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

Russian Mathematical Surveys

Feynman checkers: towards algorithmic quantum theory

Mihail Borisovich Skopenkov, Alexey Vladimirovich Ustinov

Published Jan 1, 2022• 10 citations
Publisher

Abstract

We survey and develop the most elementary model of electron motion introduced by Feynman. In this game, a checker moves on a checkerboard by simple rules, and we count the turns. Feynman checkers are also known as a one-dimensional quantum walk or an Ising model at imaginary temperature. We solve mathematically a Feynman problem from 1965, which was to prove that the discrete model (for large time, small average velocity, and small lattice step) is consistent with the continuum one. We study asymptotic properties of the model (for small lattice step and large time) improving the results due to Narlikar from 1972 and to Sunada and Tate from 2012. For the first time we observe and prove concentration of measure in the small-lattice-step limit. We perform the second quantization of the model. We also present a survey of known results on Feynman checkers. Bibliography: 53 titles.

DOI
10.1070/rm10025

Related papers

Feynman checkers: towards algorithmic quantum theory

Михаил Борисович Скопенков, A. Ustinov

We survey and develop the most elementary model of electron motion introduced by R$.$Feynman. In this game, a checker moves on a checkerboard by simple rules, and we count the turns. Feynman checkers are also known as a one-dimensional quantum walk or an Ising model at imaginary temperature. We solve mathematically a problem by R$.$Feynman from 1965, which was to prove that the discrete model (for large time, small average velocity, and small lattice step) is consistent with the continuum one. We study asymptotic properties of the model (for small lattice step and large time) improving the results by J$.$Narlikar from 1972 and by T$.$Sunada-T$.$Tate from 2012. For the first time we observe and prove concentration of measure in the small-lattice-step limit. We perform the second quantization of the model.

arXiv:2007.12879•Jul 25, 2020•1 citation

Feynman checkers: the probability of direction reversal

Il'ya Igorevich Bogdanov

We study the most elementary model of electron motion introduced by R.Feynman in 1965. It is a game, in which a checker moves on a checkerboard by simple rules, and we count the turnings. The model is also known as one-dimensional quantum walk. In his publication, R.Feynman introduces a discrete version of path integral and poses the problem of computing the limit of the model when the lattice step and the average velocity tend to zero and time tends to infinity. We get a nontrivial advance in the problem on the mathematical level of rigor in a simple particular case; even this case requires methods not known before. We also prove a conjecture by I.Gaidai-Turlov, T.Kovalev, and A.Lvov on the limit probability of direction reversal in the model generalizing a recent result by A.Ustinov.

arXiv:2010.04583•Oct 9, 2020•3 citations

Feynman checkers: Number-theoretic properties

Fedor Kuyanov, Alexey Slizkov

We study Feynman checkers, an elementary model of electron motion introduced by Feynman. In this model, a checker moves on a checkerboard, and we count the turns. Feynman checkers are also known as a one-dimensional quantum walk. We prove some new number-theoretic results in this model, for example, sign alternation of the real and imaginary parts of the electron wave function in a specific area. All our results can be stated in terms of Young diagrams, namely, we compare the number of Young diagrams with an odd and an even number of steps.

arXiv:2210.07306•Reviews in Mathematical Physics•Jun 14, 2023•3 citations

Feynman Checkerboard as a Model of Discrete Space-Time

Hanna, Edward

In 1965, Feynman wrote of using a lattice containing one dimension of space and one dimension of time to derive aspects of quantum mechanics. Instead of summing the behavior of all possible paths as he did, this paper will consider the motion of single particles within this discrete Space-Time lattice, sometimes called Feynman's Checkerboard. This empirical approach yielded several predicted emergent properties for a discrete Space-Time lattice, one of which is novel and testable.

arXiv:cs/0607018•Jul 6, 2006•1 citation

Quantum Checkers: The Development and Analysis of a Quantum Combinatorial Game

Marien Raat, Luuk van den Nouweland, Matthias Müller-Brockhausen +2 more

This paper develops and analyses a novel quantum combinatorial game: quantum checkers (named Cheqqers). The concepts of superposition, entanglement, measurements and interference from quantum mechanics are integrated into the game of checkers by adding new types of legal moves. The addition of these new rules is done gradually by introducing several levels of 'quantumness'. Quantum checkers provides a framework for interpolating between a known and solved classical game and a more complex quantum game, and serves as 1) a benchmark for AI players learning to play quantum games and 2) an interesting game for human players that allows them to build intuition for quantum phenomena. We provide the initial analysis on the complexity of this game using random agents and a Monte Carlo tree search agent.

arXiv:2506.05962•IEEE Conference on Games (CoG)•Aug 19, 2025•0 citations

Quantum lattice algorithms: similarities and connections to some classic finite difference algorithms

Paul J. Dellar

Quantum lattice algorithms originated with the Feynman checkerboard model for the one-dimensional Dirac equation. They offer discrete models of quantum mechanics in which the complex numbers representing wavefunction values on a discrete spatial lattice evolve through discrete unitary operations. This paper draws together some of the identical, or at least unitarily equivalent, algorithms that have appeared in three largely disconnected strands of research. Treated as conventional numerical algorithms, they are all only first order accurate under refinement of the discrete space/time grid, but may be raised to second order by a unitary change of variables. Much more efficient implementations arise from replacing the evolution through a sequence of unitary intermediate steps with a short path integral formulation that expresses the wavefunction at each spatial point on the most recent time level as a linear combination of values at immediately preceding time levels and neighbouring spatial points. In one dimension, a particularly elegant reformulation replaces two variables at two time levels with a single variable over three time levels. The resulting algorithm is a variational integrator arising from a discrete action principle, and coincides with the Ablowitz–Kruskal–Ladik finite difference scheme for the Klein–Gordon equation.

ESAIM Proceedings and Surveys•Dec 1, 2015•3 citations
We survey and develop the most elementary model of electron motion introduced by Feynman. In this game, a checker moves on a checkerboard by simple rules, and we count the turns. Feynman checkers are also known as a one-dimensional quantum walk or an Ising model at imaginary temperature. We solve mathematically a Feynman problem from 1965, which was to prove that the discrete model (for large time, small average velocity, and small lattice step) is consistent with the continuum one. We study asymptotic properties of the model (for small lattice step and large time) improving the results due to Narlikar from 1972 and to Sunada and Tate from 2012. For the first time we observe and prove concentration of measure in the small-lattice-step limit. We perform the second quantization of the model. We also present a survey of known results on Feynman checkers. Bibliography: 53 titles.
We survey and develop the most elementary model of electron motion introduced by R...Feynman. In this game, a checker moves on a checkerboard by simple rules, and we count the turns. Feynman checkers are also known as a one-dimensional quantum walk or an Ising model at imaginary temperature. We solve mathematically a problem by R...Feynman from 1965, which was to prove that the discrete model (for large time, small average velocity, and small lattice step) is consistent with the continuum one. We study asymptotic properties of the model (for small lattice step and large time) improving the results by J...Narlikar from 1972 and by T...Sunada-T...Tate from 2012. For the first time we observe and prove concentration of measure in the small-lattice-step limit. We perform the second quantization of the model.
We study the most elementary model of electron motion introduced by R.Feynman in 1965. It is a game, in which a checker moves on a checkerboard by simple rules, and we count the turnings. The model is also known as one-dimensional quantum walk. In his publication, R.Feynman introduces a discrete version of path integral and poses the problem of computing the limit of the model when the lattice step and the average velocity tend to zero and time tends to infinity. We get a nontrivial advance in the problem on the mathematical level of rigor in a simple particular case; even this case requires methods not known before. We also prove a conjecture by I.Gaidai-Turlov, T.Kovalev, and A.Lvov on the limit probability of direction reversal in the model generalizing a recent result by A.Ustinov.
We study Feynman checkers, an elementary model of electron motion introduced by Feynman. In this model, a checker moves on a checkerboard, and we count the turns. Feynman checkers are also known as a one-dimensional quantum walk. We prove some new number-theoretic results in this model, for example, sign alternation of the real and imaginary parts of the electron wave function in a specific area. All our results can be stated in terms of Young diagrams, namely, we compare the number of Young diagrams with an odd and an even number of steps.
This paper develops and analyses a novel quantum combinatorial game: quantum checkers (named Cheqqers). The concepts of superposition, entanglement, measurements and interference from quantum mechanics are integrated into the game of checkers by adding new types of legal moves. The addition of these new rules is done gradually by introducing several levels of 'quantumness'. Quantum checkers provides a framework for interpolating between a known and solved classical game and a more complex quantum game, and serves as 1) a benchmark for AI players learning to play quantum games and 2) an interesting game for human players that allows them to build intuition for quantum phenomena. We provide the initial analysis on the complexity of this game using random agents and a Monte Carlo tree search agent.
Quantum lattice algorithms originated with the Feynman checkerboard model for the one-dimensional Dirac equation. They offer discrete models of quantum mechanics in which the complex numbers representing wavefunction values on a discrete spatial lattice evolve through discrete unitary operations. This paper draws together some of the identical, or at least unitarily equivalent, algorithms that have appeared in three largely disconnected strands of research. Treated as conventional numerical algorithms, they are all only first order accurate under refinement of the discrete space/time grid, but may be raised to second order by a unitary change of variables. Much more efficient implementations arise from replacing the evolution through a sequence of unitary intermediate steps with a short path integral formulation that expresses the wavefunction at each spatial point on the most recent time level as a linear combination of values at immediately preceding time levels and neighbouring spatial points. In one dimension, a particularly elegant reformulation replaces two variables at two time levels with a single variable over three time levels. The resulting algorithm is a variational integrator arising from a discrete action principle, and coincides with the Ablowitz–Kruskal–Ladik finite difference scheme for the Klein–Gordon equation.