Classical Circuits Can Simulate Quantum Aspects

Jinyang Li, Qiang Guan, Dingwen Tao +1 more

The rapid advancement of quantum hardware brings a host of research opportunities and the potential for quantum advantages across numerous fields. In this landscape, quantum circuit simulations serve as an indispensable tool by emulating quantum behavior on classical computers. They offer easy access, noise-free environments, and real-time observation of quantum states. However, the sustainability aspect of quantum circuit simulation is yet to be explored. In this paper, we introduce for the first time the concept of environmental impact from quantum circuit simulation. We present a preliminary model to compute the CO2e emissions derived from quantum circuit simulations. Our results indicate that large quantum circuit simulations (43 qubits) could lead to CO2e emissions 48 times greater than training a transformer machine learning model.

Jesse Van Griensven Thé, Roydon Andrew Fraser, José R. Rosas-Bustos

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

George Biswas

The Gottesman–Knill theorem asserts that quantum circuits composed solely of Clifford gates can be efficiently simulated classically. This theorem hinges on the fact that Clifford gates map Pauli strings to other Pauli strings, thereby allowing for a structured simulation process using classical computations. In this work, we break down the step-by-step procedure of the Gottesman–Knill theorem in a beginner-friendly manner, leveraging concepts such as matrix products, tensor products, commutation, anti-commutation, eigenvalues, and eigenvectors of quantum mechanical operators. Through detailed examples illustrating superposition and entanglement phenomena, we aim to provide a clear understanding of the classical simulation of Clifford gate-based quantum circuits. While we do not provide a formal proof of the theorem, we offer intuitive physical insights at each stage where necessary, empowering readers to grasp the fundamental principles underpinning this intriguing aspect of quantum computation. Quanta 2024; 13: 20–27.

M. Van den Nest

The quantum Fourier transform (QFT) is sometimes said to be the source of various exponential quantum speed-ups. In this paper we introduce a class of quantum circuits which cannot outperform classical computers even though the QFT constitutes an essential component. More precisely, we consider normalizer circuits. A normalizer circuit over a finite Abelian group is any quantum circuit comprising the QFT over the group, gates which compute automorphisms and gates which realize quadratic functions on the group. We prove that all normalizer circuits have polynomial-time classical simulations. The proof uses algorithms for linear diophantine equation solving and the monomial matrix formalism introduced in our earlier work. We subsequently discuss several aspects of normalizer circuits. First we show that our result generalizes the Gottesman-Knill theorem. Furthermore we highlight connections to Shor's factoring algorithm and to the Abelian hidden subgroup problem in general. Finally we prove that quantum factoring cannot be realized as a normalizer circuit owing to its modular exponentiation subroutine.

Steven Sensarn