Causal Indefiniteness Reduces Query Complexity for Boolean Function Evaluation

Research demonstrates a Boolean function exhibiting reduced query complexity through causally indefinite computation. This advantage, achievable with classical-deterministic processes, represents a polynomial separation from standard deterministic models, confirming the potential of causal indefiniteness to enhance computational efficiency.

The established framework of computational complexity relies on a sequential application of operations, yet recent theoretical work explores the potential benefits of relinquishing this constraint. Researchers are now investigating ‘causally indefinite’ computation, where the order of operations is not pre-defined, potentially offering advantages in specific tasks. A team led by Alastair A. Abbott, Mehdi Mhalla, and Pierre Pocreau, affiliated with the Université Grenoble Alpes, Inria, CNRS, and Grenoble INP, detail their investigation into this area in a paper entitled ‘Classical and Quantum Query Complexity of Boolean Functions under Indefinite Causal Order’. They demonstrate a Boolean function where causal indefiniteness yields a demonstrable reduction in query complexity, a result amplified to achieve a polynomial separation, and extend these findings to quantum systems.

Demonstrating a Quantum Advantage via Causal Indefiniteness in Query Complexity

This research investigates the potential for computational benefits arising from causal indefiniteness, a departure from the conventional assumption of a fixed sequential order in computational processes. Researchers demonstrate a polynomial separation in query complexity – a measure of how many times a function must be interrogated to compute its output – for a specific Boolean function. This suggests a genuine computational advantage from embracing causal indefiniteness and opens new avenues for quantum algorithm design. The study employs the framework of Boolean function query complexity to explore whether computations performed without a defined causal structure can outperform those with a definite order, establishing a theoretical foundation for exploring non-deterministic computational models.

The core of the work centres on constructing a computational model that allows operations to occur in a superposition of orders, contrasting with traditional models where each operation follows a predetermined sequence. Researchers establish that established lower bounds in deterministic query complexity – such as polynomial and certificate bounds – remain valid even within this more generalised framework, ensuring the analysis operates within established theoretical constraints. Subsequently, the team identifies a Boolean function for which causal indefiniteness enables a reduction in the number of queries required for computation, demonstrating a step towards harnessing the power of non-deterministic computation.

The research constructs a quantum supermap – a generalisation of a quantum channel allowing for multiple computational pathways – and applies it to the computation of the Boolean function (f_{6q}). By abandoning the assumption of a fixed operational order, the computation explores multiple pathways simultaneously, leveraging a supermap constructed from base components and quantum gates designed to induce this non-deterministic behaviour. Analysis confirms that this scheme achieves a demonstrable reduction in query complexity for (f_{6q}), representing a potential computational advantage over standard quantum algorithms.

Researchers achieve this separation by constructing a specific quantum supermap, denoted (S_{f6q}), applied to three copies of a query oracle, generating a quantum state containing information about the function’s value, (f_{6q}(x)). Measurement of auxiliary registers then reveals which qubit encodes the result, highlighting the critical link between measurement outcomes and the location of the function’s value within the output state, confirming successful computation.

While the initial computation isn’t entirely ‘clean’ – the output state also contains information about the input – a post-processing channel effectively isolates the desired result, demonstrating a pathway towards refining the computational process. However, the computation exhibits a limitation: it is not ‘clean’, as the output state contains residual information about the input, rather than solely the result of the function. This necessitates a post-processing channel to extract the desired output, adding complexity and hindering further optimisation.

The lack of cleanliness currently prevents recursive composition and amplification of the quantum advantage to its full potential, highlighting the need for further refinement. Future research will focus on developing techniques to improve the cleanliness of the computation, enabling recursive composition and unlocking the full potential of causal indefiniteness.

Researchers first explore a classical-deterministic analogue, providing a rigorous foundation for understanding the quantum advantage and identifying the critical role of computational cleanliness. This approach allows for a systematic investigation of the benefits and limitations of causally indefinite computation, paving the way for future advancements in the field.

This work provides evidence that embracing causal indefiniteness can lead to demonstrably more efficient quantum computations for certain problems, establishing a theoretical advantage and suggesting that exploring causally indefinite models represents a promising avenue for advancing quantum computational power. This work represents a significant step towards harnessing the power of non-deterministic computation and developing new quantum algorithms that can outperform classical algorithms.