Читать книгу Principles of Superconducting Quantum Computers - Daniel D. Stancil - Страница 21

We refer to a transformation from one quantum state to another as a gate. The effect of a single qubit gate is to change α and β into a new mixture α′ and β′:

(1.12)

This can be written as a matrix equation

(1.13)

(1.14)

Since the length of the state vector must always be unity, we are only allowed to use matrices U that conserve the length of the vector. In other words, ⟨ψ′|ψ′⟩ = ⟨ψ|ψ⟩ = 1. This puts a very important constraint on the matrix U:

(1.15)

using the following observation:

(1.16)

Since ⟨ψ|ψ⟩ = 1, we conclude that

(1.17)

where I is the identity matrix

(1.18)

Matrices that satisfy this requirement are called unitary matrices. We can view these matrices as performing an operation on a qubit by changing the mixture of basis states. Consequently, the matrices U are also referred to as unitary operators.

The identity matrix I can be considered to be the simplest “gate” and leaves the state vector unchanged. Classically, the NOT gate is the only non-trivial single-bit gate. In contrast, there are many non-trivial single qubit quantum gates (technically, the number of 2×2 unitary matrices is unlimited). The most common non-trivial single qubit gates are the Pauli-X (X), Pauli-Y (Y), Pauli-Z (Z), and Hadamard (H) gates defined as follows:

(1.19)

(1.20)

(1.21)

(1.22)

To get an understanding of what these gates do, consider applying an X gate to the “ground” state |0⟩:

(1.23)

Similarly,

(1.24)

We see then that the X gate is a “bit flip” gate, and transforms |0⟩ into |1⟩ and vice versa. This, then is the analog of the classical NOT gate. You should verify the following results from applying Y,Z, and H gates:

(1.25)

(1.26)

(1.27)

In addition, it is interesting to note that each one of these matrices is its own Hermitian conjugate. Consequently, these four gates have the property that applying them twice gives the identity matrix:

(1.28)

Note that while X,Y, and Z gates transform between the |0⟩ and |1⟩ states, the Hadamard gate actually creates a superposition state, and therefore will prove to be particularly useful. The states resulting from applying H to the basis states are given their own names:

(1.29)

(1.30)

Multiple gates can be sequentially applied to a qubit by matrix multiplication:

(1.31)

This expression says that we start with the ground state, then apply a Hadamard gate, followed by a Pauli-Y and then a Pauli-Z. Note that this process conceptually moves from right to left. This computation can be represented by a quantum circuit diagram as shown in Figure 1.3. Note that the process moves from left to right on the circuit diagram: we begin with the ground state on the left, then apply a Hadamard gate, a Pauli-Y gate, and a Pauli-X.

Figure 1.3 Circuit representation of Eq. (1.31). In a quantum circuit diagram, the operation goes from left to right, while the matrix expression is shown going from right to left. The final box is a measurement in the standard basis, resulting in a classical bit.

Finally, after performing a quantum calculation, we generally measure each qubit. The symbol for a measurement is shown as the last element in Figure 1.3. This action collapses the final state onto one of the basis states. The outcome of a measurement is a classical bit that is stored in a classical register. “Quantum wires” are denoted by solid lines, and “classical wires” are denoted with double lines. A quantum wire simply represents a time-interval over which the state is kept unchanged.