Quantum Signal Processing Product Decomposition for Stable Qubit Rotations

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Solution Overview

Problem

Classical preprocessing for quantum signal processing, specifically the computation of interspersing single-qubit rotations, is numerically unstable and inefficient, particularly for long sequences, due to the use of unrealistic computational models and the instability of polynomial expansions.

Innovation Solution

A polynomial time algorithm on a Turing machine is developed to efficiently compute interspersing single-qubit rotations by using rational Laurent polynomials and Fourier transforms to stabilize the numerical calculations, avoiding trigonometric polynomials and ensuring precision through rational approximations and root finding.

Engineering Contradictions & Design Principles

VSEngineering Contradiction Analysis

1Reliability

If polynomial expansions are used to compute interspersing single-qubit rotations, then the computation can be performed, but numerical instability occurs particularly for long sequences

Engineering Contradiction:

Improvenumerical stabilityVSAvoidsequence length

Core Design Contradiction:

ReliabilityVSDuration of action of moving object

Solution Approach 1:

The patent transforms the computational approach by changing parameters from polynomial expansions to product decompositions of SU(2)-valued functions. This parameter change stabilizes numerical calculations for long sequences by avoiding the inherent instability of polynomial expansion methods while maintaining computational feasibility.

Inventive Principle:

Principle #35Parameter changes

Solution Approach 2:

The patent replaces the mechanical computational system of polynomial expansions with a substitute system based on product decompositions of unitary functions. This substitution eliminates numerical instability while preserving the ability to compute interspersing single-qubit rotations, enabling reliable processing of long sequences.

Inventive Principle:

Principle #28Mechanics substitution (Replace mechanical system)

2Productivity

If strong arithmetic model of computation is assumed, then polynomial time complexity can be achieved, but the model is unrealistic and too powerful

Engineering Contradiction:

Improvecomputation efficiencyVSAvoidcomputational model realism

Core Design Contradiction:

ProductivityVSAdaptability or versatility

Solution Approach 1:

The patent changes the computational model parameters from an unrealistic strong arithmetic model to a realistic Turing machine model. Despite this restrictive change, the patent achieves polynomial time complexity of poly(n log(1/ε)) through efficient product decomposition algorithms, demonstrating that high productivity can be achieved with realistic computational assumptions.

Inventive Principle:

Principle #35Parameter changes

Data Source

PatentEP3797388B1Product decomposition of periodic functions in quantum signal processing

Publication Date: 2025.09.10 MICROSOFT TECHNOLOGY LICENSING LLC

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AI summary

In some embodiments, one or more unitary-valued functions are generated by a classical computer generating using projectors with a predetermined number of signiﬁcant bits. A quantum computing device is then conﬁgured to implement the one or more unitary-valued functions. In further embodiments, a quantum circuit description for implementing quantum signal processing that decomposes complex-valued periodic functions is generated by a classical computer, wherein the generating further includes representing approximate polynomials in a Fourier series with rational coeﬃcients. A quantum computing device is then conﬁgured to implement a quantum circuit deﬁned by the quantum circuit description.