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A couple of questions regarding the conversion between Jordan-Wigner (JW) and Bravyi-Kitaev (BK) states in Qiskit.

The JW $\rightarrow$ BK conversion matrix I refer to below is the one from equation (29) here. (I'm not referring to equation (24) in Ibid. for it contains a typo.)

  1. Is there a way to generate a circuit which would map JW-encoded states to BK-encoded states? (The circuit is actually very simple since it multiplies the input $n$-qubit vector by a $n\times n$ matrix $\operatorname{mod} 2$; since the matrix is upper-triangular, it is rather trivial to construct it out of CNOTs.)

  2. Assuming that the answer to the previous question is NO: is there a way to, at least, generate the matrix implementing the conversion between the Jordan-Wigner and Bravyi-Kitaev states, for a given number of qubits $n$?

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1 Answer 1

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I have not found such functionality in Qiskit. However, one can use the openfermion function openfermion.transforms._encoder_bk():

def _encoder_bk(n_modes):
    """ Helper function for bravyi_kitaev_code that outputs the binary-tree
    (dimension x dimension)-matrix used for the encoder in the
    Bravyi-Kitaev transform.

    Args:
    n_modes (int): length of the matrix, the dimension x dimension

    Returns (numpy.ndarray): encoder matrix
    """
    reps = int(numpy.ceil(numpy.log2(n_modes)))
    mtx = numpy.array([[1, 0], [1, 1]])
    for repetition in numpy.arange(1, reps + 1):
        mtx = numpy.kron(numpy.eye(2, dtype=int), mtx)
        for column in numpy.arange(0, 2 ** repetition):
            mtx[2 ** (repetition + 1) - 1, column] = 1
    return mtx[0:n_modes, 0:n_modes]

With the aid of this function, one constructs the JW → BK circuit as follows:

def jw_to_bk_circuit_qiskit(n):
    """
    Creates a Qiskit circuit which performs the JW -> BK transformation on qubits.
    One needs this function to prepare initial states when using BK encoding.
    :param n: Number of qubits.
    :return:
    """

    # Initializing qubits:
    qc = qiskit.QuantumCircuit( n )

    # JW->BK encoder (lower-triangluar!) matrix
    bk_encoder = openfermion.transforms._encoder_bk( n )
    # print(bk_encoder)
    # Matrix multiplication mod 2 using CNOTs (starting from the bottom row)
    for i in reversed( range( 1, n ) ):
        for j in range( i  ):
            if bk_encoder[i, j] == 1:
                qc.cx(j,i)

    return qc

Not that it is crucial to perform the matrix multiplication using CNOTs starting from the bottom row, since the encoder matrix is lower-triangular.

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  • $\begingroup$ What do you mean by Matrix multiplication mod 2 using CNOTs ? $\endgroup$
    – kevin
    Aug 20, 2022 at 9:41
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    $\begingroup$ Every bitstring is a vector with components in $\{0,1\}$. Define their scalar product in a usual way. Calculate the result $\mod 2$. Similarly define the matrix product. The circuit above for a given input bitstring outputs this bitstring multiplied by the JW$\leftrightarrow$BK encoder matrix. $\endgroup$
    – mavzolej
    Aug 21, 2022 at 10:25
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    $\begingroup$ Such an implementation relies crucially on the encoder matrix being lower-triangular. If one calculates the resulting vector, starting from the last entry, then for $j$-th output component, the input components larger than $j$ are not used, and so they can store the final values, obtained on earlier steps. $\endgroup$
    – mavzolej
    Aug 21, 2022 at 10:39
  • $\begingroup$ I am actually confused as in how you come up with the logic inside the for loop inside the two functions respectively. $\endgroup$
    – kevin
    Aug 21, 2022 at 14:22
  • $\begingroup$ if I have an input QASM file already mapped in Jordan-Wigner, how would I replace the argument n inside jw_to_bk_circuit_qiskit(n) ? $\endgroup$
    – kevin
    Aug 21, 2022 at 14:37

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