Equivalence Checking of Quantum Circuits with the ZX-Calculus (2208.12820v1)

Abstract: As state-of-the-art quantum computers are capable of running increasingly complex algorithms, the need for automated methods to design and test potential applications rises. Equivalence checking of quantum circuits is an important, yet hardly automated, task in the development of the quantum software stack. Recently, new methods have been proposed that tackle this problem from widely different perspectives. One of them is based on the ZX-calculus, a graphical rewriting system for quantum computing. However, the power and capability of this equivalence checking method has barely been explored. The aim of this work is to evaluate the ZX-calculus as a tool for equivalence checking of quantum circuits. To this end, it is demonstrated how the ZX-calculus based approach for equivalence checking can be expanded in order to verify the results of compilation flows and optimizations on quantum circuits. It is also shown that the ZX-calculus based method is not complete$\unicode{x2014}$especially for quantum circuits with ancillary qubits. In order to properly evaluate the proposed method, we conduct a detailed case study by comparing it to two other state-of-the-art methods for equivalence checking: one based on path-sums and another based on decision diagrams. The proposed methods have been integrated into the publicly available QCEC tool (https://github.com/cda-tum/qcec) which is part of the Munich Quantum Toolkit (MQT).

• The paper improves equivalence checking by addressing floating-point inaccuracies and input/output permutations using enhanced ZX-calculus techniques.
• The paper introduces a phase rounding strategy to mitigate numerical errors in quantum circuit verification.
• The paper demonstrates that ZX-calculus outperforms path-sum methods for Clifford circuits while noting its limitations for universal circuits.

Overview of Equivalence Checking of Quantum Circuits with the ZX-Calculus

The paper explores the potential of using the ZX-calculus for the equivalence checking of quantum circuits, addressing substantial challenges due to the complex nature of quantum computing. With the rapid development of quantum computers and increasingly complex algorithms, verifying the equivalence of quantum circuits, especially post-compilation, has become paramount.

The ZX-calculus is a graphical notation that aids in reasoning about quantum computations. It uses diagrammatic transformations that correspond to algebraic manipulations of quantum states. The authors argue that while existing approaches, primarily based on classical computing paradigms, strive to tackle equivalence checking, the ZX-calculus offers a unique perspective by leveraging quantum mechanics' inherent graphical properties.

Key Contributions and Methodology

1. Improvement Over State-of-the-Art: The paper expands on the existing ZX-calculus frameworks to address issues like inaccuracies due to floating-point arithmetic, input/output permutations during the compilation process, and ancillary qubits. These are crucial for practical equivalence checking but were not adequately addressed in prior ZX-calculus applications.
2. Handling Inaccuracies: The paper proposes strategies to handle minor numerical inaccuracies often encountered in complex quantum computations. By introducing a methodology to round phases close to multiples of $\pi/2$, it allows the ZX-calculus to consider circuits that may be equivalent modulo small numerical errors.
3. Permutations and Ancillary Qubits: Through the adaptation of input/output permutations, the research integrates the ZX-calculus with real-world compilation flows where logical-to-physical qubit mappings alter. Additionally, the representation of ancillas—the constant qubit states crucial for certain quantum computations—within the ZX-calculus was innovatively handled.
4. Theoretical Insights: The completeness of the ZX-calculus in the context of Clifford circuits is affirmed through proof. However, the paper highlights that for universal quantum circuits, especially those beyond the Clifford group, the ZX-calculus is not yet complete for equivalence checking.

Empirical Evaluation

In an exhaustive empirical paper, the authors compare the ZX-calculus approach against other state-of-the-art methodologies like the path-sum and decision diagram methods using a range of benchmarks. Noteworthy observations include:

• Clifford Circuits: ZX-calculus is demonstrated to be effective and outperforms path-sum-based methods considerably in terms of runtime, showing orders of magnitude improvement for Clifford circuits.
• Decision Diagram-Based Methods: Although decision diagrams are canonically complete, their performance can degrade with numerical inaccuracies and non-reversible quantum operations. The ZX-calculus presents a robust alternative in these scenarios, particularly for circuits with arbitrary rotation gates.

However, both methods serve complementary roles, with each excelling under different conditions. For instance, in circuits with structured operations such as reversible functions, decision diagrams have shown efficiency, which is not matched by the ZX-calculus approach for larger complex circuits.

Future Implications

The findings in this paper indicate a significant step forward in quantum circuit verification efforts. By broadening the applicability of the ZX-calculus, the research opens up possibilities for more integrated tools that combine multiple methods for comprehensive quantum software verification. Future work will likely delve into improving the scalability of the ZX-calculus and exploring its potential in conjunction with other quantum reasoning frameworks. The integration of existing methods could inspire new algorithms tailored specifically for the evolving demands of quantum hardware.

In summary, the paper underscores the potential and current limitations of ZX-calculus-based equivalence checking, proposing enhanced methodologies for practical and theoretical improvements in quantum computing verification tasks. The comprehensive analysis and comparison with contemporary techniques underscore the ZX-calculus as a pivotal tool, particularly in handling the nuances of advanced quantum circuits.