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arXiv:2403.18813 · Lecture notes in computer science

Equivalence Checking of Quantum Circuits by Model Counting

Jingyi Mei, Tim J. Coopmans, Marcello Bonsangue, Alfons Laarman

Published Jan 1, 2024• 15 citations
arXivHTMLPDF

Abstract

Abstract Verifying equivalence between two quantum circuits is a hard problem, that is nonetheless crucial in compiling and optimizing quantum algorithms for real-world devices. This paper gives a Turing reduction of the (universal) quantum circuits equivalence problem to weighted model counting (WMC). Our starting point is a folklore theorem showing that equivalence checking of quantum circuits can be done in the so-called Pauli-basis. We combine this insight with a WMC encoding of quantum circuit simulation, which we extend with support for the Toffoli gate. Finally, we prove that the weights computed by the model counter indeed realize the reduction. With an open-source implementation, we demonstrate that this novel approach can outperform a state-of-the-art equivalence-checking tool based on ZX calculus and decision diagrams.

DOI
10.1007/978-3-031-63501-4_21

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Abstract Verifying equivalence between two quantum circuits is a hard problem, that is nonetheless crucial in compiling and optimizing quantum algorithms for real-world devices. This paper gives a Turing reduction of the (universal) quantum circuits equivalence problem to weighted model counting (WMC). Our starting point is a folklore theorem showing that equivalence checking of quantum circuits can be done in the so-called Pauli-basis. We combine this insight with a WMC encoding of quantum circuit simulation, which we extend with support for the Toffoli gate. Finally, we prove that the weights computed by the model counter indeed realize the reduction. With an open-source implementation, we demonstrate that this novel approach can outperform a state-of-the-art equivalence-checking tool based on ZX calculus and decision diagrams.
Quantum circuit compilation comprises many computationally hard reasoning tasks that nonetheless lie inside #P\mathbf{P}P and its decision counterpart in PP\mathbf{PP}PP. The classical simulation of general quantum circuits is a core example. We show for the first time that a strong simulation of universal quantum circuits can be efficiently tackled through weighted model counting by providing a linear encoding of Clifford+T circuits. To achieve this, we exploit the stabilizer formalism by Knill, Gottesmann, and Aaronson and the fact that stabilizer states form a basis for density operators. With an open-source simulator implementation, we demonstrate empirically that model counting often outperforms state-of-the-art simulation techniques based on the ZX calculus and decision diagrams. Our work paves the way to apply the existing array of powerful classical reasoning tools to realize efficient quantum circuit compilation; one of the obstacles on the road towards quantum supremacy.
Abstract Quantum circuit compilation comprises many computationally hard reasoning tasks that lie inside # P{\textsf{P}}P P and its decision counterpart in PP{\textsf{PP}}PP PP . The classical simulation of universal quantum circuits is a core example. We show for the first time that a strong simulation of universal quantum circuits can be efficiently tackled through weighted model counting by providing a linear-length encoding of Clifford+Tcircuits. To achieve this, we exploit the stabilizer formalism by Knill, Gottesmann, and Aaronson by reinterpreting quantum states as a linear combination of stabilizer states. With an open-source simulator implementation, we demonstrate empirically that model counting often outperforms state-of-the-art simulation techniques based on the ZX calculus and decision diagrams. Our work paves the way to apply the existing array of powerful classical reasoning tools to realize efficient quantum circuit compilation; one of the obstacles on the road towards quantum supremacy.
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—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).
We present Quokka#, a versatile, open-source Python library for quantum circuit analysis. Quokka# reduces various simulation, verification, and synthesis tasks to weighted model counting (#SAT). It supports universal quantum circuits and a wide variety of gates. Quokka# provides multiple encodings based on different algebraic bases and equivalence-checking methods, enabling key performance trade-offs. Moreover, the new version of Quokka# adds approximate equivalence checking, which is crucial in its synthesis algorithms, since it enables translation between arbitrary gate sets. Its synthesis engine is depth-optimal, making it well-suited to real-world quantum computing. This paper demonstrates the design, extensibility, and use of Quokka#.
Parameterised quantum circuits (PQCs) hold great promise for demonstrating quantum advantages in practical applications of quantum computation. Examples of successful applications include the variational quantum eigensolver, the quantum approximate optimisation algorithm, and quantum machine learning. However, before executing PQCs on real quantum devices, they undergo compilation and optimisation procedures. Given the inherent error-proneness of these processes, it becomes crucial to verify the equivalence between the original PQC and its compiled or optimised version. Unfortunately, most existing quantum circuit verifiers cannot directly handle parameterised quantum circuits; instead, they require parameter substitution to perform verification. In this paper, we address the critical challenge of equivalence checking for PQCs. We propose a novel compact representation for PQCs based on tensor decision diagrams. Leveraging this representation, we present an algorithm for verifying PQC equivalence without the need for instantiation. Our approach ensures both effectiveness and efficiency, as confirmed by experimental evaluations. The decision-diagram representations offer a powerful tool for analysing and verifying parameterised quantum circuits, bridging the gap between theoretical models and practical implementations.
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. However, there is no established baseline on which to judge current and future progress in equivalence checking of quantum circuits. In order to close this gap, we conduct a detailed case study of two of the most promising equivalence checking methodologies---one based on decision diagrams and one based on the ZX-calculus---and compare their strengths and weaknesses.