Fermion-to-qubit mappings are used to represent fermionic modes on quantum computers, an essential first step in many quantum algorithms for electronic structure calculations.In this work, we present a formalism to design flexible fermion-to-qubit mappings from ternary trees.We intuitively discuss the connection between the structure of the generating trees and certain properties of the resulting mapping, such as the Pauli weight and the delocalization of mode occupation.Moreover, we introduce a recipe that guarantees that Fock basis states are mapped to computational basis states in qubit space, a desirable property for many applications in quantum computing.Based on this formalism, we introduce the Bonsai algorithm, which takes as input the potentially limited topology of the qubit connectivity of a quantum device and returns a tailored fermion-to-qubit mapping that reduces the SWAP overhead compared to other paradigmatic mappings.We illustrate the algorithm by producing mappings for the heavy-hexagon topology widely used in IBM quantum computers.The resulting mappings have a favorable Pauli-weight scaling O( √ N ) on this connectivity while ensuring that no SWAP gates are necessary for single-excitation operations.
Fermion-to-qubit mappings are used to represent fermionic modes on quantum computers, an essential first step in many quantum algorithms for electronic structure calculations.In this work, we present a formalism to design flexible fermion-to-qubit mappings from ternary trees.We intuitively discuss the connection between the structure of the generating trees and certain properties of the resulting mapping, such as the Pauli weight and the delocalization of mode occupation.Moreover, we introduce a recipe that guarantees that Fock basis states are mapped to computational basis states in qubit space, a desirable property for many applications in quantum computing.Based on this formalism, we introduce the Bonsai algorithm, which takes as input the potentially limited topology of the qubit connectivity of a quantum device and returns a tailored fermion-to-qubit mapping that reduces the SWAP overhead compared to other paradigmatic mappings.We illustrate the algorithm by producing mappings for the heavy-hexagon topology widely used in IBM quantum computers.The resulting mappings have a favorable Pauli-weight scaling O( √ N ) on this connectivity while ensuring that no SWAP gates are necessary for single-excitation operations.
Fermion-to-qubit mappings are used to represent fermionic modes on quantum computers, an essential first step in many quantum algorithms for electronic structure calculations. In this work, we present a formalism to design flexible fermion-to-qubit mappings from ternary trees. We discuss in an intuitive manner the connection between the generating trees' structure and certain properties of the resulting mapping, such as Pauli weight and the delocalisation of mode occupation. Moreover, we introduce a recipe that guarantees Fock basis states are mapped to computational basis states in qubit space, a desirable property for many applications in quantum computing. Based on this formalism, we introduce the Bonsai algorithm, which takes as input the potentially limited topology of the qubit connectivity of a quantum device and returns a tailored fermion-to-qubit mapping that reduces the SWAP overhead with respect to other paradigmatic mappings. We illustrate the algorithm by producing mappings for the heavy-hexagon topology widely used in IBM quantum computers. The resulting mappings have a favourable Pauli weight scaling O(N) on this connectivity, while ensuring that no SWAP gates are necessary for single excitation operations.
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.
Quantum simulation of strongly correlated fermionic systems is among the most promising near- term applications of quantum computing, but its practical efficiency depends critically on the choice of fermion-to-qubit mapping and on the connectivity of the underlying hardware. In this work we address this problem in the context of the two-dimensional Hubbard model, simulated on IBM superconducting quantum processors with heavy-hexagon connectivity. We numerically benchmark the Jordan-Wigner, Bravyi-Kitaev, and Bonsai transformations, evaluating their Pauli weight and SWAP overhead across seven heavy-hexagon chips of increasing size. We show that, while the Bravyi-Kitaev mapping initially exhibits a lower Pauli weight, this advantage is eliminated once routing costs are taken into account, confirming the Bonsai mapping as the most hardware-efficient baseline transformation for this architecture. We then use the Bonsai mapping to construct the qubit Hamiltonian of the 2D spinful Fermi-Hubbard model, introducing a simulated annealing algorithm that optimizes the assignment of Majorana strings to lattice sites, reducing the cost function by nearly 50% percent. Finally, we simulate on quantum hardware the time evolution of fermionic states up to 6x6 lattices, confirming the viability of the Bonsai encoding for hardware-aware large-scale two-dimensional simulations.
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.
Mapping fermionic systems to qubits on a quantum computer is often the first step for algorithms in quantum chemistry and condensed matter physics. However, it is difficult to reconcile the many different approaches that have been proposed, such as those based on binary matrices, ternary trees, and stabilizer codes. This challenge is further exacerbated by the many ways to describe them -- transformation of Majorana operators, action on Fock states, encoder circuits, and stabilizers of local encodings -- making it challenging to know when the mappings are equivalent. In this work, we present a graphical framework for fermion-to-qubit mappings that streamlines and unifies various representations through the ZX-calculus. To start, we present the correspondence between linear encodings of the Fock basis and phase-free ZX-diagrams. The commutation rules of scalable ZX-calculus allows us to convert the fermionic operators to Pauli operators under any linear encoding. Next, we give a translation from ternary tree mappings to scalable ZX-diagrams, which not only directly represents the encoder map as a CNOT circuit, but also retains the same structure as the tree. Consequently, we graphically prove that ternary tree transformations are equivalent to linear encodings, a recent result by Chiew et al. The scalable ZX representation moreover enables us to construct an algorithm to directly compute the binary matrix for any ternary tree mapping. Lastly, we present the graphical representation of local fermion-to-qubit encodings. Its encoder ZX-diagram has the same connectivity as the interaction graph of the fermionic Hamiltonian and also allows us to easily identify stabilizers of the encoding.
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.