The passage you provided offers a clear explanation of a qubit in quantum computing. Let’s break down some key concepts:

**Qubit Definition:**- A qubit, short for “quantum bit,” is the basic unit of quantum information.
- It is the quantum analogue of the classical binary bit, which is represented physically with a two-state device.

**Quantum Mechanical Nature:**- A qubit is a two-state or two-level quantum-mechanical system.
- It is one of the simplest quantum systems that illustrates the peculiarities of quantum mechanics.

**Examples of Qubit Systems:**- The spin of an electron is given as an example, where the two levels are spin up and spin down.
- Another example involves the polarization of a single photon, where the two spin states (left-handed and right-handed circular polarization) can be measured as horizontal and vertical linear polarization.

**Contrast with Classical Bits:**- In a classical system, a bit would exist in one state or the other (0 or 1).

**Superposition in Quantum Mechanics:**- Quantum mechanics allows the qubit to exist in a coherent superposition of both states simultaneously.
- This property of superposition is fundamental to quantum mechanics and is a key aspect of quantum computing.

In essence, the ability of qubits to exist in multiple states at the same time (superposition) distinguishes quantum computing from classical computing. This property enables quantum computers to perform certain types of computations more efficiently than classical computers for specific problems, such as factoring large numbers and simulating quantum systems.

## Etymology

The etymology of the term “qubit” is attributed to Benjamin Schumacher, and it was coined during a conversation with William Wootters. In the acknowledgments of his 1995 paper, Schumacher mentions that the term “qubit” was created in jest during this discussion. Etymology refers to the study of the origin and history of words, including how they are coined and the circumstances surrounding their creation. In this case, the term “qubit” was playfully coined in the context of a conversation between Schumacher and Wootters.

## Bit versus qubit

This passage provides a comparison between classical bits, which are the fundamental units of classical information represented as 0 or 1, and qubits, the quantum counterparts. Here’s a breakdown of the key points:

**Classical Bits:**- In classical computers, information is represented using binary digits, or bits, which can be in the state of 0 or 1.
- A processed bit in classical computer technologies is implemented by one of two levels of low DC voltage.
- Switching between these two levels involves passing through a “forbidden zone” between logic levels, and this transition must occur as fast as possible due to the inability of electrical voltage to change instantaneously.

**Qubits in Quantum Computing:**- A qubit is the quantum version of a bit, and it can exist in a superposition of both 0 and 1 states simultaneously.
- The measurement of a qubit can result in one of two outcomes: 0 or 1, similar to a classical bit.
- Unlike classical bits, a qubit’s general state, according to quantum mechanics, can be a coherent superposition of both 0 and 1.
- Measurement of a qubit disturbs its state and destroys its coherence, unlike classical bits, which remain undisturbed after measurement.

**Information Capacity:**- While it is possible to fully encode one bit in one qubit, a qubit has the potential to hold more information. For example, through superdense coding, a qubit can carry up to two bits of information.

**Quantum State Description:**- In classical physics, a system of n components requires only n bits for a complete description of its state.
- In quantum physics, a system of n qubits requires 2n complex numbers or can be described as a single point in a 2n-dimensional vector space.

This passage highlights some of the fundamental differences between classical and quantum information representation, emphasizing the unique properties and capabilities of qubits in quantum computing.

## Standard representation

This passage provides an overview of the quantum state representation of a qubit in quantum mechanics and introduces the concept of qubit basis states and quantum registers. Let’s break down the key points:

**Quantum State of a Qubit:**- The general quantum state of a qubit can be represented by a linear superposition of its two orthonormal basis states.
- The basis states are denoted as $∣0⟩$ and $∣1⟩$, represented as column vectors $[1,0]$ and $[0,1]$ respectively.
- In Dirac notation, these basis states are written as $∣0⟩$ and $∣1⟩$, pronounced as “ket 0” and “ket 1.”

**Computational Basis:**- The basis states $∣0⟩$ and $∣1⟩$ together form the computational basis.
- They span the two-dimensional linear vector (Hilbert) space of the qubit.

**Product Basis States and Quantum Registers:**- Basis states can be combined to form product basis states.
- A set of qubits taken together is referred to as a quantum register.
- For example, two qubits can be represented in a four-dimensional linear vector space with product basis states $∣00⟩$, $∣01⟩$, $∣10⟩$, and $∣11⟩$.

**Representation of n Qubits:**- In general, n qubits are represented by a superposition state vector in a 2^n-dimensional Hilbert space.

This explanation illustrates the foundational principles of representing qubit states, the computational basis, and how multiple qubits can be combined to form quantum registers. The concept of quantum superposition allows qubits to exist in a combination of basis states, enabling the parallel processing capabilities that quantum computing leverages.

## Qubit states

This passage provides an overview of the representation of a pure qubit state, describing it as a coherent superposition of basis states and introducing the concept of probability amplitudes. Let’s break down the key points:

**Pure Qubit State:**- A pure qubit state ($∣ψ⟩$) is a coherent superposition of the basis states $∣0⟩$ and $∣1⟩$.
- It can be described by a linear combination:
$∣ψ⟩=α∣0⟩+β∣1⟩$

where $α$ and $β$ are probability amplitudes, and they are complex numbers.

**Probability Amplitudes and Measurement:**- When measuring the qubit in the standard basis, the Born rule is applied.
- The probability of obtaining outcome $∣0⟩$ (with value “0”) is $∣α_{2}$, and the probability of obtaining outcome $∣1⟩$ (with value “1”) is $∣β_{2}$.
- The absolute squares of the amplitudes correspond to probabilities.

**Normalization Constraint:**- The probability amplitudes $α$ and $β$ must satisfy the normalization constraint:
$∣α_{2}+∣β_{2}=1$

- This constraint is a result of the second axiom of probability theory.

- The probability amplitudes $α$ and $β$ must satisfy the normalization constraint:
**Interpretation of Probability Amplitudes:**- Probability amplitudes encode more than just the probabilities of measurement outcomes.
- The relative phase between $α$ and $β$ is responsible for quantum interference phenomena, as observed in experiments like the two-slit experiment.

In summary, this passage introduces the mathematical representation of a pure qubit state, the role of probability amplitudes, and the constraints imposed by the normalization condition. The probabilistic nature of quantum measurements is central to understanding quantum states, and the phase information in probability amplitudes gives rise to quantum interference effects.

This passage discusses the Bloch sphere representation of a qubit, including the use of Hopf coordinates and the visualization of quantum states on the Bloch sphere. It also introduces the concept of mixed states and their representation on the Bloch sphere.

**Bloch Sphere Representation:**- The probability amplitudes ($α$ and $β$) for a qubit state $∣ψ⟩=α∣0⟩+β∣1⟩$ can be expressed using spherical coordinates on the Bloch sphere.
- The Bloch sphere provides a visual representation of the possible quantum states for a single qubit.
- The angles $θ$ and $φ$ in the spherical coordinates represent the degrees of freedom for the qubit state.

**Hopf Coordinates:**- The Hopf coordinates are introduced as a way to remove one degree of freedom due to the normalization constraint.
- The choice of Hopf coordinates is described, and the physically observable relative phase is emphasized.

**Bloch Sphere Visualization:**- A classical bit is likened to being at either the “North Pole” or the “South Pole” of the Bloch sphere, corresponding to the basis states $∣0⟩$ and $∣1⟩$.
- Any point on the surface of the Bloch sphere can represent a pure qubit state. For example, the state $(∣0⟩+∣1⟩)/2 $ lies on the equator at the positive X-axis.

**Degrees of Freedom:**- The Bloch sphere’s surface represents the observable state space of pure qubit states and has two local degrees of freedom ($θ$ and $φ$).
- A mixed qubit state, resulting from interactions, quantum noise, and decoherence, is represented by points inside the Bloch sphere, introducing a third degree of freedom ($r$).

**Mixed States:**- A pure qubit state is represented by a single ket, while a mixed state is a statistical combination or “incoherent mixture” of different pure states.
- Mixed qubit states are represented by points inside the Bloch sphere, and they have three degrees of freedom.

**Quantum Error Correction:**- Quantum error correction is mentioned as a technique to maintain the purity of qubits.

In summary, the passage provides insights into the geometric representation of qubit states on the Bloch sphere, the role of Hopf coordinates, the visualization of quantum states, and the concept of mixed states.

## Operations on qubits

This passage outlines various types of physical operations that can be performed on qubits in the context of quantum computing:

**Quantum Logic Gates:**- Quantum logic gates are fundamental building blocks for constructing quantum circuits in quantum computers.
- These gates operate on a set of qubits (a register) and perform reversible unitary transformations on the quantum state vector.
- Mathematically, the operation involves multiplying the unitary matrix of the quantum gate with the quantum state vector, resulting in a new quantum state.

**Quantum Measurement:**- Quantum measurement is an irreversible operation that provides information about the state of a single qubit.
- The result of the measurement alters the quantum state, and coherence is lost.
- The outcome is probabilistic, with the state collapsing to either $∣0⟩∣0⟩$ or $∣1⟩∣1⟩$ with specific probabilities determined by the magnitudes of the probability amplitudes ($α$ and $β$).
- Measurement on an entangled qubit may collapse the state of other entangled qubits.

**Initialization or Re-initialization:**- Initialization involves setting the qubit to a known value, often $∣0⟩∣0⟩$.
- This operation collapses the quantum state similarly to measurement.
- Initialization can be implemented logically (e.g., using a measurement followed by a Pauli-X gate) or physically (e.g., lowering the energy of a superconducting phase qubit to its ground state).

**Sending Qubits Through Quantum Channels:**- Qubits can be sent through quantum channels to remote systems or machines, representing an input/output (I/O) operation.
- This operation is potentially part of a larger quantum network.

These operations highlight the key features of quantum computing, including reversible unitary transformations, probabilistic outcomes in measurements, the collapse of quantum states, and the potential for quantum communication through channels. Quantum computation relies on the principles of quantum mechanics to perform complex calculations and tasks that classical computers may find challenging or impractical.

## Quantum entanglement

This passage provides insights into the concept of quantum entanglement and its application in quantum computing. Here’s a breakdown of the key points:

**Quantum Entanglement:**- Quantum entanglement is a nonlocal property of two or more qubits that allows them to exhibit higher correlation than is possible in classical systems.
- Multiple qubits can be entangled, and their states become interconnected in a way that the state of one qubit is dependent on the state of another, even if they are physically separated.

**Bell State and Equal Superposition:**- The passage introduces the concept of a Bell state, specifically the $∣_{+}⟩$ Bell state.
- In this state, two entangled qubits are in an equal superposition of the product states $∣00⟩$ and $∣11⟩$.
- The probabilities of measuring either $∣00⟩$ or $∣11⟩$ are equal, demonstrating the entangled nature of the qubits.

**Measurement and Correlation:**- When one of the entangled qubits is measured, the outcome is probabilistic.
- Due to entanglement, the measurement result for one qubit is correlated with the measurement result of the other qubit, regardless of the physical separation between them.
- This correlation holds true even though each qubit individually does not reveal whether it is in state $∣0⟩$ or $∣1⟩$.

**Controlled Gates and Bell State Construction:**- Controlled gates, such as the CNOT gate, play a role in constructing entangled states like the $∣_{+}⟩$ Bell state.
- The CNOT gate performs the NOT operation on the target qubit only when the control qubit is in state $∣1⟩$.

**Applications of Bell States:**- Bell states, including $∣_{+}⟩$, are essential for various quantum algorithms and protocols, such as superdense coding, quantum teleportation, and entangled quantum cryptography.

**Unique Properties of Quantum Entanglement:**- Quantum entanglement allows multiple states to be acted on simultaneously, a feature not possible with classical bits.
- Entanglement is considered a unique resource for quantum computation, and many quantum computing and communication protocols leverage it.

**Challenges in Quantum Computing:**- The passage notes that noise in quantum gates is a major challenge in quantum computing, limiting the reliable execution of large quantum circuits.

In summary, quantum entanglement is a distinctive feature of qubits, enabling correlations and behaviors that are not achievable with classical bits. The passage highlights the use of entanglement in quantum algorithms and the challenges associated with noise in quantum gates.

## Quantum register

This passage introduces the concepts of qudits and qutrits in the context of quantum information processing:

**Qudits:**- The term “qudit” refers to the unit of quantum information that can be realized in suitable d-level quantum systems.
- A qubit register that can be measured to N states is essentially equivalent to an N-level qudit.
- Qudits can be thought of as analogous to integer types in classical computing.
- They can be mapped to arrays of qubits, but not all qudits can be mapped to arrays of qubits, especially when the d-level system is not a power of 2.
- Example: A 5-level qudit, where the system has 5 distinguishable states.

**Advancements in Qudits:**- In 2017, scientists at the National Institute of Scientific Research constructed a pair of qudits, each with 10 different states, providing more computational power than 6 qubits.
- In 2022, researchers at the University of Innsbruck developed a universal qudit quantum processor with trapped ions, demonstrating progress in realizing and manipulating qudits.

**Qutrits:**- Similar to the qudit, a qutrit is the unit of quantum information that can be realized in suitable 3-level quantum systems.
- This is analogous to the classical information unit “trit” used in ternary (base-3) computers.

**Implementation Examples:**- In 2022, researchers at Tsinghua University’s Center for Quantum Information implemented the dual-type qubit scheme in trapped ion quantum computers using the same ion species.
- This suggests ongoing research and development efforts to explore the capabilities of qudits and qutrits in different quantum systems.

In summary, qudits and qutrits represent quantum information units in d-level and 3-level quantum systems, respectively. Researchers are exploring their potential applications and building practical quantum processors that leverage these higher-dimensional quantum systems.

## Physical implementations

This passage provides an overview of qubits in quantum computing, emphasizing the diverse physical implementations and considerations related to noise and performance:

**Qubit Basis:**- Any two-level quantum-mechanical system can serve as a qubit.
- Multilevel systems are also viable if they have two effectively decoupled states, such as the ground state and first excited state of a nonlinear oscillator.

**Diverse Proposals and Implementations:**- Various proposals exist for qubit implementations.
- Physical implementations that approximate two-level systems to different degrees have been successfully realized.
- Similar to classical bits, where multiple physical systems can represent bits in the same computer, a quantum computer is likely to use diverse combinations of qubits in its design.

**Noise in Quantum Implementations:**- All physical implementations of qubits are affected by noise.
- The T1 lifetime and T2 dephasing time are used to characterize physical implementations and represent their sensitivity to noise.
- A higher T1 or T2 time alone does not necessarily imply a better-suited qubit; factors like gate times and fidelities must also be considered.

**Considerations for Quantum Computing:**- Quantum computing applications involve different implementations of qubits tailored to their specific requirements.
- Quantum sensing, quantum computing, and quantum communication are cited as examples of applications using different qubit implementations.

In summary, the passage highlights the flexibility of qubit implementations, ranging from two-level systems to more complex multilevel systems. It also underscores the importance of considering noise, T1 and T2 times, gate times, and fidelities when evaluating the suitability of qubits for quantum computing applications. Different applications may benefit from distinct qubit implementations based on their specific needs and characteristics.

## Qubit storage

This passage highlights key developments in the field of quantum computing, particularly in the area of qubit coherence and quantum data storage:

**First Coherent Transfer of Superposition State (2008):**- In 2008, a team of scientists from the U.K. and U.S. achieved a significant milestone by reporting the first relatively long and coherent transfer of a superposition state.
- The transfer occurred from an electron spin “processing” qubit to a nuclear spin “memory” qubit.
- This achievement is considered a crucial step toward the development of quantum computing.

**Quantum Data Storage Breakthrough (2013):**- In 2013, a modification of similar systems, utilizing charged (rather than neutral) donors, significantly extended the coherence time of quantum data storage.
- The reported coherence time reached 3 hours at very low temperatures and 39 minutes at room temperature.
- This breakthrough marked a substantial improvement in maintaining quantum coherence for practical durations.

**Room Temperature Qubit Preparation (2013):**- In the same year (2013), a team of scientists from Switzerland and Australia demonstrated the room temperature preparation of a qubit based on electron spins instead of nuclear spin.
- Room temperature operation is a desirable feature for the practical implementation of quantum computing technologies.

**Exploration of Ge Hole Spin-Orbit Qubit Structure (Ongoing):**- Researchers are actively exploring ways to increase the coherence of qubits.
- One avenue of exploration involves testing the limitations of a germanium (Ge) hole spin-orbit qubit structure.
- This research aims to enhance the stability and coherence of qubits, addressing challenges in quantum computing.

These advancements reflect the progress made in extending the coherence time of qubits, a critical factor for the practical realization of quantum computing. The ongoing exploration of novel qubit structures further demonstrates the dynamic and evolving nature of quantum research.