Fidelity and Metrics

Fidelity measures how close two quantum states (or unitaries) are. For states $\rho$ and $\sigma$:

$$F(\rho, \sigma) = \text{Tr}(\sqrt{\sqrt{\rho} \sigma \sqrt{\rho}})$$

For pure states, this simplifies to $F = |\langle\psi|\phi\rangle|^2$. Fidelity ranges from 0 (orthogonal) to 1 (identical).

from qutip import *
import numpy as np
 
# Two quantum states
psi1 = basis(2, 0)
psi2 = (basis(2, 0) + basis(2, 1)).unit()
rho1 = psi1 * psi1.dag()
rho2 = psi2 * psi2.dag()
 
# Fidelity between states
F = fidelity(rho1, rho2)
print(F)  # 0.5 for these states
 
# Fidelity between unitaries
U1 = sigmax()
U2 = np.sqrt(sigmax())  # √X gate
F_unitary = average_gate_fidelity(U1, U2)

• Trace distance: $D(\rho, \sigma) = \frac{1}{2}\text{Tr}|\rho - \sigma|$ (bounded by 1)
• Hilbert-Schmidt distance: $D_{\text{HS}} = \sqrt{\text{Tr}[(\rho - \sigma)^2]}$
• Bures distance: $D_B = \sqrt{2(1 - F)}$

D = tracedist(rho1, rho2)
D_hs = (rho1 - rho2).norm()  # Hilbert-Schmidt

Fidelity is the key metric for assessing quantum gate accuracy and simulation accuracy. Higher fidelity means closer to target.