observable(fermion_ops, init_term=0, mapping='jordan_wigner', wires=None)[source]

Builds the fermionic many-body observable whose expectation value can be measured in PennyLane.

The second-quantized operator of the fermionic many-body system can combine one-particle and two-particle operators as in the case of electronic Hamiltonians \(\hat{H}\):

\[\hat{H} = \sum_{\alpha, \beta} \langle \alpha \vert \hat{t}^{(1)} + \cdots + \hat{t}^{(n)} \vert \beta \rangle ~ \hat{c}_\alpha^\dagger \hat{c}_\beta + \frac{1}{2} \sum_{\alpha, \beta, \gamma, \delta} \langle \alpha, \beta \vert \hat{v}^{(1)} + \cdots + \hat{v}^{(n)} \vert \gamma, \delta \rangle ~ \hat{c}_\alpha^\dagger \hat{c}_\beta^\dagger \hat{c}_\gamma \hat{c}_\delta\]

In the latter equations the indices \(\alpha, \beta, \gamma, \delta\) run over the basis of single-particle states. The operators \(\hat{c}^\dagger\) and \(\hat{c}\) are the particle creation and annihilation operators, respectively. \(\langle \alpha \vert \hat{t} \vert \beta \rangle\) denotes the matrix element of the single-particle operator \(\hat{t}\) entering the observable. For example, in electronic structure calculations, this is the case for the kinetic energy operator, the nuclei Coulomb potential, or any other external fields included in the Hamiltonian. On the other hand, \(\langle \alpha, \beta \vert \hat{v} \vert \gamma, \delta \rangle\) denotes the matrix element of the two-particle operator \(\hat{v}\), for example, the Coulomb interaction between the electrons.

  • The observable is built by adding the operators \(\sum_{\alpha, \beta} t_{\alpha\beta}^{(i)} \hat{c}_\alpha^\dagger \hat{c}_\beta\) and \(\frac{1}{2} \sum_{\alpha, \beta, \gamma, \delta} v_{\alpha\beta\gamma\delta}^{(i)} \hat{c}_\alpha^\dagger \hat{c}_\beta^\dagger \hat{c}_\gamma \hat{c}_\delta\).

  • Second-quantized operators contributing to the many-body observable must be represented using the FermionOperator data structure as implemented in OpenFermion. See the functions one_particle() and two_particle() to build the FermionOperator representations of one-particle and two-particle operators.

  • The function uses tools of OpenFermion to map the resulting fermionic Hamiltonian to the basis of Pauli matrices via the Jordan-Wigner or Bravyi-Kitaev transformation. Finally, the qubit operator is converted to a PennyLane observable by the function convert_observable().

  • fermion_ops (list[FermionOperator]) – list containing the FermionOperator data structures representing the one-particle and/or two-particle operators entering the many-body observable

  • init_term (float) – Any quantity required to initialize the many-body observable. For example, this can be used to pass the nuclear-nuclear repulsion energy \(V_{nn}\) which is typically included in the electronic Hamiltonian of molecules.

  • mapping (str) – Specifies the fermion-to-qubit mapping. Input values can be 'jordan_wigner' or 'bravyi_kitaev'.

  • wires (Wires, list, tuple, dict) – Custom wire mapping used to convert the qubit operator to an observable measurable in a PennyLane ansatz. For types Wires/list/tuple, each item in the iterable represents a wire label corresponding to the qubit number equal to its index. For type dict, only int-keyed dict (for qubit-to-wire conversion) is accepted. If None, will use identity map (e.g. 0->0, 1->1, …).


the fermionic-to-qubit transformed observable

Return type



>>> t = FermionOperator("0^ 0", 0.5) + FermionOperator("1^ 1", 0.25)
>>> v = FermionOperator("1^ 0^ 0 1", -0.15) + FermionOperator("2^ 0^ 2 0", 0.3)
>>> observable([t, v], mapping="jordan_wigner")
    0.2625 * I(0)
  + -0.1375 * Z(0)
  + -0.0875 * Z(1)
  + -0.0375 * (Z(0) @ Z(1))
  + 0.075 * Z(2)
  + -0.075 * (Z(0) @ Z(2))