Introduction

This chapter covers quantum subsystems and their properties. Single and multiple quantum bit systems are discussed in detail. Quantum states, protocols, operations, and transformations are also presented in this chapter.

A quantum computer processes information related to real-life problems and solves them using quantum mechanics. A classical computer uses bits that are either 0 or 1. We have to pick one of those options since it is based on the electric current in a classical computer wire. Quantum units are based on the quantum superposition principle in which 0 and 1 are superposed. This opens up possible states that are infinite and based on the 0 and 1 with weights in a continuous range. In classical computers, decimal system–based values are converted to binary numbers. Similarly, quantum units are defined using the two-dimensional Hilbert space. See Figure 3-1.

Initial Setup

You need to set up Python 3.5 to run the code samples in this chapter. You can download it from https://www.python.org/downloads/release/python-350/.

Single Qubit System

The quantum bit is the foundation for a quantum computation framework. By using quantum bits (qubits), the quantum computer performs better than a classical computer. A quantum computer stores the information in quantum bits. A quantum bit represents a quantum particle like an electron having a charge as 0 or 1 and as 0 and 1 simultaneously. A Bloch sphere is used to show the position of the quantum bit’s state. The sphere has a radius of one unit. A Hilbert space is used to model the surface of the sphere. A Hilbert space is related to an inner product of vectors in vector space. Length and angle measurements are shown in the inner product. See Figure 3-2.

Now let’s look at a code sample in Python that demonstrates a single qubit system.

Multiple Qubit System

A multiple quantum bit register has the quantum system in which the quantum bits are in superposition states. A classical register with s quantum bits has 2s unique states. The quantum system will consist of 2p states in superposition. The system can be in one, two, three, … all of the 2s states. Let’s look at a quantum system with 2 qubits and their possible states (see Figure 3-3).

Now, let’s look at a 3 qubit system and the possible states (see Figure 3-4).

The quantum states of the quantum bits in a multiple-qubit system will be a vector with complex coefficients. The real logical state is referred to as observable. The state is measured, and the result is observed. A quantum system will have filters, which are used to identify the quantum state. Different functions are modeled based on the different sets of observables.

A 2 qubit system is specified using the following equation:

q(0) + q(1) = 1

Single quantum bit measurements are used to model 2 qubit measurement operations. A quantum register can be measured partially. The unmeasured leftover is transformed using the identity quantum operation.

Quantum States

A quantum bit exists in 0, 1, or a state between 0 and 1 quantum states. This helps a quantum computer to model a huge set of possibilities through the quantum state. S bits are equivalent to 2^S multiple states. The state between 0 and 1 is referred to as quantum superposition. Superposition is modeled as a wave function. A wave function can have infinite states mathematically.

Two quantum states can be mixed with one another to create another quantum state. This is similar to the mixing of matter waves in physics.

A Bloch sphere is used to present the quantum states (see Figure 3-5). A quantum bit can be in a superposition state, as shown here:

amp1|0> + amp2|1> = |amp>  where |0> and |1> are column vectors

amp1 and amp2 are called probability amplitudes , which are complex numbers.

A superposition state can also be represented as shown here (see Figure 3-6):

|ψ⟩  =   1/√2 * (|0> + |1>)

Figure 3-6 shows the quantum bits in superposition compared to the classical bits’ 0 and 1 states.

A quantum particle’s state is modeled as a quantum bit. Figure 3-7 shows the superposition state of the quantum particle.

Now let’s look at a code sample in Python to demonstrate the superposition of the quantum bits and states.

Quantum Protocols

In this section, we look at entanglement and teleportation, which are based on quantum protocols.

Quantum Entanglement

Quantum entanglement is the interaction of multiple quantum particles with each other. The interaction leads to entanglement, and the quantum particles might separate eventually in the space.

This quantum protocol–based phenomenon was observed in 1935 by Albert Einstein, Boris Podolsky, and Nathen Rosen during a quantum mechanical experiment. They wanted to show the logical impossibility of quantum mechanics.

A change in the quantum state of the entangled system–based particle impacts the other and is detectable. The particles in an entangled system are aware of each other. See Figure 3-8.

Quantum Teleportation

Quantum teleportation is the transfer of information from a source to a destination. Classical communication networks and channels are used to transfer information. The teleportation based on quantum protocols happens when there is an entanglement of quantum particles at the source and the destination. The current record is a distance of 1,400 kilometers (about 870 miles). Quantum channels can be created between the source and the destination. The quantum state of the quantum particles is teleported from the source to the destination. See Figure 3-9.

Quantum Operations

Quantum operations can be visualized using the Bloch sphere (see Figure 3-10). The quantum state of a quantum bit can be shown using a point on the surface of the sphere. The two dials of the sphere indicate the angular and azimuthal coordinates of the sphere. The sphere’s surface can be modeled using a Hilbert space. As mentioned, a Hilbert space is based on a vector space of an inner product of two vectors. The inner product helps in the measurement of the length and angle.

Quantum gates are quantum operations using quantum bits. Quantum operations are the building blocks of the quantum circuits. Quantum circuits are the fundamental blocks on which a quantum computer is built.

Let’s start looking at the Hadamard gate first. Hadamard gate is the gate that operates on the base state from |0> to (|0>+|1>)* 1/√2 and |1> to (|0> - |1>)* 1/√2. This gate transformation results in a superposition.

Now, we look at Swap gates. A Swap gate swaps the quantum state of the gate.

An X gate is a quantum equivalent of a NOT gate. A CNOT gate is a controlled-NOT gate. A CNOT gate negates a bit; the control bit is equal to only 1. A Toffoli gate is a controlled-NOT gate, which is a quantum equivalent of an AND gate. Toffoli negates a bit if both control bits are equal to only 1. A Pauli X gate is a quantum operation that rotates a quantum bit over the x-axis with an angle of 180 degrees. A Pauli Z gate is a quantum operation that rotates a quantum bit over the z-axis with an angle of 180 degrees.

Now let’s look at some CNOT, X, Z, and Measure gate implementations to demonstrate the quantum gates. The implementations of a CNOT gate, X gate, Z gate, normalize operation, and Measure gates in Python are shown next.

Quantum Transformations

Quantum transformations are related to the quantum state transformations using quantum bits. We look at the Kronecker transformation and measure gates in this section.

Summary

In this chapter, we looked at single and multiple quantum bit systems. Quantum bits system phenomena such as entanglement, superposition, and teleportation were discussed in detail. Finally, the quantum states, gates, and transformations were presented.