Quantum Kernel Machines

Last updated: June 11, 2025

Quantum Gaussian Kernels ° in Quantum Support Vector Machines ° (citing “Gaussian Kernel ° in Quantum Learning” (Bishwas et al., 2017 ° ))

1. From the classical RBF to a quantum feature map

The classical Gaussian (a.k.a. radial-basis-function, RBF °) kernel [ K_{\mathrm{RBF}}(\mathbf{x}i,\mathbf{x}_j)= \exp!\Bigl[-|\mathbf{x}_i-\mathbf{x}_j|^{{2}/(2\sigma{2})\Bigr]} \tag{1} ] can be rewritten, via its Maclaurin expansion, as an infinite-degree polynomial kernel ° [ K{\mathrm{RBF}}(\mathbf{x}i,\mathbf{x}_j)= \sum{l=0}^{{\infty}\frac{\langle\mathbf{x}_i,\mathbf{x}_j\rangle{\,l}}{l!}} =e^{{\langle\mathbf{x}_i,\mathbf{x}_j\rangle}.} \tag{2} ]

“Gaussian Kernel in Quantum Learning” shows how to replicate (2) on a quantum computer:

1. State preparation ° – encode each data vector in amplitude form

$\displaystyle |X_k\rangle=\frac{1}{\|\mathbf{x}_k\|}\sum_{p=1}^{N}x_{k,p}|p\rangle.$

1. Inner-product estimation – obtain $\langle X_i|X_j\rangle$ with a swap test °.
2. Exponential build-up – approximate $\exp[\langle X_i|X_j\rangle]$ by truncating the Taylor series ° at degree $d$; only $d$ controlled-inner-product evaluations are required.

With the obvious identification $|X_{i/j}\rangle\longleftrightarrow\mathbf{x}_{i/j}$, the resulting quantum Gaussian kernel [ K^{{q}_{\mathrm{GK}}\bigl(|X_i\rangle,|X_j\rangle\bigr)=\exp!\bigl[\langle} X_i|X_j\rangle\bigr] \tag{3} ] or, after rescaling the input width, [ K^{{q}_{\mathrm{GK}}=\exp!\Bigl[-|X_i-X_j|{2}/(2\sigma{2})\Bigr],} \tag{4} ] is mathematically identical to (1), yet computable with quantum resources ° (Bishwas et al., 2017 ° ).

2. Runtime complexity

If all $N$ components of every $\mathbf{x}_k$ are available in quantum random-access memory (QRAM °), state loading scales as [ T_{\text{prep}}=O(\log N) \qquad\text{per vector.} ]

Let $d$ be the truncation order ° chosen so that the Taylor tail $R_d<\varepsilon$. Using swap-test–based inner-product evaluation, the total gate complexity ° per kernel entry becomes [ T_{\text{quantum}}=O!\bigl(\varepsilon^{-1} d \log N\bigr), \tag{5} ] where the $\varepsilon^{-1}$ factor comes from amplitude-estimation precision [(Bishwas et al., 2017 ° ), Sec. 3].

Classically, evaluating (2) to the same truncation order requires [ T_{\text{classical}}=O(dN) \tag{6} ] per entry. Comparing (5) and (6)

[ \boxed{T_{\text{classical}}/T_{\text{quantum}} =\Theta!\bigl(N/(\varepsilon^{-1}\log N)\bigr)}, ]

i.e. an exponential speed-up in data dimension $N$, provided QRAM is available.

3. Assembling the kernel matrix

For $M$ training examples the kernel Gram matrix ° has $M^{2}$ entries.

The cubic-in-$M$ term of SVM ° training ($O(M^{3})$) is the same in both worlds, but the bottleneck $M^{2}N$ classical kernel evaluations is replaced by $M^{2}\log N$ quantum ones, giving an exponential reduction in the feature dimension °.

4. Role of QRAM

QRAM supplies the coherent state [ \sum_{p}x_{k,p}|p\rangle \quad\text{in}\;O(\log N)\;\text{time}, ] enabling (5). Without QRAM, one must load amplitudes sequentially, forfeiting the speed-up. Hence QRAM (or another fast state-preparation scheme) is the critical hardware assumption in (Bishwas et al., 2017 ° ).

5. Implications for quantum SVMs

Replacing the classical kernel in a least-squares SVM with (3) yields a quantum LS-SVM ° whose total runtime is [ O!\bigl(M^{{3}+M{2}\varepsilon{-1}d\log} N\bigr) \tag{7} ] versus $O(M^{3}+M^{2}dN)$ classically [(Bishwas et al., 2017 ° ), Sec. 4]. As $N$ grows, (7) dominates by only a poly-log factor, providing asymptotic advantage for high-dimensional data.

6. Practical considerations

• Truncation order $d$. Because the Maclaurin coefficients decrease factorially, $d=7\text{–}9$ already makes $R_d\ll10^{-3}$ for typical inner-product magnitudes [(Bishwas et al., 2017 ° ), Fig. 3].
• Error tolerance. The amplitude-estimation overhead $\varepsilon^{-1}$ in (5) applies uniformly to every kernel entry; in practice one balances $\varepsilon$ with shot noise ° and SVM regularisation °.
• Non-parametric expressivity. The Gaussian kernel corresponds to an infinite-dimensional feature space; its quantum realisation therefore inherits the strong universal approximation properties ° prized in classical SVMs ° while scaling exponentially better in $N$.
• Hardware roadmap. The algorithm requires: QRAM of size $O(N)$, swap-test circuits of depth poly$(\log N)$, and coherent repetition $d$ times. These ingredients make the quantum Gaussian kernel a realistic target for early fault-tolerant machines.

7. Conclusion

The work of Prasanna Venkatesh et al. (Bishwas et al., 2017 ° ) provides the first end-to-end blueprint for quantum-accelerated Gaussian kernels, showing:

• a faithful reconstruction of the ubiquitous RBF kernel ° inside a quantum computer,
• exponential improvement $O(N)\rightarrow O(\log N)$ in data-dimension scaling when QRAM is available,
• seamless integration ° into LS-SVM training.

This establishes Gaussian-kernel quantum SVMs as a promising pathway to demonstrable quantum advantage ° on high-dimensional, nonlinear learning problems once scalable QRAM and low-depth swap-test primitives become experimentally accessible.