The Power of the Lorentz Quantum Computer

We analyze the power of the recently proposed Lorentz quantum computer (LQC), a theoretical model leveraging hyperbolic bits (hybits) governed by complex Lorentz transformations. We define the complexity class BLQP (bounded-error Lorentz quantum polynomial-time) and demonstrate its equivalence to the complexity class <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mrow><mi mathvariant="normal">P</mi></mrow><mrow><mo>♯</mo><mi mathvariant="normal">P</mi></mrow></msup></semantics></math></inline-formula> (the class of problems solvable by a deterministic polynomial-time Turing machine with access to a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>♯</mo><mi mathvariant="normal">P</mi></mrow></semantics></math></inline-formula> oracle). LQC algorithms are shown to solve NP-hard problems, such as the maximum independent set (MIS), in polynomial time, thereby placing NP and co-NP within BLQP. Furthermore, we establish that LQC can efficiently simulate quantum computing with postselection (PostBQP), while the reverse is not possible, highlighting LQC’s unique “super-postselection” capability. By proving BLQP <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>=</mo><msup><mrow><mi mathvariant="normal">P</mi></mrow><mrow><mo>♯</mo><mi mathvariant="normal">P</mi></mrow></msup></mrow></semantics></math></inline-formula>, we situate the entire polynomial hierarchy (PH) within BLQP and reveal profound connections between computational complexity and physical frameworks like Lorentz quantum mechanics. These results underscore LQC’s theoretical superiority over conventional quantum computing models and its potential to redefine boundaries in complexity theory.