Efficient Gradient Estimation of Variational Quantum Circuits with Lie Algebraic Symmetries
This work addresses a key bottleneck for hybrid quantum-classical optimization, enabling more scalable quantum machine learning, though it is incremental as it builds on known techniques with specific assumptions.
The paper tackles the challenge of efficient gradient estimation in variational quantum circuits by developing a framework that applies the Hadamard test under mild structural assumptions, achieving an exponential reduction in measurement cost and polynomial speedup in time compared to existing methods.
Hybrid quantum-classical optimization and learning strategies are among the most promising approaches to harnessing quantum information or gaining a quantum advantage over classical methods. However, efficient estimation of the gradient of the objective function in such models remains a challenge due to several factors including the exponential dimensionality of the Hilbert spaces, and information loss of quantum measurements. In this work, we developed an efficient framework that makes the Hadamard test efficiently applicable to gradient estimation for a broad range of quantum systems, an advance that had been wanting from the outset. Under certain mild structural assumptions, the gradient is estimated with the measurement shots that scale logarithmically with the number of parameters and with polynomial classical and quantum time. This is an exponential reduction in the measurement cost and polynomial speed up in time compared to existing works. The structural assumptions are (1) the dimension of the dynamical Lie algebra is polynomial in the number of qubits, and (2) the observable has a bounded Hilbert-Schmidt norm.