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This notebook shows how to generate data for discrete quantum walk using pennylane 0.39.0 and pennylane-catalyst 0.9.0 
# import time
# import pennylane as qml
# from catalyst import qjit


# import time
# import pennylane as qml
# from catalyst import qjit

# # run an example
# SIZE = 6
# start_time = time.time()

# s = SIZE
# wires_x = list(range(s))
# coin = [s]

# def shift_op():
#     for index in range(len(wires_x)-1, 0, -1):
#         control_values = [1] * index
#         qml.ctrl(qml.PauliX, control=wires_x[:index], control_values=control_values)(wires=wires_x[index])
#         qml.PauliX(wires_x[0])


# @qml.qnode(qml.device("lightning.qubit", wires=(wires_x + coin)))
# def circuit():
#     qml.H(coin)
#     qml.ctrl(shift_op, control=coin)()
#     qml.ctrl(qml.adjoint(shift_op), control=coin, control_values=[0])()
#     return qml.probs(wires=wires_x)

# cir = qml.transforms.decompose(circuit, gate_set={qml.CNOT, qml.RZ, qml.RY, qml.RX})
# jitted_cir = qjit(cir)
# transpilation_time = time.time()-start_time
# cx_counts = jitted_cir.mlir.count("CNOT")

# print(f"==== pennylane for {SIZE}==== time: {transpilation_time}")