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Investigating Interacting Fermionic Models with Locality-Preserving Qubit Encodings

This paper demonstrates that combining the locality-preserving Derby-Klassen fermion-to-qubit mapping with a constraint-incorporating Hamiltonian Variational Ansatz enables accurate and resource-efficient quantum simulation of two-dimensional interacting fermionic systems, offering significant advantages over Jordan-Wigner transformations by reducing operator nonlocality and circuit overhead in higher dimensions.

Original authors: Ashutosh P. Tripathi, Debasish Banerjee, Sandip Maiti, Nilmani Mathur

Published 2026-09-16

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ashutosh P. Tripathi, Debasish Banerjee, Sandip Maiti, Nilmani Mathur

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of atoms and electrons, particles called fermions obey strict rules of behavior that make them notoriously difficult to simulate on ordinary computers. When these particles interact strongly with one another, as they do in many materials that define our modern technology, the mathematical complexity of tracking them grows so rapidly that even the most powerful supercomputers eventually run out of steam. This is because the number of possible arrangements for these particles increases exponentially with every new particle added to the system. To solve this, scientists are turning to quantum computers, machines designed to mimic the behavior of nature itself. However, to make a quantum computer understand these fermions, researchers must first translate the language of particles into the language of qubits, the basic units of information in a quantum machine. This translation is not a simple one-to-one switch; it requires careful mapping to ensure that the local interactions between particles—where neighbors affect neighbors—remain local in the quantum computer. If the mapping is poor, a simple interaction between two nearby particles can become a tangled web of connections stretching across the entire machine, making the calculation slow and prone to errors.

A team of researchers at the Tata Institute of Fundamental Research in India, along with colleagues in the United Kingdom and China, has developed a new way to perform this translation for two-dimensional grids of interacting particles. They focused on a specific method called the Derby-Klassen mapping, which offers a clever trade-off: it keeps the interactions between particles simple and local, but it requires the quantum computer to hold onto a few extra, temporary pieces of information to keep the system honest. Think of it like a large group of people trying to coordinate a complex handshake; instead of everyone reaching across the room to find their partner, they use a set of local rules and a few designated observers to ensure everyone is in the right place. The researchers combined this mapping with a specialized algorithm designed to find the lowest energy states of these systems, a task that is crucial for understanding how materials conduct electricity or become magnetic. By building their quantum circuit to respect the specific rules of this new mapping, they were able to simulate the behavior of fermions on a grid with high accuracy, using fewer computational resources than traditional methods would require for larger systems.

The core of their work involved creating a digital model of a grid where particles can hop from one spot to another and push or pull on their neighbors. In the traditional approach, known as the Jordan-Wigner transformation, the rules of the game force the quantum computer to carry a long "string" of information across the entire grid every time a particle moves. As the grid gets bigger, these strings get longer, and the number of steps required to perform a calculation grows rapidly, quickly overwhelming the machine. The Derby-Klassen method avoids this by placing extra qubits on the faces of the grid squares, effectively creating a buffer zone. This allows the movement of a particle to be described by a short, local operation involving only a few qubits, rather than a long chain. However, this convenience comes with a condition: the quantum computer must stay within a specific "physical" zone, avoiding states that do not correspond to real particles. The researchers solved this by preparing the initial state of the computer using a specific sequence of operations that locks the system into the correct zone, and then ensuring that every step of their calculation respects these boundaries.

To test their approach, the team simulated a grid of particles on a computer, comparing their results against exact mathematical solutions that are known for small systems. They looked at a model where particles interact with their nearest neighbors, a scenario that mimics real-world materials. Their simulations showed that their new method could accurately reproduce the energy levels and physical properties of the system, matching the known solutions almost perfectly. They also demonstrated that their method could distinguish between different states of the system, such as those with slightly different numbers of particles, which is essential for studying how materials change under different conditions. When they compared their method to the traditional one, they found that while both could solve the problem, their approach was significantly faster and required far fewer computational steps, especially as the size of the grid increased. For a grid of sixteen sites, their method completed the calculation in just a few minutes, whereas the traditional method took over twenty minutes, and the gap widens as the system grows larger.

The researchers also explored how this method would work for more complex materials where particles have an internal property called spin, which can be thought of as a tiny magnetic orientation. They tried two different ways of arranging these spinning particles on the grid. In one arrangement, they kept the particles with opposite spins far apart, which kept the movement rules simple but made the interaction between them long and complicated. In the other, they placed opposite spins right next to each other, which made the interactions simple but complicated the movement rules. They found that both arrangements worked, but each had its own strengths and weaknesses, suggesting that the best way to set up a quantum simulation depends on the specific details of the material being studied. This flexibility is a key advantage, as it allows scientists to tailor the simulation to the hardware they have available.

The study concludes that this new framework provides a practical and efficient path forward for simulating complex materials on future quantum computers. By trading a small number of extra qubits for a massive reduction in the complexity of the interactions, the researchers have shown that it is possible to keep quantum simulations manageable even as the systems they study grow larger. Their work does not just offer a theoretical improvement; it provides a concrete recipe for building quantum circuits that are less likely to fail due to errors and less demanding on the limited resources of current quantum machines. As quantum computers continue to evolve, methods like this one will be essential for unlocking the secrets of high-temperature superconductors, exotic magnetic materials, and other phenomena that remain out of reach for classical computers. The ability to simulate these systems accurately could eventually lead to the design of new materials with revolutionary properties, transforming everything from energy storage to electronics.

Technical Summary: Investigating Interacting Fermionic Models with Locality-Preserving Qubit Encodings

Problem Statement
The numerical simulation of strongly correlated fermionic systems on classical computers is hindered by the exponential growth of the many-body Hilbert space and the fermionic sign problem, particularly in higher dimensions, at finite density, or in real time. Quantum computers offer a natural framework for such simulations, but efficient implementation requires encoding fermionic degrees of freedom into qubits while preserving the locality of the original Hamiltonian.

The standard Jordan-Wigner (JW) transformation, while exact, maps geometrically local interactions in two or more dimensions to non-local qubit operators. Specifically, local hopping terms acquire Pauli strings whose weight (the number of non-identity operators) grows with the system size, leading to substantial circuit depth and gate count overheads. While strategies like fermionic SWAP networks or alternative mappings (e.g., Bravyi-Kitaev) offer tradeoffs, they do not eliminate the inherent non-locality of the encoding in higher dimensions. Locality-preserving mappings, such as the Derby-Klassen (DK) encoding, offer a solution by transferring non-locality from operators to the state space, representing the physical fermionic Hilbert space as a constrained subspace of an enlarged qubit Hilbert space. However, the practical utility of this approach within variational frameworks, particularly regarding the tradeoff between reduced operator weight and the overhead of preparing and maintaining the physical code space, requires further investigation.

Methodology
This work investigates the utility of the DK fermion-to-qubit mapping within a Variational Quantum Eigensolver (VQE) framework for two-dimensional interacting fermionic models (specifically the spinless $t$ - $V$ model and the spinful Fermi-Hubbard model).

DK Mapping and Constraints: The authors utilize the DK mapping, which associates qubits with fermionic sites and a subset of lattice faces (plaquettes). This mapping ensures that nearest-neighbor hopping operators are represented by Pauli operators of weight at most three, and density-density interactions by weight-two operators. The physical fermionic states are defined as the simultaneous $+1$ eigenspace of a set of commuting Pauli stabilizers (loop constraints).
Hamiltonian Variational Ansatz (HVA): Instead of using generic hardware-efficient ansätze (HEA), which may explore unphysical sectors of the enlarged Hilbert space, the authors construct an HVA. The generators of the HVA are derived directly from the terms of the encoded Hamiltonian. Crucially, because the DK-encoded Hamiltonian generators commute with the stabilizer constraints, an initial state prepared in the physical code space remains within that space throughout the variational optimization.
State Preparation: A systematic procedure using Clifford circuits is employed to prepare the initial state in the physical stabilizer sector. This involves diagonalizing the stabilizer generators to map the $+1$ eigenspace to a computational basis state (e.g., $|0\rangle$ ), followed by the inverse transformation.
Symmetry Preservation: The ansatz inherently conserves particle number. This allows the authors to target specific particle-number sectors independently, facilitating the resolution of degenerate low-lying states without mixing physically distinct sectors.
Algorithms: The ground state is found using VQE, while excited states are accessed using Variational Quantum Deflation (VQD) within fixed particle-number sectors.
Simulation: Statevector simulations were performed using the CUDA-Q framework on lattices of sizes $2\times2$ , $3\times3$ , $3\times4$ , and $4\times4$ . Results were benchmarked against exact diagonalization (ED) using the QuSpin package.

Key Contributions and Results

Integration of Constraints and Variational Optimization: The paper demonstrates a practical framework combining locality-preserving mappings with constraint-preserving Clifford state preparation and symmetry-preserving variational circuits. It shows that the DK stabilizer constraints can be effectively incorporated into the initial state preparation, and the HVA preserves these constraints during optimization.
Benchmarking against Exact Diagonalization: For the spinless $t$ - $V$ model at zero chemical potential, the DK-HVA framework accurately reproduces the low-energy spectrum and physical observables (staggered occupation and susceptibility) obtained from ED. The use of particle-number conservation allows for the independent targeting of degenerate ground states (e.g., in the $N_p=4$ and $N_p=5$ sectors for a $3\times3$ lattice).
Finite Chemical Potential and Comparison with JW: At finite chemical potential ( $\mu=0.5$ $μ = 0.5$ ), where particle-hole symmetry is broken, the DK-HVA is benchmarked against JW-based circuits (both HVA and HEA).
- Performance: The DK-HVA achieves variational energies and fidelities comparable to the JW-HVA.
- Resource Efficiency: The DK mapping demonstrates a significant reduction in logical circuit resources. For a $4\times4$ lattice, the DK-HVA required approximately 12 minutes of optimization time compared to over 200 minutes for the JW-HVA and over 1000 minutes for the JW-HEA.
- Circuit Depth: The DK mapping maintains a lower CNOT count and circuit depth compared to JW, as the latter generates long Pauli strings that scale with the linear lattice dimension.
Extension to Spinful Models: The framework is extended to the spinful Fermi-Hubbard model. The authors identify a tradeoff in fermionic-mode placement:
- Separate Sublattices: Keeps hopping terms local but renders on-site interactions geometrically non-local.
- Interleaved Sites: Keeps on-site interactions local but requires next-nearest-neighbor hopping terms, increasing the weight of hopping operators.
  Preliminary results for a $3\times3$ spinful lattice show that both constructions can approach the exact ground-state energy, validating the approach despite the increased complexity.

Significance and Claims
The paper establishes that locality-preserving fermion-to-qubit mappings, when combined with constraint-preserving state preparation and symmetry-aware variational circuits, offer a practical route toward resource-efficient quantum simulation of higher-dimensional interacting fermionic systems.

The primary significance lies in the demonstrated tradeoff: the DK mapping exchanges the overhead of additional auxiliary qubits and a one-time Clifford stabilizer-preparation cost for a substantial reduction in operator non-locality and circuit depth. This advantage becomes increasingly relevant as the linear extent of the two-dimensional lattice increases, where conventional JW transformations suffer from growing Pauli string weights. The work provides a concrete, practical framework that avoids the pitfalls of generic ansätze exploring unphysical Hilbert space regions, thereby offering a viable path for simulating complex fermionic systems on near-term quantum hardware.

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