Transmon-Resonator Chain Emulation

This example demonstrates how to run an analog simulation of a chain consisting of transmon qubits and resonators using YAQS.

A Hamiltonian is initialized using a coupled transmon-resonator model. A State is prepared in a specific computational basis state. The system is evolved under a noise-free analog simulation using the Tensor Jump Method (TJM). Finally, expectation values for all computational basis states are collected and visualized.

Define the system Hamiltonian

import numpy as np
from mqt.yaqs.core.data_structures.hamiltonian import Hamiltonian

length = 3 # Qubit - resonator - qubit
qubit_dim = 2
resonator_dim = 2
w_q = 4/(2*np.pi)
w_r = 4/(2*np.pi)
alpha = 0
g = 2/(2*np.pi)

H_0 = Hamiltonian.coupled_transmon(
    length=length,
    qubit_dim=qubit_dim,
    resonator_dim=resonator_dim,
    qubit_freq=w_q,
    resonator_freq=w_r,
    anharmonicity=alpha,
    coupling=g,  # T_swap = pi/(2g)
)

Define the initial state

from mqt.yaqs.core.data_structures.state import State

# Initialize in state |100⟩: left qubit excited
state = State(
    length,
    initial="basis",
    basis_string="100",
    physical_dimensions=[qubit_dim, resonator_dim, qubit_dim],
)

Define the noise model

from mqt.yaqs.core.data_structures.noise_model import NoiseModel

gamma = 0.0  # No noise for this simulation
noise_model = NoiseModel([
    {"name": name, "sites": [i], "strength": gamma}
    for i in range(length)
    for name in ["lowering"]
])

Define the simulation parameters

from mqt.yaqs.core.data_structures.simulation_parameters import Observable, AnalogSimParams

T_swap = np.pi/(2*g)

# Measure all computational basis states
bitstrings = ["000", "001", "010", "011", "100", "101", "110", "111"]
sim_params = AnalogSimParams(
    observables=[Observable(bstr) for bstr in bitstrings], elapsed_time=T_swap, dt=T_swap/1000,
    sample_timesteps=True
)

Run the simulation

from mqt.yaqs import Simulator

sim = Simulator()
result = sim.run(state, H_0, sim_params, noise_model)

Plot the results

import matplotlib.pyplot as plt
import numpy as np

pvm_observables = result.observables
leakage = np.ones_like(result.expectation_values[0])
for measurement, values in zip(pvm_observables, result.expectation_values, strict=True):
    leakage -= values
    plt.plot(values, label=measurement.gate.bitstring)
plt.plot(leakage, label="Leakage")

plt.xlabel("Timestep")
plt.ylabel("Probability")
plt.title("Population in Computational Basis States Over Time")
plt.legend(loc="center left", bbox_to_anchor=(1, 0.5))
plt.tight_layout()
plt.show()