Causally Indefinite Computation cuts Boolean function query
Uncertain cause Computational Queries Are Simplified by Causal Indefiniteness in Quantum Computing
Causally indefinite computation can reduce query complexity, unlike typical computational models that use a fixed, sequential order of operations, according to recent studies. The study suggests that calculations without a causal structure can perform better. Traditional computational complexity assumes operations are ordered. However, the researchers found that “causally indefinite” processing, where the order is not predetermined, can benefit particular workloads. This study of non-deterministic computational models creates a new theory.
Classical Benefit: A causally indefinite classical-deterministic computer counterpart was investigated first. The authors demonstrated how causal indefiniteness simplified deterministic searches for a Boolean function (f_{6c}). For this 6-bit function, the typical deterministic query complexity is 4, meaning a sequential model needs at least 4 queries. However, a causally indefinite classical-deterministic process based on the Lugano process may calculate the function in three queries, indicating a generalised deterministic query complexity of D^{Gen}(f_{6c}) = 3. Using a recursive structure, the (f_{6c}) function was iterated to increase the constant spacing (4 vs. 3). The recursive construction yielded a polynomial separation for {f_l}_l, proving that (D^{Gen}(f_l) = O(D(f_l)^{0.792\dots})) as (D(f_l) \to \infty). In classical computing, causal indefiniteness benefits asymptotic computing.
Extension of Quantum Advantage
Building on classical findings, the researchers investigated quantum systems. They demonstrated a consistent quantum query complexity advantage for a modified 6-bit Boolean function (f_{6q}), derived from (f_{6c}). Modified Lugano process, a causally indefinite quantum supermap, computes (f_{6q}) in 3 quantum queries (Q_E^{Gen}(f_{6q})=3), while sequential quantum supermaps require 4 queries (Q_E(f_{6q})=4)). This shows that causally indefinite supermaps reduce quantum query complexity.
Method and Effects:
Boolean function query complexity was used to compare calculations with and without causal structure. The process function formalism modelled classical-deterministic processes. Indeterminate causal order quantum operations were simulated using quantum supermaps. The Lugano process, a well-known causally indeterminate classical-deterministic process that laid the groundwork for computer models, demonstrated the benefit. Lower constraints for deterministic query complexity, such as polynomial and certificate bounds, remained valid in the generalised framework of causally indefinite classical computation, helping identify candidate functions with an advantage.
Future challenges and prospects:
Even if an asymptotic polynomial advantage was obtained in the classical setting, comparable recursive procedures could not directly amplify the constant quantum advantage acquired. The causally indeterminate quantum computation's output state is not “clean” and contains leftover input information, hindering recursive composition like a subroutine. The main impediment to quantum advantage maximisation is this. Future study will focus on cleaning up these computations to perhaps maximise causal indefiniteness for recursive amplification. An asymptotic distinction in quantum query complexity is unclear in this scenario. Researchers suggest investigating partial Boolean functions, which may have even greater benefits, and more general classical processes: non-deterministic, stochastic. This paper proves that embracing causal indefiniteness reduces query complexity for specific issues in both classical and quantum worlds, paving the way for non-deterministic computation and innovative quantum algorithms. The French National Research Agency funded the study.