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The Gottesman-Knill theorem asserts that a quantum circuit composed of Clifford gates can be efficiently simulated on a classical computer. Here we revisit this theorem and extend it to quantum circuits composed of Clifford and T gates,…

Quantum Physics · Physics 2019-04-11 Sergey Bravyi , David Gosset

We start by studying the subgroup structures underlying stabilizer circuits and we use our results to propose a new normal form for stabilizer circuits. This normal form is computed by induction using simple conjugation rules in the…

Quantum Physics · Physics 2021-07-05 Marc Bataille

Simulating quantum algorithms on classical computers is challenging when the system size, i.e., the number of qubits used in the quantum algorithm, is moderately large. However, some quantum algorithms and the corresponding quantum circuits…

Computational Engineering, Finance, and Science · Computer Science 2021-04-26 Linjian Ma , Chao Yang

We propose a general method for preparing stabilizer states with reduced two-qubit gate count and depth compared to the state of the art. The method starts from a graph state representation of the stabilizer state and iteratively reduces…

Quantum circuit compilation comprises many computationally hard reasoning tasks that nonetheless lie inside #$\mathbf{P}$ and its decision counterpart in $\mathbf{PP}$. The classical simulation of general quantum circuits is a core example.…

Quantum Physics · Physics 2024-03-13 Jingyi Mei , Marcello Bonsangue , Alfons Laarman

We present an efficient algorithm to reduce the number of non-Clifford gates in quantum circuits and the number of parametrized rotations in parametrized quantum circuits. The method consists in finding rotations that can be merged into a…

Quantum Physics · Physics 2024-07-11 Vivien Vandaele , Simon Perdrix , Christophe Vuillot

We investigate the problem of evaluating the output probabilities of Clifford circuits with nonstabilizer product input states. First, we consider the case when the input state is mixed, and give an efficient classical algorithm to…

Quantum Physics · Physics 2019-10-25 Kaifeng Bu , Dax Enshan Koh

(Abridged abstract.) In this thesis we introduce new models of quantum computation to study the emergence of quantum speed-up in quantum computer algorithms. Our first contribution is a formalism of restricted quantum operations, named…

Quantum Physics · Physics 2016-11-29 Juan Bermejo-Vega

Simulation of stabilizer circuits is a well-studied problem in quantum information processing, with a number of highly optimized algorithms available. Yet, we argue that further improvements can arise from the theoretical structure of…

Building upon [arXiv:2509.01224], we present a few methods on how to simulate the non-Clifford $d=5$ magic state cultivation circuits [arXiv:2409.17595] with a sum of $\approx 8$ Clifford ZX-diagrams on average, at $0.1\%$ noise. Compared…

Quantum Physics · Physics 2026-04-03 Kwok Ho Wan , Zhenghao Zhong , Ainhoa Zapirain

The quantum stabilizer formalism became foundational for understanding error correction soon after the realization of the first useful quantum error correction codes. Stabilizers provide a way to describe sets of quantum states which are…

Quantum Physics · Physics 2025-08-25 Sean Garner , Chenxu Liu , Meng Wang , Samuel Stein , Ang Li

Simulating quantum circuits classically is an important area of research in quantum information, with applications in computational complexity and validation of quantum devices. One of the state-of-the-art simulators, that of Bravyi et al,…

Quantum Physics · Physics 2019-08-07 Hammam Qassim , Joel J. Wallman , Joseph Emerson

Simulating Clifford and near-Clifford circuits using the extended stabilizer formalism has become increasingly popular, particularly in quantum error correction. Compared to the state-vector approach, the extended stabilizer formalism can…

Quantum Physics · Physics 2026-05-18 Vu Tuan Hai , Bui Cao Doanh , Le Vu Trung Duong , Pham Hoai Luan , Yasuhiko Nakashima

The stabilizer rank of a quantum state $\psi$ is the minimal $r$ such that $\left| \psi \right \rangle = \sum_{j=1}^r c_j \left|\varphi_j \right\rangle$ for $c_j \in \mathbb{C}$ and stabilizer states $\varphi_j$. The running time of several…

Quantum Physics · Physics 2022-03-09 Shir Peleg , Amir Shpilka , Ben Lee Volk

Improving the simulation of quantum circuits on classical computers is important for understanding quantum advantage and increasing development speed. In this paper, we explore a new way to express stabilizer states and further improve the…

Quantum Physics · Physics 2022-09-12 Alexander Tianlin Hu , Andrey Boris Khesin

A quantum circuit may be strongly classically simulated with the aid of ZX-calculus by decomposing its $t$ T-gates into a sum of $2^{\alpha t}$ classically computable stabiliser terms. In this paper, we introduce a general procedure to find…

Quantum Physics · Physics 2024-08-13 Matthew Sutcliffe , Aleks Kissinger

Generic quantum-circuit simulation appears intractable for conventional computers and may be unnecessary because useful quantum circuits exhibit significant structure that can be exploited during simulation. For example, Gottesman and Knill…

Data Structures and Algorithms · Computer Science 2017-12-12 Héctor J. García , Igor L. Markov

Hamiltonian simulation is a key quantum algorithm for modeling complex systems. To implement a Hamiltonian simulation, it is typically decomposed into a list of Pauli strings, each corresponds to an RZ rotation gate with many Clifford…

Quantum Physics · Physics 2025-10-16 Yingheng Li , Xulong Tang , Paul Hovland , Ji Liu

Consumption of magic states promotes the stabilizer model of computation to universal quantum computation. Here, we propose three different classical algorithms for simulating such universal quantum circuits, and characterize them by…

Quantum Physics · Physics 2021-03-23 James R. Seddon , Bartosz Regula , Hakop Pashayan , Yingkai Ouyang , Earl T. Campbell

Modeling and simulation is essential for predicting and verifying the behavior of fabricated quantum circuits, but existing simulation methods are either impractically costly or require an unrealistic simplification of error processes. We…