QC_SoFar

 Been powering through on Qisbit, and the Hello World tutorial. So far:

                                                                       

From Gemini:

The diagram in line 4 shows a quantum circuit created using Qiskit. The circuit consists of two qubits, which are represented by the vertical lines labeled 0 and 1. 

The gates applied to the qubits are represented by the shapes in the circuit. In this case, there are two gates:

* H gate: This gate is applied to qubit 0 at the beginning of the circuit. The H gate represents a Hadamard gate, which is a fundamental quantum gate that transforms a qubit from the basis state |0> to the superposition state (|0> + |1>)/sqrt(2), and vice versa.

* CNOT gate: This gate is applied to qubits 0 and 1 at the end of the circuit. The CNOT gate is a controlled NOT gate, which is another fundamental quantum gate. It performs a NOT operation on the target qubit (qubit 1 in this case) only if the control qubit (qubit 0 in this case) is in the |1> state.

The output of the circuit is the state of the qubits after the gates have been applied. In this case, the output state is not shown explicitly in the diagram. However, it can be determined by simulating the circuit or by applying the gates mathematically.

 

I hope this explanation is helpful! Let me know if you have any other questions.

                                                                

                                                                     

                                                                          

Copilot

Certainly! Phase inversion is a concept that can be actively manipulated in quantum systems. Let’s dive into it:

Phase Inversion Basics:

Phase inversion involves changing the sign of the quantum state’s phase.

Consider a qubit state |ψ⟩ with a certain phase (e.g., eiϕ|0⟩). Inverting the phase means transforming it to -eiϕ|0⟩.

This operation doesn’t affect the probability of measurement outcomes but can impact interference patterns and quantum algorithms.

How to Achieve Phase Inversion:

Hadamard Gate (H): The Hadamard gate can create superpositions and invert phases. Applying H to |0⟩ results in |+⟩ (with a relative phase of π/2), and applying it again inverts the phase back to |0⟩.

Controlled Phase (CZ) Gate: The CZ gate inverts the phase of the |11⟩ state while leaving other states unchanged.

Phase Kickback: In some algorithms (like the quantum phase estimation algorithm), phase inversion occurs due to entanglement between control and target qubits.

Applications:

Grover’s Algorithm: Grover’s search algorithm uses phase inversion to amplify the amplitude of the correct solution state.

Quantum Fourier Transform: In the quantum phase estimation algorithm, phase inversion helps estimate eigenvalues of unitary operators.

Quantum Error Correction: Stabilizer codes use phase inversion to correct errors.

Experimental Realization:

Quantum gates (like H and CZ) are implemented using physical qubits (e.g., superconducting circuits, trapped ions).

Researchers actively work on improving gate fidelity and minimizing errors to achieve reliable phase manipulation.

In summary, phase inversion is indeed something we can actively control in quantum systems, and it plays a crucial role in quantum algorithms and protocols! 🌌🔀🚀

Quantum computing/ X gate vs CZ gate

Copilot

Using

the web

Certainly! Let’s explore the differences between the X gate and the CZ (Controlled-Z) gate in quantum computing:

X Gate (Pauli-X):

The X gate is a single-qubit operation that maps |0⟩ to |1⟩ and vice versa.

It equates to a rotation around the X-axis of the Bloch sphere by π radians.

Mathematically, it acts as follows:

X|0⟩ = |1⟩

X|1⟩ = |0⟩

CZ Gate (Controlled-Z):

The CZ gate is a multi-qubit gate with a unique behavior.

It performs a Z operation on the target qubit if and only if the control qubit is in the state |1⟩.

A Z operation imparts a phase of -1 to the target qubit when it’s in the state |1⟩, leaving it unchanged when it’s in the state |0⟩.

Mathematically, CZ maps:

α|00⟩ + β|01⟩ + γ|10⟩ + δ|11⟩ to α|00⟩ + β|01⟩ + γ|10⟩ − δ|11⟩

Clearly, the qubit roles are symmetric; it doesn’t matter which qubit is the control or target11.

Summary:

The X gate flips the state of a single qubit, while the CZ gate operates on two qubits.

X gate: |0⟩ ↔ |1⟩

CZ gate: |11⟩ ↔ -|11⟩ (phase inversion for |11⟩ state)

In circuits, CZ gates are often shown as:

───●───

        │

───●───

In summary, the X gate flips qubit states, while the CZ gate introduces phase inversion based on the control qubit’s state! 🚀🔀🔬

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We are ending up with 2 qubit at 1, and 4 at 0.