The Power of the Lorentz Quantum Computer

Area of Science:

• Theoretical Computer Science
• Quantum Computing
• Mathematical Physics

Background:

• The Lorentz quantum computer (LQC) is a novel theoretical model utilizing hyperbolic bits (hybits) and Lorentz transformations.
• Understanding the computational power of new quantum computing paradigms is crucial for advancing complexity theory.

Purpose of the Study:

• To analyze the computational power of the Lorentz quantum computer (LQC).
• To define and explore the complexity class BLQP (bounded-error Lorentz quantum polynomial-time).
• To establish the relationship between LQC, existing complexity classes, and quantum computing with postselection.

Main Methods:

• Definition of the bounded-error Lorentz quantum polynomial-time (BLQP) complexity class.
• Equivalence proof between BLQP and P♯P (polynomial-time Turing machine with a ♯P oracle).
• Analysis of LQC's ability to solve NP-hard problems and simulate quantum computing with postselection (PostBQP).

Main Results:

• Demonstrated equivalence: BLQP = P♯P.
• LQC solves NP-hard problems (e.g., Maximum Independent Set) in polynomial time, placing NP and co-NP within BLQP.
• LQC efficiently simulates PostBQP, exhibiting 'super-postselection' capabilities not present in reverse.
• Established that the entire polynomial hierarchy (PH) is within BLQP.

Conclusions:

• LQC possesses theoretical superiority over conventional quantum computing models.
• Profound connections revealed between computational complexity and Lorentz quantum mechanics.
• LQC has the potential to redefine the boundaries of complexity theory and computational power.