Quantum Kernel Matrix Computation via Reduced Density Matrices
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Solution Overview
Problem
Kernel methods for machine learning face challenges when dealing with large feature spaces, as the computational cost of estimating kernel functions becomes prohibitively expensive, and scalability issues arise due to the exponential decay of signal with the number of qubits in conventional quantum kernel methods.
Innovation Solution
The implementation of quantum computing with kernel methods involves a quantum computing device that computes a kernel matrix representing similarities among quantum data points using reduced density matrices, which is then provided to a classical processor for further processing and model construction.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If conventional quantum kernel methods are used to compute kernel functions, then quantum computing capability is utilized, but computational cost becomes prohibitively expensive and scalability deteriorates due to exponential decay of signal with number of qubits
Solution Approach 1:
The patent segments the quantum system into subsystems corresponding to different qubits and computes kernel functions based on reduced density matrices of these subsystems rather than the full quantum state. This segmentation reduces the computational complexity from exponential to polynomial in the number of qubits, resolving the scalability issue while maintaining quantum computing advantages.
Solution Approach 2:
The patent extracts only the necessary information (reduced density matrices of subsystems) from the full quantum state to compute kernel functions. By taking out only the relevant subsystem information rather than processing the entire quantum state, the computational cost is dramatically reduced while preserving the essential quantum correlations needed for kernel computation.
2Measurement precision
If kernel functions are computed using full quantum states, then accuracy is maintained, but computational cost becomes prohibitively expensive
Solution Approach 1:
The patent extracts only the necessary subsystem information (reduced density matrices) from the full quantum state to compute kernel functions. This extraction maintains prediction accuracy by preserving the essential quantum correlations while dramatically reducing computational cost from exponential to polynomial scaling with the number of qubits.
Solution Approach 2:
The patent applies local quality by computing kernel functions based on local subsystem properties (reduced density matrices of individual qubits or small groups of qubits) rather than requiring global quantum state information. This local approach maintains the necessary accuracy for prediction while reducing the overall computational burden.
Data Source
AI summary
Methods, systems, and apparatus for quantum machine learning. In one aspect, a method includes obtaining, by a quantum computing device, a training dataset of quantum data points; computing, by the quantum computing device, a kernel matrix that represents a similarity between the quantum data points included in the training dataset, comprising computing a value of a kernel function for each pair of quantum data points in the training dataset, wherein the kernel function is based on reduced density matrices for the quantum data points; and providing, by the quantum computing device, the kernel matrix to a classical processor, wherein the classical processor performs a training algorithm using the kernel matrix to construct a machine learning model.