[Paper Review] Generating quantum feature maps for SVM classifier

Bang-Shien Chen, Jann–Long Chern|arXiv (Cornell University)|Jul 23, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes two methods—genetic algorithm with multi-objective optimization and variational quantum circuits using unitary matrix decomposition—for generating quantum feature maps in quantum-enhanced support vector machines. It achieves high classification accuracy (up to 100%) while significantly reducing quantum gate cost, with optimized circuits reaching gate counts as low as 3 without sacrificing performance.

ABSTRACT

We present and compare two methods of generating quantum feature maps for quantum-enhanced support vector machine, a classifier based on kernel method, by which we can access high dimensional Hilbert space efficiently. The first method is a genetic algorithm with multi-objective fitness function using penalty method, which incorporates maximizing the accuracy of classification and minimizing the gate cost of quantum feature map circuit. The second method uses variational quantum circuit, focusing on how to contruct the ansatz based on unitary matrix decomposition. Numerical results and comparisons are presented to demonstrate how the fitness fuction reduces gate cost while remaining high accuracy and conducting circuit through unitary matrix obtains even better performance. In particular, we propose some thoughts on reducing and optimizing the gate cost of a circuit while remaining perfect accuracy.

Motivation & Objective

• To develop efficient quantum feature maps that enhance support vector machine classification using quantum kernel methods.
• To minimize quantum circuit gate cost while maintaining high classification accuracy, addressing noise and resource constraints in NISQ devices.
• To compare the performance of genetic algorithm-based optimization with variational quantum circuit approaches using unitary decomposition.
• To optimize and reduce circuit complexity through systematic gate pruning and parameter analysis.
• To demonstrate that low-depth, low-gate-cost circuits can achieve perfect classification accuracy on benchmark datasets.

Proposed method

• Uses a multi-objective genetic algorithm with penalty method to simultaneously maximize classification accuracy and minimize quantum circuit gate cost.
• Encodes quantum feature map generation as a combinatorial optimization problem over quantum circuit parameters and structure.
• Employs a fitness function that balances accuracy and circuit depth, with penalty terms for high gate counts.
• Applies variational quantum circuits with hardware-efficient ansatz and unitary decomposition-based ansatz to learn optimal feature maps.
• Uses unitary matrix decomposition to construct ansatz that can express complex quantum operations with controlled gate sequences.
• Performs circuit optimization by testing gate necessity, removing idle qubits, and eliminating redundant gates such as identity rotations and unentangled operations.

Experimental results

Research questions

• RQ1 Can a genetic algorithm with multi-objective fitness function effectively balance classification accuracy and quantum circuit gate cost in quantum feature map generation?
• RQ2 How does the performance of variational quantum circuits with unitary decomposition ansatz compare to genetic algorithm-based approaches in terms of accuracy and circuit efficiency?
• RQ3 To what extent can quantum circuits be optimized by removing redundant gates and idle qubits without affecting classification performance?
• RQ4 Can low-gate-cost circuits (e.g., gate count ≤ 3) achieve perfect classification accuracy on non-trivial datasets like moonshape and ad hoc?
• RQ5 How does the proposed method compare to existing quantum kernel methods, such as the covariant quantum kernel, in terms of accuracy and resource efficiency?

Key findings

• The genetic algorithm method achieved 100% accuracy on the moonshape dataset with only 3 gate cost, outperforming standard QSVM and ZZFeatureMap.
• The variational quantum circuit with unitary decomposition ansatz achieved the highest accuracy (97.5%) on the ad hoc dataset, though at a higher fixed gate cost of 38.
• Through systematic optimization, the circuit for the covariant quantum kernel dataset was reduced from 51 to 3 gate cost while maintaining 100% accuracy.
• Only 4 out of 14 features were essential in the covariant dataset, and further pruning reduced the circuit to just 4 rotation gates, then to 3 essential gates after necessity testing.
• The final optimized circuit for the covariant dataset required only 3 gate operations (3 rotation gates), demonstrating that high accuracy is achievable with minimal quantum resources.
• Hardware-efficient ansatz showed slightly better accuracy than genetic algorithm on complex datasets, but with higher gate cost; unitary decomposition ansatz delivered the best overall accuracy at the cost of higher fixed gate count.