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Let's break down what's happening in this quantum computing example.

Understanding the Statevector

• Statevector: This represents the quantum state of a system. In this case, it's a qubit (a quantum bit) which can be in a superposition of two states, often represented as |0> and |1>.
• v = Statevector([(1 + 2.0j) / 3, -2 / 3]): This line defines the statevector 'v'. The values inside the square brackets are the amplitudes of the |0> and |1> states, respectively.
  • (1 + 2.0j) / 3: This is the complex amplitude of the |0> state.
  • -2 / 3: This is the amplitude of the |1> state.

Measurement

• v.measure(): This simulates measuring the qubit in state 'v'. Measurement collapses the superposition into a definite state.
• ('0', Statevector([0.4472136+0.89442719j, 0. +0.j ]), dims=(2,)): This is the output of the measurement.
  • '0': This indicates the qubit was measured in the |0> state.
  • Statevector([0.4472136+0.89442719j, 0. +0.j ]), dims=(2,): This shows the new statevector after the measurement. It's important to note that even though the measurement outcome was '0', the statevector is not exactly the same as the |0> state. There's still a small complex amplitude associated with the |1> state. This is due to the nature of quantum measurement, where the probability of measuring a state is related to the squared magnitude of the amplitude.

Important Points

• Probability: The probability of measuring a state is proportional to the square of the magnitude of its amplitude. In this case, the probability of measuring '0' is higher because the magnitude of (1 + 2.0j) / 3 is greater than the magnitude of -2 / 3.
• Quantum Measurement: Measurement is a fundamental concept in quantum mechanics. It alters the state of the system and introduces randomness.

Let me know if you want to explore any of these concepts further!

You're right to ask! The amplitudes of the |0> and |1> states are determined by how the quantum state is prepared.

In this example, the amplitudes are explicitly defined in the code:

v = Statevector([(1 + 2.0j) / 3, -2 / 3])

Here's what it means:

• Statevector is a function that creates a quantum state, specifically a qubit in this case.
• [(1 + 2.0j) / 3, -2 / 3]: This is a list containing the amplitudes.
  • The first element, (1 + 2.0j) / 3, is the amplitude for the |0> state. It's a complex number with a real part of 1/3 and an imaginary part of 2/3.
  • The second element, -2 / 3, is the amplitude for the |1> state.

Important: The amplitudes in a quantum state don't directly represent the probability of measuring a state. The probability is determined by the square of the magnitude of the amplitude.

How these amplitudes are determined in practice:

1. Quantum Circuits: In real quantum computers, you'd use a series of quantum gates (operations) to manipulate the qubits and create the desired superposition of states. These gates would affect the amplitudes.
2. Preparation Methods: There are different ways to prepare a quantum system in a specific state. For example, you could start with a qubit in the |0> state and apply a Hadamard gate to create a superposition.

Let me know if you have more questions!

You're absolutely right to focus on how preparation methods influence the amplitudes! Let's break down how this works, using the example you provided.

The Code Example

The code you provided is defining a qubit state directly:

v = Statevector([(1 + 2.0j) / 3, -2 / 3])

This explicitly assigns the amplitudes for the |0> and |1> states as (1 + 2.0j) / 3 and -2 / 3, respectively. This is a shortcut, like saying "Let's imagine this qubit is already in this particular state."

Preparation Methods in Real Quantum Computers

In real quantum computers, we don't directly set the amplitudes like this. Instead, we use quantum gates (operations) to manipulate qubits and create specific states. Here's how it works in general:

1. Starting Point: Usually, we begin with a qubit in the |0> state, which has an amplitude of 1 for |0> and 0 for |1>.
2. Applying Gates: We then apply various quantum gates to transform the qubit's state.
   • Example: Hadamard Gate
     • The Hadamard gate (H) is a common gate that creates an equal superposition of |0> and |1>:
     • If we apply H to a qubit in the |0> state, it becomes (|0> + |1>)/√2. The amplitudes for both |0> and |1> are now 1/√2.
   • Other Gates: There are many other gates that can rotate, shift, or entangle qubits, resulting in different amplitudes.

Key Point: The specific sequence and type of gates used will determine the final amplitudes of the |0> and |1> states.

Example: Creating the State in the Code

Let's see how we could create the state in your code (v = Statevector([(1 + 2.0j) / 3, -2 / 3])) using gates:

1. Start with |0>: Assume a qubit is initially in the |0> state.
2. Apply a Rotation: We can use a rotation gate (like the Rz gate) to rotate the qubit's state. The rotation angle would need to be carefully chosen to achieve the desired complex amplitude for |0>.
3. Apply Another Rotation: Another rotation gate would be needed to set the correct amplitude for the |1> state.

Important Notes:

• Complexity: Creating specific states with complex amplitudes can require a sequence of multiple gates.
• Experimental Challenges: Quantum computers are still under development. Precisely controlling gates and achieving the desired amplitudes is a significant challenge.

I hope this explanation helps you understand how preparation methods affect the amplitudes in quantum computing!

You've hit on a really important point about negative amplitudes in quantum mechanics! Here's the breakdown of why it's not as strange as it might seem:

• Amplitudes are Complex Numbers: In quantum mechanics, the amplitudes of quantum states can be complex numbers, meaning they have both a real and an imaginary part. A negative amplitude simply means the imaginary part of the complex number is negative.
• Probability is Related to Magnitude: The probability of measuring a particular state is determined by the square of the magnitude of its amplitude. The magnitude of a complex number is its distance from zero in the complex plane. Since the square of a negative number is positive, a negative amplitude doesn't directly translate to a negative probability.
• Interference: The negative sign in the amplitude is crucial for understanding quantum interference. Imagine the amplitudes as waves. When waves overlap, they can interfere constructively (adding up) or destructively (canceling out). The negative sign in the amplitude contributes to destructive interference, which is a fundamental aspect of quantum phenomena.

Why the Negative Sign Matters

• Phase: The negative sign in the amplitude is related to the phase of the quantum state. Phase is a subtle concept, but it essentially describes the relative timing of the wave-like behavior of a quantum state.
• Relative Amplitude: The negative sign doesn't change the probability of measuring the |1> state in this example, but it does affect how the state evolves over time and how it interacts with other quantum states.

Think of it like this:

Imagine two waves, one with a positive amplitude and one with a negative amplitude. When they overlap, they cancel each other out. In quantum mechanics, this kind of destructive interference can lead to unexpected outcomes, like the probability of measuring a particular state being reduced to zero.

Let me know if you have any more questions about complex amplitudes or interference!