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arXiv:2506.05187 · Electronic Proceedings in Theoretical Computer Science

Classical and Quantum Query Complexity of Boolean Functions under Indefinite Causal Order

Alastair A. Abbott, Mehdi Mhalla, Pierre Pocreau

Published Aug 19, 2025• 0 citations
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Abstract

Computational models typically assume that operations are applied in a fixed sequential order.In recent years several works have looked at relaxing this assumption, considering computations without any fixed causal structure and showing that such "causally indefinite" computations can provide advantages in various tasks.Recently, the quantum query complexity of Boolean functions has been used as a tool to probe their computational power in a standard complexity theoretic framework, but no separation in exact query complexity has thus-far been found.In this paper, we investigate this problem starting with the simpler and fully classical notion of deterministic query complexity of Boolean functions, and using classical-deterministic processes -which may exhibit causal indefiniteness -as a generalised computational framework.We first show that the standard polynomial and certificate lower bounds of deterministic query complexity also hold in such generalised models.Then, we formulate a Boolean function for which causal indefiniteness permits a reduction in query complexity and show that this advantage can be amplified into a polynomial separation.Finally, with the insights gained in the classical-deterministic setting, we give a Boolean function whose quantum query complexity is reduced by causally indefinite computations.

DOI
10.4204/eptcs.426.11

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Computational models typically assume that operations are applied in a fixed sequential order.In recent years several works have looked at relaxing this assumption, considering computations without any fixed causal structure and showing that such "causally indefinite" computations can provide advantages in various tasks.Recently, the quantum query complexity of Boolean functions has been used as a tool to probe their computational power in a standard complexity theoretic framework, but no separation in exact query complexity has thus-far been found.In this paper, we investigate this problem starting with the simpler and fully classical notion of deterministic query complexity of Boolean functions, and using classical-deterministic processes -which may exhibit causal indefiniteness -as a generalised computational framework.We first show that the standard polynomial and certificate lower bounds of deterministic query complexity also hold in such generalised models.Then, we formulate a Boolean function for which causal indefiniteness permits a reduction in query complexity and show that this advantage can be amplified into a polynomial separation.Finally, with the insights gained in the classical-deterministic setting, we give a Boolean function whose quantum query complexity is reduced by causally indefinite computations.
The standard model of quantum circuits assumes operations are applied in a fixed sequential “causal” order. In recent years, the possibility of relaxing this constraint to obtain causally indefinite computations has received significant attention. The quantum switch, for example, uses a quantum system to coherently control the order of operations. Several computational and information-theoretical advantages have been demonstrated, raising questions as to whether advantages can be obtained in a more unified complexity theoretic framework. In this paper, we approach this problem by studying the query complexity of Boolean functions under general higher-order quantum computations. To this end, we generalize the framework of query complexity from quantum circuits to quantum supermaps to compare different models on an equal footing. We show that the recently introduced class of quantum circuits with quantum control of causal order cannot lead to any reduction in query complexity, and that any potential advantage arising from causally indefinite supermaps can be bounded by the polynomial method, as is the case with quantum circuits. Nevertheless, we find some functions for which the minimum error with which they can be computed using two queries is strictly lower when exploiting causally indefinite supermaps. Published by the American Physical Society 2024
Quantum theory is consistent with a computational model permitting black-box operations to be applied in an indefinite causal order, going beyond the standard circuit model of computation. The quantum switch -- the simplest such example -- has been shown to provide numerous information-processing advantages. Here, we prove that the action of the quantum switch on two nnn-qubit quantum channels cannot be simulated deterministically and exactly by any causally ordered quantum circuit that uses MMM calls to one channel and one call to the other, if M≤max⁡(2,2n−1)M \leq \max(2, 2^n-1)M≤max(2,2n−1). This demonstrates an exponential separation in quantum query complexity of indefinite causal order compared to standard quantum circuits.
Query complexity provides a basic framework to quantify the inherent computational cost of functions in the model where access to the input is limited to oracle queries. In this paper, we provide a detailed discussion of classical and quantum models of inquiry, emphasizing the structural differences and the common mathematical basis. We consider deterministic, randomized and quantum query complexities and highlight key techniques to prove lower bounds such as adversarial techniques, polynomial approximations and hybrid arguments . The main theme of this paper is the polynomial-spectral relation which underlies both the classical and the quantum models. In the classical setting, the decision tree complexity is closely related to the degree of real polynomials representing Boolean functions. In the quantum setting, the relation involves the spectral properties of the operators controlling quantum query algorithms, and gives deep insights into quantum accelerations. We study tight bounds for basic problems such as search and element distinctness, proving separations between classical and quantum query complexity. The present study uses methods from approximation theory, linear algebra and spectrum analysis to build a foundation for a complete understanding of the power and limitations of query-based computation.
It is well known that quantum, randomized and deterministic (sequential) query complexities are polynomially related for total boolean functions. We find that significantly larger separations between the parallel generalizations of these measures are possible. In particular, 1) We employ the cheatsheet framework to obtain an unbounded parallel quantum query advantage over its randomized analogue for a total function, falsifying a conjecture of [https://arxiv.org/abs/1309.6116]. 2) We strengthen 1 by constructing a total function which exhibits an unbounded parallel quantum query advantage despite having no sequential advantage, suggesting that genuine quantum advantage could occur entirely due to parallelism. 3) We construct a total function that exhibits a polynomial separation between 2-round quantum and randomized query complexities, contrasting a result of [https://arxiv.org/abs/1001.0018] that there is at most a constant separation for 1-round (nonadaptive) algorithms. 4) We develop a new technique for deriving parallel quantum lower bounds from sequential upper bounds. We employ this technique to give lower bounds for Boolean symmetric functions and read-once formulas, ruling out large parallel query advantages for them. We also provide separations between randomized and deterministic parallel query complexities analogous to items 1-3.
We study classical query algorithms with post-selection, and find that they are closely connected to rational functions with nonnegative coefficients. We show that the post-selected classical query complexity of a Boolean function is equal to the minimal degree of a rational function with nonnegative coefficients that approximates it (up to a factor of two). For post-selected quantum query algorithms, a similar relationship was shown by Mahadev and de Wolf, where the rational approximations are allowed to have negative coefficients. Using our characterisation, we find an exponentially large separation between post-selected classical query complexity and post-selected quantum query complexity, by proving a lower bound on the degree of rational approximations (with nonnegative coefficients) to the Majority function. This lower bound can be generalised to arbitrary symmetric functions, and allows us to find an unbounded separation between non-deterministic quantum and post-selected classical query complexity. All lower bounds carry over into the communication complexity setting. We show that the zero-error variants of post-selected query algorithms are equivalent to non-deterministic classical query algorithms, which in turn are characterised by nonnegative polynomials, and for some problems require exponentially more queries to the input than their bounded-error counterparts. Finally, we describe a post-selected query algorithm for approximating the Majority function, and an efficient query algorithm for approximate counting.
It is well known that quantum, randomized and deterministic (sequential) query complexities are polynomially related for total boolean functions. We find that significantly larger separations between the parallel generalizations of these measures are possible. In particular, 1) We employ the cheatsheet framework to obtain an unbounded parallel quantum query advantage over its randomized analogue for a total function, falsifying a conjecture of [https://arxiv.org/abs/1309.6116]. 2) We strengthen 1 by constructing a total function which exhibits an unbounded parallel quantum query advantage despite having no sequential advantage, suggesting that genuine quantum advantage could occur entirely due to parallelism. 3) We construct a total function that exhibits a polynomial separation between 2-round quantum and randomized query complexities, contrasting a result of [https://arxiv.org/abs/1001.0018] that there is at most a constant separation for 1-round (nonadaptive) algorithms. 4) We develop a new technique for deriving parallel quantum lower bounds from sequential upper bounds. We employ this technique to give lower bounds for Boolean symmetric functions and read-once formulas, ruling out large parallel query advantages for them. We also provide separations between randomized and deterministic parallel query complexities analogous to items 1-3.