Qubits, Qutrits and Qudits

- Overview

The quantum in "quantum computing" refers to the quantum mechanics that the system uses to calculate outputs. In physics, a quantum is the smallest possible discrete unit of any physical property. It usually refers to properties of atomic or subatomic particles, such as electrons, neutrinos, and photons.

Classical computers were programmed using bits (zeros and ones) as the unit of data. Quantum computers use so-called qubits, which can simultaneously represent combinations of zeros and ones according to a principle called superposition. 

Quantum computing is essentially harnessing and exploiting the amazing laws of quantum mechanics to process information. Traditional computers use long strings of "bits", encoding zeros or ones. 

On the other hand, quantum computers use quantum bits or quantum bits. what is the difference? 

A qubit is a quantum system that encodes zeros and ones into two distinguishable quantum states. However, because qubits have quantum behavior, we can exploit the phenomena of "superposition" and "entanglement". 

Ordinary computers use bits to store information that has only two states: zero or one. However, quantum computers allow subatomic particles to exist in more than one state at the same time, so they can exist in zero, one, or both at the same time. 

As a result, quantum bits, called "qubits," can process large amounts of information faster than ordinary computers. However, the qubits need to be synchronized using a quantum effect called entanglement, which Albert Einstein called "ghost action at a distance."

- Qubits

A qubit, or quantum bit, is the basic unit of information in quantum computing. It's the quantum version of the classical binary bit. 

Qubits are two-state quantum-mechanical systems. They can be encoded in a variety of physical systems, including atoms, particles of light, and superconducting circuits.
Qubits behave differently from bits because of the quantum properties on which they're based. 

For example, qubits have a property called spin, which is always pointing either fully up or fully down. Using the spin states of up and down, a spin qubit can be built. 0 = pointing up, 1 = pointing down. 

Quantum computing is driving new discoveries in healthcare, energy, environmental systems, smart materials, and beyond. 

Please refer to the following for more details:

• Wikipedia: Qubit

- Qubits and Quantum Superposition

A qubit is the basic unit of information in quantum computing. Qubits play a similar role in quantum computing as bits play in classical computing, but they behave very differently. Classical bits are binary and can hold only a position of 0 or 1, but qubits can hold a superposition of all possible states. 

A qubit (or quantum bit) is the quantum mechanical analogue of a classical bit. In classical computing the information is encoded in bits, where each bit can have the value zero or one. In quantum computing the information is encoded in qubits. A qubit is a two-level quantum system where the two basis qubit states are usually written as ∣0⟩ and ∣1⟩. A qubit can be in state ∣0⟩, ∣1⟩ or (unlike a classical bit) in a linear combination of both states. The name of this phenomenon is superposition.

Quantum superposition is a fundamental principle of quantum mechanics. It states that, much like waves in classical physics, any two (or more) quantum states can be added together ("superposed") and the result will be another valid quantum state; and conversely, that every quantum state can be represented as a sum of two or more other distinct states. Mathematically, it refers to a property of solutions to the Schrödinger equation; since the Schrödinger equation is linear, any linear combination of solutions will also be a solution.

- Qutrits

A qutrit is a unit of quantum information that exists as a superposition of three orthogonal quantum states. It is also known as a quantum trit. 

Qutrits are analogous to classical trits, just as qubits are analogous to classical bits. A qubit is a quantum particle with two possible states. 

Qutrits can exist in a superposition of three possible quantum states, represented as |0⟩, |1⟩, and |2⟩. They can increase the amount of information encoded in a single element and enable techniques that can dramatically decrease readout errors.

Qutrits can also be used to perform quantum computation by using three-level qutrits instead of qubits. This can substantially reduce the resource requirements of quantum computations.

- Qudits

In quantum computing, a qudit is a unit of quantum information that's described by a superposition of states, where the number of states is an integer greater than two.

Qudits are a multi-level computational unit that's an alternative to the conventional 2-level qubit. Compared to qubits, qudits provide a larger state space to store and process information. This can reduce circuit complexity, simplify the experimental setup, and enhance the algorithm efficiency.

Qudit computers may prove better at tackling complex problems than qubit computers, and may unlock more computational power with fewer components.

Controlling and programming quantum devices to process quantum information by the unit of quantum dit, i.e., qudit, provides the possibilities for:

• Noise-resilient quantum communications
• Delicate quantum molecular simulations
• Efficient quantum computations

- Qudits vs. Qubits

Here are some differences between qudits and qubits: 

• Dimensionality: A qubit is a 2-dimensional system, while a qudit is a d-dimensional system.
• State space: A qubit register's state space can be represented as a Bloch sphere, which is a 2-dimensional space. A qudit provides a larger state space for information processing.
• Resource efficiency: Qudits are more resource efficient than qubits when it comes to spanning the state space.
• Mathematical operations: Some difficulties can arise when performing mathematical operations with qudits and qubits.
• Gates: In Cirq, qudits work like qubits, but they can only be used with gates specific to their dimension.

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