts as a coherence generator and a raising operator. The h_m quantify the position-dependent degree of quantum interference on both halves of the distribution, either individually (for x >= 0) or through an inverse-Pascal combination of several h_m (for x < 0), and the ballistic O(n^2) scaling of the variance emerges from the superposition of all excited states. Numerical illustrations for n=6 and n=10 corroborate the analytical results.
Abstract We consider quantum random walks (QRW) on the integers, a subject that has been considered in the last few years in the framework of quantum computation. We show how the theory of CMV matrices gives a natural tool to study these processes and to give results that are analogous to those that Karlin and McGregor developed to study (classical) birth‐and‐death processes using orthogonal polynomials on the real line. In perfect analogy with the classical case, the study of QRWs on the set of nonnegative integers can be handled using scalar‐valued (Laurent) polynomials and a scalar‐valued measure on the circle. In the case of classical or quantum random walks on the integers, one needs to allow for matrix‐valued versions of these notions. We show how our tools yield results in the well‐known case of the Hadamard walk, but we go beyond this translation‐invariant model to analyze examples that are hard to analyze using other methods. More precisely, we consider QRWs on the set of nonnegative integers. The analysis of these cases leads to phenomena that are absent in the case of QRWs on the integers even if one restricts oneself to a constant coin. This is illustrated here by studying recurrence properties of the walk, but the same method can be used for other purposes. The presentation here aims at being self‐contained, but we refrain from trying to give an introduction to quantum random walks, a subject well surveyed in the literature we quote. For two excellent reviews, see [1, 9]. See also the recent notes [20]. © 2009 Wiley Periodicals, Inc.
Inspired by the classical spectral analysis of birth-death chains using orthogonal polynomials, we study an analogous set of constructions in the context of open quantum dynamics and related walks. In such setting, block tridiagonal matrices and matrix-valued orthogonal polynomials are the natural objects to be considered. We recall the problems of the existence of a matrix of measures or weight matrix together with concrete calculations of basic statistics of the walk, such as site recurrence and first passage time probabilities, with these notions being defined in terms of a quantum trajectories formalism. The discussion concentrates on the models of quantum Markov chains, due to S. Gudder, and on the particular class of open quantum walks, due to S. Attal et al. The folding trick for birth-death chains on the integers is revisited in this setting together with applications of the matrix-valued Stieltjes transform associated with the measures, thus extending recent results on the subject. Finally, we consider the case of non-symmetric weight matrices and explore some examples.