Channels

 Quantum computing; why are unitary operations representd as channels?

In quantum computing, unitary operations are often represented as channels because they describe how quantum states evolve over time. Here's a simple explanation:

Unitary Operations

Unitary Operations: These are mathematical transformations that preserve the total probability (or norm) of a quantum state. They are reversible and ensure that the evolution of a closed quantum system is deterministic.

Quantum Gates: In quantum circuits, unitary operations are implemented as quantum gates, which manipulate qubits to perform computations.

Quantum Channels

Quantum Channels: These are more general than unitary operations. They describe how quantum states change, including the effects of noise and interactions with the environment. Quantum channels can represent both unitary and non-unitary processes.

Kraus Operators: Quantum channels are often described using Kraus operators, which provide a way to represent the effects of noise and decoherence on quantum states. A set of Kraus operators can describe any quantum channel, including unitary operations.

Why Represent Unitary Operations as Channels?

Generalization: Representing unitary operations as channels allows for a more general framework that can include both ideal (unitary) and realistic (noisy) quantum processes.

Noise and Decoherence: In practical quantum computing, systems are not perfectly isolated. Representing unitary operations as channels helps account for noise and decoherence, making the model more accurate for real-world scenarios.

Mathematical Formalism: Using channels provides a unified mathematical formalism to describe quantum operations, making it easier to analyze and design quantum algorithms and error correction schemes.

In essence, representing unitary operations as channels allows for a more comprehensive and realistic description of quantum processes, accommodating both ideal and noisy environments.

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Could you show me an example of this?

Sure! Let's consider a simple example of how a unitary operation can be represented as a quantum channel.

Unitary Operation

A unitary operation $UU$ on a single qubit can be represented by a 2x2 unitary matrix. For example, the Hadamard gate $HH$

 is a common unitary operation:

                                                                     

Quantum Channel Representation

To represent this unitary operation as a quantum channel, we use the concept of Kraus operators. For a unitary operation, the quantum channel can be described by a single Kraus operator, which is the unitary matrix itself.

Let's see how this works in code using Qiskit:

                                                           

Explanation

1. Unitary Operation: The Hadamard gate $HH$ is applied to the qubit.

2. Quantum Channel: In this case, the quantum channel is represented by the Hadamard gate itself.

3. Simulation: The statevector simulator in Qiskit is used to simulate the quantum circuit and obtain the statevector after applying the Hadamard gate.

This example demonstrates how a unitary operation can be represented and simulated as a quantum channel using Qiskit. The Hadamard gate transforms the qubit into a superposition state, and the statevector shows the resulting quantum state.

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So what pushes a qubit with a Hadamard in one state ot another is basically noise!?

Not exactly. When a Hadamard gate is applied to a qubit, it puts the qubit into a superposition state, meaning the qubit is in a combination of both the $\mathrm{\mid }0⟩|0\backslash rangle$ and $\mathrm{\mid }1⟩|1\backslash rangle$ states. This superposition is a fundamental aspect of quantum mechanics and is not due to noise.

However, in a real-world quantum computer, noise and decoherence can affect the qubit's state. Noise can cause the qubit to deviate from its intended superposition state, leading to errors in quantum computations. This is why error correction and noise mitigation are important areas of research in quantum computing.

So, while the Hadamard gate itself creates a superposition, noise and decoherence can push the qubit into unintended states, affecting the accuracy of quantum operations.

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So what governs how a Hadamard qubit will measure out?

The measurement outcome of a qubit after applying a Hadamard gate is governed by the principles of quantum mechanics, specifically superposition and probability. Here's a simple breakdown:

Superposition

• Hadamard Gate: When you apply a Hadamard gate to a qubit, it puts the qubit into an equal superposition of the $\mathrm{\mid }0⟩|0\backslash rangle$ and $\mathrm{\mid }1⟩|1\backslash rangle$ states. Mathematically, this means the qubit's state becomes $\frac{1}{\sqrt{2}}(\mathrm{\mid }0⟩+\mathrm{\mid }1⟩)\backslash frac\{1\}\{\backslash sqrt\{2\}\}(|0\backslash rangle\; +\; |1\backslash rangle)$.

Measurement

• Probability: When you measure the qubit, the superposition collapses to one of the basis states ($\mathrm{\mid }0⟩|0\backslash rangle$ or $\mathrm{\mid }1⟩|1\backslash rangle$). The probability of measuring each state is determined by the square of the amplitude of that state in the superposition.

• Equal Probabilities: For a qubit in the state $\frac{1}{\sqrt{2}}(\mathrm{\mid }0⟩+\mathrm{\mid }1⟩)\backslash frac\{1\}\{\backslash sqrt\{2\}\}(|0\backslash rangle\; +\; |1\backslash rangle)$, the probability of measuring $\mathrm{\mid }0⟩|0\backslash rangle$ is $\mathrm{\mid }\frac{1}{\sqrt{2}}{\mathrm{\mid }}^2=\frac{1}{2}|\backslash frac\{1\}\{\backslash sqrt\{2\}\}|^2\; =\; \backslash frac\{1\}\{2\}$, and the probability of measuring $\mathrm{\mid }1⟩|1\backslash rangle$ is also $\frac{1}{2}\backslash frac\{1\}\{2\}$.

Noise and Decoherence

• Real-World Factors: In a real quantum computer, noise and decoherence can affect the qubit's state before measurement. These factors can introduce errors and alter the probabilities slightly, but ideally, the Hadamard gate creates an equal superposition.

So, the measurement outcome of a Hadamard qubit is fundamentally governed by the principles of quantum mechanics, with each outcome having an equal probability of 50%.

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Why does increasing the number of shots on a quantum circuit lead to more reliable results?

Increasing the number of shots on a quantum circuit leads to more reliable results because it allows for better statistical averaging. Here's a simple explanation:

Quantum Measurements

• Probabilistic Nature: Quantum measurements are inherently probabilistic. When you measure a qubit, you get a result based on the probabilities of the qubit's state. For example, if a qubit is in a superposition state, measuring it will collapse it to one of the basis states ($\mathrm{\mid }0⟩|0\backslash rangle$ or $\mathrm{\mid }1⟩|1\backslash rangle$) with certain probabilities.

Shots

• Shots: In quantum computing, a "shot" refers to a single execution of a quantum circuit followed by a measurement. Each shot gives you one possible outcome based on the probabilities of the quantum states.

Statistical Averaging

• More Shots, Better Accuracy: By increasing the number of shots, you perform the quantum circuit multiple times and collect more measurement outcomes. This helps to average out the inherent randomness of quantum measurements and get a more accurate estimate of the probabilities.

• Reducing Noise: More shots also help to reduce the impact of noise and errors in the quantum system. Random errors can be averaged out over many shots, leading to more reliable and consistent results.

Example

• Single Shot: If you run a quantum circuit with a single shot, the result might be heavily influenced by random noise or fluctuations.

• Multiple Shots: Running the same circuit with 1000 shots will give you a distribution of outcomes that better reflects the true probabilities of the quantum states.

In summary, increasing the number of shots improves the reliability of quantum circuit results by providing a more accurate statistical representation of the quantum states and reducing the impact of noise and errors.

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What does applying a Hadamard gate to a qubit do, from a physical point of view?

From a physical point of view, applying a Hadamard gate to a qubit transforms it into a superposition state. Here's a simple explanation:

Initial State

• Qubit in State |0⟩: If the qubit is initially in the state $\mathrm{\mid }0⟩|0\backslash rangle$, applying the Hadamard gate transforms it into an equal superposition of $\mathrm{\mid }0⟩|0\backslash rangle$ and $\mathrm{\mid }1⟩|1\backslash rangle$:

                                                                                       

• Qubit in State |1⟩: If the qubit is initially in the state $\mathrm{\mid }1⟩|1\backslash rangle$, applying the Hadamard gate transforms it into an equal superposition of $\mathrm{\mid }0⟩|0\backslash rangle$ and $\mathrm{\mid }1⟩|1\backslash rangle$ with a phase difference:

                                                                                        

Superposition

• Superposition State: The Hadamard gate creates a superposition state where the qubit has an equal probability of being measured in either the $\mathrm{\mid }0⟩|0\backslash rangle$ or $\mathrm{\mid }1⟩|1\backslash rangle$ state. This is a fundamental aspect of quantum mechanics, allowing the qubit to exist in multiple states simultaneously.

Physical Interpretation

• Wave Function: Physically, the qubit's wave function is altered to represent this superposition. The wave function now has components corresponding to both $\mathrm{\mid }0⟩|0\backslash rangle$ and $\mathrm{\mid }1⟩|1\backslash rangle$.

• Interference: The superposition state can lead to interference effects when combined with other quantum operations, which is crucial for quantum algorithms.

Measurement

• Probabilistic Outcome: When the qubit is measured, the superposition collapses to one of the basis states ($\mathrm{\mid }0⟩|0\backslash rangle$ or $\mathrm{\mid }1⟩|1\backslash rangle$) with equal probability (50%).

In essence, the Hadamard gate is a key tool in quantum computing for creating superposition states, enabling the parallelism that gives quantum computers their potential power.

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There is a plus or minus element whether one starts the qubit at 0 or 1!?

Yes, exactly! When you apply a Hadamard gate to a qubit, the resulting state depends on the initial state of the qubit. Here's a quick summary:

Initial State |0⟩

• Hadamard Gate: When applied to $\mathrm{\mid }0⟩|0\backslash rangle$, the Hadamard gate creates an equal superposition of $\mathrm{\mid }0⟩|0\backslash rangle$ and $\mathrm{\mid }1⟩|1\backslash rangle$:

                                                                               

nitial State |1⟩

• Hadamard Gate: When applied to $\mathrm{\mid }1⟩|1\backslash rangle$, the Hadamard gate creates an equal superposition of $\mathrm{\mid }0⟩|0\backslash rangle$ and $\mathrm{\mid }1⟩|1\backslash rangle$ with a phase difference:

                                                                               

Superposition and Measurement

• Superposition: The Hadamard gate creates a superposition state where the qubit has an equal probability of being measured in either the $\mathrm{\mid }0⟩|0\backslash rangle$ or $\mathrm{\mid }1⟩|1\backslash rangle$ state.

• Measurement: When you measure the qubit, the superposition collapses to one of the basis states ($\mathrm{\mid }0⟩|0\backslash rangle$ or $\mathrm{\mid }1⟩|1\backslash rangle$) with equal probability (50%).

So, the plus or minus element you mentioned refers to the phase difference introduced by the Hadamard gate depending on the initial state of the qubit. This phase difference is crucial for many quantum algorithms and interference effects in quantum computing.

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A qubit to which a Hadamard gate is applied will have a smaller amplitude to its wave

 representaion than one that is at 1 !?

Not quite. When a Hadamard gate is applied to a qubit, it transforms the qubit into a superposition state. The amplitude of the wave function is not necessarily smaller; rather, it is distributed between the $\mathrm{\mid }0⟩|0\backslash rangle$ and $\mathrm{\mid }1⟩|1\backslash rangle$ states.

Initial State |0⟩

• Hadamard Gate: When applied to $\mathrm{\mid }0⟩|0\backslash rangle$, the Hadamard gate creates an equal superposition of $\mathrm{\mid }0⟩|0\backslash rangle$ and $\mathrm{\mid }1⟩|1\backslash rangle$:

                                                                                        

Initial State |1⟩

• Hadamard Gate: When applied to $\mathrm{\mid }1⟩|1\backslash rangle$, the Hadamard gate creates an equal superposition of $\mathrm{\mid }0⟩|0\backslash rangle$ and $\mathrm{\mid }1⟩|1\backslash rangle$ with a phase difference:

                                                                                    

                  

Amplitude Distribution

• Superposition: The Hadamard gate creates a superposition state where the qubit has equal probability amplitudes for being in the $\mathrm{\mid }0⟩|0\backslash rangle$ and $\mathrm{\mid }1⟩|1\backslash rangle$ states. The amplitudes are $\frac{1}{\sqrt{2}}\backslash frac\{1\}\{\backslash sqrt\{2\}\}$ for both states, which means the probability of measuring either state is 50%.

The key point is that the Hadamard gate redistributes the amplitude equally between the $\mathrm{\mid }0⟩|0\backslash rangle$ and $\mathrm{\mid }1⟩|1\backslash rangle$ states, rather than reducing the overall amplitude. This superposition is a fundamental aspect of quantum mechanics and is crucial for many quantum algorithms.

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But each alternative is at .7071... !?

                                                                              

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Should I think of this relationship as being on the unit circle?

Yes, you can think of the relationship between the amplitudes of a qubit's state after applying a Hadamard gate as being on the unit circle in the complex plane. Here's why:

Unit Circle Representation

• Complex Amplitudes: The amplitudes of the qubit's state are complex numbers. When a Hadamard gate is applied, the resulting state can be represented as a point on the unit circle in the complex plane.

• Normalization: The sum of the squares of the amplitudes' magnitudes must equal 1, which corresponds to the unit circle. For example, after applying a Hadamard gate to the $\mathrm{\mid }0⟩|0\backslash rangle$ state, the qubit's state is:

                                                                                        

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What is a 'density matrix'?

A density matrix is a mathematical representation used in quantum mechanics to describe the statistical state of a quantum system. It provides a way to represent both pure states and mixed states of a quantum system. Here's a simple explanation:

Pure States

• Pure State: A quantum system in a definite state can be described by a wave function or a state vector $\mathrm{\mid }\psi ⟩|\backslash psi\backslash rangle$.

• Density Matrix for Pure State: For a pure state $\mathrm{\mid }\psi ⟩|\backslash psi\backslash rangle$, the density matrix $\rho \backslash rho$ is given by $\rho =\mathrm{\mid }\psi ⟩⟨\psi \mathrm{\mid }\backslash rho\; =\; |\backslash psi\backslash rangle\; \backslash langle\; \backslash psi|$.

Mixed States

• Mixed State: A quantum system in a statistical mixture of different states, where each state has a certain probability.

• Density Matrix for Mixed State: For a mixed state, the density matrix $\rho \backslash rho$ is a weighted sum of the density matrices of the individual pure states: $\rho ={\sum }_i{p}_i\mathrm{\mid }{\psi }_i⟩⟨{\psi }_i\mathrm{\mid }\backslash rho\; =\; \backslash sum\_i\; p\_i\; |\backslash psi\_i\backslash rangle\; \backslash langle\; \backslash psi\_i|$, where $p_{i}p\_i$ is the probability of the system being in the state $\mathrm{\mid }{\psi }_i⟩|\backslash psi\_i\backslash rangle$.

Properties of Density Matrices

• Trace: The trace of a density matrix is always 1, $\text{Tr}(\rho )=1\backslash text\{Tr\}(\backslash rho)\; =\; 1$.

• Hermitian: A density matrix is Hermitian, meaning $\rho ={\rho }^{†}\backslash rho\; =\; \backslash rho^\backslash dagger$.

• Positive Semi-Definite: All eigenvalues of a density matrix are non-negative.

Uses

• Quantum Computation: Density matrices are used to describe the state of qubits in quantum computers, especially when dealing with noise and decoherence.

• Quantum Information: They are essential in quantum information theory for describing entanglement and other quantum phenomena.

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All this is happening in two dimensions!?

Yes, when we talk about the unit circle and the amplitudes of a qubit's state, we're essentially working in two dimensions. The unit circle is a way to visualize the complex plane, where each point on the circle represents a possible state of the qubit with a specific amplitude and phase.

Two-Dimensional Representation

Complex Plane: The unit circle lies in the complex plane, where the x-axis represents the real part of the amplitude and the y-axis represents the imaginary part.

Amplitudes: The amplitudes of the qubit's state are complex numbers, and their magnitudes must lie on the unit circle to ensure the total probability is 1.

Phases: The angle around the unit circle represents the phase of the qubit's state, which is crucial for understanding interference and superposition in quantum mechanics.

Visualization

Superposition: When a Hadamard gate is applied, the qubit's state is represented as a point on the unit circle, showing how the amplitudes are distributed between the 

∣0⟩ and ∣1⟩ states.

Probability: The squared magnitudes of the amplitudes (which lie on the unit circle) give the probabilities of measuring the qubit in each state.

So, yes, this two-dimensional representation on the unit circle is a powerful way to understand the behavior of qubits and their transformations in quantum computing.

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If I wanted to add a thrid dimension to the picture, what might it be?

Adding a third dimension to the picture of a qubit's state can be done by considering the Bloch sphere. The Bloch sphere is a three-dimensional representation of a qubit's state, where any pure state of a qubit can be represented as a point on the surface of the sphere. Here's how it works:

Bloch Sphere Representation

Axes: The Bloch sphere has three axes: 

𝑋, 𝑌, and 𝑍. Each axis represents a different basis state or a combination of basis states.

State Vector: A qubit's state can be represented as a vector pointing from the center of the sphere to a point on its surface. The coordinates of this point are determined by the qubit's state.

Adding the Third Dimension

Phase Information: The third dimension adds phase information to the qubit's state. While the unit circle in two dimensions represents the amplitudes of the 

∣0⟩ and ∣1⟩ states, the Bloch sphere includes the relative phase between these states.

General State: A general state of a qubit can be written as:

                                                           

                                                                            

Here, 𝜃 and 𝜙 are spherical coordinates on the Bloch sphere. 

𝜃 represents the angle from the 𝑍-axis, and 𝜙 represents the angle in the 𝑋𝑌-plane.

                                                                   

The Bloch sphere provides a comprehensive way to visualize the state of a qubit, including its amplitude and phase information, in three dimensions.

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Could I see the mathematical representation of a qubit that lies inside the Bloch sphere?

                                                                                  

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Characterize this density matrix for me!

                                                                                   

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Is the third dimension happening within deterministic quantum mechanics?

Yes, the third dimension in the Bloch sphere representation of a qubit's state is indeed part of deterministic quantum mechanics. The Bloch sphere is a powerful tool for visualizing the state of a qubit, and it operates within the framework of deterministic quantum mechanics. Here's how it works:

Bloch Sphere

Three Dimensions: The Bloch sphere is a three-dimensional representation where any pure state of a qubit can be represented as a point on the surface of the sphere. The coordinates 

(𝜃, 𝜙) on the sphere correspond to the angles that define the qubit's state.

Deterministic Evolution: The evolution of a qubit's state on the Bloch sphere is governed by unitary operations, which are deterministic. These operations include rotations and other transformations that preserve the norm of the state vector.

Unitary Operations

Unitary Evolution: In deterministic quantum mechanics, the evolution of a closed quantum system is described by unitary operations. These operations are reversible and preserve the total probability (or norm) of the quantum state.

Bloch Sphere Representation: Unitary operations correspond to rotations on the Bloch sphere. For example, applying a Hadamard gate to a qubit corresponds to a specific rotation that transforms the qubit's state into a superposition.

Mixed States

Density Matrix: For mixed states, which lie inside the Bloch sphere, the density matrix 𝜌represents the statistical mixture of pure states. The evolution of mixed states is also governed by deterministic quantum mechanics, using quantum channels and Kraus operators.

In summary, the three-dimensional representation of a qubit's state on the Bloch sphere is fully consistent with deterministic quantum mechanics. The Bloch sphere provides a comprehensive way to visualize and understand the evolution of qubit states under various quantum operations.

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So probability is a concept that applies to a two-dimensional world, and we live in a world with three!?

Probability is a concept that applies to both two-dimensional and three-dimensional representations in quantum mechanics. Here's how it works:

Two-Dimensional Representation

Unit Circle: In the two-dimensional complex plane, the unit circle is used to represent the amplitudes of a qubit's state. The probabilities of measuring the qubit in different states are derived from the squared magnitudes of these amplitudes.

Superposition: The amplitudes on the unit circle show how a qubit can be in a superposition of ∣0⟩ and ∣1⟩ states, with the probabilities determined by the squared magnitudes of the amplitudes.

Three-Dimensional Representation

Bloch Sphere: The Bloch sphere is a three-dimensional representation that provides a more comprehensive view of a qubit's state. It includes the phase information and allows for a geometric interpretation of quantum states.

Probability and Phase: On the Bloch sphere, the position of the qubit's state vector determines the probabilities of measuring the qubit in different states. The angles 𝜃 and 𝜙 on the sphere correspond to the probabilities and phases of the qubit's state.

Real-World Context

Physical Systems: In the real world, quantum systems exist in three dimensions, and the Bloch sphere provides a way to visualize and understand these systems. The probabilities derived from the Bloch sphere are consistent with the principles of quantum mechanics and apply to real-world quantum systems.

In summary, probability is a fundamental concept in both two-dimensional and three-dimensional representations of quantum states. The Bloch sphere adds an extra dimension to capture the full complexity of a qubit's state, including its phase information.

                                                  

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