Parallel Quantum Hamiltonian Learning (PQHL) Using QSPE to achieve CRLB
QSPE and the block-diagonal structure of parallel-learnable Hamiltonians allow the Parallel Quantum Hamiltonian Learning (PQHL) technique to achieve the Cramér–Rao Lower Bound (CRLB) saturated optimum precision.
Parallel Quantum Hamiltonian Learning is a novel metrology method that accurately characterizes a quantum system's Hamiltonian, notably complex many-body systems.
This methodology addresses the inadequacies of past approaches, which often failed because they required prior knowledge of the Hamiltonian structure or were confined to learning fundamental one- or two-qubit systems.
A breakdown of the process follows.
Choose the Best Subroutines
Due to Quantum Signal Processing Estimation, PQHL is optimum.
The challenge is simplified by the learning procedure, which breaks down the system into various invariant subspaces using the structure of a parallel-learnable Hamiltonian.
QSPE is a sophisticated metrology technology that can reach the Heisenberg limit for regular activities like calibrating two-qubit gates. By applying QSPE to each 2x2 invariant subspace, the approach ensures accurate parameter estimation.
This high precision allows the following estimators for the entire Hamiltonian parameters to saturate the CRLB. The global learning process is synthesized from optimal QSPE sub-routines.
Fast Precision Scaling
The estimated precision (variance) is compared to the theoretically achievable minimal bound to verify CRLB saturation.
PQHL uses the QSPE technique, which scales angle variance well. Variance reduces far faster than the standard Heisenberg limit.
In particular, variance reduction scales as O(1/d^4), inverse of shot count and fourth power of repetition cycles (depth), faster than the standard Heisenberg limit scaling of O(1/d^3).
The overall variance of the final Hamiltonian parameters, as obtained by typical post-processing, matches this accelerated scaling. The estimated variance matches the predicted ideal variance from the CRLB, confirming the optimal performance of typical post-processing steps in learning algorithms.
Dissociation and Strength
PQHL uses parameter decoupling to achieve and maintain optimal precision against noise.
Fourier Domain Inference: The QSPE inference step works in the Fourier domain, which naturally separates measurement parameters. This decoupling prevents system noise like time-dependent coherent errors (such as drift on a local field parameter) or decoherence from affecting the accuracy of predicting other independent parameters, such as interaction terms.
Noise Resilience: Modern quantum gear can withstand depolarizing noise, SPAM mistakes, and time-dependent coherent noise, maintaining theoretical precision in noisy conditions.
