Latin quantum squares were quickly accepted by a community of theoretical physicists and mathematicians interested in their unusual properties. Last year, French mathematical physicists Ion Nechita and Jordi Pillet created a quantum version of Sudoku—SudoQ. Instead of using the integers 0 through 9, in SudoQ, each row, column, and subset has nine perpendicular vectors.
These advances have led to Adam Burchardt, a postdoctoral researcher at the Jagiellonian University in Poland, and his colleagues to revisit Euler’s old puzzle of 36 officers. What if, they wondered, Euler’s officers were made quantum?
In the classical version of the problem, each entry is an officer with a clearly defined rank and regiment rank. It is useful to think of the 36 officers as colorful pieces, whose ranks can be king, queen, piece, bishop, knight or pawn, and the regiment is red, orange, yellow, green, blue or purple. But in the quantum version, officers are made up of ranks and regiment levels. For example, an officer might be the superposition of a red king and an orange queen.
Importantly, the quantum states that make up these officers have a special relationship known as entanglement, which deals with the correlations between different entities. For example, if a red king is entangled with an orange queen, even if both the king and queen are in stacks of multiple legions, observing the red king will tell you right away. that the queen is orange. It is because of the special nature of entanglement that officers along each line can be perpendicular to each other.
The theory seems to work, but to prove it, the authors had to build a 6 x 6 array filled with quantum officers. A large number of configurations and problems can mean that they have to resort to computer help. The researchers took a quasi-classical solution (an arrangement of 36 classical officers with only a repeating number of ranks and regiments in a row or column) and applied an algorithm that adjusted the alignments. aligned towards a true quantum solution. The algorithm works like solving a Rubik’s Cube by brute force, where you fix the first row, then the first column, the second column, etc. As they repeat the algorithm over and over, the array of puzzles rotates around the date. as close to being a correct solution. Eventually, the researchers reached a point where they could see the sample and fill in the few remaining entries by hand.
In a sense, Euler was proven wrong – although in the 18th century he could not have known about the capabilities of quantum officers.
“They closed the book on this, it was very good,” Nechita said. “It’s a very nice result, and I love how they got it.”
One surprising feature of their solution, according to co-author Suhail Rather, a physicist at the Indian Institute of Technology Madras in Chennai, is that officer ranks are only entangled in successive ranks (king vs. queens, troops with bishops, knights with pawns) and regiments. with neighboring regiments. Another surprise is that the coefficients appear in the entries of the quantum Latin square. These coefficients are numbers that tell you basically how much weight to give different terms in a superposition. Oddly enough, the ratio of the coefficients achieved by the algorithm is Φ, or 1.618…, the famous golden ratio.
