Quantum Discord and Coherence in DQC1 for Machine Learning

By Dr. Sampath LonkaReviewed by Susha Cheriyedath, M.Sc.Nov 19 2023

In a recent publication in the journal Scientific Reports, researchers explored the potential advantages offered by quantum discord and quantum coherence within the deterministic quantum computing with one qubit (DQC1) model in supervised machine learning.

Background

Recent developments in controlling decoherence and noise have enabled the advancement of intermediate-scale quantum devices featuring hundreds of qubits. Although lacking fault tolerance, these devices exhibit significant computational prowess compared to classical supercomputers, attributed to the support of quantum entanglement. As quantum hardware continues to progress, its significance is anticipated to grow substantially in domains such as quantum chemistry, quantum simulations, and quantum machine learning.

The utilization of quantum hardware in intricate computations, like kernel function estimation, is proposed to attain a quantum advantage in machine learning. Despite the susceptibility of entanglement to noise, there is a critical need to explore quantum correlations that are less sensitive to noise.

The DQC1 model employs a single qubit to engage with a notably mixed quantum state, enabling the estimation of computationally intensive functions—an idea referred to as the "power of one qubit." DQC1 generates quantum discord, a robust form of weak quantum correlation, utilizing pure qubit coherence. Quantum discord, being more resilient to noise than entanglement, holds the potential to provide a quantum advantage in noisy situations, particularly in tasks related to quantum illumination.

The literature on DQC1 in machine learning contexts is limited, focusing on addressing the parity learning problem and proposing applications in kernel-based machine learning. The present study explores the application of the DQC1 model in the context of supervised machine learning for the efficient estimation of kernel functions, implementing the study on IBM hardware. Through an examination of quantum discord, coherence consumption, and the impact of hardware noise, the DQC1 protocol, despite an increased need for gates, mitigates measurement errors by focusing on the measurement of only one qubit, ultimately achieving the best classification accuracy.

Quantum discord and coherence in DQC1

Initially designed for nuclear magnetic resonance quantum information processing, the DQC1 model finds implementation across diverse physical settings. The circuit involves a control qubit and n target qubits in a mixed state. By adjusting parameters, the purity of the control qubit is modulated, influencing computational outcomes.

The evolution of the DQC1 circuit, governed by an arbitrary unitary matrix applied to target qubits, effectively estimates the trace of this matrix. Obtaining the matrix of the control qubit through the tracing out of target qubits facilitates trace estimation via measurements. Quantum coherence and discord in DQC1 are explored as alternative quantum resources. Coherence, defined rigorously, is efficiently consumed by DQC1, linking it to the trace of the unitary matrix. Discord, a measure of quantum correlations, is demonstrated to be produced through coherence consumption.

Transitioning to supervised machine learning, the study introduces support vector machines (SVM) and the kernel method. For non-linearly separable data, SVM employs a kernel function, and the DQC1 model demonstrates efficiency in estimating complex kernel functions.

The implementation on IBM hardware follows, utilizing the quantum processor. The circuit involves a control qubit and two target qubits, demonstrating robustness against hardware noise. The study draws parallels with the projected kernel method, emphasizing DQC1's efficiency in requiring only control qubit measurements. The experiments showcase the accuracy of the Qiskit simulator and IBM hardware, validating the practicality and potential of the DQC1 model in quantum machine learning applications.

Dynamics of control qubit coherence

Researchers investigated the impact of control qubit coherence, hardware noise, coherence consumption, and quantum discord in the implemented setting. Control qubit purity's role is explored by repeating the learning task with varying states. In simulations, accuracy peaks at the qubit's maximal purity, while hardware noise degrades accuracy.

Using datasets "make-moon" and "make-circle," critical values for accuracy change with noise strength. Kernel differences between simulation and IBM hardware are attributed to noise, influencing coherence consumption and discord generation. The coherence consumption is minimized along diagonal axes, and quantum discord is successfully estimated.

Conclusion

In summary, the authors established an upper bound on the generalization error of the fidelity quantum kernel model, linked to the average purity of encoded states. They noted that a noisier encoding process adversely affects training performance.

Empirical results across three datasets corroborate that enhancing control qubit purity improves DQC1 kernel model accuracy. This suggests a potential avenue for theoretical studies connecting coherence consumption to generalization errors. The study advocates exploring the computational prowess of a single qubit as a universal classifier in broader contexts and encourages further exploration of quantum coherence and discord integration within machine learning.