Quantum Gate

Table of Contents

• Quantum Logic Gates
  • Common Quantum Logic Gates in Matrix Form
  • Multi-qubit Quantum Gates
• API Introduction
  • Interface Overview
  • Querying Qubits
  • Gate Properties
  • Matrix Operations
  • Gate Type and Parameters

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Quantum Logic Gates

In classical computing, the most basic unit is the bit, and the fundamental control mechanism is the logic gate. We can achieve our control of circuits through combinations of logic gates. Similarly, the way we manipulate qubits is through quantum logic gates. By using quantum logic gates, we consciously induce the evolution of quantum states. Thus, quantum logic gates form the foundation of quantum algorithms.

Quantum logic gates are represented by unitary matrices. The most common quantum gates operate in the space of one or two qubits, just as common classical logic gates operate on one or two bits.

Common Quantum Logic Gates in Matrix Form

Gate	Name	Matrix
I	I	\[\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\]
H	Hadamard	\[\begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \end{bmatrix}\]
T	T	\[\begin{bmatrix} 1 & 0 \\ 0 & \exp\left(\frac{i\pi}{4}\right) \end{bmatrix}\]
S	S	\[\begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix}\]
X	Pauli-X	\[\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\]
Y	Pauli-Y	\[\begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}\]
Z	Pauli-Z	\[\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\]
X1	X1	\[\begin{bmatrix} \frac{1}{\sqrt{2}} & -\frac{i}{\sqrt{2}} \\ -\frac{i}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix}\]
Y1	Y1	\[\begin{bmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix}\]
Z1	Z1	\[\begin{bmatrix} \exp\left(-\frac{i\pi}{4}\right) & 0 \\ 0 & \exp\left(\frac{i\pi}{4}\right) \end{bmatrix}\]
RX	RX	\[\begin{bmatrix} \cos\left(\frac{\theta}{2}\right) & -i\sin\left(\frac{\theta}{2}\right) \\ -i\sin\left(\frac{\theta}{2}\right) & \cos\left(\frac{\theta}{2}\right) \end{bmatrix}\]
RY	RY	\[\begin{bmatrix} \cos\left(\frac{\theta}{2}\right) & -\sin\left(\frac{\theta}{2}\right) \\ \sin\left(\frac{\theta}{2}\right) & \cos\left(\frac{\theta}{2}\right) \end{bmatrix}\]
RZ	RZ	\[\begin{bmatrix} \exp\left(-\frac{i\theta}{2}\right) & 0 \\ 0 & \exp\left(\frac{i\theta}{2}\right) \end{bmatrix}\]
U1	U1	\[\begin{bmatrix} 1 & 0 \\ 0 & \exp(i\theta) \end{bmatrix}\]
U2	U2	\[\begin{bmatrix} \frac{1}{\sqrt{2}} & -\frac{\exp(i\lambda)}{\sqrt{2}} \\ \frac{\exp(i\phi)}{\sqrt{2}} & \frac{\exp(i\lambda + i\phi)}{\sqrt{2}} \end{bmatrix}\]
U3	U3	\[\begin{bmatrix} \cos\left(\frac{\theta}{2}\right) & -\exp(i\lambda)\sin\left(\frac{\theta}{2}\right) \\ \exp(i\phi)\sin\left(\frac{\theta}{2}\right) & \exp(i\lambda + i\phi)\cos\left(\frac{\theta}{2}\right) \end{bmatrix}\]
U4	U4	\[\begin{bmatrix} u_0 & u_1 \\ u_2 & u_3 \end{bmatrix}\]

Multi-qubit Quantum Gates

Gate	Name	Matrix
CNOT	CNOT	\[\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix}\]
CR	CR	\[\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \exp(i\theta) \end{bmatrix}\]
iSWAP	iSWAP	\[\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos(\theta) & i\sin(\theta) & 0 \\ 0 & i\sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\]
SWAP	SWAP	\[\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}\]
CZ	CZ	\[\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -1 \end{bmatrix}\]
CU	CU	\[\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & u_0 & u_1 \\ 0 & 0 & u_2 & u_3 \end{bmatrix}\]
RXX	RXX	\[\begin{bmatrix} \cos(\theta/2) & 0 & 0 & -i\sin(\theta/2) \\ 0 & \cos(\theta/2) & -i\sin(\theta/2) & 0 \\ 0 & -i\sin(\theta/2) & \cos(\theta/2) & 0 \\ -i\sin(\theta/2) & 0 & 0 & \cos(\theta/2) \end{bmatrix}\]
RYY	RYY	\[\begin{bmatrix} \cos(\theta/2) & 0 & 0 & i\sin(\theta/2) \\ 0 & \cos(\theta/2) & -i\sin(\theta/2) & 0 \\ 0 & -i\sin(\theta/2) & \cos(\theta/2) & 0 \\ i\sin(\theta/2) & 0 & 0 & \cos(\theta/2) \end{bmatrix}\]
RZZ	RZZ	\[\begin{bmatrix} \exp(-i\theta/2) & 0 & 0 & 0 \\ 0 & \exp(i\theta/2) & 0 & 0 \\ 0 & 0 & \exp(i\theta/2) & 0 \\ 0 & 0 & 0 & \exp(-i\theta/2) \end{bmatrix}\]
RZX	RZX	\[\begin{bmatrix} \cos(\theta/2) & 0 & -i\sin(\theta/2) & 0 \\ 0 & \cos(\theta/2) & 0 & i\sin(\theta/2) \\ -i\sin(\theta/2) & 0 & \cos(\theta/2) & 0 \\ 0 & i\sin(\theta/2) & 0 & \cos(\theta/2) \end{bmatrix}\]
Toffoli	Toffoli	\[\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 \\ \end{bmatrix}\]

API Introduction

QPanda3 encapsulates all quantum logic gates as APIs for user access, returning values of type QGate. For example, if you want to use the Hadamard gate, you can obtain it as follows:

Python

h = H(0)

As you can see, the H function only takes one qubit. Similarly, if you want to use the RX gate, you can obtain it like this:

Python

rx = RX(0, 3.14)

As shown, the RX gate takes two parameters: the first is the target qubit, and the second is the rotation angle. You can also use the same method for the RY and RZ gates.

The usage of two-qubit quantum logic gates is similar to that of single-qubit gates, but the input parameters differ. For example, using the CNOT gate:

Python

cnot = CNOT(0, 1)

The CNOT gate takes two parameters: the first is the control qubit, and the second is the target qubit.

Interface Overview

Here we will introduce all the interfaces related to the QGate class. Firstly, we need to know where to import the QGate class from

Python

from pyqpanda3.core import *

Querying Qubits

• Get Number of Qubits

  Using the qubits_num function to get this gate num of qubits . Example:

  Python

  print("Number of qubits:", gate.qubits_num())

• Get Qubits

  Using the qubits function to get this gate all qubits. Example:

  Python

  print("Qubits:", gate.qubits())

• Get Control Qubits

  Using the control_qubits function to get this gate control_qubits. Example:

  Python

  print("Control qubits:", gate.control_qubits())

• Get Target Qubits

  Using the target_qubits function to get this gate target_qubits. Example:

  Python

  print("Target qubits:", gate.target_qubits())

Gate Properties

• Dagger Operation

  As mentioned at the beginning of this chapter, all quantum logic gates are unitary matrices, and you can also perform conjugate transpose operations on quantum logic gates. The QGate type has two member functions that can perform conjugate transpose operations: dagger.

  The dagger function creates a copy of the current quantum logic gate and updates the copied gate's dagger flag. For example:

  Python

  rx_dagger = RX(0, 3.14).dagger()

• Raise Gate to a Power

  Using the power function to get this gate raised to a power. Example:

  Python

  powered_gate = gate.power(2)

• Control Operations

  In addition to the conjugate transpose operation, you can also add control qubits to the quantum logic gates. After adding control qubits, whether the current quantum logic gate executes depends on the quantum state of the control qubits. If the control qubit's state is \(|1\rangle\), the current quantum logic gate can be executed; if the state is \(|0\rangle\), the gate will not execute. The QGate type has two member functions to help you add control qubits: control and add_control.

  The add_control function creates a copy of the current quantum and adds control qubits to the current quantum logic gate. For example:

  Python

  gate = RX(0, 3.14)
  gate = gate.add_control(1)

  The control function creates a copy of the current quantum logic gate and set control qubits to the copied gate. For example:

  Python

  rx_control = RX(0, 3.14).control(1)

• Clear Control Qubits

  The clear_control function clear control qubits of gate. For example:

  Python

  gate.clear_control()

Matrix Operations

• Get Matrix Representation

  Using the matrix function to obtain a matrix.

  Note

  The meaning of the expanded parameter is whether the matrix is obtained based on the order of qubits,for example, the matrix of CNOT (1,0) is equivalent to CNOT (0,1) when the parameter is False.

  Example:

  Python

  gate = CNOT(1,0)
  maxtrix = gate.matrix(False)
  matrix_expanded = gate.matrix(True)

Gate Type and Parameters

• Get Gate Type

  Using the gate_type function to get gate type. Example:

  Python

  print("Gate type:", CNOT(0,1).gate_type())

• Get and Set Parameters

  Using the parameters function to get gate parameters. Using the set_parameters function to set gate parameters. Example:

  Python

  gate = U3(0,3.14,0,0)
  gate.set_parameters([0.5, 1.0,2.0])
  print("Gate parameters:", gate.parameters())

• Get Gate Name

  Using the name function to get gate name.

  Example:

  Python

  print("Gate name:", CNOT(0,1).name())