Support Vector Machines with Quantum Kernels (QSVM)

Support Vector Machines (SVMs) are powerful supervised learning models used for classification and regression tasks. A primary strength of SVMs lies in their use of the "kernel trick." Instead of explicitly mapping data into a high-dimensional feature space, SVMs rely on a kernel function $k(x, x')$ that computes the inner product of data points $x$ and $x'$ in that feature space. This allows SVMs to find complex, non-linear decision boundaries efficiently.

Classical SVMs utilize kernels like the linear, polynomial, or Radial Basis Function (RBF) kernel. The core idea behind the Quantum Support Vector Machine (QSVM) is straightforward: replace the classical kernel function with a quantum kernel computed using a quantum feature map. As discussed earlier in this chapter, a quantum feature map $\phi$ encodes classical data $x$ into a quantum state $|\phi(x)\rangle$ residing in a potentially vast Hilbert space $\mathcal{H}$. The quantum kernel is then defined based on the similarity between these quantum states, typically as the squared overlap (fidelity):

$$k_q(x, x') = |\langle\phi(x)|\phi(x')\rangle|^2$$

Other definitions based on the inner product $\langle\phi(x)|\phi(x')\rangle$ are also sometimes used, ensuring the resulting kernel matrix is positive semi-definite, a requirement for the underlying SVM optimization problem.

Why Use a Quantum Kernel?

The motivation for QSVM stems from the hypothesis that quantum feature maps can map classical data into Hilbert spaces where the data becomes more easily separable, potentially offering an advantage over classical kernels for certain datasets. The structure of the quantum feature space $\mathcal{H}$, influenced by entanglement and quantum interference generated by the feature map circuit $U_\phi(x)$, might capture complex correlations in the data that are inaccessible to standard classical kernels. If a quantum feature map provides a representation where a linear separator (in $\mathcal{H}$) performs significantly better than what's achievable with classical kernels in their respective feature spaces, QSVM could outperform classical SVMs.

However, proving such an advantage is non-trivial and depends heavily on the choice of the feature map and the specific problem domain. The potential for quantum advantage here lies entirely within the representation power of the quantum feature space accessed via the kernel, not in speeding up the SVM optimization itself.

The QSVM Algorithm

The QSVM algorithm operates as a hybrid quantum-classical procedure:

1. Quantum Kernel Computation: The computationally intensive part delegated to the quantum computer (or simulator) is the estimation of the kernel matrix $K$, where each entry $K_{ij}$ is given by $k_q(x_i, x_j)$ for pairs of training data points $(x_i, x_j)$.
2. Classical SVM Optimization: Once the quantum kernel matrix $K$ is estimated, it is fed into a standard classical SVM solver.

Let's break down the kernel computation step further. To estimate a single kernel entry $k_q(x_i, x_j) = |\langle\phi(x_i)|\phi(x_j)\rangle|^2$, we need a quantum circuit that measures the overlap or fidelity between the states $|\phi(x_i)\rangle = U_\phi(x_i)|0\rangle^{\otimes n}$ and $|\phi(x_j)\rangle = U_\phi(x_j)|0\rangle^{\otimes n}$. Common techniques include:

• Hadamard Test: Uses an auxiliary qubit to estimate the real and imaginary parts of the inner product $\langle\phi(x_i)|\phi(x_j)\rangle$.
• SWAP Test: Employs controlled-SWAP gates to estimate the squared overlap $|\langle\phi(x_i)|\phi(x_j)\rangle|^2$ directly, though often less practical for fidelity estimation compared to other methods.
• Compute-Uncompute Method (Overlap Test): A frequently used approach involves constructing a circuit that prepares $|\phi(x_i)\rangle$, applies the inverse of the feature map for $x_j$, $U_\phi(x_j)^\dagger$, and measures the probability of returning to the all-zero state $|0\rangle^{\otimes n}$. This probability is exactly $|\langle 0|^{\otimes n} U_\phi(x_j)^\dagger U_\phi(x_i)|0\rangle^{\otimes n}|^2 = |\langle\phi(x_j)|\phi(x_i)\rangle|^2$.

Workflow for training a Quantum Support Vector Machine (QSVM). Quantum resources are used to estimate the kernel matrix entries, which are then processed by a classical SVM algorithm. Prediction also requires quantum kernel evaluations between the new data point and the support vectors.

Each estimation requires multiple measurements (shots) on the quantum computer to obtain a statistically reliable estimate of the probability. Calculating the full $N \times N$ kernel matrix for a training set of size $N$ requires $O(N^2)$ calls to the quantum kernel estimation subroutine.

Mathematical Formulation Recap

Recall the dual formulation of the classical soft-margin SVM for binary classification:

$$\max_{\alpha} \sum_{i=1}^N \alpha_i - \frac{1}{2} \sum_{i=1}^N \sum_{j=1}^N \alpha_i \alpha_j y_i y_j k(x_i, x_j)$$

subject to:

$$\sum_{i=1}^N \alpha_i y_i = 0$$ $$0 \le \alpha_i \le C, \quad \text{for } i=1, \dots, N$$

Here, $y_i \in \{-1, +1\}$ are the class labels, $\alpha_i$ are the Lagrange multipliers (non-zero only for support vectors), $C$ is the regularization parameter, and $k(x_i, x_j)$ is the kernel function.

In QSVM, we simply replace $k(x_i, x_j)$ with the quantum kernel $k_q(x_i, x_j)$ estimated using one of the quantum circuit methods mentioned above. The optimization problem remains a convex quadratic program that is solved classically using standard numerical optimization libraries.

Once the optimal $\alpha_i^*$ and the bias $b^*$ are found, the prediction for a new data point $x_{new}$ is given by:

$$y_{pred} = \text{sign} \left( \sum_{i=1}^N \alpha_i^* y_i k_q(x_i, x_{new}) + b^* \right)$$

Notice that prediction also requires computing quantum kernel entries, specifically between the new data point $x_{new}$ and all the support vectors $x_i$ (those with $\alpha_i^* > 0$).

Implementation Considerations

• Feature Map Choice: The success of QSVM hinges critically on designing an effective quantum feature map $U_\phi(x)$. A poorly chosen map might lead to a feature space where data is less separable than with classical kernels, or it might suffer from issues like kernel concentration (discussed in the previous section).
• Kernel Estimation Accuracy: Since quantum measurements are probabilistic, $k_q(x_i, x_j)$ is estimated with finite precision, dependent on the number of shots. Measurement noise and gate errors on real quantum hardware further degrade this accuracy. The impact of inaccuracies in the kernel matrix on the SVM solution needs careful consideration.
• Computational Cost: Estimating the $N^2$ entries of the kernel matrix can be demanding, especially if many shots are needed per entry for sufficient accuracy. This often limits QSVM application to relatively small datasets in the near term.
• Classical Baseline: It's essential to compare QSVM performance against well-tuned classical SVMs with standard kernels (like RBF) to assess whether the quantum approach provides any practical benefit for a given problem.

QSVM represents a way to integrate quantum feature spaces into a well-established machine learning framework. While practical advantages remain an active area of research, understanding its mechanism provides valuable insight into the potential and challenges of quantum kernel methods. The next section details the implementation and challenges when applying QSVM to complex datasets.