Circuits

 I got mail: Medium sent me a very promising looking collection

of ready-made Qiskit circuits. There's even an adder in the lot. Can't wait to try it:

https://medium.com/qiskit/a-guide-to-the-qiskit-circuit-library-36ee0f189956

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So there I was in bed nine-ish last night, with a laptop on my tray. Just chatting with

Meta to review things in my mind, when the electricity went out. Totally, everywhere.

We had been rained on steadily with the remnants of hurricane Debby all day. Crawled 

around the appartment with my tablet as the only light source. Decided to forget about 

candles and all that, and just opted to go to sleep.

Turns out Meta kept a copy of our chat. Below:

Quantum Computing; why do we consider that a qubit in superposition contains all the states at once rather than none of them?

In quantum computing, a qubit (quantum bit) in superposition is considered to contain all possible states simultaneously, rather than none of them, due to the principles of quantum mechanics. Here's why:

Superposition: A qubit can exist in a superposition of states, meaning it can represent both 0 and 1 at the same time. This is unlike a classical bit, which can only be 0 or 1.

Wave function: The qubit's state is described by a wave function, which encodes the probability of each possible state. When in superposition, the wave function is a linear combination of the wave functions for each possible state.

Measurement: When we measure a qubit in superposition, the wave function collapses to one of the possible states. This is known as wave function collapse.

Interpretations: There are different interpretations of quantum mechanics, but most agree that the qubit in superposition contains all possible states simultaneously. This is because:

Mathematically, the wave function describes all possible states at once.

Experimentally, measurements on qubits in superposition show statistical patterns consistent with all states being present.

Practically, considering all states simultaneously allows for quantum parallelism, enabling quantum computers to solve certain problems more efficiently than classical computers.

Considering a qubit in superposition to contain none of the states is not supported by the mathematical formalism and experimental evidence of quantum mechanics.

Would you like me to elaborate on any of these points?

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Is it just 0 or 1 and or all states in between as well?

A qubit in superposition can represent not just 0 and 1, but also any linear combination of 0 and 1, which includes all states in between. This means a qubit can exist in a superposition of:

0 and 1 (e.g., 0 and 1 simultaneously)

Any linear combination of 0 and 1, such as:

25% chance of being 0 and 75% chance of being 1

50% chance of being 0 and 50% chance of being 1 (equal superposition)

Any other combination, like 37.5% chance of being 0 and 62.5% chance of being 1

These intermediate states are often represented as complex numbers, using the Bloch sphere representation:

a|0+ b|1

where 'a' and 'b' are complex coefficients satisfying the normalization condition |a|^2 + |b|^2 = 1.

This allows for a continuous range of states, enabling quantum computers to explore an exponentially large solution space simultaneously.

Think of it like a color palette: a classical bit can only be red (0) or blue (1), but a qubit can be any shade of purple, representing all possible combinations of red and blue.

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What about + and -, and i values?

In addition to the 0 and 1 states, a qubit in superposition can also have coefficients with positive (+), negative (-), and imaginary (i) values.

Positive and negative coefficients:

+: Represents a "positive" probability amplitude (think of it like a "yes" or "true" bias)

-: Represents a "negative" probability amplitude (think of it like a "no" or "false" bias)

These signs affect how the qubit behaves when measured or combined with other qubits.

Imaginary coefficients (i):

i: Represents a "rotated" or "orthogonal" probability amplitude (think of it like a "maybe" or "superposed" state)

Imaginary coefficients allow for more subtle control over the qubit's behavior and are crucial for many quantum algorithms.

Example:

A qubit in superposition with complex coefficients:

a|0+ bi|1

where:

a is a real number (positive or negative)

b is an imaginary number (positive or negative, multiplied by i)

|a|^2 + |b|^2 = 1 (normalization condition)

This qubit exists in a superposition of 0 and 1, with both real and imaginary components influencing its behavior.

Bloch sphere representation:

The Bloch sphere is a useful visualization tool for qubits. It represents the qubit's state as a point on a 3D sphere, where:

The x-axis represents the real part of the coefficient (a)

The y-axis represents the imaginary part of the coefficient (b)

The z-axis represents the relative phase between the 0 and 1 states

This allows for a geometric understanding of how qubits behave and interact.

Would you like more information on qubits or quantum computing?

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How do imaginary components influence behavior?

Imaginary components in qubit coefficients influence behavior in several ways:

1. Phase relationships:

Imaginary coefficients introduce relative phases between the 0 and 1 states. This affects how the qubit interferes with itself or other qubits, leading to:

Constructive interference (reinforcement)

Destructive interference (cancellation)

2. Rotation in the Bloch sphere:

Imaginary coefficients rotate the qubit's state around the Bloch sphere, changing its orientation and behavior. This rotation:

Affects measurement outcomes

Influences entanglement with other qubits

3. Quantum gates and operations:

Imaginary coefficients impact how quantum gates and operations affect the qubit. For example:

Hadamard gate (H): Creates superposition with equal real and imaginary components

Pauli-Y gate (Y): Rotates the qubit around the y-axis, introducing imaginary components

Phase shift gate (S): Adds a relative phase between 0 and 1 states, influencing interference

4. Quantum algorithms:

Imaginary components play a crucial role in many quantum algorithms, such as:

Shor's algorithm (factorization): Relies on precise control of phase relationships

Grover's algorithm (search): Utilizes imaginary components for quantum interference

Quantum simulation: Often requires complex coefficients to accurately model physical systems

5. Entanglement and correlations:

Imaginary coefficients influence entanglement and correlations between qubits, enabling:

Quantum teleportation

Superdense coding

Quantum cryptography

In summary, imaginary components in qubit coefficients introduce phase relationships, rotations, and complex behavior, enabling the unique capabilities of quantum computing.

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How do imaginary numbers influence entanglement?

Imaginary numbers play a crucial role in entanglement by introducing complex phases that correlate the states of entangled qubits. Here's how:

1. Relative phases:

Imaginary coefficients create relative phases between the states of entangled qubits. These phases determine the correlations between the qubits.

2. Entanglement swapping:

Imaginary numbers enable entanglement swapping, where the entanglement between two qubits is transferred to two other qubits, even if they are separated by large distances.

3. Quantum correlations:

Imaginary components introduce quantum correlations between entangled qubits, allowing for:

Quantum teleportation (transferring a qubit's state from one location to another)

Superdense coding (encoding multiple classical bits in a single qubit)

Quantum cryptography (secure communication using entangled qubits)

4. Bell states:

Imaginary numbers are used to create Bell states, which are maximally entangled states of two qubits. Bell states are essential for quantum teleportation, superdense coding, and quantum cryptography.

5. Entanglement measurement:

Imaginary components influence the measurement outcomes of entangled qubits, allowing for the detection of entanglement and the verification of quantum correlations.

Mathematical representation:

Entanglement is often represented using complex coefficients, like:

|Φ= a|00+ bi|11

where a and b are complex numbers, and |00and |11represent the states of the entangled qubits.

The imaginary components (bi) introduce the relative phase between the states, enabling the correlations and quantum behavior characteristic of entanglement.

In summary, imaginary numbers are essential for creating and manipulating entanglement, enabling the unique capabilities of quantum computing and quantum information processing.

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Do we have complex coefficients just in cases of entanglement?

No, complex coefficients are not exclusive to entanglement. They appear in various quantum systems, including:

1. Single qubits:

Complex coefficients describe the superposition of a single qubit, like:

a|0+ bi|1

where a and b are complex numbers.

2. Quantum harmonic oscillators:

Complex coefficients describe the states of quantum harmonic oscillators, like photons or phonons.

3. Quantum field theory:

Complex coefficients are used to describe the states of particles in quantum field theory, like electrons or quarks.

4. Quantum many-body systems:

Complex coefficients appear in the study of quantum many-body systems, like the Hubbard model or the Heisenberg model.

5. Quantum optics:

Complex coefficients describe the states of light, like coherent states or squeezed states.

6. Quantum computing:

Complex coefficients are used in quantum algorithms, like Shor's algorithm or Grover's algorithm, even when entanglement is not explicitly involved.

Complex coefficients are a fundamental aspect of quantum mechanics, allowing for the description of:

Superposition

Interference

Phase relationships

Quantum correlations

Entanglement is just one of the many phenomena that rely on complex coefficients to describe the quantum behavior of systems.

In summary, complex coefficients are a general feature of quantum mechanics, appearing in various contexts beyond entanglement.

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So what is the role of real coefficients in quantum computing ie values on the x-axis?

In quantum computing, real coefficients (values on the x-axis) play a crucial role in:

1. Probability amplitudes:

Real coefficients represent the probability amplitudes of a qubit's states. The square of the real coefficient gives the probability of measuring the corresponding state.

2. Classical probabilities:

Real coefficients correspond to classical probabilities, which are essential for:

Quantum measurement outcomes

Quantum error correction

Quantum information processing

3. Quantum gates:

Real coefficients are used in quantum gate operations, such as:

Pauli-X gate (bit flip): swaps the real coefficients of |0and |1states

Pauli-Z gate (phase flip): changes the sign of the real coefficient of |1state

4. Quantum algorithms:

Real coefficients are used in various quantum algorithms, like:

Quantum Approximate Optimization Algorithm (QAOA): uses real coefficients to parameterize the quantum circuit

Variational Quantum Eigensolver (VQE): uses real coefficients to optimize the quantum state

5. Quantum state preparation:

Real coefficients are used to prepare specific quantum states, like:

|0= (1, 0) (real coefficient 1, imaginary coefficient 0)

|1= (0, 1) (real coefficient 0, imaginary coefficient 1)

6. Quantum measurement:

Real coefficients influence the measurement outcomes of qubits, determining the probabilities of observing |0or |1.

In summary, real coefficients in quantum computing represent probability amplitudes, classical probabilities, and play a crucial role in quantum gate operations, algorithms, state preparation, and measurement outcomes.

The x-axis (real coefficients) and y-axis (imaginary coefficients) together form the complex plane, enabling the description of quantum states and operations.