Shift rule for gradient determination in parameterised quantum evolutions
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Existing methods for determining derivatives in parameterised quantum evolutions, such as the stochastic parameter shift rule (SPSR), are inadequate for noisy intermediate-scale quantum (NISQ) computing due to approximation errors and technical limitations, especially when the B-component of the Hamiltonian cannot be switched off, requiring time-consuming re-designs or high drive strengths.
Innovation Solution
A hybrid quantum-classical computing system that estimates derivatives using a phase-correction-scaled derivative calculation, employing a subset of shift values and weighting factors to handle perturbed quantum evolutions without re-designing the quantum system, allowing for exact derivative estimation.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If the stochastic parameter shift rule (SPSR) is used to determine derivatives in parameterised quantum evolutions, then the derivative can be estimated, but approximation errors are introduced and the quantum system requires time-consuming re-design or high drive strengths when the B-component of the Hamiltonian cannot be switched off
Solution Approach 1:
The patent extracts and isolates the problematic B-component of the Hamiltonian from the parameterised quantum evolution. By separating the total Hamiltonian H(θ) = Aθ + B into parameter-dependent and parameter-independent parts, the method applies the parameter shift rule only to the Aθ component while treating B as a static perturbation, thereby eliminating approximation errors associated with treating the full Hamiltonian.
Solution Approach 2:
The patent segments the quantum evolution into distinct components: the parameterised unitary evolution e^(-iAθ) that admits exact parameter shift rules, and the static B-component that is treated as a perturbation. This segmentation allows the derivative estimation to be performed exactly on the parameterised part while accounting for the B-component through its action on the evolved state, rather than requiring re-design of the quantum system.
2Adaptability or versatility
If the B-component of the Hamiltonian is included in the parameterised quantum evolution, then the quantum system can be controlled, but the existing parameter shift rule becomes inapplicable and requires modification beyond simple parameter value changes
Solution Approach 1:
The patent applies the parameter shift rule partially to only the Aθ component of the Hamiltonian rather than attempting to apply it to the full H(θ) = Aθ + B. This partial application allows the method to maintain exact derivative estimation for the parameterised part while treating the B-component as a static perturbation that does not require modification of the quantum evolution beyond parameter value changes.
3Speed
If the drive strength is increased to compensate for the B-component, then the quantum evolution can proceed, but the system requires high drive strengths which may not be technically feasible
Solution Approach 1:
The patent introduces an intermediary mathematical framework that separates the parameterised evolution e^(-iAθ) from the B-component perturbation. This intermediary approach allows the derivative to be calculated exactly for the parameterised part while accounting for B through its action on the evolved state, eliminating the need to increase drive strength to compensate for the B-component.
Data Source
AI summary
A hybrid computing system configured to estimate a derivative of a parameter-dependent physical quantity that is dependent on a first control parameter θ of a parameterised quantum evolution executed by the quantum computing system. The classical computing system determines, based on an input bounding value and phase-correction value, a subset multiset of shift-values which define an equivalent subset multiset of trial control parameter values and corresponding weighting values and summation factors such that the trial control parameter values obey a targeted probability distribution. The quantum computing system executes the parameterised quantum evolution at the trial control parameter values and measures a parameter-dependent physical quantity for each trial control parameter value. On the classical computing system, the measurement results from these measurements are combined with the weighting values and summation factors to calculate at an estimate of a phase-correction-scaled derivative of the parameter-dependent physical quantity.