Expressibility and Entanglement Capability of the Parameterized Quantum Circuits

Question

I am trying to calculate the expressibility and entangling capability of a quantum state resulting from a circuit as defined in reference I. One of my attempts was to follow reference II which gives us some Python code using Qiskit like this:

def random_unitary(N):
    """
        Return a Haar distributed random unitary from U(N)
    """

    Z = np.random.randn(N, N) + 1.0j * np.random.randn(N, N)
    [Q, R] = sp.linalg.qr(Z)
    D = np.diag(np.diagonal(R) / np.abs(np.diagonal(R)))
    return np.dot(Q, D)

def haar_integral(num_qubits, samples):
    """
        Return calculation of Haar Integral for a specified number of samples.
    """

    N = 2**num_qubits
    randunit_density = np.zeros((N, N), dtype=complex)

    
    zero_state = np.zeros(N, dtype=complex)
    zero_state[0] = 1
    
    for _ in range(samples):
      A = np.matmul(zero_state, random_unitary(N)).reshape(-1,1)
      randunit_density += np.kron(A, A.conj().T) 

    randunit_density/=samples

    return randunit_density
    
def pqc_integral(num_qubits, ansatze, size, samples):
    """
        Return calculation of Integral for a PQC over the uniformly sampled 
        the parameters θ for the specified number of samples.
    """

    N = num_qubits
    randunit_density = np.zeros((2**N, 2**N), dtype=complex)

    for _ in range(samples):
      params = np.random.uniform(-np.pi, np.pi, size)
      ansatz = ansatze(params, N)
      result = execute(ansatz, 
                       backend=Aer.get_backend('statevector_simulator')).result()
      U = result.get_statevector(ansatz, decimals=5).reshape(-1,1)
      randunit_density += np.kron(U, U.conj().T)

    return randunit_density/samples

To create the ansatz test, reference II did:

def ansatz2(params, num_qubits): 

    params = np.array(params).reshape(1)
    ansatz = QuantumCircuit(num_qubits, num_qubits)
    ansatz.h(0)
    ansatz.cx(0, 1)
    ansatz.rx(params[0], 0)

    return ansatz

And to instantiate:

np.linalg.norm(haar_integral(2, 2048) - pqc_integral(2, ansatz2, 1, 2048))

However, the code is not compiling and the output is:

AttributeError: 'Statevector' object has no attribute 'reshape'

My question is: does anyone know how to solve this problem? is there any other way to coherently calculate the expressibility of a circuit? I tried to use medium fidelity but had issues with sanity checks.

References: (I) https://arxiv.org/abs/1905.10876 (II) https://obliviateandsurrender.github.io/blogs/expr.html

Answer 1

The underlying issue from that error is that the Statevector class being returned by get_statevector() doesn't have the reshape method defined on it. Based on the code it looks like you're expecting a numpy array object instead of Statevector. A Statevector object can be used in place of a numpy array for numpy functions. So you can do something like:

np.reshape(result.get_statevector(ansatz, decimals=5), (-1,1))

or alternatively cast the statevector to an array:

np.asarray(result.get_statevector(ansatz, decimals=5)).reshape(-1,1)

either approach does basically the same thing and will let you call reshape on the statevector which should unblock you on that failure.

I didn't run the full code example though, so I can't say whether that's enough to fix everything. But that's the immediate error your

Answer 2

Use StateVector class to get the state vector. Then convert the Statevector into array to do matrix operations.

U = Statevector(result.get_statevector()) 
U_array = U.data.reshape(-1, 1)
U_conj_transpose = U_array.conj().T 

This would work.