In the Shadow of the Hadamard Test: Using the Garbage State for Good and Further Modifications

Area of Science:

• Quantum Information Science
• Quantum Computing Algorithms

Background:

• The Hadamard test is crucial for quantum algorithms but typically ignores system register information.
• Classical shadows offer efficient quantum state representation, developed independently of the Hadamard test.

Purpose of the Study:

• To improve the Hadamard test by integrating classical shadows for enhanced quantum state analysis.
• To leverage classical shadows on the work register to extract additional information during the Hadamard test.

Main Methods:

• Combining the Hadamard test with classical shadows on the n-qubit work register.
• Utilizing measurements of an auxiliary qubit and classical shadows for comprehensive state characterization.

Main Results:

• Substantial improvements to the Hadamard test by exploiting classical shadows.
• Access to additional quantum state features like fidelity, energy, and purity using classical shadows.
• Demonstrated application in statistical phase estimation and potential for broader algorithm use.

Conclusions:

• The combined Hadamard test and classical shadows framework offers significant advantages for intermediate quantum applications.
• This integrated approach enhances information extraction from quantum systems, settling exploitation for current quantum devices.
• The framework is general and applicable to various quantum algorithms, augmenting capabilities with anticontrolled unitaries.