Quantum Algorithms in a Superposition of Spacetimes (2403.02937v3)

Abstract: Quantum computers are expected to revolutionize our ability to process information. The advancement from classical to quantum computing is a product of our advancement from classical to quantum physics -- the more our understanding of the universe grows, so does our ability to use it for computation. A natural question that arises is, what will physics allow in the future? Can more advanced theories of physics increase our computational power, beyond quantum computing? An active field of research in physics studies theoretical phenomena outside the scope of explainable quantum mechanics, that form when attempting to combine Quantum Mechanics (QM) with General Relativity (GR) into a unified theory of Quantum Gravity (QG). QG is known to present the possibility of a quantum superposition of causal structure and event orderings. In the literature of quantum information theory, this translates to a superposition of unitary evolution orders. In this work we show a first example of a natural computational model based on QG, that provides an exponential speedup over standard quantum computation (under standard hardness assumptions). We define a model and complexity measure for a quantum computer that has the ability to generate a superposition of unitary evolution orders, and show that such computer is able to solve in polynomial time two of the fundamental problems in computer science: The Graph Isomorphism Problem ($\mathsf{GI}$) and the Gap Closest Vector Problem ($\mathsf{GapCVP}$), with gap $O\left( n \sqrt{n} \right)$. These problems are believed by experts to be hard to solve for a regular quantum computer. Interestingly, our model does not seem overpowered, and we found no obvious way to solve entire complexity classes that are considered hard in computer science, like the classes $\mathbf{NP}$ and $\mathbf{SZK}$.

• The paper defines a new computational model that leverages a superposition of spacetime geometries to achieve polynomial-time solutions for GI and GapCVP.
• It introduces the Order Interference (OI) oracle, enabling a superposition of unitary execution orders to significantly boost quantum computational power.
• The findings challenge traditional quantum limits and suggest that principles from quantum gravity can drive future advances in solving complex computational problems.

Quantum Algorithms in a Superposition of Spacetimes: An Analysis

This paper explores a novel computational model inspired by theories of quantum gravity, particularly focused on the superposition of spacetime geometries. The authors aim to decipher whether advanced theories from physics could not only enhance quantum computing but propel computation beyond the conventional limits set by current quantum computers.

Key Findings and Results

The core contribution of the paper is the definition of a unique computational model that leverages the concept of a superposition of event orders, grounded in quantum gravity principles. The paper builds upon the idea that a superposition of spacetime geometries could lead to a superposition of event orders, directly affecting computational processes. The suggested model, termed computable order interference, offers a substantial potential speedup over standard quantum computation, under typical complexity assumptions.

Specifically, the paper shows that this new model can solve the Graph Isomorphism Problem (GI) and the Gap Closest Vector Problem (GapCVP) in polynomial time, which are both considered difficult for traditional quantum computers. Notably, the model achieves this by allowing quantum computers to perform a superposition of unitary execution orders, enhancing their computational power.

Computational Implications

The implications of introducing quantum gravitational correlations, without necessarily inducing entanglement, are significant. The paper proposes a computational model that makes pragmatic use of these correlations to amplify the functional capability of quantum systems. This approach is depicted using the Order Interference (OI) oracle, which enables a quantum computer to handle unitary orders and choices more efficiently, potentially solving prominent problems in polynomial time that are typically deemed hard.

The reduction techniques introduced, particularly from fundamental problems like GI and GapCVP to Sequentially Invertible Statistical Difference Problem (SISD), open up new avenues in computational complexity theory. This signifies a potential paradigm shift in how quantum computations could be structured, leveraging the findings from quantum gravity studies.

Theoretical and Future Implications

The theoretical framework suggested in this paper also raises questions about the inherent limits of computational barriers posed by classical and quantum mechanics. While the current model does not overtly simplify NP-complete problems or entire complexity classes such as SZK, it stretches the known boundaries of quantum computational models and invites further research into the capabilities of quantum computers in handling computational complexity.

This work prompts a reevaluation of the interrelation between advanced quantum theories and computational practice. It suggests an iterative process where insights from theoretical physics could continually reshape computational methods, particularly with quantum gravity’s unknowns potentially harboring untapped computational advancements.

Conclusion

In conclusion, this paper offers a compelling examination of how concepts from quantum gravity could be translated into practical computational models. While showcasing how a superposition of unitary orders transforms quantum computation, it bridges theoretical physics and computer science, suggesting new realms of efficiency and complexity reduction. The advancement prompts further exploration into the synthesis of physical theories and computation, with a focus on potentially redefining what is computationally feasible in a quantum-enhanced future.