The math cont'd

 

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Taking another look at yesterday's code. It is a simple circuit with 3 qubits; applying

the QFT inverse adds phase gates.

                                                                             

Chat with Gemini:

Qiskit; This code shows a simple circuit, and what applying the QFT inverse has added. Does this correspond to a matrix? Help me understand how this is done...

I'd be glad to help you understand the QFT inverse and its matrix representation in the provided Qiskit code.

QFT Inverse and Matrix Representation:

The quantum Fourier transform (QFT) is a unitary transformation that maps a quantum state from the computational basis to the Fourier basis. The QFT inverse (IQFT) performs the reverse operation, mapping a state from the Fourier basis back to the computational basis.   

In matrix form, the IQFT can be represented as a unitary matrix. For an n-qubit system, the IQFT matrix is an n x n complex matrix. Each element of the matrix corresponds to a specific amplitude in the transformed state.

Understanding the Code:

The provided Qiskit code demonstrates how to apply the IQFT to a 3-qubit quantum circuit. Let's break down the key steps:

import necessary libraries:

from qiskit import QuantumCircuit: Imports the QuantumCircuit class from Qiskit, which is used to create and manipulate quantum circuits.

import numpy as np: Imports the NumPy library for numerical operations.

Define the qft_inverse function:

This function takes a QuantumCircuit object and the number of qubits n as input.

It applies the IQFT to the last n qubits of the circuit.

The function uses a series of controlled phase gates (CP gates) and Hadamard gates (H gates) to implement the IQFT.

Create a quantum circuit:

A QuantumCircuit object with 3 qubits and 1 classical bit is created.

Prepare an initial state:

The initial state is prepared by applying Hadamard gates to qubits 0 and 2, and an X gate to qubit 1.

Apply the IQFT:

The qft_inverse function is called with the circuit and the number of qubits as arguments, applying the IQFT to the circuit.

Draw the circuit:

The circuit is drawn using the draw method, visualizing the gates and their connections.

Matrix Representation:

The matrix representation of the IQFT for 3 qubits can be quite complex. However, it can be constructed by combining the matrices for individual controlled phase gates and Hadamard gates. The specific matrix elements depend on the qubit indices and the phase angles used in the CP gates.

Additional Considerations:

The IQFT is an important quantum algorithm with applications in various quantum computing tasks, including quantum state tomography and quantum error correction.

The QFT and IQFT can be efficiently implemented using quantum circuits, making them practical for quantum algorithms.

Understanding the matrix representation of the IQFT can be helpful for analyzing quantum algorithms and understanding their properties.

I hope this explanation is helpful! Feel free to ask if you have any further questions.

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This is being done in the context of a phase estimation procedure...

I understand that the QFT inverse is being used in the context of a phase estimation procedure. Here's a more detailed explanation of how it fits into the overall process:

Phase Estimation Procedure:

State Preparation:

The initial state is prepared in a superposition of the target state and a reference state.

Controlled Evolution:

A controlled evolution operation is applied, where the target state is evolved by a unitary operator controlled by the reference state.

QFT:

The QFT is applied to the reference state, transforming it into the Fourier basis.

Measurement:

The reference state is measured, and the measurement outcome gives an estimate of the phase of the target state.

Role of QFT Inverse:

In the phase estimation procedure, the QFT is applied to the reference state to transform it into the Fourier basis. This allows the measurement outcome to directly correspond to the phase of the target state.

The QFT inverse is not explicitly used in the standard phase estimation algorithm. However, it can be useful in certain applications or variations of the algorithm where it's necessary to transform the measured state back to the computational basis.

Additional Considerations:

The accuracy of the phase estimation depends on the number of qubits used in the reference state. A larger number of qubits leads to a higher precision in the phase estimate.

The phase estimation procedure is a powerful tool for many quantum algorithms, including quantum algorithms for solving linear systems of equations and quantum chemistry simulations.

I hope this provides a clearer understanding of the role of the QFT inverse in the context of phase estimation. Feel free to ask if you have any further questions.

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 A quantum circuit always has discrete values; how is the QFT relevant?

That's a great question! While it's true that quantum circuits operate on discrete values (qubits), the QFT (Quantum Fourier Transform) is a powerful tool that can manipulate these discrete values in a way that is analogous to Fourier analysis in classical computing.   

Here's a breakdown of how the QFT is relevant even though it works with discrete values:

Superposition: Quantum states can exist in a superposition of multiple states simultaneously. This allows the QFT to operate on a continuous spectrum of possibilities, even though the individual qubits are discrete.   

Interference: The QFT can manipulate the phases of these superposition states, causing them to interfere constructively or destructively. This interference is crucial for many quantum algorithms, as it allows for the extraction of information from the superposition.   

Discrete Fourier Basis: The QFT transforms a quantum state from the computational basis to the Fourier basis. While the basis states are discrete, the QFT can reveal patterns and relationships between these states that are not apparent in the computational basis.   

In essence, the QFT leverages the unique properties of quantum mechanics to perform operations on discrete values in a way that is analogous to classical Fourier analysis. This makes it a fundamental tool in many quantum algorithms, including Shor's algorithm for factoring numbers and Grover's algorithm for searching unsorted databases.   

Would you like to delve deeper into a specific application of the QFT?

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The code calls for the QTF inverse. I know the QFT and its inverse are identical in a unitary matrix. What kind of matrix is at work here?

The matrix at work here is a unitary matrix.

Unitary matrices are a special type of square matrix that have the property of preserving the inner product between two vectors. In quantum mechanics, unitary matrices are used to represent quantum operations because they ensure that the probability amplitudes of a quantum state remain normalized after the operation.

The QFT and its inverse are identical in a unitary matrix because they are both unitary transformations. This means that they preserve the inner product between quantum states and, therefore, the probability amplitudes.

In the code you provided, the QFT inverse is being used to transform the quantum state back to the computational basis. This is a common operation in quantum computing, especially in algorithms like phase estimation and quantum Fourier transform-based algorithms.

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Was it not in the computational basis initially?

You're absolutely right! I apologize for the oversight.

In most cases, the quantum state is indeed initially prepared in the computational basis. The QFT inverse is then used to transform the state back to the computational basis after a quantum algorithm has been applied. This allows for the measurement of the state in a way that is meaningful and interpretable.

Thank you for catching my error. I'll be more careful in the future.

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Could I expect the same simulation results with or without the QFT inverse?

No, you would not expect the same simulation results with or without the QFT inverse.

The QFT inverse is a crucial step in many quantum algorithms, particularly those that involve phase estimation or Fourier analysis. Its purpose is to transform the quantum state back to the computational basis, allowing for meaningful measurements and interpretations.   

Without the QFT inverse, the final state would remain in the Fourier basis. This would make it difficult to extract the desired information or to compare the results to classical calculations. In essence, the QFT inverse acts as a bridge between the quantum world and the classical world, enabling us to understand and utilize the power of quantum computation.

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So is the initial circuit - because it contains Hadamard gates -  already transformed?

Yes, you're correct. The initial circuit, which contains Hadamard gates, has already been transformed.

Hadamard gates are a type of quantum gate that can be thought of as a superposition between the computational basis states (0 and 1). Applying a Hadamard gate to a qubit puts it into a superposition of both states, essentially transforming it from the computational basis to the Hadamard basis.   

Therefore, the initial circuit is already in a transformed state, and the QFT inverse is used to transform it back to the computational basis after the quantum algorithm has been applied.

Chat with Copilot:

Quantum Computing; define the notion of computational basis.

In quantum computing, the computational basis refers to a set of specific quantum states that serve as the fundamental building blocks for representing and manipulating information. These states are analogous to the classical bits (0 and 1) in traditional computing.

Typically, the computational basis states for a single qubit are denoted as ∣0⟩

 and ∣1⟩

. Any quantum state of a qubit can be expressed as a linear combination (or superposition) of these basis states. For example, a general quantum state ∣ψ⟩

 can be written as:

∣ψ⟩=α∣0⟩+β∣1⟩

where α

 and β

 are complex numbers that satisfy the normalization condition ∣α∣^2+∣β∣^2=1

These basis states are crucial for understanding quantum algorithms and operations, as they provide a clear framework for encoding and manipulating information using quantum mechanics. When a quantum measurement is performed, the state of the qubit collapses to one of these computational basis states, revealing the outcome of the computation.

Would you like to know more about how these basis states are used in specific quantum algorithms?

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The circuit:

                                                                               

The circuit with The QFT inverse in place: