Fermion To Qubit Mappings Jordan Wigner

Short answer: To simulate electrons or other fermions on a quantum computer, their states and operations must first be mapped onto qubits. The two most commonly used schemes are the Jordan-Wigner and Bravyi-Kitaev transformations. Jordan-Wigner is the simplest but produces long strings of Pauli operators, and research comparing the schemes finds that Bravyi-Kitaev is typically at least about as efficient and often needs fewer gates. The choice affects resource estimates for simulations, not whether the simulation is correct.

Why a mapping is needed

A 2018 paper in the Journal of Chemical Theory and Computation, also on arXiv (1812.02233), notes that classically intractable electronic structure calculations are often cited as a principal application of quantum computing. It says most techniques need a scheme for mapping electronic states and operations onto states of and operations upon qubits. The two most commonly used are the Jordan-Wigner and Bravyi-Kitaev transformations, though other constructions have been examined. The paper describes the usual workflow: a mapping, then a Trotter-Suzuki approximation to form an evolution operator implementable on the device, then phase estimation to find the ground state energy of the molecule.

The cost of Jordan-Wigner

A 2017 Physical Review A paper on fermionic lattice models says the simplest method, the Jordan-Wigner transformation, yields strings of Pauli operators acting on an extensive number of qubits. This overhead can hinder implementation on qubit-based quantum simulators, especially in the analog context. The authors review and analyze alternatives, including two approaches by Bravyi and Kitaev and the auxiliary fermion transformation. They reformulate the Bravyi-Kitaev transform in terms of a classical data structure and generalize it for a further locality improvement for local fermionic models on a rectangular lattice. Their conclusions are that the most compact encoding of fermionic operators uses ancilla qubits with the auxiliary fermion scheme, and that without ancillas a variant of the Bravyi-Kitaev transform gives the most compact mapping for Hubbard-like models.

Comparing across molecules

The 2018 paper observes that earlier comparisons of Jordan-Wigner and Bravyi-Kitaev had been limited to individual small molecules. The authors discuss the resource implications of Bravyi-Kitaev, specifically the number of quantum gates required, for small systems that might be simulatable on near-future devices and for systems large enough that classical simulation is intractable. Using 86 molecular systems, they report that Bravyi-Kitaev is typically at least approximately as efficient as Jordan-Wigner and gives substantially reduced gate count estimates when limited circuit optimizations are applied. These are gate-count estimates, not results from running the circuits.

What this does not tell you

The papers compare mappings by operator locality and estimated gate counts. They do not show that either mapping lets a quantum computer outperform classical methods for chemistry, and they say the quantum benefit depends on algorithms and hardware beyond the mapping. The best choice may differ by model, as the lattice-model paper shows.

FAQ

What is a fermion-to-qubit mapping?

A scheme that encodes fermionic states and operations as qubit states and operations.

Which mappings are most common?

Jordan-Wigner and Bravyi-Kitaev, per the 2018 paper.

What is the downside of Jordan-Wigner?

It produces Pauli strings acting on an extensive number of qubits.

Is Bravyi-Kitaev always better?

Across 86 molecules it was typically at least approximately as efficient and often needed fewer gates after limited optimization. For lattice models, other schemes were more compact in some cases.

Do these mappings give a quantum advantage?

The papers do not claim that. They compare resource costs.