Complex-Valued Quantum Similarity

• Complex-valued quantum similarity is the quantification of overlaps between quantum states by retaining both amplitude and phase in complex Hilbert spaces.
• The methodology employs complex inner products and Schatten norms to accurately measure state similarities and distinguish between different quantum ensembles.
• Applications include quantum machine learning and attention mechanisms, with empirical evidence showing improved classification and kernel evaluation performance.

Complex-valued quantum similarity is the study, quantification, and exploitation of similarity relations between quantum states, operations, or data representations in complex Hilbert space. It encompasses both foundational questions in quantum information—such as the distinction between real and complex quantum structures and the optimal distinguishability of Haar-random quantum states—as well as applied domains including quantum machine learning, complex-valued neural networks, and quantum self-attention mechanisms. Unlike classical or real-valued metrics, complex-valued quantum similarity explicitly retains both amplitude and phase structure and often leverages the full complex inner product or its modulus as a quantitative measure.

1. Mathematical Foundations: Complex-Valued Inner Products and Norms

The core object in complex-valued quantum similarity is the complex inner product $\langle \psi | \phi \rangle$ between pure states in a Hilbert space. For normalized quantum states $\ket{\psi_i} = \sum_j (a_j + i b_j)\ket{j}$ and $\ket{\psi_k} = \sum_j (c_j + i d_j)\ket{j}$, the inner product is

$$\langle \psi_k | \psi_i \rangle = \sum_j (a_j c_j + b_j d_j) + i \sum_j (b_j c_j - a_j d_j).$$

This quantity is linear in $\ket{\phi}$, conjugate-linear in $\ket{\psi}$, and encodes both the overlap (real part) and phase relation (imaginary part) (Chen et al., 24 Mar 2025).

The modulus $|\langle \psi | \phi \rangle|^2$ produces a real-valued measure (the fidelity) commonly used in quantum information. More sophisticated tasks retain the full complex value, which can be decomposed as

$$S(\psi,\phi) = \Re \langle\psi|\phi\rangle + i\, \Im \langle\psi|\phi\rangle$$

by suitable measurement protocols, such as Hermitian observable measurements or photon-number interference in optical settings (Li et al., 26 Feb 2026).

For quantum operations (unitaries), similarity is quantified by the normalized Schatten 2-norm of the difference: $$\left\| U_1 - U_2 \right\|_{S_2} = \sqrt{ \frac{1}{d} \mathrm{Tr} \left( (U_1-U_2)^\dagger (U_1-U_2) \right) }$$ serving as a complex-valued metric for functional proximity of quantum channels (Chen et al., 2022). This norm relates directly to average-case functional fidelity.

2. Distinguishability and Quantum Designs: Real vs. Complex Haar-Random States

Complex-valued quantum similarity underlies the fundamental problem of distinguishing between real-valued and complex-valued quantum ensembles. For $t$ copies of a $d$-dimensional Haar-random state, define the real-Haar and complex-Haar $t$-copy density operators

$$R^{(O)}_{t,d} = \int_{O\in O(d)} \left(O|0\rangle\langle 0| O^T \right)^{\otimes t}\, d\mu_O, \quad R^{(U)}_{t,d} = \int_{U\in U(d)} \left(U|0\rangle\langle 0| U^\dagger \right)^{\otimes t}\, d\mu_U.$$

Their trace distance quantifies optimal distinguishability: $$D_{t,d} = \frac12 \| R^{(O)}_{t,d} - R^{(U)}_{t,d} \|_1,$$ where $D_{t,d}$ has closed-form expressions involving irreducible representations and admits well-characterized asymptotics:

• For $t\ll \sqrt{d}$, $D_{t,d}\sim \frac{t(t-1)}{2d}\to 0$.
• For $t\sim \alpha\sqrt{d}$, $D_{t,d}\to 1-e^{-\alpha^2/2}$.
• For $t\gg \sqrt{d}$, $D_{t,d}\to 1$.

This crossover delineates a fundamental "complex-valued quantum similarity" transition between indistinguishability and perfect separability (Nemoz et al., 22 Jul 2025).

These results yield lower bounds on the approximation of Haar moments by real $t$-designs: any real $t$-design ensemble has total variation distance at least $D_{t,d}$ from the complex Haar ensemble, enforcing $t=o(\sqrt{d})$ for small error. Imaginarity detection—that is, distinguishing real from complex states—requires $\Omega(\sqrt{d})$ copies.

3. Efficient Estimation Protocols and Information-Theoretic Limits

Complex-valued quantum similarity metrics can be estimated optimally both algorithmically and physically. For Hilbert-space vectors, photonic protocols use bosonic interference via balanced beam splitters and parity measurements. If $\rho^{(A)}$ and $\rho^{(B)}$ are the density matrices of two registers, then interference plus measurement yields

$$\mathrm{Tr}[ \rho^{(A)} \rho^{(B)} ] = \mathrm{Tr}[ U_{\mathrm{BS}}^\dagger ( \rho^{(A)} \otimes \rho^{(B)} ) U_{\mathrm{BS}} \hat{\Pi} ]$$

with $\hat{\Pi}$ the total photon-number parity operator (Li et al., 26 Feb 2026). For pure inputs, this extracts $|\langle \psi | \phi \rangle|^2$.

The sample complexity for precision $\epsilon$ is $O(\epsilon^{-2})$, independent of Hilbert-space dimension, saturating the Helstrom bound for state discrimination.

For quantum channels, the Schatten-2 estimator is realized by quantum sampling circuits: random preparation of sampling states, Hadamard tests for $$\Re \langle x(\theta)|U_1U_2^\dagger|x(\theta) \rangle$$, and estimation of the norm via repeated sampling. The sample complexity retains $O(\epsilon^{-2})$ scaling and is independent of system size (Chen et al., 2022).

4. Complex-Valued Quantum Similarity in Machine Learning Architectures

Quantum and quantum-inspired machine learning frameworks exploit complex-valued similarity both as a mathematical construct and as an architectural primitive.

Generalized Quantum Similarity Learning (GQSim) encodes classical data as quantum states via parameterized circuits and defines similarity as the (possibly asymmetric) trace overlap or partial trace overlap between embeddings: $$\mathcal{S}^m_{\theta, \eta}(x, \tilde{x}) = \mathrm{Tr} \left\{ \ket{0^m}\bra{0^m} \mathrm{Tr}_{n-m}[ | \phi^{\theta, \eta}_{x, \tilde{x}} \rangle \langle \phi^{\theta, \eta}_{x, \tilde{x}} | ] \right\}$$ with $$|\phi^{\theta, \eta}_{x, \tilde{x}} \rangle = U_\theta(x)^\dagger V_\eta(\tilde{x}) |0^n\rangle$$. The full complex nature of the similarity is crucial for expressivity and for discriminating datasets distinguished only by phase relations. Asymmetry and intransitivity are permitted, enabling richer similarity structures suitable for heterogeneous or directed data (Radha et al., 2022).

Quantum Complex-Valued Self-Attention Model (QCSAM) defines the attention coefficient as the full complex-valued inner product: $$w_{i,k} = \langle K_i | Q_k \rangle = \sum_j (c_j - i d_j)(a_j + i b_j)$$ and utilizes the Complex Linear Combination of Unitaries (CLCUs) to construct complex-weighted combinations of states. This approach statistically outperforms real-valued-only quantum attention mechanisms on standard benchmarks, confirming the utility of retaining phase information (Chen et al., 24 Mar 2025).

5. Interpretability and Representation: Complex Quantum Similarity in Linguistic and Statistical Models

Complex-valued quantum similarity is conceptually leveraged in models such as the Complex-valued Network for Matching (CNM), where language units are embedded as quantum pure states and sentences as mixed states. The network computes measurement probabilities via Born’s rule and matches sequences by comparing resulting feature-activation distributions. Max-pooled measurement statistics define real vectors, which are then compared by cosine similarity, aligning with quantum measurement interpretations (Li et al., 2019).

This demonstrates a pipeline from complex-valued quantum similarity to interpretable, physically rooted matching criteria, even in classical data domains.

6. Applications and Empirical Observations

Practical applications of complex-valued quantum similarity span:

• Quantum data classification: Kernel methods with quantum kernels $K(x,y) = |\langle \psi(x) | \psi(y) \rangle|^2$ for SVMs and other classifiers (Li et al., 26 Feb 2026).
• Quantum circuit learning: Variational circuit synthesis with Schatten-2 loss or direct overlap-based loss functions (Chen et al., 2022).
• Graph completion and generative tasks: Use of learned quantum similarity as a criterion for link prediction or for generating maximally similar data in heterogeneous spaces (Radha et al., 2022).
• Quantum neural attention mechanisms: Enhanced accuracy across multiple datasets, as phase-rich complex similarity measures capture fine-grained distinctions lost by real-valued overlaps (Chen et al., 24 Mar 2025).

Empirical studies consistently demonstrate that retaining and utilizing the complex-valued structure leads to superior classification, learning, and data-structuring accuracy compared to metrics that discard phase.

7. Theoretical Limits, Resource Requirements, and Future Directions

Analysis of distinguishability between real and complex Haar ensembles provides sharp asymptotic regimes for when complex structure becomes operationally significant (i.e., $t \gtrsim \sqrt{d}$ for near-perfect distinguishability) (Nemoz et al., 22 Jul 2025). Lower bounds for state design and imaginarity testing are imposed by these results.

Experimentally, information-theoretic optimality is achieved in photonic implementations: the O($\epsilon^{-2}$) scaling and dimension-independence are realized on platforms such as Prakash-1, enabling direct quantum kernel evaluation and efficient online training (Li et al., 26 Feb 2026).

A plausible implication is that advances in the precision, scalability, and architecture of quantum devices will further enhance the exploitation of complex-valued quantum similarity in both foundational and practical contexts, particularly in high-dimensional, phase-sensitive quantum learning applications.