Lorentz Quantum Computer Model Outperforms Conventional Quantum Computing, Study Suggests

The Lorentz Quantum Computer (LQC) is a theoretical model that could potentially outperform conventional quantum computers. The LQC model uses a different type of bit, the hyperbolic bit (hybit), which evolves according to complex Lorentz transformations. The model introduces a complexity class BLQP for LQC, which includes all problems that can be solved by LQC in polynomial time with bounded error. The authors demonstrate that LQC can solve problems in the complexity classes NP, coNP, PH, PP, and PP in polynomial time. They also introduce two important logic gates, CV and CCV, central to LQC’s power.

What is the Power of Lorentz Quantum Computer?

The Lorentz Quantum Computer (LQC) is a theoretical model of computing that has been recently proposed. This model is not physically sound at the current stage, but it is interesting and potentially very useful theoretically. The LQC model has a different type of bit, known as the hyperbolic bit (hybit), which evolves according to complex Lorentz transformations. This model was inspired by physics, and it may become physically sound when the future physical theory of unifying quantum mechanics and gravity falls within the framework of Lorentz quantum mechanics.

The LQC model is more powerful than the conventional quantum computer as it can exponentially speed up the Grover search. In this work, the authors explore systematically the power of LQC. They introduce a complexity class BLQP (bounded-error Lorentz quantum polynomial-time) for LQC. It consists of all the problems that can be solved by LQC in polynomial time with bounded error. As the conventional quantum computer is a special case of LQC, BQP (bounded-error quantum polynomial-time) is clearly a subset of BLQP.

How Does LQC Compare to Conventional Quantum Computers?

The authors present LQC circuits for algorithms that solve in polynomial time the problem of maximum independent set, which is NP2 hard, and all the problems in the complexity classes NP and coNP. This means that both NP and coNP are subsets of BLQP. They further find LQC algorithms that can solve in polynomial time problems in the classes of PH (polynomial hierarchy), PP (probabilistic polynomial-time), and PP. Along with these algorithms, they explain the source of computing power of LQC.

The authors make a detailed comparison between LQC and quantum computing with postselection. LQC is shown to be able to simulate the postselection efficiently. In contrast, a capability of LQC, which they call superpostselection, cannot be simulated by quantum computer with postselection. So the class PostBQP, defined for quantum computing with postselection, is included in BLQP. However, they cannot prove that PostBQP is a true subset of BLQP.

What are the Basics of LQC?

The theoretical model of Lorentz Quantum Computer (LQC) was proposed and discussed extensively in a previous reference. It is based on the Lorentz quantum mechanics, a theory generalized from the Bogoliubov-de Genne equation, which describes the dynamics of bosonic Bogoliubov quasiparticles. One key feature of these systems is that they have two branches of excitations. Although both are involved in the dynamics, only one of them, i.e., bosonic Bogoliubov quasiparticle, is regarded as physical, while the other with negative energy is regarded unphysical and unobservable. LQC takes advantage of this feature by introducing hyperbolic bits or hybits for brevity, for which only one of its basis is observable.

What are the Important Logic Gates in LQC?

The authors introduce two important logic gates, CV gate and CCV gate. These two gates play central roles in their efficient algorithms for solving various problems such as MIS, NP, PP, and PP, and underscore why LQC is much more powerful than the conventional quantum computer.

How Does LQC Compare to Quantum Computing with Postselection?

The authors compare and discuss the relation between LQC and quantum computing with postselection. LQC is shown to be able to simulate the postselection efficiently. In contrast, a capability of LQC, which they call superpostselection, cannot be simulated by quantum computer with postselection. So the class PostBQP, defined for quantum computing with postselection, is included in BLQP. However, they cannot prove that PostBQP is a true subset of BLQP.