In the landscape of quantum computing, the interaction between qubits is harnessed through two-qubit gates. In this chapter, we venture into the realm of entangling gates, focusing on three prominent ones: the CNOT gate, the Controlled Phase gate, and the SWAP gate. These gates facilitate controlled operations between qubits, enabling essential entanglement and manipulation processes.

Introduction to Two-Qubit Gates

Two-qubit gates are fundamental in creating entanglement and implementing quantum algorithms. They allow qubits to interact and exchange information, laying the foundation for quantum parallelism and computational advantages. Among the diverse set of two-qubit gates, the CNOT, Controlled Phase, and SWAP gates hold significant roles.

CNOT Gate: Entangling Qubits

The CNOT (Controlled-NOT) gate, an iconic gate in quantum computing, entangles two qubits. The gate performs a NOT operation on the target qubit (flips its state) if and only if the control qubit is in the state |1⟩. This conditional flipping generates entanglement, a cornerstone of quantum algorithms and quantum error correction codes.

Controlled Phase Gate: Inducing Phase Changes

The Controlled Phase gate, also known as the CPHASE gate or the Controlled-Z gate, introduces a phase shift to the target qubit’s state if the control qubit is in the state |1⟩. Geometrically, it manifests as a rotation around the Z-axis of the Bloch sphere, modulating the quantum state’s phase. This gate’s ability to create controlled phase changes is crucial for quantum algorithms such as quantum Fourier transform.

SWAP Gate: Qubit Exchange

The SWAP gate, as the name implies, exchanges the states of two qubits. It effectively swaps the quantum information between the qubits, making them trade places. The SWAP gate is vital for rearranging qubit states and implementing algorithms that involve qubit permutations, as well as for certain quantum error correction techniques.

Entanglement and Quantum Gates

Two-qubit gates are instrumental in entangling qubits, a pivotal aspect of quantum computation. Entanglement endows quantum computers with the ability to perform complex computations more efficiently than classical counterparts. The entanglement generated by gates like the CNOT and Controlled Phase underpins the power of quantum algorithms, from Shor’s algorithm to quantum annealing.

Summary: Navigating Entangling Dynamics

In this chapter, we’ve journeyed through the landscape of two-qubit gates, exploring the CNOT, Controlled Phase, and SWAP gates. We’ve witnessed their roles in creating entanglement, inducing phase changes, and facilitating qubit exchange. Armed with this understanding, we’re poised to delve further into the interconnected world of multi-qubit gates and quantum algorithms that harness their potential.


By Yogev