[Paper Review] Ultrafast Hybrid Fermion-to-Qubit mapping

Oliver O'Brien, Sergii Strelchuk|arXiv (Cornell University)|Nov 29, 2022

Quantum Computing Algorithms and Architecture5 references4 citations

TL;DR

This paper introduces two novel families of fermion-to-qubit mappings—Hybrid and Hybrid+—that achieve unprecedented reductions in auxiliary qubit overhead while preserving locality. By combining the low connectivity demands of Jordan-Wigner with the locality advantages of Bravyi-Kitaev at different scales, the Hybrid mapping reduces gate counts by up to 98% compared to Jordan-Wigner and achieves a record-low 1.016 qubits per fermion, outperforming existing schemes on near-term quantum architectures.

ABSTRACT

Fermion-to-qubit mappings play a crucial role in representing fermionic interactions on a quantum computer. Efficient mappings translate fermionic modes of a system to qubit interactions with a high degree of locality while using few auxiliary resources. We present a family of locality-preserving fermion-to-qubit mappings that require fewer auxiliary qubits than all existing schemes known to date. One instance requires only 1.016 qubits-per-fermion compared to 1.25 for the best-known locality-preserving mapping by Y.-A. Chen and Y. Xu [PRX Quantum 4, 010326 (2023)]. Our family of mappings (parameterised by integer $n$) establishes a direct trade-off between the number of auxiliary qubits ($\frac{1}{n^2}$) and the circuit length ($O(\log n)$). Furthermore, we present a non-local variant that combines the strengths of the Jordan-Wigner and Bravyi-Kitaev mappings to give 98\% shorter circuits than the Jordan-Wigner mapping. This is achieved by applying seemly incompatible mappings at different scales, making it possible for their respective strengths to complement each other.

Motivation & Objective

• To develop fermion-to-qubit mappings that minimize auxiliary qubit overhead while preserving locality in quantum simulations.
• To overcome the trade-off between gate count and ancilla qubit usage in existing mappings by introducing a scalable hybrid framework.
• To enable efficient simulation of fermionic systems on near-term quantum computers with limited connectivity and qubit counts.
• To establish a direct, tunable trade-off between ancilla qubit count and circuit depth through a parameterized family of mappings.
• To explore the potential of combining seemingly incompatible mappings (Jordan-Wigner and Bravyi-Kitaev) at different scales to achieve superior performance.

Proposed method

• Proposes a parametrized Hybrid mapping family indexed by integer $ n $, where each $ n imes n $ cell uses Bravyi-Kitaev mapping internally and Jordan-Wigner between cells.
• Employs a hierarchical decomposition: local fermionic interactions are mapped via Bravyi-Kitaev within $ n imes n $ cells, while inter-cell interactions use Jordan-Wigner to reduce connectivity overhead.
• Derives circuit depth scaling as $ O(rac{N}{2n} + rac{n}{2}) $ under limited connectivity and $ O(rac{N}{2n^2} + rac{n}{2}) $ under all-to-all connectivity.
• Introduces the Hybrid+ variant that uses $ 1 + rac{1}{n^2} $ qubits per fermion mode, interpolating between AQM and Bravyi-Kitaev by embedding Bravyi-Kitaev within cells and eliminating non-local gates on roots.
• Analyzes performance across lattice sizes from $32\times32$ to $128\times128$, identifying optimal $ n $-values based on trade-offs between ancilla count and interaction qubit count.
• Validates results via numerical simulations on realistic quantum architectures, comparing gate counts and interaction qubit usage against Jordan-Wigner, Bravyi-Kitaev, and super-compact mappings.

Experimental results

Research questions

• RQ1 Can a hybrid mapping combining Jordan-Wigner and Bravyi-Kitaev mappings at different scales achieve superior circuit efficiency while reducing ancilla overhead?
• RQ2 What is the optimal trade-off between number of auxiliary qubits and circuit depth in locality-preserving fermion-to-qubit mappings?
• RQ3 How does the performance of the Hybrid mapping compare to existing mappings (e.g., Jordan-Wigner, Bravyi-Kitaev, super-compact) on near-term quantum hardware with limited connectivity?
• RQ4 Can a parametrized family of mappings be constructed that allows tunable control over ancilla count and circuit depth while maintaining local Pauli string structure?
• RQ5 Does the Hybrid+ mapping achieve a significant reduction in ancilla qubits compared to the best-known local mappings without sacrificing locality or increasing circuit depth excessively?

Key findings

• The Hybrid mapping reduces circuit depth by up to 98% compared to the Jordan-Wigner mapping by combining low-connectivity Jordan-Wigner between cells with high-locality Bravyi-Kitaev within cells.
• For lattices up to $164\times164$, the Hybrid mapping outperforms both Jordan-Wigner and Bravyi-Kitaev in terms of interaction qubit count, even under all-to-all connectivity.
• The Hybrid+ mapping achieves a qubit-to-fermion ratio of 1.0625 for $n=4$ and 1.0156 for $n=8$, significantly below the 1.25 ratio of the best-known local mapping (super-compact scheme).
• The Hybrid+ mapping reduces ancilla qubit count by 94% compared to the super-compact mapping for $n=8$, with ancilla scaling as $O(1/n^2)$.
• Numerical simulations show that the optimal cell size $n$ increases with lattice size, from $4\times4$ at $32\times32$ to $16\times16$ at $128\times128$, indicating scalability.
• The Hybrid mapping achieves $O(\log n)$ growth in average interaction qubit count under all-to-all connectivity, while maintaining $O(\frac{N}{2n^2})$ gate count scaling.