Genuinely quantum solutions of the game Sudoku and their cardinality

Paczos, Jerzy; Wierzbiński, Marcin; Rajchel-Mieldzioć, Grzegorz; Burchardt, Adam; Życzkowski, Karol

doi:10.1103/physreva.104.042423

Cited by 10 publications

(8 citation statements)

References 22 publications

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“…3d shows the quantum pattern with the maximal cardinality, C = d 2 = 16. It satisfies the rules of a quantum Sudoku [41] [40] for an arbitrary dimension d.…”

Section: Quantum Latin Squaresmentioning

confidence: 90%

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9 × 4 = 6 × 6: Understanding the Quantum Solution to Euler’s Problem of 36 Officers

Życzkowski

Bruzda

Rajchel-Mieldzioć

et al. 2023

J. Phys.: Conf. Ser.

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The famous combinatorial problem of Euler concerns an arrangement of 36 officers from six different regiments in a 6×6 square array. Each regiment consists of six officers each belonging to one of six ranks. The problem, originating from Saint Petersburg, requires that each row and each column of the array contains only one officer of a given rank and given regiment. Euler observed that such a configuration does not exist. In recent work, we constructed a solution to a quantum version of this problem assuming that the officers correspond to superpositions of quantum states. In this paper, we explain the solution which is based on a partition of 36 officers into nine groups, each with four elements. The corresponding quantum states are locally equivalent to maximally entangled two-qubit states, hence each quantum officer is represented by a superposition of at most 4 classical states. The entire quantum combinatorial design involves 9 Bell bases in nine complementary 4–dimensional subspaces.

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“…3d shows the quantum pattern with the maximal cardinality, C = d 2 = 16. It satisfies the rules of a quantum Sudoku [41] [40] for an arbitrary dimension d.…”

Section: Quantum Latin Squaresmentioning

confidence: 90%

“…It is possible to show [40] that in dimensions d = 2 and d = 3 all quantum Latin squares satisfy the relation C = d, so they are apparently quantum. For d = 4 there exist QLS with cardinality greater than four, which are genuinely quantum and are not equivalent to any classical solution.…”

Section: Quantum Latin Squaresmentioning

confidence: 99%

9 × 4 = 6 × 6: Understanding the Quantum Solution to Euler’s Problem of 36 Officers

Życzkowski

Bruzda

Rajchel-Mieldzioć

et al. 2023

J. Phys.: Conf. Ser.

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show abstract

“…It is possible to show [39] that in dimensions d = 2 and d = 3 all quantum Latin squares satisfy the relation C = d, so they are apparently quantum. For d = 4 there exist QLS with cardinality greater than four, which are genuinely quantum and are not equivalent to any classical solution.…”

Section: Quantum Latin Squaresmentioning

confidence: 99%

“…It satisfies the rules of a quantum Sudoku [40], so it offers a constellation of 3 × 4 = 12 different orthogonal measurements in H 4 . Further examples of QLS(d) with maximal cardinality C = d 2 were constructed in [39] for an arbitrary dimension d.…”

Section: Quantum Latin Squaresmentioning

confidence: 99%

9 $\times$ 4 = 6 $\times$ 6: Understanding the quantum solution to the Euler's problem of 36 officers

Życzkowski¹,

Bruzda²,

Rajchel-Mieldzioć³

et al. 2022

Preprint

Self Cite

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The famous combinatorial problem of Euler concerns an arrangement of 36 officers from six different regiments in a 6 × 6 square array. Each regiment consists of six officers each belonging to one of six ranks. The problem, originating from Saint Petersburg, requires that each row and each column of the array contains only one officer of a given rank and given regiment. Euler observed that such a configuration does not exist. In recent work, we constructed a solution to a quantum version of this problem assuming that the officers correspond to quantum states and can be entangled. In this paper, we explain the solution which is based on a partition of 36 officers into nine groups, each with four elements. The corresponding quantum states are locally equivalent to maximally entangled two-qubit states, hence each officer is entangled with at most three out of his 35 colleagues. The entire quantum combinatorial design involves 9 Bell bases in nine complementary 4-dimensional subspaces.

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“…By associating with each number l ∈ [d] in a classical Latin square of order d and the computational basis element |l ∈ C d , we get a quantum Latin square for which the elements in every row or column form a computational basis, and we call it a classical quantum Latin square. Moreover, if a quantum Latin square is equivalent to a classical one, then we also call it a classical quantum Latin square, otherwise, it is a non-classical quantum Latin square [33] or a genuinely quantum Latin square [35].…”

Section: Lemma 23 ([20]mentioning

confidence: 99%

Quantum combinatorial designs and $k$-uniform states

Zang,

Facchi,

Tian

2021

Preprint

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Goyeneche et al. [Phys. Rev. A 97, 062326 (2018)] 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 2-and 3-uniform states.2000 Mathematics Subject Classification.

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Genuinely quantum solutions of the game Sudoku and their cardinality

Cited by 10 publications

References 22 publications

9 × 4 = 6 × 6: Understanding the Quantum Solution to Euler’s Problem of 36 Officers

9 × 4 = 6 × 6: Understanding the Quantum Solution to Euler’s Problem of 36 Officers

9 $\times$ 4 = 6 $\times$ 6: Understanding the quantum solution to the Euler's problem of 36 officers

Quantum combinatorial designs and $k$-uniform states

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