Quantum Cardinality Estimation (QCardEst)

• Quantum Cardinality Estimation (QCardEst) is a quantum approach to estimate dimensions or distinct counts in complex systems, leveraging amplitude amplification and variational circuits.
• Grover-based algorithms in QCardEst use a two-phase process—rough guessing and adaptive refinement—to achieve high accuracy while reducing circuit complexity.
• Hybrid quantum machine learning and combinatorial methods in QCardEst enhance database query optimization and state verification, outperforming classical estimators in efficiency.

Quantum Cardinality Estimation (QCardEst) concerns the estimation of the dimension, population, or distinct count (“cardinality”) within a quantum context, including quantum state collections, quantum games, and quantum data processing. In database systems, QCardEst is the process of predicting the number of result rows for SQL queries via quantum computational models, critical for query optimization and resource allocation. In quantum information science, QCardEst extends to the certification of quantum sources, analysis of genuinely quantum solutions, and cardinality estimation via algorithms exploiting quantum mechanical properties, such as amplitude amplification and variational quantum circuits. This article details the principal methodologies, foundational results, algorithmic frameworks, and practical impact of QCardEst, spanning Grover-based counting, quantum machine learning hybrids, statistical estimators, and specialized structures in quantum combinatorial games.

1. Grover-Based Quantum Cardinality Estimation Algorithms

Quantum approximate counting, epitomized by the BHMT algorithm, leverages Grover iterations to estimate the cardinality $K$ of marked elements in a set of size $N$ with a relative error $\varepsilon$ using only $$O\left(\frac{1}{\varepsilon}\sqrt{\frac{N}{K}}\right)$$ queries (Aaronson et al., 2019). Aaronson and Rall present a QFT-free alternative based solely on Grover-type rotations and classical amplification, divided into a two-phase procedure:

1. Rough Guess Phase: For each $k$, perform $r_k \approx 1.05^k$ Grover iterations, progressively attaining bounds $$\theta_{\min} \approx 0.9 \cdot 1.05^{-k_\text{end}}$$ and $\theta_{\max} \approx 1.65 \cdot \theta_{\min}$ for the rotation angle $\theta=\arcsin\sqrt{K/N}$ associated with the amplitude of marked states.
2. Amplification/Refinement Phase: Iterative update of $\theta_{\min}$ and $\theta_{\max}$ via adaptive query selection, employing the Rotation Lemma to ensure sharp binary discrimination, until $\theta_{\max}/\theta_{\min} \leq 1+\varepsilon'$. The cardinality estimate is $\widehat{K}=N \cdot \sin^2(\theta_{\max})$, guaranteeing $K(1-\varepsilon) < \widehat{K} < K(1+\varepsilon)$ with high probability.

Key traits include elimination of the quantum Fourier transform (QFT), reduced circuit complexity, efficient use of Grover diffusion operators, and adaptability for amplitude estimation. The model is theoretically significant for QCardEst, making Grover-exclusive algorithms optimal in queries, circuit depth, and compatibility with near-term hardware constraints.

2. Quantum Machine Learning Approaches to Cardinality Estimation

Recent advancements define QCardEst in the context of data-driven quantum machine learning for SQL query metrics. The QCardEst algorithm described in (Winker et al., 10 Sep 2025) utilizes a hybrid quantum–classical network:

• Encoding: Each query joining $n$ tables is mapped into $n$ qubits, with table identifiers and selectivities encoded via axis-specific rotations on each qubit.
• VQC Structure: The encoding layer prepares quantum states, followed by several parametric gate layers (RY, RZ, etc.), and a measurement stage yielding a probability vector $\mathbf{x}$.
• Classical Post-Processing: The output vector is mapped to predicted cardinality using differentiable functions—linear, rational, threshold, or place-value layers—such as $v=s\cdot x_0$ or $v = \log((x_0+\varepsilon)/(x_1+\varepsilon))$.

This architecture achieves low quantum resource overhead by matching qubit count to the number of tables, ensuring feasibility on extant QPUs. The model is trained via Adam optimization in a quantum–classical loop. Comparative results show the system outperforms classical estimators in benchmarks (e.g., 6.37× improvement over PostgreSQL and 3.47× over MSCN on JOB-light).

The correction method QCardCorr applies a learned quantum factor $g(q)$ to classical estimators $f(q)$: $f(q) \cdot g(q) \approx t(q)$, further reducing error. Models employing threshold or rational layers yield maximal accuracy improvements (up to 8.66× over PostgreSQL for STATS).

3. Quantum State Identity Testing and Cardinality Scaling

Testing the identity of quantum state collections directly addresses QCardEst in quantum information science (Fanizza et al., 2021). The methodology estimates the mean squared Hilbert–Schmidt distance $\mathcal{M}^{2}_{HS}(\rho)$ between $d$-dimensional states $\{\rho_i\}$ in a collection of cardinality $N$, utilizing sample access distributions $p_i$.

• Estimator Construction: An unbiased quantum observable $\mathcal{D}_M$ over $M$ copies evaluates $\mathcal{M}^{2}_{HS}(\rho)$, efficiently computing pairwise distances with swap/transpose operators.
• Sample Complexity: Statistical distinction between identical and far cases is achieved (with high probability) using $M = O(\sqrt{N} d / \varepsilon^2)$ samples; a lower bound is proved via the Holevo–Helstrom theorem.

This paradigm showcases sublinear scaling in $N$ analogously to classical property testing, providing practical mechanisms for state stability certification and rapid diagnostics in high-dimensional quantum systems, with direct implications for quantum communication, metrology, and error synthesis.

4. Genuinely Quantum Cardinality in Quantum Combinatorial Games

Quantum cardinality in combinatorial games, exemplified by genuinely quantum solutions in SudoQ (Paczos et al., 2021), extends QCardEst to the enumeration of distinct quantum state vectors in grid-like patterns. Solutions of size $N^2$ manifest cardinality $|\mathcal{C}| > N^2$ through overcomplete vector configurations:

• State Construction: Vectors $|\psi_{ij}\rangle = U_i |e_j\rangle$ use unitaries $U_i$ to generate ensembles exceeding the classical cell-bound.
• Orthogonality and Welch Bound: The maximum inner product among distinct vectors is bounded below by $(|\mathcal{C}| - N)/[N(|\mathcal{C}| - 1)]$, with maximal cardinality constructions saturating the Welch bound.
• Mutually Unbiased Bases (MUBs): For prime $N$, maximal cardinality $N^4$ is achievable with $N+1$ MUBs, resulting in $N^3$ orthogonal measurements of optimal discrimination geometry.

These measures highlight the capacity for robust, noise-resilient estimation schemes and connect quantum cardinality estimation to foundational structures in quantum state geometry, cryptographic design, and measurement theory.

5. Encoding and Expressibility in Quantum Cardinality Models

The process of encoding classical data or structured inputs (such as SQL queries) into quantum states is crucial for effective QCardEst. As demonstrated in (Uotila, 2023) and (Winker et al., 10 Sep 2025):

• Compositional Encoding: Context-free grammars are mapped to DisCoPy diagrams, transformed through functors to pregroup diagrams, and subsequently into quantum circuits (e.g., via lambeq with IQPAnsatz).
• Expressibility & Entanglement: Expressibility is measured via KL divergence between circuit output state distributions and Haar measure, with values ($0.017$ for binary classification, $0.032$ for 4-class) indicating strong representation power at low class cardinality. Meyer–Wallach entanglement measures (e.g., $0.517$ for binary) quantify circuit capability in producing complex, entangled states.

Performance comparisons reveal that while binary QCardEst tasks match commercial estimators, multi-class tasks show accuracy limitations, indicating a trade-off between expressibility and circuit complexity as the number of classes increases.

6. Practical Implications, Performance, and Limitations

Quantum Cardinality Estimation has direct application in:

• Database Query Optimization: Quantum-enhanced cardinality estimation drives improved join ordering and resource allocation, with hybrid models (QCardEst/QCardCorr) outperforming traditional database estimators by factors up to $8.66\times$.
• Quantum Information Certification: Sublinear-sample tests enable rapid assessment of system homogeneity and error threshold monitoring.
• Quantum Game Theory & Measurement: Overcomplete quantum configurations facilitate optimal state discrimination and estimation in combinatorial and cryptographic settings.

Limitations of QCardEst include the need for adaptive measurements in Grover-based algorithms, additional classical post-processing overhead, and scalability challenges for fine-grained multi-class regression tasks. Choice of classical post-processing layer significantly affects output distribution and error correction capability.

In summary, QCardEst encapsulates a spectrum of quantum cardinality techniques, from algorithmic reductions of circuit complexity in Grover-based methods, to machine learning hybrids and overcomplete quantum state constructions, each of which advances efficient estimation and optimization in quantum computation and information processing contexts.