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= Spin Qubits = Spin qubits are a type of qubit which utilize the spin of a system to store and process quantum information. In order to build quantum computers capable of large-scale calculations, scalable systems that are compatible with current technology and can reliably encode quantum information must be realized. The manipulation of the spin states through the application of magnetic or electric fields and quantum gate operations in spin qubits allows for the processing of quantum information and the formation of a foundation for scalable quantum computing technologies. While various types of qubits, such as charge, superconducting , and topological , have been studied to meet this challenge, spin qubits are especially advantageous in their degree of compatibility with classical electronic devices. The implementation of spin qubits in classically used electronic devices, such as semiconductor materials, is beneficial due to the pre-existing fabrication methods for this technology and the potential for integration into classical circuits. Despite this advantage, many challenges in the implementation of spin qubits remain. While spin qubits generally possess long coherence times compared to other qubits, they remain susceptible to decoherence due to noise or impurities in the semiconductor material. Research into the high-fidelity control and readout of spin qubits also remains ongoing due to the difficult nature of maintaining precise control over and accurately reading out the states of qubits.

Description
In spin qubits, quantum information is encoded depending on the orientation of the qubit's spin, either spin-up or spin-down. These spin-up and -down states correspond to the quantum states $$|0\rangle$$ and $$|1\rangle$$, which are denoted on the Bloch sphere. Unlike in a classical bit, where information can only be encoded in these two states, qubits can exist in a superposition of the spin-up and -down states. The addition of superposition states allows quantum computers to perform calculations more efficiently by processing information in parallel.

Coherence
The ability for quantum bits to exist in superposition states allows for quantum computers to process information in parallel, rather than sequentially. While this term is also used in classical computing to refer to the implementation of multiple parallel classical processors, in this context, it describes the quantum effect of multiple computational paths arising from interference effects. The amount of time that a qubit remains in a superposition state before decohering to a classical state is referred to as the coherence time. The development of qubits possessing long coherence times is important for the successful implementation of quantum technologies. The loss of coherence, or decoherence, occurs through interactions of the quantum system with the environment. Effects such as fluctuations in the magnetic field, spin-orbit coupling, material impurities, and thermal fluctuations can cause changes in the spin states and lead to decoherence.

Magnetic fields are used to induce an energy splitting between the up and down states of spin qubits, but any fluctuations in these magnetic fields could result in changes to this energy splitting. Magnetic field pulses are also used to implement gate operations and transitions between states, but fluctuations in the field could lead to a loss of resonance and lead to decoherence. These magnetic field fluctuations could also couple the qubit to nearby nuclear or electron spins in the environment which could cause the spin to flip, contributing to noise and decoherence. Spin-orbit coupling is the interaction between an electron's spin and orbital motion occurring in electron spin qubits. When the spin qubit's spin and orbital degrees of freedom are coupled, the qubit becomes more sensitive to fluctuations in the environment which lead to decoherence. For instance, vibrations or impurities in the material could more easily lead to transitions between states and decoherence. The qubit would also become more susceptible to decoherence from fluctuations in magnetic and electric fields caused by charge impurities or other factors. Spin-orbit coupling effects are most prominent in semiconductor-based spin qubit systems and several factors such as the design of gate operations and the choice of materials with specific symmetries that minimize this effect are studied to lengthen coherence time.

Decoherence is also an important factor to consider in nuclear spin qubits. Although nuclear spin qubits typically have longer coherence times than electron spin qubits, environmental interactions and material impurities cause these systems to decohere. The nuclear spin of the qubit interacts with other nuclear and electron spins in the environment. Interactions with other nuclear spins can affect the coherence of the nuclear spin qubits if they become entangled and hyperfine interactions with electron spins become a factor in decoherence when there are fluctuations in the electron spin via spin flips. Any impurities in the material of the device can also affect the decoherence as any defects can introduce additional nuclear spins.

The introduction of thermal noise to a system can also disrupt qubit states and lead to decoherence. When the qubit system is introduced to high temperatures, the qubit can be excited into higher energy states and this results in spin state transitions. The motion that is induced by the thermal noise can lead to uncertainty in the energy levels of the states and make the precise control and manipulation of the states more difficult. When combined with effects such as spin-orbit coupling and interactions with external spin states, this thermal excitation also causes spin flips and leads to decoherence. Current research aims to mitigate thermal effects by either isolating the device to maintain a thermally stable environment or by creating operations which are robust against these effects.

Semiconductor quantum dot devices
Spin qubits are implemented in semiconductor materials through the use of quantum dots, nanostructures that three-dimensionally confine electrons. Due to this three-dimensional confinement of a single electron, quantum dots are ideal for the manipulation of an isolated electron spin. The semiconductor material chosen to form the quantum dots depends on the desired band gap. The bandgap refers to the difference in energy between the valence and conduction bands and is used to define the potential landscape. Semiconductor materials of different sized bandgaps are layered to form a heterostructure for the purpose of electron confinement. Materials with smaller bandgaps are used to laterally confine the dots, creating a well where the electron resides, while vertical confinement is achieved through the placement of larger bandgap materials above and below the quantum dot to create a potential barrier. This vertical confinement of the electron can also arise from the application of gate voltages, typically patterned through electron beam lithography. The potential well can then be tuned through the application of these external gate voltages. The amount of confinement, or size of the quantum dot, determines the discrete energy levels. In practice, quantum dots are formed through nanofabrication techniques such as epitaxial growth and electron beam lithography. In the epitaxial growth process, thin layers of the material are deposited with well-defined orientations and thickness. This technique allows for the precise control over the formation of quantum dots. Electron beam lithography uses a beam of electrons to etch the substrate, resulting in high resolution patterning. The use of these well-defined nanofabrication techniques in the formation of quantum dots is beneficial to the integration of spin qubits into existing electronic devices.

Types of spin qubits
While the most common area of research in spin qubits focuses on the manipulation of an electron's spin, or an electron spin qubit, nuclear spin qubits have also proven to be promising in the development of quantum technologies. Although both electron and nuclear spin qubits rely on the intrinsic spins of particles for the processing of information, there are a few key differences in their properties that should be noted.

Electron spin qubits
Electron spin qubits are qubits which utilize the spin of electrons in their function. These spin qubits are typically implemented through the three-dimensional confinement of an electron in semiconductor quantum dot devices. The electrons in donor atoms within a semiconductor lattice can also be used as spin qubits. The manipulation of the electron's spin, important in processing information, is performed through the application of magnetic fields and pulses. The control of these magnetic fields allows for gate operations to be performed on the spin states. While electron spin qubits are advantageous in their use of semiconductor materials, which are compatible with current technologies, reliable qubit operations are highly sensitive to the precision of quantum dot fabrication. Electron spins also have strong interactions with external fields, meaning that these qubits are susceptible to decoherence and have shorter coherence times than nuclear spin qubits.

Nuclear spin qubits
Nuclear spin qubits utilize the intrinsic spins of atomic nuclei, necessarily nuclei with non-zero net spin, in their function. Nuclei such as phosphorus-31 and nitrogen-14 are typically used in nuclear spin qubit applications. The manipulation of nuclear spin is employed through the manipulation of hyperfine interactions or the interaction between the magnetic moment of the nucleus and a magnetic field. External pulses are applied to the system as gate operations to control the spin states where hyperfine coupling constants determine the parameters of the gates. The application of these pulses can also be optimized to preserve the coherence of the system. However, it is difficult to maintain precise control over hyperfine interactions, which makes device implementation challenging. Nuclear spins also have weaker interactions with external fields, which means that these qubits are more stable than electron spin qubits and are less likely to decohere. Because of this, nuclear spin qubits have longer coherence times than electron spin qubits and are promising in quantum computing applications.

Applications
Spin qubits have many potential applications in the quantum computing field. Due to the potential use of spin qubits in large-scale, fault tolerant quantum computers, the applications of spin qubits overlap with those of quantum computers. Several of these applications include quantum algorithms such as Shor's and Grover's algorithm, simulation, cryptography, machine learning and optimization problems. Research in these fields is ongoing and challenges in their realization still remain.

Mathematical formulation
The mathematical formulation of spin qubits relies on foundational concepts in quantum mechanics. The spin of a particle, or collection of particles in the case of nuclear spin qubits, is a property that represents an intrinsic form of angular momentum. For the spin 1/2 systems of interest in spin qubits, the spin exists in either a spin-up or spin-down state. These states are represented by wavefunctions, which contain information about the physical system. The Pauli matrices, representations of the spin operators, span the observables in the two-dimensional Hilbert space.

$$\vec{S}=\frac{\hbar}{2} \sigma{}$$

where

$$S_x=\frac{\hbar}{2}\sigma_x, \quad S_y=\frac{\hbar}{2}\sigma_y, \quad S_z=\frac{\hbar}{2}\sigma_z$$

The Hamiltonian describing spin qubits represents the total energy of the system and depends on the conditions of the qubit environment.

Zeeman Interaction
The implementation of spin qubits in quantum computing applications depends on the ability to manipulate the intrinsic spin of the qubit. However, the energy levels of spin qubits are typically degenerate in the absence of an external magnetic field. This means that states corresponding to different spin orientations, either spin-up or spin-down, are indistinguishable. The ability to distinguish spin states of the system, and thereby manipulate the spin qubit into a superposition state, is crucial for the applications of spin qubits. For this purpose, an external static magnetic field is applied to the qubits in order to induce the Zeeman effect. The presence of this magnetic field causes an energy level splitting between the spin-up and spin-down states, where the energy levels are proportional to both the strength of the magnetic field and the spin magnetic moment.

The Hamiltonian for an electron spin qubit in an external static magnetic field is expressed generally as:$$\hat{H}_{electron}=-\gamma \vec{B} \cdot \vec{S} $$where
 * $$\gamma$$ is the gyromagnetic ratio
 * $$\vec{B}$$ is the static external magnetic field
 * $$\vec{S}$$ is the electron spin operator

The dot product between the spin operator and the external magnetic field varies depending on the chosen quantization axis for the spin. Typically, both the quantization axis and the external magnetic field are chosen along the z-axis. In the case of the quantization axis being chosen along z, the Hamiltonian can also be expressed as:

$$\hat{H}_{electron}=-\frac{\hbar}{2}\omega_e\sigma_z$$

where


 * $$\hbar$$ is the reduced Planck's constant
 * $$\omega_e$$ is the Larmor frequency associated with electron spin
 * $$\sigma_z$$ is the Pauli-z matrix

It should be noted that the Larmor frequency, $$\omega_e$$, is the product of the gyromagnetic ratio and the amplitude of the static magnetic field.

Similarly, the Hamiltonian for a nuclear spin qubit in an external static magnetic field is expressed as:

$$\hat{H} _{nuclear} = -\gamma_n \vec{B} \cdot \vec{I} $$

where


 * $$\gamma_n$$ is the gyromagnetic ratio of the nucleus
 * $$\vec{B}$$ is the static external magnetic field
 * $$\vec{I}$$ is the nuclear spin operator

While similar in form, there are a few key differences between the interaction of the electron spin and nuclear spin qubits with a static, external field. The Zeeman term of the Hamiltonian is what describes the coupling of either the electron or nucleus's magnetic moment to the magnetic field. Since the magnetic moment of an electron is much stronger than that of a nucleus, the electron spin qubit will possess a Hamiltonian which depends more dominantly on the Zeeman interaction.

In the context of spin qubits, the Zeeman effect plays a crucial role in the definition of energy levels for both electron and nuclear spin qubits. The separation of these energy levels allows for the manipulation and readout of quantum information from spin qubits.

Oscillating magnetic field
Oscillating, rather than static, magnetic fields are applied to spin qubits as a means of performing quantum gate operations. The oscillating field induces a rotation in the spin of a qubit through the application of pulses at specific frequencies to manipulate the state of the system. When the frequency of the oscillating field is chosen to be on resonance, the magnetic field induces controlled transitions between the spin states.

The general addition of this oscillating term to the Hamiltonian, with the assumption of z as the quantization axis, can be written as:

$$\hat{H}_ {electron} = \hat{H}_{static} + \hat{H}_{oscillating} = -\frac {\hbar}{2}\omega_e\sigma_z -\frac{\hbar}{2} B_1 (\cos ( \theta t)\sigma_x + \sin(\theta t)\sigma_y) $$

where

$$ is the amplitude of the oscillating magnetic field The angle $$\theta$$ is typically the associated Larmor frequency which is a product of the gyromagnetic ratio and the amplitude of a static magnetic field.
 * $$\hbar$$ is the reduced Planck's constant
 * $$B_1
 * $$t$$ is time
 * $$\sigma_x, \sigma_y$$ are the Pauli x and y matrices

Different implementations of specific gates, such as the Hadamard, Pauli X, Y, and Z gates, using an oscillating or driving field depend on the specifications of the system.

The master equation
An important factor in the understanding of spin qubit devices is the evolution of the Hamiltonian during decoherence. The Lindblad master equation is used to describe the dynamics of the system as it interacts with the environment, leading to decoherence.

$$\dot\rho=-{i\over\hbar}[H,\rho]+\sum_{i}^{} \gamma_i\left(L_i\rho L_i^\dagger -\frac{1}{2} \left\{L_i^\dagger L_i, \rho\right\} \right)$$

where


 * $$H$$ is the system's Hamiltonian
 * $$\rho$$ is the system's density matrix
 * $$\gamma_i$$ are damping rates ($$\gamma_i \geq 0$$)
 * $$L_i$$ are a set of jump operators that describe dissipative dynamics
 * $$\{a, b\} = ab + ba$$ is the anticommutator

During decoherence, the Hamiltonian of a system doesn't change, but the interactions with the environment affect the density matrix $$\rho$$. The Lindblad jump operators $$L_i$$ model these dissipative processes that occur in open quantum systems and lead to decoherence. Non-unitary, irreversible processes such as energy relaxation and dephasing are modelled in this equation.

Spin shuttling
The largest quantum computer to date contains only a few hundred qubits, but in order for quantum computation to have practical applications, millions of qubits are required. In a simple, 2-qubit device, the qubits are close enough for information to be easily exchanged but the application of larger devices require qubits to be separated by larger distances. Current devices implement systems of wires connecting each qubit, but this quickly becomes impractical for increasing numbers of qubits. Spin shuttling is a promising alternative as it allows for efficient information exchange within a quantum dot device. Shuttling is the controlled transport of an electron spin through an array of quantum dots, which changes the electrochemical potential for each dot but preserves the spin qubit’s quantum information.

Due to the electron's lighter mass and stronger interaction with external fields, as opposed to the nuclear spins' weaker interactions, electron spin qubits are exclusively used for spin shuttling methods. While there are methods such as nuclear magnetic resonance (NMR) employed in the efficient control and manipulation of nuclear spin qubits, this section focuses on electron spin shuttling in quantum dot devices.

The development of spin shuttling techniques present many advantages in quantum technologies including more precise control over the qubits, improved coherence, and scalability. The precise control of qubits is generally achieved through the application of magnetic or electric fields, pulses and adiabatic transitions, which minimize transitions to undesired states. The manipulation of the strength of those fields and frequency of the pulses allows for specific operations to be performed. This careful control over the shuttling process allows for the extension of coherence times, useful in quantum systems. The architecture of spin shuttling techniques, or the quantum dot arrays, enable the transfer of quantum information between qubits at a larger scale and allows for the potential of larger quantum systems.

Experimental setup
The experimental process involved in spin shuttling varies depending on the specific parameters of the experiment. However, the general process consists of quantum dot formation, application of external fields, and state preparation. The creation of the quantum dot arrays is performed through nanofabrication techniques with the deposition of thin semiconductor layers, followed by lithography or etching techniques for patterning. Gate electrodes are placed along the quantum dot array in order to control the electrons and manipulate their spin through the voltage applied. External fields are then used to control the spin states of the qubits through the manipulation of the potential energy landscape.

Shuttling process
The actual process of spin shuttling can be implemented through several different techniques, including the control of electric and magnetic fields, the application of pulses, and the manipulation of gate electrodes. Electric field control is the method of controlling the movement of electrons and the information encoded in their spin states through the application of an electric field. In the process of spin shuttling, the electric field is used to selectively shuttle electrons between the quantum dots within the array. When an electric field is applied to the quantum dot array, the field changes the energy levels of the quantum dots and changes the potential energy landscape to create potential barriers and wells. These barriers confine the electron within the quantum dot, but, depending on the height of the barrier, the electron can possess enough energy to escape the dot and move into adjacent dots. Researchers can adjust the electric field applied to the quantum dot array in such a way that a well defined path between quantum dots is created for an electron to be shuttled through. The information contained in the spin state is preserved through these transitions between quantum dots. This precise control over the potential landscape of the quantum dot array helps to mitigate any unwanted interactions. Similarly, magnetic field control allows for the control over the spin states of electrons during the spin shuttling process. In the presence of a magnetic field, several processes that induce precession and rotation occur in spin qubits. For instance, electrons undergo a Larmor precession in the presence of a magnetic field due to their magnetic moment. This Larmor precession is the precession of the electron's spin vector along the axis of the magnetic field with a frequency proportional to the field's strength. Thus, the manipulation of the magnetic field's amplitude can be used to selectively cause spin rotations. The application of magnetic pulses can also be used to selectively manipulate spin states so that all other states remain unchanged. In this way, magnetic fields can be used to exactly control the trajectory of a spin state during the process of spin shuttling.

While both of these processes aim to establish control over the movement of spin states through a quantum dot array without the loss of quantum information, the mitigation of decoherence is not ensured. Adiabatic processes in both electric and magnetic field control techniques are key for the maintenance of coherence. The adiabatic process is a process by which parameters are varied slowly enough to ensure that the system remains in an instantaneous eigenstate. Without this adiabaticity, there is a risk of spin flips, decoherence and a loss of accuracy in both electric and magnetic field control. In the presence of an electric field along with either spin-orbit coupling or spin-phonon processes, there is a risk of the system being excited to higher energy levels which would result in a change in the spin state. Through adiabatic evolution, the energy levels are adjusted slowly enough so that the original spin state is preserved. Similarly, the application of magnetic fields induces a spin rotation which could cause transitions to higher energy spin states. With the slow variation of the magnetic field strength, the original spin states are preserved.

History
The concept of utilizing electron spins as qubits in semiconductor quantum dots was first proposed in 1998 by researchers Daniel Loss and David P. DiVincenzo in the paper entitled “Quantum Computation with Quantum Dots”. This paper laid the groundwork for the development of spin qubits as they are implemented today. While this theoretical idea was proposed in 1998, it took several years for the experimental control of spin qubits to be achieved. In order for the experimental realization of spin qubit control, several advancements in technology had to occur.

Nanofabrication techniques, such as electron beam lithography and molecular beam epitaxy, paved the way for the physical creation of semiconductor quantum dots through the development of precise fabrication and manipulation of structures at the nanoscale. Quantum dots themselves serve as necessary components in harnessing the electron spin through the three-dimensional confinement of electrons. The ability to control the size of the quantum dots is crucial in this confinement.

Jason Petta et al. demonstrated the first use of qubits in quantum dots in 2004. This work proved fundamental for the implementation of single-qubit operations and quantum computation as a whole. Following this, Frank Koppens et al. was able to show the first demonstration of coherent rotations in electron spins. After these initial implementations, research has been focused on further advancements in the scalability of spin qubits.