Quantum Gradient Circuit Using Finite-Difference Phase Oracles

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Solution Overview

Problem

Existing quantum gradient algorithms, such as those described by Jordan and Gilýen, only guarantee a complexity scaling advantage for a specific class of smooth functions (Gevrey class G1/2 functions, limiting their applicability to real-world phenomena that do not fit this category.

Innovation Solution

The development of quantum circuits implementing Simulation-Free Quantum Gradient (SFQG) methods that provide a complexity scaling advantage over classical computations for functions without requiring them to be Gevrey class G1/2 functions, using a phase oracle defined by a finite difference approximation with an order greater than zero and employing quantum computing systems to determine gradients with improved accuracy.

Engineering Contradictions & Design Principles

VSEngineering Contradiction Analysis

1Reliability

If quantum gradient algorithms (Jordan and Gilýen) are applied to Gevrey class G1/2 functions, then complexity scaling advantage is guaranteed, but applicability is limited to specific smooth functions only

Engineering Contradiction:

Improvecomplexity scaling advantage guaranteeVSAvoidapplicability to real-world functions

Core Design Contradiction:

ReliabilityVSAdaptability or versatility

Solution Approach 1:

The patent changes the smoothness parameter requirement from Gevrey class G1/2 (which requires very strong smoothness conditions) to a broader class of functions with polynomial decay of derivatives. This parameter relaxation allows the quantum gradient algorithm to maintain complexity scaling advantage while applying to a much wider range of functions including those that model real-world phenomena.

Inventive Principle:

Principle #35Parameter changes

Solution Approach 2:

Instead of requiring functions to satisfy strict Gevrey class conditions to achieve quantum advantage, the patent inverts the approach by designing the algorithm to work with polynomially bounded derivative growth. This inversion expands the function class from a narrow smooth category to a broader category that includes many practical functions while preserving the quantum speedup.

Inventive Principle:

Principle #13The other way round (Inversion)

2Measurement precision

If higher order finite-difference approximation is used in phase oracle, then gradient estimation accuracy is improved, but computational complexity increases

Engineering Contradiction:

Improvegradient estimation accuracyVSAvoidquantum circuit complexity

Core Design Contradiction:

Measurement precisionVSDevice complexity

Solution Approach 1:

The patent uses second-order finite-difference approximation (excessive action beyond minimal first-order) to achieve gradient estimation accuracy with polynomial decay rates. This partial excess in approximation order provides sufficient accuracy for practical applications while keeping the quantum circuit complexity manageable through efficient implementation of the phase oracle.

Inventive Principle:

Principle #16Partial or excessive action

Data Source

PatentUS20260057280A1Quantum circuit for gradient estimation of non-gevrey class g1/2 function

Publication Date: 2026.02.26 INTERNATIONAL BUSINESS MACHINE CORPORATION

AI summary

A quantum circuit is configured to implement a quantum gradient algorithm when executed on qubits of a quantum computing system. The quantum gradient algorithm includes a phase oracleOSfmdefined by a finite difference approximation with an order greater than zero, and a complexity of the quantum gradient algorithm scales as (√{square root over (k)}/ϵ). The quantum circuit is repeatedly executed on qubits of a quantum computing system to determine a k-dimensional gradient of a function ƒ(x) within an error ϵ at point x0, where ƒ(x) is not a Gevrey class G1/2 function.