Superconducting Qubit Dynamics

Model a transmon as a driven anharmonic oscillator and simulate single- and two-qubit gate dynamics, including the cross-resonance interaction.

A driven transmon

import qalgora
from qalgora import operators
import numpy as np

w, alpha = 5.0, -0.3                   # transmon frequency and anharmonicity (GHz)
amp, w_d = 0.1, 5.0                    # drive amplitude and drive frequency

# transmon as a 3-level system with anharmonicity alpha
b = operators.boson.annihilate(0)
H0 = w * operators.boson.create(0) * b + 0.5 * alpha * \
     operators.boson.create(0) * operators.boson.create(0) * b * b

def drive(t):
    return amp * np.cos(w_d * t)

H = H0 + operators.scalar(drive) * (b + operators.boson.create(0))

Cross-resonance gate

Driving one qubit at a neighbour's frequency generates an entangling interaction — the basis of the CR gate on fixed-frequency transmons. Build H_cr as a two-transmon Hamiltonian driven at the target qubit's frequency (illustrative fragment — supply H_cr, schedule and psi0).

# H_cr: two coupled transmons with the control driven at the target frequency
result = qalgora.evolve(H_cr, dimensions={0: 3, 1: 3},
                        schedule=schedule, initial_state=psi0)

import qalgora
from qalgora import operators
import numpy as np

w, alpha = 5.0, -0.3                   # transmon 频率与非谐性 (GHz)
amp, w_d = 0.1, 5.0                    # 驱动幅度与驱动频率

# 把 transmon 当作带非谐性 alpha 的三能级系统
b = operators.boson.annihilate(0)
H0 = w * operators.boson.create(0) * b + 0.5 * alpha * \
     operators.boson.create(0) * operators.boson.create(0) * b * b

def drive(t):
    return amp * np.cos(w_d * t)

H = H0 + operators.scalar(drive) * (b + operators.boson.create(0))

# H_cr 两个耦合 transmon 控制比特以目标频率驱动
result = qalgora.evolve(H_cr, dimensions={0: 3, 1: 3},
                        schedule=schedule, initial_state=psi0)