Quantum computers hold great promise for efficiently simulating Fermionic systems, benefiting fields like quantum chemistry and materials science. To achieve this, algorithms typically begin by choosing a Fermion-to-qubit mapping to encode the Fermionic problem in the qubits of a quantum computer. In this work, we introduce ‘Treespilation,’ a technique for efficiently mapping Fermionic systems using a large family of favourable tree-based mappings while minimising a generic cost function to reduce quantum simulation overhead. We use this technique to minimise the number of CNOT gates required to simulate approximate chemical groundstate circuits and observe significant reductions, up to 74%, in CNOT counts on full connectivity. For devices with limited qubit connectivity, we observe similar reductions in CNOT counts, often surpassing the full connectivity CNOT count for circuits encoded with the Jordan-Wigner mapping. We observed similar reductions when applied to reducing the Pauli weight of Hubbard model Hamiltonians.
Abstract Using a quantum computer to simulate fermionic systems requires fermion-to-qubit transformations. Usually, lower Pauli weight of transformations means shallower quantum circuits. Therefore, most existing transformations aim for lower Pauli weight. However, in some cases, the circuit depth depends not only on the Pauli weight but also on the coefficients of the Hamiltonian terms. In order to characterize the circuit depth of these algorithms, we propose a new metric called weighted Pauli weight, which depends on Pauli weight and coefficients of Hamiltonian terms. To achieve smaller weighted Pauli weight, we introduce a novel transformation, Huffman-code-based ternary tree (HTT) transformation, which is built upon the classical Huffman code and tailored to different Hamiltonians. We tested various molecular Hamiltonians and the results show that the weighted Pauli weight of the HTT transformation is smaller than that of commonly used mappings. At the same time, the HTT transformation also maintains a relatively small Pauli weight. The mapping we designed reduces the circuit depth of certain Hamiltonian simulation algorithms, facilitating faster simulation of fermionic systems.
This paper introduces the Hamiltonian-Adaptive Ternary Tree (HATT) framework to compile optimized Fermion-to-qubit mapping for specific Fermionic Hamiltonians. In the simulation of Fermionic quantum systems, efficient Fermion-toqubit mapping plays a critical role in transforming the Fermionic system into a qubit system. HATT utilizes ternary tree mapping and a bottom-up construction procedure to generate Hamiltonian aware Fermion-to-qubit mapping to reduce the Pauli weight of the qubit Hamiltonian, resulting in lower quantum simulation circuit overhead. Additionally, our optimizations retain the important vacuum state preservation property in our Fermion-toqubit mapping and reduce the complexity of our algorithm from
O(N4) to
O(N3). Evaluations on various Fermionic systems demonstrate
5∼25% reduction in Pauli weight, gate count, and circuit depth, alongside excellent scalability to larger systems. Experiments on the Ionq device also show the advantages of HATT in noise resistance in quantum simulations.
Routing is the task of permuting qubits in such a way that quantum operations can be parallelized maximally, given constraints on the hardware geometry. When simulating fermions in the Jordan-Wigner encoding with qubits, a one-dimensional nearest-neighbor-connected geometry is effectively imposed on the system, independently of the underlying hardware, which means that naively, an
O(N) depth routing overhead is incurred. Recently, Maskara et al. [arXiv:2509.08898] demonstrated that this routing overhead can be reduced to
O(logN) by decomposing general fermion routing into
O(logN) interleave permutations of depth
O(1), using
Θ(N) ancillary qubits and employing measurements and feedforward. Here, we exhibit an alternative construction that achieves the same asymptotic performance. We also generalize the result in two ways. Firstly, we show that fermion routing can be performed in depth
O(log2N) \emph{without} ancillas, measurements, or feedforward. Secondly, we construct efficient mappings with
O(log2N) depth between all product-preserving ternary tree fermionic encodings, thereby showing that fermion routing in any such encoding can be done efficiently. While these results assume all-to-all connectivity, they also imply upper bounds for fermion routing in devices with limited connectivity by multiplying the fermion routing depth by the worst-case qubit routing depth.
Efficient simulation of interacting fermionic systems is a key application of near-term quantum computers, but is hindered by the overhead required to encode fermionic operators on qubit hardware. Here, we consider models with
N fermionic modes in which each participates in at most a constant number
d of interactions and study the circuit depth required to implement the Trotterized time evolution on qubit hardware with all-to-all connectivity. We introduce an encoding that augments each physical fermionic mode with a small number of auxiliary fermions, enabling the removal of Jordan--Wigner strings. Although the preparation of the auxiliary fermion state incurs an initial overhead, this state remains invariant under time evolution. As a result, long-time evolution can be implemented with asymptotically optimal circuit depth, reducing a previously multiplicative
O(logN) overhead to an additive overhead. Our results thus establish that the simulation of sparse fermionic models on qubit hardware matches the performance achievable on ideal fermionic hardware up to constant factors and
O(dN) ancillary qubits.
A compelling application of quantum computers with thousands of qubits is quantum simulation. Simulating fermionic systems is both a problem with clear real-world applications and a computationally challenging task. In order to simulate a system of fermions on a quantum computer, one has to first map the fermionic Hamiltonian to a qubit Hamiltonian. The most popular such mapping is the Jordan-Wigner encoding, which suffers from inefficiencies caused by the high weight of some encoded operators. As a result, alternative local encodings have been proposed that solve this problem at the expense of a constant factor increase in the number of qubits required. Some such encodings possess local stabilizers, i.e., Pauli operators that act as the logical identity on the encoded fermionic modes. A natural error mitigation approach in these cases is to measure the stabilizers and discard any run where a measurement returns a -1 outcome. Using a high-performance stabilizer simulator, we classically simulate the performance of a local encoding known as the Derby-Klassen encoding and compare its performance with the Jordan-Wigner encoding and the ternary tree encoding. Our simulations use more complex error models and significantly larger system sizes (up to
18×18) than in previous work. We find that the high sampling requirements of postselection methods with the Derby-Klassen encoding pose a limitation to its applicability in near-term devices and call for more encoding-specific circuit optimizations.