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The Power of Lorentz Quantum Computer

Qi Zhang, Biao Wu

TL;DR

An associated computational complexity class termed bounded-error Lorentz quantum polynomial-time (BLQP) is introduced, demonstrating its equivalence to the complexity class P (probabilistic polynomial-time), all within polynomial time.

Abstract

We demonstrate the superior capabilities of the recently proposed Lorentz quantum computer (LQC) compared to conventional quantum computers. We introduce an associated computational complexity class termed bounded-error Lorentz quantum polynomial-time (BLQP), demonstrating its equivalence to the complexity class ${\text P}^{\sharp \text{P}}$. We present LQC algorithms that efficiently solve the problem of maximum independent set, PP (probabilistic polynomial-time), and consequently ${\text P}^{\sharp \text{P}}$, all within polynomial time. Additionally, we show that the quantum computing with postselection proposed by Aaronson can be efficiently simulated by LQC, but not vice versa.

The Power of Lorentz Quantum Computer

TL;DR

An associated computational complexity class termed bounded-error Lorentz quantum polynomial-time (BLQP) is introduced, demonstrating its equivalence to the complexity class P (probabilistic polynomial-time), all within polynomial time.

Abstract

We demonstrate the superior capabilities of the recently proposed Lorentz quantum computer (LQC) compared to conventional quantum computers. We introduce an associated computational complexity class termed bounded-error Lorentz quantum polynomial-time (BLQP), demonstrating its equivalence to the complexity class . We present LQC algorithms that efficiently solve the problem of maximum independent set, PP (probabilistic polynomial-time), and consequently , all within polynomial time. Additionally, we show that the quantum computing with postselection proposed by Aaronson can be efficiently simulated by LQC, but not vice versa.
Paper Structure (16 sections, 78 equations, 8 figures)

This paper contains 16 sections, 78 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Theoretical Model of Lorentz Quantum Computer
  3. LQC algorithms for the maximum independent set
  4. ${\text{P}}^{\sharp \text{P}}$ $\subseteq$BLQP
  5. LQC algorithm for PP
  6. LQC algorithm for ${\text{P}}^{\sharp \text{P}}$
  7. BLQP $=$ ${\text{P}}^{\sharp \text{P}}$
  8. proof of BLQP$\subseteq$${\text{P}}^{\sharp \text{P}}$ from the perspective of computational problem
  9. proof of BLQP$\subseteq$${\text{P}}^{\sharp \text{P}}$ by decision problem
  10. Comparison between LQC and quantum computing with postselection
  11. Simulation of postselection by LQC
  12. Super-postselection by LQC
  13. Summary
  14. State of the oracle qubit, the auxiliary qubit, and the hybit
  15. Possible values of $\eta=\beta/\alpha$
  16. ...and 1 more sections

Figures (8)

  • Figure 1: The hierarchy diagram for major complexity classes. For two connecting classes, the class below is included within the class above. BLQP is a complexity class defined for Lorentz quantum computer in parallel to BQP for conventional quantum computer. This diagram without BLQP can be found at www.complexityzoo.com.
  • Figure 2: (a) Single qubit gates $H$ and $T$; (b) single hybit gates $\tau$ and $T$.
  • Figure 3: Four different controlled-$\sigma_z$ gates
  • Figure 4: (a) Two-bit logical $CV$ gate; (b) a simple way to realize $CV$ using the controlled-$\sigma_z$ gate and the $\tau$ gate for $\chi=2\ln(\sqrt{2}+1)$.
  • Figure 5: (a) Three-bit logical CCV gate; (b) the circuit that implements the CCV gate with four $\tau$ gates and four controlled-$\sigma_z$ gates for $\chi=4\ln(\sqrt{2}+1)$.
  • ...and 3 more figures