Spin Qubits

Spin Qubits

Spin qubits encode quantum information in the spin state of electrons or nuclei:

The spin interacts with magnetic fields:

where is the g-factor, is the Bohr magneton, and is the magnetic field.

Spin qubits offer long coherence times, small size, and compatibility with existing semiconductor fabrication.

Quantum Dots

Quantum dots confine individual electrons in semiconductor nanostructures:

• Gate-defined dots: electrostatic confinement in GaAs or Si/SiGe
• Self-assembled dots: epitaxially grown InAs/GaAs
• Si/Si MOS: silicon metal-oxide-semiconductor dots

A single electron in a quantum dot has a well-defined spin, manipulated by:

• Microwave pulses: spin resonance (ESR)
• Electric fields: electric dipole spin resonance (EDSR)
• Exchange coupling: two-qubit gates between neighboring dots

Silicon Qubits

Silicon spin qubits are the most promising for scaling:

Advantages:

• Isotopically purified Si-28: removes nuclear spin noise
• Small size: ~100 nm, compatible with CMOS fabrication
• Fast gates: exchange coupling enables ~ns gates
• Long coherence: ms in purified Si

Intel, UNSW, and QuTech are leading silicon qubit development.

Exchange Coupling

The exchange interaction between two neighboring spins:

where is the exchange coupling strength, controlled by gate voltage.

By pulsing :

• : SWAP gate
• : gate (entangling)

The exchange interaction is all-to-all within a quantum dot array, enabling flexible connectivity.

Single-Qubit Gates

Single-qubit gates on spin qubits:

1. ESR (electron spin resonance): microwave pulse at the Larmor frequency
2. EDSR (electric dipole spin resonance): electric field modulation coupled to spin via spin-orbit interaction
3. Global control: uniform microwave field + local detuning

Gate fidelities >99.9% have been demonstrated in silicon spin qubits.

Scaling Challenges

Scaling spin qubits to millions:

• Wire count: each qubit needs control lines
• Cross-talk: neighboring qubits interfere
• Yield: fabrication defects
• Operating temperature: millikelvin

Solutions: CMOS-compatible fabrication, cryo-CMOS control, multiplexed addressing.

Python: Spin Qubit Physics

import numpy as np

def larmor_frequency(B, g=2.0):
    # Compute Larmor frequency for spin in magnetic field.
    mu_B = 9.274e-24  # Bohr magneton (J/T)
    h = 6.626e-34     # Planck constant (J*s)
    return g * mu_B * B / h

def exchange_gate(J, t):
    # Exchange gate for given J and time.
    # S = sigma/2
    Sx = np.array([[0,1],[1,0]], dtype=complex) / 2
    H = J * (np.kron(Sx, Sx))
    return scipy.linalg.expm(-1j * H * t)

import scipy.linalg
B = 0.5  # Tesla
print(f"Larmor frequency: {larmor_frequency(B)/1e9:.3f} GHz")

J = 2 * np.pi * 1e6  # 1 MHz exchange
U = exchange_gate(J, 1/J)
print(f"Exchange gate fidelity: {np.abs(np.trace(U @ U.conj().T))/4:.4f}")

Silicon Qubit Advantages

1. CMOS compatibility: leverage existing semiconductor fabrication
2. Small size: ~100 nm, enabling high qubit density
3. Fast gates: exchange coupling enables ~ns two-qubit gates
4. Long coherence: ms in isotopically purified Si-28
5. Identical qubits: atomic qubits are identical by nature

Silicon Qubit Roadmap