Assembling furniture usually starts with dumping all the parts on the floor and sorting them into piles. For quantum computers, the role of such parts is played by Pauli strings, the simplest building blocks used to assemble a description of electrons in a molecule. But there is a way to build a cabinet without taking it apart into boards: Japanese physicists from RIKEN and the University of Tokyo showed that a quantum circuit can be built directly from whole blocks, preserving their internal logic.

Usually, to simulate the behavior of electrons on a quantum computer, physicists first rewrite fermionic operators (electron creation and annihilation operators) as long sums of Pauli strings. Each string is a separate operation, and each one requires its own gates. As a result, the circuit bloats, and the connections between parts that were obvious in the original description disappear. The authors of the new work took a different path: they diagonalized ladder string pairs directly, using entangled states of many qubits (GHZ states) and parity encoding. For a two-body operator that would require 16 Pauli strings in the standard approach, their circuit contains just one multi-controlled rotation and needs no Trotterization (approximating the exponential of a sum by a product of exponentials).

Then the real optimization begins. The authors noticed that three two-body operators sharing four indices can be combined into a single circuit, where gates cancel through commutation and local rewrites. As a result, the number of CX gates (the basic operations that entangle qubits) for such a triple drops from 36 to 12. And if you order a set of such triples, changing indices in a minimal way, the scaling of the gate count changes from quadratic to linear. It is as if, instead of screwing in each slat anew every time, you assembled whole cabinet sections and connected them together.