Parity and Other Fermionic Encodings
Beyond Jordan-Wigner and Bravyi-Kitaev, encodings that trade qubits, locality, and error resilience in different ways.
The encoding spectrum
Mapping fermions to qubits requires storing two kinds of information: the occupation of each orbital and the parity (running sum) needed to fix exchange signs. Different encodings distribute this information differently, and each choice sets operator locality and circuit cost.
The parity encoding
In the parity encoding, qubit p stores the cumulative parity of orbitals 0 through p, rather than the bare occupation. This makes parity checks local (a single qubit) but occupation readout non-local (differences of adjacent qubits). It is, in a sense, the mirror image of Jordan-Wigner: JW makes occupation local and parity non-local, parity encoding does the reverse. Bravyi-Kitaev interpolates between them.
Qubit-saving encodings
Symmetries can eliminate qubits. Because total particle number and spin are conserved, two qubits can often be removed by fixing those symmetry sectors (the qubit-tapering technique of Bravyi et al.). This is standard in VQE chemistry to shrink circuits without losing physics.
- Jordan-Wigner: local occupation, O(N) parity strings.
- Parity: local parity, non-local occupation.
- Bravyi-Kitaev: balanced, O(log N) locality.
- Superfast / Verstraete-Cirac: extra qubits for constant locality on sparse graphs.
- Qubit tapering: removes qubits by exploiting conserved symmetries.
Error-resilient encodings
Some encodings, notably the Bravyi-Kitaev superfast and related fermion-to-qubit codes, embed the Hamiltonian on a graph with redundancy that can detect certain errors, blurring the line between encoding and error correction. These are attractive for fault-tolerant chemistry where locality and error resilience both matter.
Choosing an encoding
There is no universal best encoding. The optimal choice depends on the Hamiltonian's connectivity, the hardware's native gates and qubit topology, and whether the goal is a shallow near-term circuit or a low-gate-count fault-tolerant one. Encoding selection is an optimization step in the simulation pipeline, often coupled with orbital ordering and swap networks.