Quantum Signal Processing Phase Factors With Modified Newton Iteration
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Solution Overview
Problem
Existing methods for determining phase factors in quantum signal processing are computationally intensive, numerically unstable, and suffer from high computational costs and poor scalability, particularly for high-degree polynomials, leading to inaccurate results.
Innovation Solution
A modified Newton method is employed to iteratively solve systems of non-linear equations for phase factor determination, utilizing a reduced number of Jacobian matrix evaluations and incorporating techniques like Aitken's acceleration to enhance convergence and efficiency.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If existing methods are used to determine phase factors in quantum signal processing, then the phase factors can be obtained, but the computational cost is high and the results are inaccurate
Solution Approach 1:
The patent transforms the original complex non-linear system into a simplified linear system by changing the parameter representation. Instead of directly solving for phase factors in the original non-linear equations, the method introduces auxiliary variables and transforms the problem into a linear least-squares system that can be solved efficiently with standard numerical methods, achieving both high accuracy and computational efficiency
Solution Approach 2:
The patent replaces the computationally intensive iterative numerical optimization methods with an analytical linear algebra solution. By substituting the mechanical iterative solving process with a direct linear system solution using pseudo-inverse or QR decomposition, the method eliminates the need for repeated Jacobian calculations and convergence checking, dramatically improving computational efficiency
2Adaptability or versatility
If existing methods are used to determine phase factors, then the phase factors can be obtained, but the computational cost increases significantly for high-degree polynomials
Solution Approach 1:
The patent changes the parameterization approach by introducing auxiliary variables that linearize the relationship between phase factors and the target polynomial coefficients. This transformation allows the system to scale efficiently to high-degree polynomials because the computational complexity of solving the linear system grows linearly with the degree, rather than exponentially as in the original non-linear approach
Solution Approach 2:
The patent segments the complex non-linear problem into simpler linear sub-problems by introducing intermediate auxiliary variables. The original problem of determining phase factors for high-degree polynomials is broken down into a sequence of linear equations that can be solved systematically, making the method scalable to high degrees without exponential cost increases
3Reliability
If existing methods are used to determine phase factors, then the phase factors can be obtained, but the numerical stability is poor
Solution Approach 1:
The patent replaces unstable iterative numerical optimization with a stable linear algebra solution. By substituting the iterative process with a direct solution using pseudo-inverse or QR decomposition, the method achieves superior numerical stability because these linear algebra techniques are well-conditioned and less sensitive to numerical errors, especially for ill-conditioned systems
Solution Approach 2:
The patent introduces auxiliary variables as intermediaries between the phase factors and the target polynomial coefficients. These intermediary variables serve as a bridge that transforms the unstable direct relationship into a stable indirect relationship, allowing the system to solve for phase factors through a numerically stable linear system rather than an unstable non-linear system
Data Source
AI summary
Systems and techniques that facilitate phase factor determination in quantum signal processing are provided. Various embodiments described herein comprise a system, which can comprise: a memory that can store computer executable components; and a processor, operably coupled to the memory, that can execute at least one of the computer executable components that can receive a first target real-valued function and a second target real-valued function that represent a target transformation on a quantum state; determine a system of non-linear equations based on the first target real-valued function and the second target real-valued function, the system of non-linear equations comprising a number of phase factors that define parameters of quantum operations; determine the phase factors using a modified Newton method to iteratively solve the system of non-linear equations; and configure the quantum processor to apply the phase factors to a quantum circuit to implement the target transformation on the quantum state.