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(...) and affect all downstream algorithms.

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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 installed PennyLane/qchem backend:

• does not support the mapping argument, or

• fails during Hamiltonian construction

the code will:

1. retry without the mapping argument

2. optionally retry using method="openfermion"

Implication:

Mapping selection is best-effort, and actual behaviour may depend on the installed backend. If mapping-specific construction is unsupported, the Hamiltonian falls back to the backend default (typically Jordan–Wigner).

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

• JW typically deepest

• BK / parity can be shallower

2. Measurement cost

• more non-local terms → more measurements

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

3. Optimization landscape

Mappings 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

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

• if results appear identical across mappings:

  • your backend may have fallen back to the default mapping

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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