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Advanced Quantum Technologies

Genuinely Quantum Solutions of Orthogonal Sudoku Squares

Yajuan Zang, Hailong Zhang, Zihong Tian, Shao‐Ming Fei

Published May 1, 2026• 0 citations
Publisher

Abstract

ABSTRACT Quantum Sudoku is the generalized version of the popular game Sudoku. A quantum Sudoku square is a special kind of quantum Latin square. A genuinely quantum solution of quantum Sudoku squares has been solved completely. In this paper, we introduce the notion of orthogonality of two quantum Sudoku squares. We establish three novel constructions of mutually orthogonal quantum Sudoku squares including Lorch construction, Pedersen‐Vis construction and direct product construction. As a result, we determine the genuinely quantum solution of a pair of orthogonal Sudoku squares of size for any positive integer . In general, we show that the resulting squares naturally define an absolutely maximally entangled state . Furthermore, we also show that the mutually orthogonal quantum Sudoku squares give rise to mutually unbiased bases based on maximally entangled states in for any prime power .

DOI
10.1002/qute.70293

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ABSTRACT Quantum Sudoku is the generalized version of the popular game Sudoku. A quantum Sudoku square is a special kind of quantum Latin square. A genuinely quantum solution of quantum Sudoku squares has been solved completely. In this paper, we introduce the notion of orthogonality of two quantum Sudoku squares. We establish three novel constructions of mutually orthogonal quantum Sudoku squares including Lorch construction, Pedersen‐Vis construction and direct product construction. As a result, we determine the genuinely quantum solution of a pair of orthogonal Sudoku squares of size for any positive integer . In general, we show that the resulting squares naturally define an absolutely maximally entangled state . Furthermore, we also show that the mutually orthogonal quantum Sudoku squares give rise to mutually unbiased bases based on maximally entangled states in for any prime power .
We expand the quantum variant of the popular game Sudoku by introducing the notion of cardinality of a quantum Sudoku (SudoQ), equal to the number of distinct vectors appearing in the pattern. Our considerations are focused on the genuinely quantum solutions---the solutions of size N2{N}^{2}N2 that have cardinality greater than N2{N}^{2}N2, and therefore cannot be reduced to classical counterparts by a unitary transformation. We find the complete parametrization of the genuinely quantum solutions of a $4\ifmmode\times\else\texttimes\fi{}4$ SudoQ game and establish that in this case the admissible cardinalities are 4, 6, 8, and 16. In particular, a solution with the maximal cardinality equal to 16 is presented. Furthermore, the parametrization enabled us to prove a recent conjecture of Nechita and Pillet [I. Nechita and J. Pillet, Quantum Inf. Comput. 21, 781 (2021)] for this special dimension. In general, we proved that for any NNN it is possible to find an ${N}^{2}\ifmmode\times\else\texttimes\fi{}{N}^{2}$ SudoQ solution of cardinality N4{N}^{4}N4, which for a prime NNN is related to a set of NNN mutually unbiased bases of size N2{N}^{2}N2. Such a construction of N4{N}^{4}N4 different vectors of size NNN yields a set of N3{N}^{3}N3 orthogonal measurements.
We introduce orthogonal quantum Latin squares, which restrict to traditional orthogonal Latin squares, and investigate their application in quantum information science. We use quantum Latin squares to build maximally entangled bases, and show how a pair of mutually unbiased maximally entangled bases can be constructed in square dimension from orthogonal quantum Latin squares. We also compare our construction to an existing construction due to Beth and Wocjan and show that ours is strictly more general.
We expand the quantum variant of the popular game Sudoku by introducing the notion of cardinality of a quantum Sudoku (SudoQ), equal to the number of distinct vectors appearing in the pattern. Our considerations are focused on the genuinely quantum solutions, which are the solutions of size N2N^2N2 that have cardinality greater than N2N^2N2, and therefore cannot be reduced to classical counterparts by a unitary transformation. We find the complete parameterization of the genuinely quantum solutions of 4×44 \times 44×4 SudoQ game and establish that in this case the admissible cardinalities are 4, 6, 8 and 16. In particular, a solution with the maximal cardinality equal to 16 is presented. Furthermore, the parametrization enabled us to prove a recent conjecture of Nechita and Pillet for this special dimension. In general, we proved that for any NNN it is possible to find an N2×N2N^2 \times N^2N2×N2 SudoQ solution of cardinality N4N^4N4, which for a prime NNN is related to a set of NNN mutually unbiased bases of size N2N^2N2. Such a construction of N4N^4N4 different vectors of size NNN yields a set of N3N^3N3 orthogonal measurements.
Abstract Goyeneche et al [2018 Phys. Rev. A 97 062326] introduced several classes of quantum combinatorial designs, namely quantum Latin squares, quantum Latin cubes, and the notion of orthogonality on them. They also showed that mutually orthogonal quantum Latin arrangements can be entangled in the same way in which quantum states are entangled. Moreover, they established a relationship between quantum combinatorial designs and a remarkable class of entangled states called k -uniform states, i.e. multipartite pure states such that every reduction to k parties is maximally mixed. In this article, we put forward the notions of incomplete quantum Latin squares and orthogonality on them and present construction methods for mutually orthogonal quantum Latin squares and mutually orthogonal quantum Latin cubes. Furthermore, we introduce the notions of generalized mutually orthogonal quantum Latin squares and generalized mutually orthogonal quantum Latin cubes, which are equivalent to quantum orthogonal arrays of size d 2 and d 3 , respectively, and thus naturally provide two- and three-uniform states.
We introduce several classes of quantum combinatorial designs, namely quantum Latin squares, cubes, hypercubes, and a notion of orthogonality between them. A further introduced notion, quantum orthogonal arrays, generalizes all previous classes of designs. We show that mutually orthogonal quantum Latin arrangements can be entangled in the same way in which quantum states are entangled. Furthermore, we show that such designs naturally define a remarkable class of genuinely multipartite highly entangled states called kkk-uniform, i.e., multipartite pure states such that every reduction to kkk parties is maximally mixed. We derive infinitely many classes of mutually orthogonal quantum Latin arrangements and quantum orthogonal arrays having an arbitrary large number of columns. The corresponding multipartite kkk-uniform states exhibit a high persistency of entanglement, which makes them ideal candidates to develop multipartite quantum information protocols.
Quantum combinatorial designs are gaining popularity in quantum information theory. Quantum Latin squares can be used to construct mutually unbiased maximally entangled bases and unitary error bases. Here we present a general method for constructing quantum Latin arrangements from irredundant orthogonal arrays. As an application of the method, many new quantum Latin arrangements are obtained. We also find a sufficient condition such that the improved quantum orthogonal arrays [10] are equivalent to quantum Latin arrangements. We further prove that an improved quantum orthogonal array can produce a quantum uniform state.