[Paper Review] Equivalence, Identity, and Unitarity Checking in Black-Box Testing of Quantum Programs

Peixun Long, Jianjun Zhao|arXiv (Cornell University)|Jul 4, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper proposes novel black-box testing algorithms for quantum programs, using the Swap Test for equivalence checking, a simplified repeated execution method for identity checking, and a theoretical unitarity condition for unitarity verification. The methods demonstrate high effectiveness and efficiency in experimental evaluations, enabling rigorous verification of quantum program behavior without internal access.

ABSTRACT

Quantum programs exhibit inherent non-deterministic behavior, which poses more significant challenges for error discovery compared to classical programs. While several testing methods have been proposed for quantum programs, they often overlook fundamental questions in black-box testing. In this paper, we bridge this gap by presenting three novel algorithms specifically designed to address the challenges of equivalence, identity, and unitarity checking in black-box testing of quantum programs. We also explore optimization techniques for these algorithms, including specialized versions for equivalence and unitarity checking, and provide valuable insights into parameter selection to maximize performance and effectiveness. To evaluate the effectiveness of our proposed methods, we conducted comprehensive experimental evaluations, which demonstrate that our methods can rigorously perform equivalence, identity, and unitarity checking, offering robust support for black-box testing of quantum programs.

Motivation & Objective

• Address fundamental gaps in black-box testing of quantum programs, where inherent non-determinism and measurement-induced collapse hinder error detection.
• Provide rigorous, black-box methods to verify core quantum program properties: equivalence, identity, and unitarity.
• Develop practical algorithms that do not require access to internal circuit structure, enabling testing under real-world constraints.
• Optimize algorithms for performance and scalability, particularly for equivalence and unitarity checking.
• Validate the proposed methods through comprehensive theoretical analysis and experimental evaluation on quantum simulators.

Proposed method

• Leverage the Swap Test to compare outputs of two quantum programs on Pauli basis states, estimating their fidelity to determine equivalence.
• Introduce a streamlined approach for identity checking by repeatedly applying the program and measuring output state overlap with input.
• Establish a necessary and sufficient condition for unitarity based on trace preservation and isometry, enabling formal verification of unitary operations.
• Design optimized variants of the equivalence and unitarity checking algorithms to improve runtime and reduce resource usage.
• Use parameter tuning strategies to balance accuracy and performance, particularly in fidelity estimation and measurement repetition counts.
• Implement all methods within a quantum simulator framework (Q#) to evaluate practical feasibility and scalability.

Experimental results

Research questions

• RQ1 How can two quantum programs be checked for functional equivalence in a black-box setting without access to their internal structure?
• RQ2 How can we determine whether a quantum program implements the identity operation using only input-output behavior?
• RQ3 What formal criterion can be used to verify whether a quantum program implements a unitary transformation based solely on observable outputs?

Key findings

• The proposed equivalence checking method achieves high accuracy in detecting functional equivalence between quantum programs using the Swap Test on Pauli states.
• The identity checking method successfully identifies identity operations with minimal overhead through repeated execution and state overlap measurement.
• The theoretical unitarity condition enables precise verification of unitary behavior, forming a solid foundation for algorithmic implementation.
• Experimental results show that the methods are effective and efficient, with execution times under 2 seconds for small-scale programs (e.g., 4-qubit circuits).
• The optimized versions of equivalence and unitarity checking algorithms significantly reduce runtime and resource consumption, improving scalability.
• The methods remain effective even with limited qubit availability, though performance degrades on large-scale programs due to qubit count constraints.