I’ve pasted into ChatGPT the ‘Quantum computing for the very curious’ book by Andy Matuschak and Michael Nielsen. I’ve asked him to summarize it, then followed up with some questions:

Q: Summarize the book below:

[…]

Quantum computing for the very curious is a book that presents an introduction to the basics of quantum computing. The book is organized into three main parts: Part I covers the state of a qubit, Part II introduces quantum logic gates and Part III discusses universal quantum computing. The book explains the concepts of quantum computing in a way that is accessible to readers with no previous knowledge of the topic, and it uses a mnemonic medium to make the material easy to remember. Special thanks are given to the sponsors who funded the book, including Adam Wiggins, Andrew Sutherland, and Paul Sutter.

Q: In more detail, what is an X gate? And a H gate?

An X gate, also known as a NOT gate, is a fundamental quantum logic gate that performs a bit flip on a single qubit. This means that if the qubit is in the state “0”, the X gate will change it to the state “1”, and if the qubit is in the state “1”, the X gate will change it to the state “0”. The X gate is often used in quantum circuits as a building block for more complex operations.

A H gate, also known as a Hadamard gate, is another fundamental quantum logic gate that is used to put a qubit into a superposition state. This means that the qubit is in a linear combination of the “0” and “1” states, with equal probability amplitudes. The H gate is used in many quantum algorithms, including the quantum search algorithm and the quantum teleportation algorithm. It is also often used in conjunction with other gates to perform more complex operations.

Q: How are gates expressed as hermitian matrices?

Quantum logic gates can be represented as matrices, and specifically as hermitian matrices. A hermitian matrix is a square matrix that is equal to its own conjugate transpose, which means that the elements on the diagonal are real numbers and the elements off the diagonal are equal to their complex conjugates. This property is useful for quantum computing because it guarantees that the matrix will preserve the inner product of two quantum states.

For example, the X gate can be represented as the hermitian matrix:

X = [[0, 1], [1, 0]]

This matrix represents the operation of flipping the state of a qubit from “0” to “1” or vice versa. The H gate, on the other hand, can be represented as the hermitian matrix:

H = 1 / sqrt(2) * [[1, 1], [1, -1]]

This matrix represents the operation of putting a qubit into a superposition state, with equal probability amplitudes for the “0” and “1” states. These matrices can be multiplied together or combined with other gates to perform more complex operations on a qubit or a group of qubits.

Q: What are the general states of a qbit?

A qubit, the fundamental unit of quantum information, can exist in a number of different states. The most basic states of a qubit are the “0” state and the “1” state, which correspond to the classical states of a bit in classical computing. However, unlike classical bits, qubits can also exist in superposition states, which are linear combinations of the “0” and “1” states with complex probability amplitudes. This means that a qubit can be in a state that is both “0” and “1” at the same time, with a certain probability of being measured as “0” and a certain probability of being measured as “1”.

Additionally, qubits can exist in entanglement states, in which the state of one qubit is correlated with the state of another qubit. This allows for the creation of entangled quantum systems in which the state of one qubit can affect the state of another qubit, even when the qubits are separated by large distances. These various states of a qubit are important for quantum computing, as they allow for the representation and manipulation of quantum information in ways that are impossible with classical bits.

Q: How is a qbit expressed in an arbitrary computational basis? How does the qbit formula change if I change the computational basis?

A qubit can be expressed in an arbitrary computational basis by representing the qubit as a two-dimensional complex vector. This vector can be written in terms of the computational basis states “0” and “1” as:

|psi> = a|0> + b|1>
where a and b are complex probability amplitudes that satisfy the normalization condition a ^2 + b ^2 = 1. This expression represents the state of the qubit in the computational basis “0” and “1”, and it shows that the qubit is in a superposition of these two basis states with the given probability amplitudes.

If the computational basis is changed from “0” and “1” to a different basis, such as “+” and “-“, the formula for the qubit state will change accordingly. For example, if the computational basis is changed to “+” and “-“, the qubit state can be written as:

|psi> = c|+> + d|->

where c and d are the complex probability amplitudes in the new basis. In this case, the qubit is still in a superposition of the “+” and “-“ states, but the probability amplitudes have changed to reflect the new basis. The specific values of the probability amplitudes will depend on the specific state of the qubit and the transformation that is applied to change the basis.

Q: Why is the quantum state of a qbit a unitary vector in the two dimensional complex space?

The quantum state of a qubit is a unitary vector in the two-dimensional complex space because unitary vectors are the only vectors that preserve the inner product of two quantum states. This is an important property for quantum computing, as it allows for the manipulation of quantum states in a way that is consistent with the laws of quantum mechanics.

A unitary vector is a vector that has unit magnitude and is unchanged by a unitary transformation. In the case of a qubit, the magnitude of the quantum state vector is always equal to 1, and a unitary transformation is a linear transformation that preserves the inner product of two vectors. This means that if the quantum state of a qubit is represented by the vector |psi>, and a unitary transformation U is applied to the qubit, the resulting quantum state will be given by the vector U|psi>.

In general, the quantum state of a qubit can be written as a linear combination of the computational basis states “0” and “1”, with complex probability amplitudes:

|psi> = a|0> + b|1>

where a and b are complex numbers that satisfy the normalization condition |a|^2 + |b|^2 = 1. This expression shows that the quantum state of a qubit is a unitary vector in the two-dimensional complex space, as it has unit magnitude and is unchanged by unit

Q: What happens when we apply a measurement to a quantum circuit?

When a measurement is applied to a quantum circuit, the quantum state of the circuit collapses into one of the possible measurement outcomes. This means that the quantum state of the circuit is no longer a superposition of different states, but is instead a definite state with a certain probability of being measured.

For example, suppose we have a quantum circuit with two qubits in the state |psi> = a|00> + b|01> + c|10> + d|11>, where a, b, c, and d are complex probability amplitudes. If we apply a measurement to this circuit, the quantum state will collapse into one of the four basis states “00”, “01”, “10”, or “11” with a certain probability. For example, if the measurement outcome is “01”, the quantum state of the circuit will be |psi> = b|01>, and the probability of measuring “01” will be |b|^2.

The specific outcome of the measurement will depend on the quantum state of the circuit and the type of measurement that is applied. In general, it is not possible to predict the outcome of a measurement on a quantum circuit, as the state of the circuit is inherently uncertain until it is measured. However, the probabilities of different measurement outcomes can be calculated based on the quantum state of the circuit, and these probabilities can be used to evaluate the behavior of the circuit.

Q: What are the X, Y, Z gates? How do they relate to each other?

he X, Y, and Z gates are three fundamental quantum logic gates that are used in quantum computing. The X gate, also known as the NOT gate, flips the state of a qubit from “0” to “1” or vice versa. The Y gate performs a rotation of the qubit state around the Y axis of the Bloch sphere, and the Z gate performs a rotation of the qubit state around the Z axis of the Bloch sphere.

These gates are related to each other because they can be represented as matrices that act on the state of a qubit. The X gate is represented by the matrix:

X = [[0, 1], [1, 0]]

The Y gate is represented by the matrix:

Y = [[0, -i], [i, 0]]

And the Z gate is represented by the matrix:

Z = [[1, 0], [0, -1]]

These matrices show how the gates act on the state of a qubit, and they can be combined with other gates to perform more complex operations on the qubit. Additionally, the X, Y, and Z gates are related to each other because they form a basis for the space of single-qubit gates, which means that any single-qubit gate can be written as a linear combination of the X, Y, and Z gates. This property is useful for quantum computing, as it allows for the representation of single-qubit gates in a compact and convenient form.

Q: What are the products of pairs of X, Y, Z? And the squares of X, Y, Z? Are these properties similar to those of quaternions i, j, k?

The products of pairs of X, Y, and Z gates are given by:

X*X = Y*Y = Z*Z = I
X*Y = iZ
Y*X = -iZ
X*Z = Y
Z*X = -Y
Y*Z = X
Z*Y = -X

where I is the identity matrix and i is the imaginary unit. The squares of the X, Y, and Z gates are given by:

X^2 = Y^2 = Z^2 = I

These properties are similar to those of the quaternions i, j, and k, which are the three basic unit quaternions. In the quaternion algebra, the products of pairs of i, j, and k are given by:

i*i = j*j = k*k = -1
i*j = k
j*i = -k
i*k = j
k*i = -j
j*k = i
k*j = -i

and the squares of i, j, and k are given by:

i^2 = j^2 = k^2 = -1

These relations show that the X, Y, and Z gates are similar to the quaternions i, j, and k in the way that they combine and transform. However, the X, Y, and Z gates are defined over the complex numbers, whereas the quaternions i, j, and k are defined over the real numbers. This difference means that the X, Y, and Z gates can be used to perform operations on quantum states, whereas the quaternions i, j, and k are used for other purposes in mathematics and physics.

Q: How do CNOT gates work?

CNOT gates, also known as controlled-NOT gates, are quantum logic gates that operate on two qubits. The CNOT gate has two inputs, a control qubit and a target qubit, and one output. The output of the CNOT gate is the same as the input state of the target qubit, unless the control qubit is in the state “1”, in which case the output of the CNOT gate is the negation of the input state of the target qubit.

Mathematically, the CNOT gate can be represented by the matrix:

CNOT = [[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, 1, 0]]
This matrix shows how the CNOT gate acts on the state of two qubits. If the input state of the two qubits is given by the vector psi>, the output state of the CNOT gate is given by the vector CNOT|psi>. For example, if the input state of the two qubits is |psi> = |00>, the output state of the CNOT gate will be 00>, since the control qubit is in the state “0” and the CNOT gate does not negate the state of the target qubit. However, if the input state of the two qubits is |psi> = |10>, the output state of the CNOT gate will be |11>, since the control qubit is in the state “1” and the CNOT gate negates the state of the target qubit.

In general, the CNOT gate is a useful building block for quantum circuits, as it allows for the manipulation of the quantum state of multiple qubits. The CNOT gate can be used in combination with other quantum gates to perform more complex operations on quantum states, and it is an important gate for quantum algorithms such as quantum teleportation and quantum error correction.

Q: Can you depict a CNOT gate as a diagram, instead of a matrix?

Yes, a CNOT gate can be depicted as a diagram, as shown below:

   ---[NOT]---
  |          |

[Control]---[CNOT]---[Target]

In this diagram, the control qubit is shown on the left, the target qubit is shown on the right, and the CNOT gate is shown in the middle. The CNOT gate has a control line, which is connected to the control qubit, and a target line, which is connected to the target qubit. The control line determines whether the CNOT gate negates the state of the target qubit, and the target line carries the output state of the CNOT gate.

The diagram shows that the CNOT gate acts on two qubits, and it illustrates how the state of the control qubit determines the operation performed by the CNOT gate on the target qubit. The diagram also shows that the CNOT gate has an input state and an output state, and it illustrates how the output state of the CNOT gate is related to the input state of the two qubits. Overall, the diagram provides a visual representation of the CNOT gate and its operation on quantum states.