Optimised Fermion-Qubit Encodings for Quantum Simulation with Reduced Transpiled Circuit Depth

TL;DR

This paper tackles the challenge of efficiently simulating fermionic systems on quantum hardware by optimizing fermion-qubit encodings without altering their underlying tree structure. It introduces TOPP-HATT, a deterministic, topology-preserving optimisation that reduces Pauli-weight and coefficient-weight across Majorana-string ternary-tree encodings and device-derived subgraphs. The method yields substantial qDRIFT circuit-depth reductions for water in STO-3G and demonstrates benefits across standard encodings, Hamiltonian-optimised trees, and device connectivity layouts. With public code and data, the approach offers a broadly applicable co-design tool to lower circuit depth and gate error in near-term quantum simulations.

Abstract

Simulation of fermionic Hamiltonians with gate-based quantum computers requires the selection of an encoding from fermionic operators to quantum gates, the most widely used being the Jordan-Wigner transform. Many alternative encodings exist, with quantum circuits and simulation results being sensitive to choice of encoding, device connectivity and Hamiltonian characteristics. Non-stochastic optimisation of the ternary tree class of encodings to date has targeted either the device or Hamiltonian. We develop a deterministic method which optimises ternary tree encodings without changing the underlying tree structure. This enables reduction in Pauli-weight without ancillae or additional swap-gate overhead. We demonstrate this method for a variety of encodings, including those which are derived from the qubit connectivity graph of a quantum computer. Across a suite of standard encoding methods applied to water in STO-3G basis, including Jordan-Wigner, our method reduces qDRIFT circuit depths on average by $27.7\%$ and $26.0\%$ for untranspiled and transpiled circuits respectively.

Figures (16)

• Figure 1: Graphical representation of Majorana-string encodings.
• Figure 2: Pairwise Non-trivial overlap.
• Figure 3: Ternary tree structures of a) Jiang-Kalev-Mruczkiewicz-Neven (JKMN) b)Jordan-Wigner (JW) c) Bravyi-Kitaev (BK) d) Parity (PE) encodings jiang_optimal_2020 for four modes. Nodes are shown in black and leaves in red. Each node is enumerated with a qubit index, while each leaf has an associated Majorana operator $\gamma_i$. Edges between nodes show the Pauli operator associated to the edge. By convention outward edges point downward and $\hat{X}$, $\hat{Y}$,$\hat{Z}$ are arranged as left, centre and right respectively.
• Figure 4: Enumerations of the four mode Jordan-Wigner encoding. a) The naive enumeration, in which fermionic mode index and qubit index are equal, and increasing with distance from the root node. b) An altered fermionic mode enumeration, in which the Majorana-operators assigned to fermionic modes $1$ and $2$ have been swapped. c) An altered qubit enumeration, in which the indices of qubits $1$ and $2$ have been swapped.
• Figure 5: Pauli-weight and coefficient-scaled Pauli-weight of 1000 random enumerations of fermionic modes for the water molecule in an STO-3G basis, generated using the 'ferrmion' software package.michael_williams_de_la_bastida_ucl-ccsferrmion_2025