From quantamagazine
Machine Heart Compilation
Editors: Chen Ping, Du Wei
Quanta play their magic in solving mathematical problems.
In 1779, the famous Swiss mathematician Leonhard Leonhard Euler once asked the question: how should 6 officers of 6 different ranks from different 6 army regiments be selected from each of the 6 army regiments in a platoon of 6 rows and 6 columns, so that the 6 officers in each row happen to be from different corps and have different ranks. History calls this problem the "thirty-six officer problem." After the thirty-six officer issues were raised, they were not resolved for a long time.
Image source: irishtimes.com
This puzzle is easy to solve when there are 5 ranks and 5 legions, or 7 ranks and 7 legions. But Euler did not find a solution for the thirty-six officers, who concluded that such an arrangement was impossible, although it was impossible to give a rigorous proof.
More than a century later, in 1901, the French mathematician Gaston Gaston Tarry proved that there was really no way to arrange Euler's 36 officers in a 6×6 square without repeating, and he wrote all possible arrangements of the 6x6 square, proving that the 36 officer problem was impossible. By 1960, mathematicians had used computers to prove that there was a solution to any number of ranks and legions, except for 6 ranks and 6 corps.
For more than 200 years, this puzzle has attracted countless mathematicians. They made a "Rubik's Cube," which consisted of a set of integers arranged in a square, with the sums of each row, each column, and each main diagonal equal; in addition, researchers made a "Latin square," a phalanx of n × n, in which there are exactly nine different elements in this n× n square, each of which appears only once in the same row or column.
At present, a Latin phalanx, Sudoku, is popular, and there are no duplicate symbols in Sudoku. The Euler Thirty-Six Officers' Question requires an "orthogonal Latin phalanx" that satisfies two sets of attributes, such as rank and legion, both of which satisfy the rules of the Latin phalanx.
A five-by-five grid can be filled with five pieces of different levels and five different colors so that no row or column has duplicate levels or colors.
Although Euler believes that there is no such 6×6 square, this conclusion is changing.
In a paper submitted to Physical Review Letters, "Thirty-six entangled officers of Euler: Quantum solution to a classically impossible problem," a team of quantum physicists from the Indian Institute of Technology (Madras Polytechnic Campus), Jagiellonian University, and other institutions demonstrated that 36 officers could be arranged in a way that met Euler's standards. As long as officers can have a quantum mix of rank and legion. This is the latest research in the quantum version of the Rubik's Cube and the Latin phalanx, which is not only an interesting game, but can also be applied to quantum communication and quantum computing.
Address: https://arxiv.org/pdf/2104.05122.pdf
Gemma De las Cuevas, a quantum physicist at the University of Innsbruck who did not participate in the study, said: "I think their paper is very interesting, it introduces a lot of quantum magic. Not only that, but you can feel their love for the subject throughout the paper."
The introduction of the concept of quantum Latin squares
In quantum mechanics, objects such as electrons can be in the "superposition" of multiple possible states, which can be here and there, or they can be magnetically oriented up and down. Quantum objects remain in an intermediate or indeterminate state before being measured, and remain in a state after measurement. Quantum Latin squares can also be in a quantum state of quantum superposition. Mathematically, a quantum state is represented by a vector that has a length and direction like an arrow. An overlay is an arrow that combines multiple vectors. Also, similar to the requirement that the symbols along the Latin square matrix are not repeated for each row and each column, the quantum states along the quantum Latin square matrix for each row or column must also correspond to vectors perpendicular to each other.
Later, the special properties of quantum Latin squares intrigued a group of theoretical physicists and mathematicians, and the concept was quickly adopted. In 2020, French mathematical physicists Ion Nechita and Jordi Pillet created a quantum version of the sudoku game SudoQ. They don't use integers between 0 and 9, but instead have 9 vertical vectors for each row, column, and square in SudoQ.
Ion Nechita。
These advances led Adam Burchardt, a postdoctoral fellow at jagiellonian University in Poland (a co-author of this work), and colleagues to revisit Euler's age-old puzzle about the official front of the 36th Army. They wondered, what if the officer in Euler's problem was in a quantum state?
Adam Burchardt。
In the classic version of the question, each entry is an officer with a definite rank and corps. It is helpful to think of these 36 officers as colorful pieces, whose rank could be king, queen, car, elephant, horse, or soldier (chess). The regiment to which these officers belong can be indicated by red, orange, yellow, green, or purple. But in the quantum version, officers are formed by the superposition of ranks and legions, for example an officer can be a superposition of the Red King and the Orange Queen.
Crucially, the quantum states that make up these officers have entangled relationships that involve correlations between different entities. For example, if a red king is entangled with an orange queen, then even if the king and queen are in a superposition of multiple legions, we observe that the king is red, and we immediately know that the queen is orange. It is precisely because of the special nature of entanglement that officers along each line can be vertical.
Implement the real solution with approximate solutions and algorithms
The above theory seems to work, but to prove it, the researchers must construct a quantum-state officer's 6×6 square matrix. The large number of possible configurations and entanglements meant that they had to resort to computers. So the researchers inserted a classical approximation solution (an arrangement of 36 classical officers, with ranks and regiments of only a few officers in a row or column being duplicated) and applied an algorithm that adjusts the arrangement to a true quantum solution. The algorithm works a bit like playing a Rubik's Cube with brute force, fixing the first row first, then the first column, the second column, and so on. As they repeated the algorithm over and over again, the 36th Army's official puzzles were getting closer and closer to being truly solved.
Eventually, the researchers got this pattern and manually filled in the remaining few entries.
In a sense, Euler proved wrong, although in the 18th century he could not have known the possibility of the existence of quantum officers.
"They shut down the book on the subject, which is good," Ion Nechita said. "It's a very beautiful result and I love the way they get it."
According to co-author Anderson Suhair, a physicist at the Madras Institute of Technology in Chennai, India, According to Lasser, a surprising feature of their solution was that the officer rank was entangled only with the adjacent ranks (King and Queen, White Chariot and Bishop, Knight and Chess piece). Regiments with adjacent groups. Another surprise is the coefficients that appear in the quantum Latin square. These coefficients are essentially numbers that tell you how much weight is assigned to different terms in the overlay. Curiously, the ratio of the coefficients employed by the algorithm is Φ, which is 1.618..., the famous golden ratio.
Also known as absolute maximum entangled state (AME, Absolutely Maximally Entangled state), this is a problem with the arrangement of quantum objects that is important in many applications, including quantum error correction, such as the way redundant information is stored in quantum computers so that even if the data is corrupted, the information can be preserved. In AME, there should be a strong correlation between the measurements of quantum objects: in the case of a coin toss, if two people (Alice, Bob) toss a tangled coin, where Alice tossed a coin and got a positive, he would know that Bob was the opposite, and vice versa. Two coins can be entangled to the maximum, three can be, but four can't: if two people join the coin toss together, Alice will never know what Bob got.
However, new research proves that if you have a set of four entangled dice instead of coins, they can be entangled to the greatest extent possible. The arrangement of the six-sided dice is equivalent to a 6×6 quantum Latin square. Because of the golden ratio in the solution, the researchers refer to it as the "Golden AME."
Researchers have started with classical error-correcting codes to design other AME and found similar quantum versions. But the newly discovered gold AME is different, it doesn't have the classic crypto simulation. Burchardt thinks the findings could be the new first class of quantum error correction codes.
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