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Mathematical Background
Quantum Channels
Quantum channels describe how quantum states evolve under noise. Each channel transforms the Bloch vector r = (x, y, z) differently:
Bit Flip Channel
Effect: Flips qubit with probability p
Transform: (x, y, z) → (x, (1-2p)y, (1-2p)z)
Physics: Preserves X-component, shrinks Y and Z
Phase Flip Channel
Effect: Applies phase flip with probability p
Transform: (x, y, z) → ((1-2p)x, (1-2p)y, z)
Physics: Preserves Z-component, shrinks X and Y
Depolarizing Channel
Effect: Mixes state with maximally mixed state
Transform: (x, y, z) → (sx, sy, sz) where s = 1 - 4p/3
Physics: Uniform shrinking toward center
Amplitude Damping
Effect: Models energy loss (T₁ relaxation)
Transform: (x, y, z) → (√(1-γ)x, √(1-γ)y, (1-γ)z + γ)
Physics: Drives states toward |0⟩ (north pole)
Phase Damping
Effect: Models dephasing (T₂ relaxation)
Transform: (x, y, z) → (e^(-γ)x, e^(-γ)y, z)
Physics: Preserves populations, destroys coherence
Key Metrics
Fidelity
Measures how similar the transformed states are to the original states. Higher values (closer to 1) indicate better preservation of quantum information.
Average Distance
Average Euclidean distance between original and transformed Bloch vectors. Lower values indicate less distortion.
Volume Ratio
Ratio of transformed to original state space volume. Values < 1 indicate contraction (information loss).