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arXiv:0705.0017 · Digest of technical papers/Digest of technical papers - IEEE/ACM International Conference on Computer-Aided Design

Checking equivalence of quantum circuits and states

George F. Viamontes, Igor L. Markov, John P. Hayes

Published Nov 1, 2007• 70 citations
arXivHTMLPDF

Abstract

Among the post-CMOS technologies currently under investigation, quantum computing (QC) holds a special place. QC offers not only extremely small size and low power, but also exponential speed-ups for important simulation and optimization problems. It also poses new CAD problems that are similar to. but more challenging, than the related problems in classical (non-quantum) CAD. such as determining if two states or circuits are functionally equivalent. While differences in classical states are easy to detect, quantum states, which are represented by complex-valued vectors, exhibit subtle differences leading to several notions of equivalence. This provides flexibility in optimizing quantum circuits, but leads to difficult new equivalence-checking issues for simulation and synthesis. We identify several different equivalence-checking problems and present algorithms for practical benchmarks, including quantum communication and search circuits, which are shown to be very fast and robust for hundreds of qubits.

DOI
10.1109/iccad.2007.4397246

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Among the post-CMOS technologies currently under investigation, quantum computing (QC) holds a special place. QC offers not only extremely small size and low power, but also exponential speed-ups for important simulation and optimization problems. It also poses new CAD problems that are similar to. but more challenging, than the related problems in classical (non-quantum) CAD. such as determining if two states or circuits are functionally equivalent. While differences in classical states are easy to detect, quantum states, which are represented by complex-valued vectors, exhibit subtle differences leading to several notions of equivalence. This provides flexibility in optimizing quantum circuits, but leads to difficult new equivalence-checking issues for simulation and synthesis. We identify several different equivalence-checking problems and present algorithms for practical benchmarks, including quantum communication and search circuits, which are shown to be very fast and robust for hundreds of qubits.
In the not-so-distant future, quantum computing will change the way we tackle certain problems. It promises to dramatically speed-up many chemical, financial, cryptographical, and machine-learning applications. However, in order to capitalize on those promises, complex design flows composed of steps, such as compilation, decomposition, mapping, or transpilation need to be employed before being able to execute a conceptual quantum algorithm on an actual device. This results in many descriptions at various levels of abstraction which may significantly differ from each other. The complexity of the underlying design problems makes it ever more important to not only provide efficient solutions for the single steps but also to verify that the originally intended functionality is preserved throughout all levels of abstraction. This motivates methods for equivalence checking of quantum circuits. However, most existing methods for this are inspired by equivalence checking in the classical realm and have merely been extended to support quantum circuits (i.e., circuits which do not only rely on 0's and 1's but also employ superposition and entanglement). In this work, we propose an advanced methodology that takes the different paradigms of quantum circuits not only as a burden but as an opportunity. In fact, the proposed methodology explicitly utilizes characteristics unique to quantum computing in order to overcome the shortcomings of existing approaches. We show that by exploiting the reversibility of quantum circuits, complexity can be kept feasible in many cases. Moreover, we show that, in contrast to the classical realm, simulation is very powerful in verifying quantum circuits. Experimental evaluations confirm that the resulting methodology allows one to conduct equivalence checking dramatically faster than ever before-in many cases just a single simulation run is sufficient. An implementation of the proposed equivalence checking flow is publicly available at https://iic.jku.at/eda/research/quantum_verification/.
Quantum computing is gaining considerable momentum through the recent progress in physical realizations of quantum computers. This led to rather sophisticated design flows in which the originally specified quantum functionality is compiled through different abstractions. This increasingly raises the question whether the respectively resulting quantum circuits indeed realize the originally intended function. Accordingly, efficient methods for equivalence checking are gaining importance. However, existing solutions still suffer from significant shortcomings such as their exponential worst case performance and an increased effort to obtain counterexamples in case of non-equivalence. In this work, we propose an improved DD-based equivalence checking approach which addresses these shortcomings. To this end, we utilize decision diagrams and exploit the fact that quantum operations are inherently reversible - allowing for dedicated strategies that keep the overhead moderate in many cases. Experimental results confirm that the proposed strategies lead to substantial speed-ups - allowing to perform equivalence checking of quantum circuits factors or even magnitudes faster than the state of the art.
The rapid rate of progress in the physical realization of quantum computers sparked the development of elaborate design flows for quantum computations on such devices. Each stage of these flows comes with its own representation of the intended functionality. Ensuring that each design step preserves this intended functionality is of utmost importance. However, existing solutions for equivalence checking of quantum computations heavily struggle with the complexity of the underlying problem and, thus, no conclusions on the equivalence may be reached with reasonable efforts in many cases. In this work, we uncover the power of simulation for equivalence checking in quantum computing. We show that, in contrast to classical computing, it is in general not necessary to compare the complete representation of the respective computations. Even small errors frequently affect the entire representation and, thus, can be detected within a couple of simulations. The resulting equivalence checking flow substantially improves upon the state of the art by drastically accelerating the detection of errors or providing a highly probable estimate of the operations' equivalence.
As several compilation and optimization techniques have been proposed, equivalence checking for quantum circuits has become essential in design flows. The state-of-the-art to this problem observed that even small errors substantially affect the entire quantum system. As a result, it exploited random simulations to prove the non-equivalence of two quantum circuits. However, when errors occurred close to outputs, it was hard for the work to prove the non-equivalence of some non-equivalent quantum circuits under a limited number of simulations. In this work, we propose a novel simulation-based approach using a set of specially designed stimuli. The simulation runs of the proposed approach is linear rather than exponential to the number of quantum bits of a circuit. According to the experimental results, the success rate of our approach is 100% (100%) under a simulation run (execution time) constraint for a set of benchmarks, while that of the state-of-the-art is only 69% (74%) on average. Our approach also achieves a speedup of 26 on average.
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.
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).