Trace-Induced Quantum Kernels

• Trace-induced quantum kernels are similarity measures derived from quantum state embeddings using the trace of operator products, unifying global fidelity and local projected approaches.
• Generalized kernels combine fundamental Lego kernels, allowing precise control over expressivity, inductive bias, and quantum resource trade-offs.
• They underpin diverse applications from quantum time series analysis to multi-output regression, balancing performance with reduced quantum circuit demands.

A trace-induced quantum kernel is a family of similarity measures between inputs, constructed from quantum state embeddings where kernel values are given by the trace of operator products applied to density matrices. This framework encompasses global fidelity kernels, various local or projected quantum kernels, and operator-valued or entangled kernels. Trace-induced quantum kernels play a central role in quantum machine learning for encoding similarity in quantum feature spaces, with implications for expressivity, sample complexity, and quantum resource requirements (Gan et al., 2023, Shin et al., 26 Mar 2025, Huusari et al., 2021).

1. Mathematical Foundation and Definitions

Given a classical input $x\in\mathcal{X}$, a trace-induced quantum kernel is constructed by defining a data-embedding channel $U(x)$ which generates an $n$-qubit density operator: $\rho(x) = U(x)\rho_0 U(x)^\dagger,$ where $\rho_0$ is the initial state.

A trace-induced quantum kernel is defined as: $k(x,x') = \mathrm{Tr}[ F(\rho(x))\, G(\rho(x')) ],$ where $F$ and $G$ are Hermitian-preserving linear superoperators. Significant special cases include:

• Global Fidelity Quantum Kernel (GFQK):

$$k_{\mathrm{global}}(x,x') = \mathrm{Tr}[\rho(x)\rho(x')],$$

which measures the Hilbert-Schmidt inner product (state overlap) of embedded density operators (Gan et al., 2023, Shin et al., 26 Mar 2025).

• Local Projected Quantum Kernels (LPQK):

For a subset $s \subset \{1,\dots,n\}$:

$$k_{s}(x,x') = \mathrm{Tr}_s [\rho_s(x)\rho_s(x')], \quad \text{with} \quad \rho_s(x) = \mathrm{Tr}_{\bar s}[\rho(x)].$$

Averaging over all subsets $|s|=S$ yields the $S$-LPQK.

These constructions unify widely used quantum kernels and form the basis for systematic kernel design in quantum machine learning (Gan et al., 2023).

2. Generalization via Lego Kernels

All trace-induced quantum kernels can be expressed as (nonnegative) combinations of fundamental "Lego" kernels. Given an orthonormal operator basis $\{A_i\}_{i=1}^{4^n}$ (e.g., Pauli basis), a single Lego kernel is: $$k_i(x,x') = \mathrm{Tr}[\rho(x)A_i]\, \mathrm{Tr}[\rho(x')A_i].$$ A generalized trace-induced quantum kernel (GTQK) is a positive combination of Lego kernels: $$k_W(x,x') = \sum_{i=1}^{4^n} w_i\, k_i(x,x') = \sum_{i=1}^{4^n} w_i\, \mathrm{Tr}[\rho(x)A_i]\,\mathrm{Tr}[\rho(x')A_i],$$ with $w_i\ge 0$ and $\sum_i w_i^2=1$. All global and local quantum kernels correspond to specific choices of $w_i$ (Gan et al., 2023):

• GFQK: $w_i = 1/2^n$ (all features "on").
• $s$-LPQK: $w_i = 1/2^{|s|}$ if $A_i$ is supported on $s$, and zero otherwise.

A feature map perspective yields: $$k_W(x,x') = \langle \Phi_W(x), \Phi_W(x') \rangle, \;\; \Phi_W(x) = \left(\sqrt{w_i}\, \mathrm{Tr}[\rho(x) A_i ]\right)_{i=1}^{4^n}.$$

3. Expressivity, Inductive Bias, and Generalization

The representational power of a trace-induced quantum kernel is determined by the set and weights of the underlying Lego (operator) features. The associated RKHS is

$$\mathcal{H}_W = \left\{ f(\cdot) = \sum_{i=1}^{4^n} \beta_i \sqrt{w_i}\, \mathrm{Tr}[\rho(\cdot)A_i] : \sum_i \beta_i^2 < \infty \right\}.$$

The number $p = \#\{i: w_i \neq 0\}$ of active Lego features controls model complexity:

• $p=4^n$ yields full GFQK expressivity.
• Reduced $p$ (via sparse $w_i$) improves generalization by restricting capacity.

Under regularized empirical risk minimization, features with smaller $w_i$ are penalized more. The empirical Rademacher complexity of the RKHS $\mathcal{H}_W$ scales as $O(\sqrt{\sqrt{p}/N})$, so statistical complexity increases as $p^{1/4}$ (Gan et al., 2023). This explicitly links inductive bias and generalization behavior to kernel structure.

4. Systematic Kernel Complexity and Resource Trade-offs

The trace-induced kernel formalism enables a stepwise approach to increasing kernel complexity, which directly correlates with quantum resource costs:

• $H$-body LPQKs: Restrict Lego kernels to Pauli operators of weight $\leq H$:

$$k^{(H)}(x,x') = \sum_{i:\, \mathrm{wt}(P_i)\leq H} \frac{1}{\sqrt{d_H}}\, \mathrm{Tr}[\rho(x)P_i]\, \mathrm{Tr}[\rho(x')P_i],\; d_H = \sum_{h=0}^{H} \binom{n}{h}3^h.$$

This construction systematically expands capacity while controlling measurement costs.

• Resource scaling:
  • GFQK evaluation for $N$ data points requires $O(N^2)$ circuit executions, each of twice the embedding circuit depth.
  • $H$-body LPQKs need only $O(N)$ executions, with circuit depth identical to $U(x)$, since all required observables can be measured via classical shadows or similar protocols.

Empirical results on benchmark datasets demonstrate that 2-body LPQKs with $p\sim 150$ features can match GFQK classification accuracy while requiring orders of magnitude fewer gate applications and measurement shots (Gan et al., 2023).

5. Entangled, Operator-Valued, and Tensor-Kernel Perspectives

Trace-induced quantum kernels are a specific instance of the broader class of entangled tensor kernels (ETKs) (Shin et al., 26 Mar 2025). An ETK has the form

$K_C(x,y) = \langle F(x) | C | F(y) \rangle,$

where $F(x)$ is a tensor product of local feature maps, and $C$ is a positive semidefinite "core" matrix.

All embedding quantum kernels with fidelity structure,

$K(x,y) = \mathrm{Tr}[\rho(x)\rho(y)],$

can be written as ETKs, where $F(x)$ represents a trigonometric feature (e.g., $$\otimes_{k} [1, \cos\, \phi_k(x), \sin\, \phi_k(x)]$$), and $C_T$ encodes the data-independent circuit structure. The entangling structure of $C_T$ introduces global correlations not present in simple product kernels, which governs inductive bias, expressivity (Fourier spectrum), and complexity.

In the context of operator-valued kernels (Huusari et al., 2021), the partial trace construction yields matrix-valued kernels encoding output structure or input-output "entanglement". The Choi–Kraus decomposition translates these into finite-rank operator sums, compatible with classical kernel alignment and regression pipelines.

6. Applications in Time Series and Generative Models

Trace-induced quantum kernels are applied beyond standard supervised learning tasks:

• Quantum Time Series Kernels: When data is generated by a quantum hidden Markov model (QHMM), a trace-based kernel can be induced by mapping sequences $x$ to quantum states $\rho_x$, then defining similarity via a function of the trace-norm distance:

$$K(x,y) = \exp\left( -\gamma\, d_{\mathrm{tr}}(\rho_x, \rho_y)^2 \right),\;\; d_{\mathrm{tr}}(\rho, \sigma) = \tfrac12 \|\rho - \sigma\|_1,$$

which is guaranteed positive semidefinite (Mercer kernel) (Markov et al., 2023). This approach outperforms classical RBF kernels in experiments on sequence classification tasks, with improved separation as the Hilbert space dimension increases.

• Multivariate Outputs / Multi-output Regression: Operator-valued, trace-induced kernels enable modeling input-output correlations in problems with structured or high-dimensional labels. Entangled kernels learned via Riemannian optimization outperform ordinary kernel ridge regression and input-output separable kernels, especially when the number of outputs approaches or exceeds the sample size (Huusari et al., 2021).

7. Implementation Guidelines and Dequantization Strategies

Practical implementation of trace-induced quantum kernels involves choices and trade-offs:

• Select a problem-tailored circuit ansatz $U(x)$ and kernel bandwidth (data scaling).
• Start with low-weight ($H$-body) LPQKs to limit quantum resource requirements, and increase complexity if performance gains justify it.
• Use classical shadows or other efficient tomography techniques to estimate requisite observable expectation values.
• Regularize kernel complexity and model parameters (e.g., SVM penalty) via cross-validation.
• If lower-body LPQKs achieve comparable performance to GFQK on the target task, this yields significant resource savings.

For dequantization and classical simulation, one may approximate the entangled core tensor $C_T$ using low-bond-dimension matrix product operator (MPO) techniques, learn surrogate ETKs, or use randomized (Fourier) feature strategies—each exploiting the insight that kernel expressivity and computational hardness are encoded in the spectrum and structure of $C_T$ (Shin et al., 26 Mar 2025).

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Key references:

• “A Unified Framework for Trace-induced Quantum Kernels” (Gan et al., 2023)
• “Quantum Time Series Similarity Measures and Quantum Temporal Kernels” (Markov et al., 2023)
• “Entangled Kernels—Beyond Separability” (Huusari et al., 2021)
• “New perspectives on quantum kernels through the lens of entangled tensor kernels” (Shin et al., 26 Mar 2025)