Equivalence Checking of Parameterized Quantum Circuits: Verifying the Compilation of Variational Quantum Algorithms (2210.12166v1)

Abstract: Variational quantum algorithms have been introduced as a promising class of quantum-classical hybrid algorithms that can already be used with the noisy quantum computing hardware available today by employing parameterized quantum circuits. Considering the non-trivial nature of quantum circuit compilation and the subtleties of quantum computing, it is essential to verify that these parameterized circuits have been compiled correctly. Established equivalence checking procedures that handle parameter-free circuits already exist. However, no methodology capable of handling circuits with parameters has been proposed yet. This work fills this gap by showing that verifying the equivalence of parameterized circuits can be achieved in a purely symbolic fashion using an equivalence checking approach based on the ZX-calculus. At the same time, proofs of inequality can be efficiently obtained with conventional methods by taking advantage of the degrees of freedom inherent to parameterized circuits. We implemented the corresponding methods and proved that the resulting methodology is complete. Experimental evaluations (using the entire parametric ansatz circuit library provided by Qiskit as benchmarks) demonstrate the efficacy of the proposed approach. The implementation is open source and publicly available as part of the equivalence checking tool QCEC (https://github.com/cda-tum/qcec) which is part of the Munich Quantum Toolkit (MQT).

• The paper presents a multi-stage framework that integrates symbolic analysis via ZX-calculus with numerical instantiation to robustly check circuit equivalence.
• It introduces linear system instantiation for parameter verification, enhancing the reliability of variational quantum algorithm compilations on NISQ devices.
• Experimental results on Qiskit circuits demonstrate significant scalability and orders-of-magnitude performance improvements over traditional methods.

Equivalence Checking of Parameterized Quantum Circuits

The paper "Equivalence Checking of Parameterized Quantum Circuits" by Tom Peham, Lukas Burgholzer, and Robert Wille addresses a crucial challenge in quantum computing: verifying the equivalence of parameterized quantum circuits, particularly in the context of variational quantum algorithms. With the increasing deployment of these algorithms in conjunction with noisy intermediate-scale quantum (NISQ) devices, the accurate compilation of parameterized circuits is vital yet non-trivial. This paper introduces new methodological contributions to verify the compilation of such circuits efficiently and comprehensively.

Summary and Methodological Contributions

Variational quantum algorithms operate using parameterized quantum circuits whose parameters are iteratively optimized within a classical optimization routine. These circuits require compilation into a form compatible with quantum hardware gate sets and topology—a process prone to errors. Existing equivalence checking methods are inadequate for parameterized circuits due to the added complexity of variable parameters. This paper proposes a multi-stage methodology capable of addressing these challenges by verifying equivalence through a combination of symbolic and numerical approaches.

1. Symbolic Equivalence via ZX-Calculus: The authors leverage the ZX-calculus—an established graphical language for reasoning about quantum processes—to initially handle parameterized circuits. This calculus facilitates an equational reasoning framework that can, in certain cases, conclusively determine the equivalence of parameterized circuits via diagrammatic transformations. The key advantage is that this method operates in a purely symbolic space, abstracting over specific parameter values.
2. Instantiation Through Linear Systems: When symbolic equivalence through ZX-calculus is inconclusive, the method progresses to a second stage where parameters are instantiated. This is achieved by constructing and solving linear systems derived from the parameterized gates. The goal is to instantiate parameters in ways that simplify subsequent equivalence checks—specifically targeting cases where non-equivalence arises from parameterized gates.
3. Comprehensive Verification with Random Instantiation: The methodology incorporates a fail-safe mechanism by employing random parameter instantiation followed by equivalence checking with conventional methods. While random instantiation poses a computational challenge, the authors convincingly demonstrate that it statistically minimizes false positives due to the nature of real and analytic functions used within quantum circuits.

Experimental Validation and Performance

The approach has been rigorously tested on the Qiskit library of parameterized quantum circuits, demonstrating scalability across significant numbers of qubits and circuit depths. The evaluations demonstrate that this methodology can verify equivalences efficiently compared to random instantiation approaches, showing significant performance improvements—sometimes up to several orders of magnitude—against state-of-the-art methods. Notably, the proposed method can handle the largest quantum architectures currently available, confirming its practical applicability.

Implications and Future Developments

The implications of this research extend significantly into practical applications of quantum computing where reliable circuit compilation is paramount. This work facilitates the development of robust quantum algorithms by ensuring that compilation processes do not introduce errors. The integration of ZX-calculus with parameter instantiation enriches the landscape of quantum circuit verification by offering a robust framework adaptable to variational circuits and potentially other parameterized quantum models.

Future research directions could explore extending this methodology to encompass broader classes of quantum circuits, including those with nonlinear parameter dependencies. Further, the integration of this verification methodology into existing quantum compilers could automate and streamline the verification process, thereby minimizing manual oversight and enhancing reliability in quantum software stacks.

In conclusion, this paper offers an impactful advancement in the verification of parameterized quantum circuits, ensuring that NISQ-based implementations can be realized with higher confidence in their computational fidelity. The comprehensive approach and its successful application on real-world circuit libraries underscore the methodology's potential to become a staple tool in quantum software development.