Working with fermionic circuits

Important

The concepts in this guide are currently available only in the Python API. Equivalent functionality will be made available through the C API in a future release.

This guide explains how to use the FermionicCircuit to implement quantum algorithms directly in fermionic space, rather than mapping to qubits first.

Why use fermionic circuits

Consider the traditional workflow for implementing the time evolution of a fermionic Hamiltonian on a qubit-based architecture:

  1. Define the fermionic Hamiltonian (for example, \(H = \sum_{pq} h_{pq} c^\dagger_p c_q\)).

  2. Map it to a qubit operator (for example, by using the Jordan-Wigner mapping).

  3. Implement the time evolution in qubit space (using Trotterization or other methods).

This approach has a fundamental limitation: early mapping to qubits discards problem-aware knowledge about the fermionic structure. Standard optimization passes cannot exploit fermionic symmetries or allowed term reorderings, resulting in suboptimal circuits with unnecessary depths and gate counts.

The FermionicCircuit enables a better workflow:

  1. Define the fermionic Hamiltonian (for example, \(H = \sum_{pq} h_{pq} c^\dagger_p c_q\)).

  2. Implement the time evolution directly in fermionic space using fermionic gates.

  3. Map to qubit space as part of the transpilation process.

This approach addresses the limitation by keeping problem-aware knowledge in fermionic space, where optimization passes can exploit fermionic structure and commutation relations to reduce circuit depth. The fermionic circuit describes what computation to perform, while transpilation handles how to map it to qubits, providing a cleaner separation of concerns.

Generic mode indexing

Both the operators module and the FermionicCircuit use generic mode-based indexing that makes no assumptions about the nature of the modes. A mode is simply an abstract index labeling a fermionic degree of freedom, with no inherent semantics. This design choice mirrors the operators module and ensures maximum flexibility. For example, you can implement time evolution for any operator implementing the OperatorTrait protocol, regardless of its mathematical representation.

Build a fermionic circuit

The example below implements the time evolution discussed in the previous section. It constructs a FermionicCircuit by specifying the number of fermionic modes, then adds fermionic gates from the qiskit_fermions.circuit.library to implement the time evolution.

The example also demonstrates how to incorporate domain knowledge into an operator by using operator term grouping. By assigning group indices to the Hamiltonian terms, structural information gets preserved that optimization passes can exploit throughout the transpilation stack.

>>> from qiskit_fermions.circuit import FermionicCircuit
>>> from qiskit_fermions.circuit.library import Evolution
>>> from qiskit_fermions.operators import FermionOperator, cre, ann
>>>
>>> # Create a circuit with 4 fermionic modes
>>> circuit = FermionicCircuit(4)
>>>
>>> # Define a simple fermionic Hamiltonian
>>> hamiltonian = FermionOperator.from_terms([
...     ([cre(0), ann(2)], 0.5),
...     ([cre(2), ann(0)], 0.5),
...     ([cre(1), ann(3)], 0.5),
...     ([cre(3), ann(1)], 0.5),
... ])
>>> # Add some problem structure by grouping our Hamiltonian terms
>>> hamiltonian.groups = [0, 0, 1, 1]
>>>
>>> # Add an evolution gate to implement exp(-i * t * H)
>>> evolution = Evolution(4, hamiltonian, time=1.0)
>>> circuit.append(evolution, circuit.register)

// The C API for FermionicCircuit is not available yet.

(png, hires.png, pdf)

A simple FermionicCircuit with a single time evolution gate.

(png, hires.png, pdf)

A decomposed FermionicCircuit with several time evolution gates.

Notice how the operator term grouping is preserved even in simple operations like decomposition. This demonstrates how structural information flows through the circuit stack. To understand the full transpilation to qubits, refer to the Transpiling fermionic circuits guide.

Transpile fermionic circuits

To implement the quantum algorithm represented by your FermionicCircuit it must be transpiled to a QuantumCircuit. You can learn how to do this in the Transpile fermionic circuits guide.

Next steps