Core Concepts
The paper establishes that the problems of StateEq and ContainsEntryk in phase-free ZH calculus are NP#P-complete, and that circuit extraction in phase-free ZH is #P-hard.
Abstract
The paper focuses on the computational complexity of problems related to the phase-free ZH calculus, a graphical language for quantum computation reasoning.
Key highlights:
- The authors show that two problems in phase-free ZH calculus are NP#P-complete:
- StateEq: Determining if there exists a computational basis state on which two given diagrams equalize.
- ContainsEntryk: Checking if the matrix representation of a given diagram contains an entry equal to a given number.
- The authors also prove that circuit extraction, the problem of finding an ancilla-free circuit equivalent to a given phase-free ZH diagram, is #P-hard.
- The proofs involve crafting an artificial NPC=P[1]-complete problem called SAT&Compare#SAT, which combines SAT and Compare#SAT problems.
- The hardness results hold not only for phase-free ZH, but also generalize to other graphical calculi like ZX, ZH, and ZW, as phase-free ZH diagrams can be represented in those calculi.