How to repeatedly initialize a fixed quantum circuit, and the initialization vector is random？

Question

I need to repeatedly input one random quantum state after another into a quantum circuit that has been designed and assigned parameters, and these quantum states will be used as input. I have tried

Statevector()

https://qiskit.org/documentation/locale/ja_JP/stubs/qiskit.quantum_info.Statevector.html?highlight=statevector#qiskit.quantum_info.Statevector

Custom()

https://qiskit.org/documentation/stable/0.19/stubs/qiskit.aqua.components.initial_states.Custom.html

initialize()

https://qiskit.org/documentation/stubs/qiskit.circuit.QuantumCircuit.initialize.html?highlight=initialize#qiskit.circuit.QuantumCircuit.initialize

, but I found Although these functions can generate new state vectors, they cannot be inserted into already designed quantum circuits, resulting in the same circuit results every time. How should I design it? The three functions mentioned above lack the inplace property and cannot generate a new circuit copy.

{ParameterVectorElement(aa[0]): 0.2379597082677889, ParameterVectorElement(aa[1]): 1.4010239667773599, ParameterVectorElement(aa[2]): 0.10401258494998716, ParameterVectorElement(aa[3]): 0.3326296026269176, ParameterVectorElement(bb[0]): 1.6443757859109702, ParameterVectorElement(bb[1]): 0.8877583818970967, ParameterVectorElement(bb[2]): 0.07150986168780663, ParameterVectorElement(bb[3]): 1.720576457948209, ParameterVectorElement(cc[0]): 1.9974865102845014, ParameterVectorElement(cc[1]): 1.4917958623436192, ParameterVectorElement(cc[2]): 0.7199744574903324, ParameterVectorElement(cc[3]): 2.635969165530908}
WARNING:qiskit.aqua.components.initial_states.custom:The provided state_vector is ignored in favor of the provided custom circuit.
ParameterView([]) [0.00449711 0.         0.         ... 0.         0.         0.        ]
C:\Users\15691\AppData\Local\Temp\ipykernel_347040\162399.py:121: DeprecationWarning: The return type of saved density matrices has been changed from a `numpy.ndarray` to a `qiskit.quantum_info.DensityMatrix` as of qiskit-aer 0.10. Accessing numpy array attributes is deprecated and will result in an error in a future release. To continue using saved result objects as arrays you can explicitly cast them using  `np.asarray(object)`.
  t = np.array(density_matrix.real)
[[0.09660034 0.09656075 0.11104839 0.11262035 0.09307498 0.09300402
  0.10599975 0.10750024]
 [0.09656075 0.0984683  0.11262035 0.11646331 0.09300402 0.0948085
  0.10750024 0.11116849]
 [0.11104839 0.11262035 0.1565212  0.16014388 0.10599975 0.10750024
  0.14853878 0.15202993]
 [0.11262035 0.11646331 0.16014388 0.16699589 0.10750024 0.11116849
  0.15202993 0.15858678]
 [0.09307498 0.09300402 0.10599975 0.10750024 0.09142649 0.09132274
  0.10308861 0.10454789]
 [0.09300402 0.0948085  0.10750024 0.11116849 0.09132274 0.09306052
  0.10454789 0.10811539]
 [0.10599975 0.10750024 0.14853878 0.15202993 0.10308861 0.10454789
  0.14355167 0.14698168]
 [0.10750024 0.11116849 0.15202993 0.15858678 0.10454789 0.10811539
  0.14698168 0.15337558]]
{ParameterVectorElement(aa[0]): 0.2379597082677889, ParameterVectorElement(aa[1]): 1.4010239667773599, ParameterVectorElement(aa[2]): 0.10401258494998716, ParameterVectorElement(aa[3]): 0.3326296026269176, ParameterVectorElement(bb[0]): 1.6443757859109702, ParameterVectorElement(bb[1]): 0.8877583818970967, ParameterVectorElement(bb[2]): 0.07150986168780663, ParameterVectorElement(bb[3]): 1.720576457948209, ParameterVectorElement(cc[0]): 1.9974865102845014, ParameterVectorElement(cc[1]): 1.4917958623436192, ParameterVectorElement(cc[2]): 0.7199744574903324, ParameterVectorElement(cc[3]): 2.635969165530908}
WARNING:qiskit.aqua.components.initial_states.custom:The provided state_vector is ignored in favor of the provided custom circuit.
ParameterView([]) [0.01146697 0.         0.         ... 0.         0.         0.        ]
[[0.09660034 0.09656075 0.11104839 0.11262035 0.09307498 0.09300402
  0.10599975 0.10750024]
 [0.09656075 0.0984683  0.11262035 0.11646331 0.09300402 0.0948085
  0.10750024 0.11116849]
 [0.11104839 0.11262035 0.1565212  0.16014388 0.10599975 0.10750024
  0.14853878 0.15202993]
 [0.11262035 0.11646331 0.16014388 0.16699589 0.10750024 0.11116849
  0.15202993 0.15858678]
 [0.09307498 0.09300402 0.10599975 0.10750024 0.09142649 0.09132274
  0.10308861 0.10454789]
 [0.09300402 0.0948085  0.10750024 0.11116849 0.09132274 0.09306052
  0.10454789 0.10811539]
 [0.10599975 0.10750024 0.14853878 0.15202993 0.10308861 0.10454789
  0.14355167 0.14698168]
 [0.10750024 0.11116849 0.15202993 0.15858678 0.10454789 0.10811539
  0.14698168 0.15337558]]
{ParameterVectorElement(aa[0]): 0.2379597082677889, ParameterVectorElement(aa[1]): 1.4010239667773599, ParameterVectorElement(aa[2]): 0.10401258494998716, ParameterVectorElement(aa[3]): 0.3326296026269176, ParameterVectorElement(bb[0]): 1.6443757859109702, ParameterVectorElement(bb[1]): 0.8877583818970967, ParameterVectorElement(bb[2]): 0.07150986168780663, ParameterVectorElement(bb[3]): 1.720576457948209, ParameterVectorElement(cc[0]): 1.9974865102845014, ParameterVectorElement(cc[1]): 1.4917958623436192, ParameterVectorElement(cc[2]): 0.7199744574903324, ParameterVectorElement(cc[3]): 2.635969165530908}

Answer 1

You can use QuantumCircuit.compose() method to prepend an Initialize instruction to your circuit. compose() takes a parameter called inplace. If True, the circuit will be modified. Otherwise, a new circuit will be created and returned.

from qiskit import QuantumCircuit
from qiskit.extensions import Initialize
from qiskit.quantum_info import random_statevector
from IPython.display import display

# Create a sample 2-qubit circuit:
circ = QuantumCircuit(2)
circ.h(0)
circ.cx(0, 1)

# Repeatedly initialize the circuit with a random quantum state:
for m in range(5):
    # Create a 2-qubit random statevector:
    state = random_statevector(2 ** 2)
    # Create an "Initialize" instruction and prepend it to the circuit:
    new_circ = circ.compose(Initialize(state), front = True, inplace = False)
    print('Circuit No.', m + 1)
    display(new_circ.draw(fold = -1))

The result:

Circuit No. 1
     ┌────────────────────────────────────────────────────────────────────────────────────┐┌───┐     
q_0: ┤0                                                                                   ├┤ H ├──■──
     │  Initialize(-0.58349+0.2304j,0.35503+0.12377j,-0.31208+0.16129j,-0.36705-0.45493j) │└───┘┌─┴─┐
q_1: ┤1                                                                                   ├─────┤ X ├
     └────────────────────────────────────────────────────────────────────────────────────┘     └───┘
Circuit No. 2
     ┌─────────────────────────────────────────────────────────────────────────────────────┐┌───┐     
q_0: ┤0                                                                                    ├┤ H ├──■──
     │  Initialize(-0.49122+0.28218j,0.22847-0.11786j,0.10971+0.66447j,-0.084787-0.39018j) │└───┘┌─┴─┐
q_1: ┤1                                                                                    ├─────┤ X ├
     └─────────────────────────────────────────────────────────────────────────────────────┘     └───┘
Circuit No. 3
     ┌───────────────────────────────────────────────────────────────────────────────────────┐┌───┐     
q_0: ┤0                                                                                      ├┤ H ├──■──
     │  Initialize(0.41866+0.81892j,0.1204-0.051467j,0.060036-0.00024577j,-0.13383-0.33975j) │└───┘┌─┴─┐
q_1: ┤1                                                                                      ├─────┤ X ├
     └───────────────────────────────────────────────────────────────────────────────────────┘     └───┘
Circuit No. 4
     ┌────────────────────────────────────────────────────────────────────────────────────┐┌───┐     
q_0: ┤0                                                                                   ├┤ H ├──■──
     │  Initialize(-0.26621+0.53027j,-0.13784+0.57166j,0.1804-0.15494j,-0.30107-0.39365j) │└───┘┌─┴─┐
q_1: ┤1                                                                                   ├─────┤ X ├
     └────────────────────────────────────────────────────────────────────────────────────┘     └───┘
Circuit No. 5
     ┌───────────────────────────────────────────────────────────────────────────────────────┐┌───┐     
q_0: ┤0                                                                                      ├┤ H ├──■──
     │  Initialize(-0.043747+0.10925j,0.23058+0.92953j,0.21934+0.00061991j,0.095399-0.1084j) │└───┘┌─┴─┐
q_1: ┤1                                                                                      ├─────┤ X ├
     └───────────────────────────────────────────────────────────────────────────────────────┘     └───┘

Answer 2

consider control flow, WhileLoopOp or ForLoopOp, and random_statevector, for the task, will make your life easier. and run the circuit with .run(qc)

Example:

from qiskit.quantum_info import random_statevector
from qiskit.circuit import ForLoopOp,QuantumCircuit
from qiskit.providers.aer import AerSimulator

def rand_init(qubit_num):
    loop_qc = QuantumCircuit(2)
    loop_qc.initialize(random_statevector(2**qubit_num),[0,1])
    return loop_qc


qc = QuantumCircuit(2, 2)

# method 1
#loop_op = ForLoopOp(range(4),None,rand_init(qc.num_qubits))
#qc.append(loop_op,[0,1])

# method 2
#qc.for_loop(range(4),None,rand_init(qc.num_qubits), [0,1], [])

# method 3(recommend)
with qc.for_loop(range(5)) as i:
    qc.append(rand_init(qc.num_qubits),[0,1])
    #extra_example#
    #qc.measure(0, 0)
    #qc.break_loop().c_if(0, True)

qc.draw(cregbundle=False)

output:
     ┌───────────┐
q_0: ┤0          ├
     │  For_loop │
q_1: ┤1          ├
     └───────────┘
c_0: ═════════════
                  
c_1: ═════════════

qc.measure_all(add_bits=False)
job = AerSimulator(method="statevector").run(qc) 
job.result().get_counts()
output:
{'01': 188, '11': 165, '00': 400, '10': 271}

For more example, check the qiskit test file: https://github.com/Qiskit/qiskit-terra/blob/ea0266769802a57de2ca823b1b996b668ec81178/test/python/circuit/test_control_flow.py