Quantum Algorithms in a Superposition of Spacetimes
Omri Shmueli
TL;DR
This work introduces a quantum-computation model grounded in quantum gravity’s indefinite causal structure by defining computable order interference (OI) and an Order Interference (OI) oracle. It proves that Graph Isomorphism (GI) and Gap Closest Vector Problem at gap O(n√n) can be solved in polynomial time within the model (GI ∈ BQP^{OI} and GapCVP_{O(n√n)} ∈ BQP^{OI}), while not trivially solving NP or SZK. Central to the results is the Sequentially Invertible Statistical Difference (SISD) problem and its polynomial-time solvability under OI, together with reductions GI ≤ SISD_{poly, O(log n)} and GapCVP ≤ SISD_{poly, O(log n)}. The paper provides a polynomial-time quantum algorithm that uses an OI oracle to generate multiple copies of output-distribution states and employs repeated swap tests to distinguish distributions, embedding a new complexity class BQP^{OI} and opening questions about the full power of OI-based computation. Overall, the work forges a link between quantum-gravity-inspired information processing and fundamental computational problems, presenting a potential pathway to new algorithmic regimes while outlining clear boundaries on which complexity classes remain unaddressed by the model.
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 $\mathbf{SZK}$.