The paper by Sergey Bravyi and Alexei Kitaev explores the possibility of universal quantum computation (UQC) using a limited set of elementary operations, specifically Clifford unitaries, the creation of the state \(|0\rangle\), and qubit measurement in the computational basis. Additionally, they allow the creation of a one-qubit ancilla in a mixed state \(\rho\), which is treated as a parameter of the model. The goal is to determine for which \(\rho\) UQC can be efficiently simulated. To achieve this, the authors construct purification protocols that consume multiple copies of \(\rho\) and produce a single output qubit with higher polarization. These protocols allow increasing the polarization only along certain "magic" directions. If the polarization of \(\rho\) along a magic direction exceeds a threshold value (about 65%), the purification asymptotically yields a pure state, called a magic state. The authors show that Clifford group operations combined with magic state preparation are sufficient for UQC. The paper also discusses the connection between their results and the Gottesman-Knill theorem, which states that operations from the Clifford group alone are not sufficient for UQC. The authors provide detailed proofs and constructions for two specific distillation schemes for \(T\)-type and \(H\)-type magic states, demonstrating that UQC can be efficiently simulated when the initial fidelity between \(\rho\) and these states exceeds certain threshold values. The threshold fidelities for \(T\)-type and \(H\)-type magic states are approximately 0.910 and 0.927, respectively. The paper concludes with a discussion on the practical implications and potential improvements to these threshold values, as well as open problems in the field.The paper by Sergey Bravyi and Alexei Kitaev explores the possibility of universal quantum computation (UQC) using a limited set of elementary operations, specifically Clifford unitaries, the creation of the state \(|0\rangle\), and qubit measurement in the computational basis. Additionally, they allow the creation of a one-qubit ancilla in a mixed state \(\rho\), which is treated as a parameter of the model. The goal is to determine for which \(\rho\) UQC can be efficiently simulated. To achieve this, the authors construct purification protocols that consume multiple copies of \(\rho\) and produce a single output qubit with higher polarization. These protocols allow increasing the polarization only along certain "magic" directions. If the polarization of \(\rho\) along a magic direction exceeds a threshold value (about 65%), the purification asymptotically yields a pure state, called a magic state. The authors show that Clifford group operations combined with magic state preparation are sufficient for UQC. The paper also discusses the connection between their results and the Gottesman-Knill theorem, which states that operations from the Clifford group alone are not sufficient for UQC. The authors provide detailed proofs and constructions for two specific distillation schemes for \(T\)-type and \(H\)-type magic states, demonstrating that UQC can be efficiently simulated when the initial fidelity between \(\rho\) and these states exceeds certain threshold values. The threshold fidelities for \(T\)-type and \(H\)-type magic states are approximately 0.910 and 0.927, respectively. The paper concludes with a discussion on the practical implications and potential improvements to these threshold values, as well as open problems in the field.