Indecision, quantum mechanics. Now even Euler, the god of mathematics, has to obey this sentence, because his puzzles are now solved by quantum mechanics. More than 240 years ago, Euler posed a 36 officer problem: each of the 6 legions has 6 officers of different ranks, the 36 officers are arranged in a 6 x 6 square, and no duplicate ranks or legions appear in any row or column, May I?
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A hint of familiarity? Isn’t it kind of like a Sudoku game?
In fact, these two problems are similar, that is, fill in n numbers in an n×n square, so that each number can only appear once in a row and a column (mathematically called “Latin square”) . It’s just that Sudoku also added a 3×3 grid limit.
After the continuous efforts of mathematicians, the Euler 36 officer problem was finally proved – impossible. Coincidentally, this problem has a solution if you replace it with 5×5 or 7×7, or any natural number other than 6 and greater than 2.
△ A solution of a 5×5 Latin square matrix (Source: Quanta Magazine)
However, in the quantum world, the “heterogeneous” problem of 6×6 has also been solved.
Quantum Officer
Since it cannot be solved in the classical world, physicists have moved their “brains”-if the six officers are all “quantum officers”, can the question be answered?
We assume that Officer 36 is in a quantum superposition state:
Each officer is in a superposition of multiple corps and multiple ranks.
It’s like Schrödinger’s cat, which can be both dead and alive at the same time.
Last year, two French physicists, Ion Nechita and Jordi Pillet, ripped apart the issue.
They created a quantum version of SudoQ, which replaces the 9 numbers with 9 mutually perpendicular vectors. This quantum Sudoku also has a solution. This inspired later people to solve Euler’s problem.
From classical to quantum
Recently, a group of quantum physicists from the Indian Institute of Technology and Jagiellonian University in Poland found the answer to Euler’s problem along the lines of quantum sudoku.
For ease of presentation, let’s begin by representing officers as playing cards. The card points A, K, Q, J, 10, 9 represent the legion; suits ?, ?, ?, ?, ?, ? represent the military rank.
In each lattice, we can put not only one playing card, but also the quantum entangled state of two playing cards.
If ?A and ?K are entangled, then no matter how the state is superimposed, as long as we observe that A’s suit is ?, we also immediately know that K’s suit is ?.
Because of this particularity of entanglement, more possibilities are created.
Due to the existence of a large number of entangled states in quantum officers, the amount of calculation is too large, and we must rely on the help of computers.
Physicists first find an approximate solution to a 6×6 classical arrangement, that is, a row or column with only a small number of repeating points and suits.
The computer then brute-forces it, fixing the first row, and so on. Repeat over and over until you get close to the real solution. Finally, a human finds a suitable pattern, fills in the remaining cells by hand, and finds a solution:
△ A solution to the 36 officer problem
A peculiarity of their solution, says Suhail Rather, a physicist at the Indian Institute of Technology, Chennai, and one of the authors of the paper, is that officers’ corps are only entangled with adjacent corps.
What is even more amazing is that the ratio of the coefficients of the two quantum states in the square, that is, the weight of the superposition of the quantum states, happens to be the famous golden ratio ratio of 0.618.
more than games
You might ask, what’s the use of solving this problem?
In fact, this is not just a game, it has an important role in quantum computing.
The solution to this problem is called the absolute maximum entangled state (AME), an arrangement of quantum states that is important in quantum error correction.
Previously, scientists started with classical error-correcting codes and found similar quantum error-correcting codes to design other AMEs.
But the AME found through the Euler 36 officer problem is different, he doesn’t have the classic encryption simulation.
So Adam Burchardt, another author of the paper, thinks they have even created an entirely new quantum error-correcting code.