This paper introduces the Heisenberg representation of quantum computation, which describes the evolution of operators rather than states. This formalism is particularly useful for understanding quantum operations, especially those involving error correction and communication protocols. The paper discusses the Pauli group, a set of tensor products of Pauli matrices, which is sufficient to describe a wide range of quantum effects. However, it falls short of the full power of quantum computation. The Heisenberg representation is analogous to the Heisenberg picture in quantum mechanics, where operators evolve in time rather than states. This approach is especially useful for describing quantum circuits and analyzing the behavior of quantum gates. The paper explains how the Heisenberg representation can be used to analyze quantum circuits, including examples of quantum teleportation and error correction. The paper also discusses the Clifford group, a set of quantum gates that transform elements of the Pauli group into other elements of the Pauli group. The Clifford group is important in quantum computing because it allows for efficient descriptions of quantum circuits. The paper provides examples of how the Heisenberg representation can be used to analyze circuits involving the Clifford group. The paper concludes by discussing the importance of the Heisenberg representation in quantum computing, particularly in the context of quantum error correction and fault-tolerant quantum computation. It also highlights the limitations of the Clifford group and the need for additional gates to achieve universal quantum computation. The paper emphasizes the importance of the Heisenberg representation in understanding and analyzing quantum circuits, and its potential applications in quantum computing and communication.This paper introduces the Heisenberg representation of quantum computation, which describes the evolution of operators rather than states. This formalism is particularly useful for understanding quantum operations, especially those involving error correction and communication protocols. The paper discusses the Pauli group, a set of tensor products of Pauli matrices, which is sufficient to describe a wide range of quantum effects. However, it falls short of the full power of quantum computation. The Heisenberg representation is analogous to the Heisenberg picture in quantum mechanics, where operators evolve in time rather than states. This approach is especially useful for describing quantum circuits and analyzing the behavior of quantum gates. The paper explains how the Heisenberg representation can be used to analyze quantum circuits, including examples of quantum teleportation and error correction. The paper also discusses the Clifford group, a set of quantum gates that transform elements of the Pauli group into other elements of the Pauli group. The Clifford group is important in quantum computing because it allows for efficient descriptions of quantum circuits. The paper provides examples of how the Heisenberg representation can be used to analyze circuits involving the Clifford group. The paper concludes by discussing the importance of the Heisenberg representation in quantum computing, particularly in the context of quantum error correction and fault-tolerant quantum computation. It also highlights the limitations of the Clifford group and the need for additional gates to achieve universal quantum computation. The paper emphasizes the importance of the Heisenberg representation in understanding and analyzing quantum circuits, and its potential applications in quantum computing and communication.