Two and three qubits quantum gates

Summary

In this video on quantum computing, we explore multi-qubit gates, including the CNOT and Toffoli gates, which are key to creating entangled states. We compare classical and quantum gates, highlighting the differences in their reversibility and how quantum gates act on qubits. The CNOT gate is demonstrated in both simple swaps and Bell state generation, showing its importance in creating entanglement for quantum computation. The Toffoli gate, an extension of the CNOT gate for three qubits, is also discussed, showcasing its role in more complex quantum circuits. The video concludes by emphasizing the potential of quantum gates in solving problems beyond classical computing.

Takeaways

• 😀 Quantum gates act on qubits and change their state, unlike classical gates which manipulate bits with fixed outputs based on Boolean logic.
• 😀 Unlike classical gates, quantum gates are reversible, and their number of inputs equals the number of outputs, maintaining the quantum system's consistency.
• 😀 Classical gates such as XOR and NAND are not reversible because it’s impossible to determine the inputs from the output in certain cases.
• 😀 Quantum gates, being unitary operators, can be undone using the adjoint version of the same gate.
• 😀 In a multi-qubit system, the state vector is represented by a tensor product of individual qubit vectors, leading to exponentially increasing state vector dimensions.
• 😀 The CNOT (Controlled NOT) gate is a two-input quantum gate that flips the target qubit based on the control qubit’s state, similar to the classical XOR gate.
• 😀 A CNOT gate can be used to create entanglement between qubits, a crucial property for many quantum algorithms.
• 😀 Swapping the states of two qubits can be achieved by applying a sequence of three CNOT gates with alternating control and target qubits.
• 😀 Bell states are maximally entangled two-qubit states, and can be generated using a combination of single-qubit gates (like the Hadamard gate) and the CNOT gate.
• 😀 The Toffoli gate, or controlled-controlled-NOT gate, is a three-qubit gate where two qubits act as controls and one as the target, flipping the target qubit only if both control qubits are in the |1> state.

Q & A

What is the main focus of the video?

The video focuses on multi-qubit gates, particularly the CNOT gate, and how they can be used to manipulate qubits, swap qubits, and generate maximally entangled states. It also introduces the Toffoli gate, a three-qubit gate.

What is the key difference between classical and quantum gates?

The key difference is that classical gates have a fixed output for a given set of inputs, while quantum gates have the same number of inputs and outputs and are reversible. Quantum gates are unitary operators that change the state of qubits, conserving the number of qubits during computation.

Why are quantum gates reversible, unlike classical gates in some cases?

Quantum gates are reversible because they are unitary operators. This means they can be undone by applying the adjoint (inverse) of the same gate, ensuring that no information is lost during the operation.

What does the tensor product represent in quantum systems?

The tensor product represents the state of a composite system of multiple qubits. It combines the state vectors of individual qubits into a higher-dimensional vector representing the combined system.

What is the function of the CNOT gate in quantum computing?

The CNOT gate, or controlled-NOT gate, is a two-input gate where one qubit (the control qubit) determines whether the second qubit (the target qubit) undergoes an XOR operation. This gate is crucial for creating entanglement and implementing multi-qubit quantum operations.

How does the CNOT gate affect the basis states of a two-qubit system?

The CNOT gate flips the target qubit (the second qubit) only when the control qubit (the first qubit) is in the state |1⟩. It leaves the qubits unchanged if the control qubit is in the state |0⟩.

How does the swap operation work using CNOT gates?

The swap operation can be performed by applying three CNOT gates in sequence, alternating the control and target qubits with each operation. This results in the qubits swapping their states, effectively exchanging their information.

What are Bell states and why are they important?

Bell states are a set of maximally entangled two-qubit states. They are important because they are used in quantum computing to demonstrate entanglement, which is a key resource for many quantum algorithms and protocols, such as quantum teleportation and quantum cryptography.

How can the Phi-plus Bell state be generated?

The Phi-plus Bell state can be generated by applying a Hadamard (H) gate to the first qubit, followed by a CNOT gate with the first qubit as the control and the second qubit as the target.

What is the Toffoli gate, and how does it differ from the CNOT gate?

The Toffoli gate, or controlled-controlled-NOT (CCNOT) gate, is a three-qubit gate that operates with two control qubits and one target qubit. The target qubit is flipped only if both control qubits are in the state |1⟩. It is an extension of the CNOT gate, which uses only one control qubit.