Playing Sudoku in the Quantum World: physicists have found the answer to a mathematical puzzle judged to be unsolved

Mathematician Euler put forward a problem of 36 officers similar to 6 × 6 Sudoku: 36 officers of six different ranks were selected from each of the six legions, and these 36 officers were arranged into a phalanx. Can the legions and ranks of officers in each row and column belong to different legions and ranks? later, mathematicians proved that similar problems of order 5 and order 7 could be solved, but there was no solution at order 6. Later, a group of physicists wondered: if every officer is in a superposition of two legions and two ranks, is there a solution to this problem? they really found a quantum solution.

Sudoku games are popular all over the world, whether you like to play or not, at least you have heard of the rules of this game: a 9 × 9 grid is divided into nine 3 × 3 "palaces". Fill the number 1x9 into these squares and make sure there are no duplicate numbers in each row, column and house. In general, a Sudoku game will give some hints, and the rest of the numbers need to be filled in by the player's reasoning. It is such a simple rule that has derived a lot of problem-solving skills, attracting countless players to enjoy it.

The predecessor of Sudoku can be traced back to Europe in the 18th century, when mathematician Leonhard Euler summed up a popular crossword puzzle called "Latin phalanx" (Latin square). The rule of the game is to fill in n Latin letters in the square grid of order n (similar to the number 1x 2 in order 2 Sudoku and 1 # 3 in order 3 Sudoku), so that the letters in each row and column will not repeat. This square matrix is not limited to the ninth order and has no palace restrictions, but it retains the most basic "no repetition per row and column" requirement of Sudoku.

But what fascinates Euler is a more complex version of the Latin phalanx. Euler considered filling each grid with a Latin letter and a Greek letter so that the letters in each line and column would not be repeated, and the Greek-Latin pairs in each grid would not be repeated. This kind of square is called "Greek-Latin square matrix" (Graeco-Latin square), and its essence is to combine two orthogonal Latin square matrices (orthogonal Latin squares) into one square matrix. The "orthogonal" here means that the ordered pairs of the corresponding lattices of the two matrices are not repeated. If you want to try, the elements in the grid don't have to be Greek and Latin letters, you can also use a combination of playing cards, or even ordered pairs.

The unsolved problem of 36 officers

After a careful study of the Greek-Latin square, Euler found an interesting phenomenon: the Greek-Latin square of order 7 can be constructed, but the Greek-Latin matrix of order 2 and order 6 can not be constructed. The problem of order 2 is easy to deal with, through the exhaustive method, we can see that such a Greek-Latin matrix does not exist, while the problem of order 6 is relatively complex. Euler repeated the question in more popular language: 36 officers of six different ranks were selected from each of the six legions, and the 36 officers were lined up in a phalanx, so that the officers in each row and column belonged to different legions and ranks.

Euler believes that this "36 officer problem" is unsolvable, that is, there is no sixth-order Greco-Latin phalanx. And he guessed that all Greek-Latin squares with orders divided by 4 and 2 did not exist, that is to say, 2meme, 6pm, 10pm, 14. The Greek-Latin phalanx of order does not exist.

More than a century later, in 1901, the French mathematician Gaston Tarry proved by the method of exhaustion that the elements in the lattice of the 6th-order square matrix constructed according to the rules are always repeated, and the 6th-order Greco-Latin matrix does not exist. In 1959, mathematicians proved that Euler's further conjecture was not valid, that is to say, all Greek-Latin matrices of order 2 and 6 existed. At this point, the question about the original version of Sudoku has been answered mathematically.

Quantum solution

In the 21st century, a group of physicists rediscovered Euler's 36-officer problem. Although the problem has been mathematically determined, they have opened up a brain from the point of view of physics: if these 36 officers are in a quantum superposition state, each officer "partly" belongs to one regiment and one rank, and "partly" belongs to another regiment and another rank, is there a solution to this question?

Along this line of thinking, some physicists modified the construction rules of the Greco-Latin square matrix and gave a quantum version of the Sudoku game. In quantum mechanics, the state of an object can be expressed by a vector. In the quantum version 36 officer problem, the regiment to which each officer belongs can be expressed as a vector in a six-dimensional space, and the rank can be expressed as a vector in another six-dimensional space. Because officers can be in various superposition states, these vectors can be different, and their 6 × 6 square matrix can easily meet the requirement of "the vectors of each row and column are different", but it is of no research value. Physicists are interested in whether the vectors of each row and column constitute a set of standard orthogonal bases of the space to which they belong.

To understand the so-called "standard orthogonal basis", you can make an analogy. In the three-dimensional space that we are familiar with, we can establish a Cartesian coordinate system, and the unit vectors along the direction of the x _ ray _ y _ Z axis in the coordinate system form a set of standard orthogonal bases. These three vectors satisfy: pairwise perpendicular in direction and unit length in size. The problem of 36 officers can be similarly understood, which means that the vectors representing the legions and ranks of officers in the 6 × 6 square should be vertical in pairs of each row and column, and the size is in unit length.

In fact, the 6-dimensional space representing the legion and the 6-dimensional space representing the rank can be expanded to a 36-dimensional space, and the regiment and rank of each officer can be represented by a vector in this 36-dimensional space. The 6 × 6 square matrix of these vectors still needs to be satisfied: the vectors of each row and column are vertical, and the size is unit length.

The researchers say that the quantum solution of this ancient Sudoku problem is equivalent to an absolute maximally entangled state (Absolutely Maximally Entangled state) of a four-particle subsystem. This entangled state can be applied to many scenarios such as error correction in quantum computing, such as storing redundant information in this state in a quantum computer, even if the data is damaged. This ancient mathematical problem, which originated from Euler, got a new answer in physics 243 years later. Perhaps this is just a fun brain for theoretical physicists, but it benefits researchers in the fields of quantum communication and quantum computing. Scientific progress often takes place in such games.