Mappings

Fermion-to-Qubit Mappings

Scope

This document describes the fermion-to-qubit mappings supported in this repository, their conceptual differences, and their practical impact on VQE, QPE, and related workflows.

Mappings are applied during Hamiltonian construction via common.hamiltonian.build_hamiltonian(...). Hartree–Fock reference bits use the same encoding. UCC and ADAPT circuits apply their excitation gates in the occupation basis and then encode the prepared state with a reversible CNOT network. Consequently, corresponding noiseless UCC circuits represent the same physical state and energy across mappings. Generic hardware-efficient ansatzes operate directly on the chosen qubit register; they do not necessarily represent equivalent physical states across mappings.

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Overview

Electronic-structure Hamiltonians are naturally expressed in terms of fermionic creation and annihilation operators. Quantum circuits, however, operate on qubits.

A fermion-to-qubit mapping transforms the fermionic Hamiltonian into an equivalent qubit operator:

\[ H_{\text{fermionic}} \;\longrightarrow\; H_{\text{qubit}}. \]

This transformation preserves the spectrum but changes the operator structure, which directly impacts:

• Pauli string length and locality

• circuit depth

• measurement cost

• optimization behaviour

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Supported Mappings

This repository supports the following mappings (via PennyLane-qchem):

• jordan_wigner (default)

• bravyi_kitaev

• parity

Mapping selection is exposed through:

• CLI: --mapping

• Python API: mapping="..."

Example:

vqe -m H2 --mapping bravyi_kitaev

from common.hamiltonian import build_hamiltonian

H, n_qubits, hf = build_hamiltonian("H2", mapping="parity")

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Jordan–Wigner (JW)

Idea

Maps fermionic modes directly to qubits using occupation-number encoding.

Each fermionic operator is represented using strings of Pauli operators with Z-chains enforcing fermionic anti-commutation.

Structure

Creation operator (schematically):

\[ a^\dagger_j ;\sim; X_j \prod_{k<j} Z_k \]

Properties

• simple and explicit mapping

• direct correspondence between orbitals and qubits

• produces long Pauli strings (length grows with system size)

Practical impact

• deeper circuits due to long operator strings

• higher measurement cost

• very stable and widely used baseline

Usage in this repository

• default mapping

• used as the baseline for comparisons

• always available via PennyLane backends

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Bravyi–Kitaev (BK)

Idea

Encodes both occupation and parity information in a balanced way using a binary tree structure.

Properties

• reduces average Pauli string length relative to JW

• logarithmic scaling of parity information

• more complex encoding than JW

Practical impact

• shorter Pauli strings on average

• can reduce circuit depth

• may improve optimization stability in some cases

Usage in this repository

• available via mapping="bravyi_kitaev"

• useful for comparing operator locality effects

• backend support may vary depending on PennyLane/qchem version

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Parity Mapping

Idea

Encodes cumulative parity rather than direct occupation.

Closely related to Jordan–Wigner but reorganizes information to expose symmetry structure.

Properties

• parity-based encoding

• can lead to more structured Hamiltonians

• often used in conjunction with symmetry reduction (not applied here)

Practical impact

• may reduce effective operator complexity

• can change optimization landscape

• behaviour depends strongly on molecule and ansatz

Usage in this repository

• available via mapping="parity"

• no explicit symmetry tapering is applied in this codebase

• primarily used for comparative studies

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Implementation Details (Repository-Specific)

Mapping entrypoint

Mappings are applied in:

common.hamiltonian.build_molecular_hamiltonian(...)

via:

qml.qchem.molecular_hamiltonian(..., mapping=mapping)

Default behaviour

• default mapping: jordan_wigner

• mapping string is normalized to lowercase

• passed through to PennyLane-qchem

Backend fallback behaviour

If the primary backend fails, the optional OpenFermion retry preserves the requested mapping. Unsupported mappings raise an error; the package never silently substitutes Jordan–Wigner. This keeps the Hamiltonian, reference state, and chemistry circuit in the same encoding.

No symmetry reduction

This repository:

• does not perform qubit tapering

• does not exploit Z₂ symmetries

Therefore:

• qubit counts are determined directly by the mapping + basis

• differences between mappings are limited to operator structure, not qubit reduction

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Effect on VQE and Other Algorithms

Mappings influence several aspects of algorithm performance:

1. Circuit depth

• longer Pauli strings → deeper circuits

• UCC circuits add an encoding CNOT network for BK / parity

• Hamiltonian measurement locality and circuit preparation costs are distinct

2. Measurement cost

• more non-local terms → more measurements

• affects shot-based simulations (less critical in analytic mode)

3. Optimization landscape

For untransformed hardware-efficient ansatzes, changing the Hamiltonian encoding can change:

• gradient magnitudes

• curvature of the loss surface

• presence of flat regions or oscillations

This can affect:

• convergence speed

• optimizer sensitivity

• stability of excited-state methods

For consistently encoded UCC circuits, the noiseless energy landscape is equivalent across mappings, up to numerical precision. Noise channels act on encoded qubits and need not preserve this equivalence.

4. Excited-state methods

For:

• QSE / EOM-QSE

• LR-VQE / EOM-VQE

mapping affects:

• operator matrices

• conditioning of overlap matrices

• numerical stability of eigenvalue problems

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Summary

Mapping	Locality	Pauli string length	Complexity	Typical role
Jordan–Wigner	Low	Long	Simple	Default baseline
Bravyi–Kitaev	Medium	Medium	Moderate	Improved locality
Parity	Medium	Medium	Moderate	Structural comparisons

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Practical Guidance

• use Jordan–Wigner for:

  • baseline experiments

  • debugging

  • reproducibility across environments

• try Bravyi–Kitaev for:

  • reducing operator length

  • improving circuit structure

• try Parity for:

  • comparative studies

  • exploring different optimization behaviour

• identical noiseless UCC energies across mappings are expected for equivalent physical circuits; compare operator structure and encoded states as well

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References

• Jordan and Wigner, Über das Paulische Äquivalenzverbot

• Bravyi and Kitaev, Fermionic Quantum Computation

• Seeley et al., The Bravyi–Kitaev transformation for quantum computation of electronic structure