Exponential quantum space advantage for Shannon entropy estimation in data streams

Abstract: Near-term quantum devices with limited qubits motivate the study of space-bounded quantum computation in the data stream model. We show that Shannon entropy estimation exhibits an exponential separation between quantum and classical space complexity in this setting. Technically, we develop a two-stage quantum streaming algorithm based on a quantum procedure with an explicitly constructed oracle derived from the streaming input. This algorithm achieves logarithmic space complexity in the accuracy parameter, whereas any classical streaming algorithm requires polynomial space. In sharp contrast, existing results for Shannon entropy estimation in the quantum query model achieve only a quadratic speedup. Our work establishes a natural problem with practical applications in computer networking that admits an exponential quantum space advantage, revealing a fundamental gap between quantum query complexity and streaming space complexity.

• The paper proves an exponential quantum-classical space separation for Shannon entropy estimation in data streams.
• It introduces a two-stage quantum streaming algorithm combining majority detection and quantum amplitude estimation to optimize performance.
• It establishes a classical lower bound via Gap Hamming Distance reduction, underscoring practical benefits in memory-constrained applications.

Exponential Quantum Space Advantage for Shannon Entropy Estimation in Streaming Data

Background and Motivation

This paper addresses a fundamental question in quantum computing: the potential for exponential quantum-classical separations in space complexity within the data stream model. Prior quantum advantages have been established primarily in time complexity, query complexity, and communication complexity. However, quantum space complexity, particularly in streaming algorithms for natural problems, remains largely unexplored. Shannon entropy estimation is a core task in information theory and deeply relevant in practical domains such as network anomaly detection and traffic analysis. While classical streaming algorithms for entropy estimation are well-established, quantum streaming algorithms—particularly those achieving exponential advantages—have not been systematically analyzed.

Computational Models and Problem Formulation

The manuscript formalizes Shannon entropy estimation in the classical and quantum streaming models over a stream $A = \langle x_1, x_2, ..., x_m \rangle$ sampled from an alphabet $[n]$. The objective is to approximate the empirical entropy

$H(p) = \sum_{i=1}^n -p_i \log p_i,$

where $p_i$ is the frequency of symbol $i$ divided by $m$. Classical streaming algorithms are assessed in terms of bits; quantum algorithms in qubits, with the quantum model permitting unitary updates and measurements on limited workspace.

Main Technical Results

Exponential Space Separation

The central result is the proof of an exponential separation between quantum and classical space complexity for entropy estimation in the streaming setting. The quantum algorithm achieves an $(\varepsilon, \delta)$ approximation using $O(\log(1/\varepsilon))$ qubits and $O(1/\varepsilon)$ passes, while the classical lower bound requires $$\Omega\left(\frac{1}{T \varepsilon^2 \log^2(1/\varepsilon)}\right)$$ bits for $[n]$0 passes—space polynomially dependent on $[n]$1.

This separation is notably stronger than results in the quantum query model, which offer only a quadratic speedup for Shannon entropy estimation [titLiW19, stocBunKT18, GilyenL20]. The quantum streaming algorithm thus achieves exponential space efficiency for an applied information-theoretic task.

Quantum Streaming Algorithm Construction

The authors construct a two-stage quantum streaming algorithm. The first stage detects the presence of a majority element using the Boyer-Moore algorithm. The second stage adapts entropy estimation, either directly (when no majority item exists) or via stream reduction (removing the majority element).

A key technical contribution is the construction of an implementable quantum oracle, explicitly derived from the streaming input, which enables quantum query procedures within the streaming paradigm. The quantum algorithm reduces entropy estimation to the expectation of a random variable computed from suffix counts in the stream. Quantum amplitude estimation is used to efficiently approximate this expectation. The space complexity is determined by the encoding size—$[n]$2 qubits.

Classical Lower Bound

By reducing the Gap Hamming Distance (GHD) problem with its known communication complexity lower bounds to entropy estimation, the authors demonstrate that any classical multi-pass streaming algorithm with multiplicative error $[n]$3 requires $[n]$4 bits of space. This reduction leverages the entropy of a constructed concatenated stream revealing GHD distance, enforcing a polynomial space requirement.

Numerical and Structural Claims

• The quantum algorithm achieves logarithmic space complexity in $[n]$5.
• Classical algorithms require nearly linear space in $[n]$6 for comparable accuracy.
• The separation persists under $[n]$7 passes; increasing passes reduces classical space, but never below exponential quantum advantage.

Implications and Future Directions

Practical implications include the prospect of deploying near-term quantum devices, constrained in qubit count, for tasks such as real-time network monitoring and anomaly detection, where classical streaming algorithms face stringent memory limitations.

Theoretical implications pertain to the realization that quantum streaming algorithms, with explicit query-to-streaming transformations, can deliver provable exponential advantages for natural problems. This opens a promising direction for investigating quantum advantages for other entropy-like measures (Rényi, Tsallis), tasks on matrix or higher-dimensional data, and extensions to single-pass streaming models.

Recent work on exponential quantum advantages in matrix processing under sampling access [zhao2026exponential] underscores the relevance and distinctness of the streaming model.

Conclusion

This paper rigorously establishes an exponential quantum-classical space complexity separation for Shannon entropy estimation in data streams (2604.18014). The explicit quantum streaming algorithm is logspace-efficient, contrasting sharply with classical polynomial space requirements. The results deepen our understanding of quantum advantages beyond traditional time and query complexity and underscore the viability of quantum algorithms for real-world, memory-constrained data analysis. Future research may extend these techniques to broader classes of information-theoretic and streaming problems, advancing both the practical utility and foundational knowledge of quantum computation.