A recipe for local simulation of strongly-correlated fermionic matter on quantum computers: the 2D Fermi-Hubbard model

TL;DR

The paper tackles the challenge of simulating strongly correlated fermionic matter, specifically the two-dimensional Fermi–Hubbard model, on near-term quantum devices. It proposes a concrete end-to-end recipe built around the Derby–Klassen compact fermion mapping to achieve strictly local qubit operations, embedding on hardware with fixed connectivity, and end-to-end workflows for state preparation, unitary time evolution via Trotterization, and measurement of local, equal-time, and unequal-time observables including Green's functions. Key contributions include a detailed mapping framework, explicit unitary decompositions into Pauli strings, a toric-code–based vacuum for physical subspace, and a thorough resource estimate for a $6\times8$ lattice under different connectivities, complemented by extensive error-mitigation strategies (measurement/post-selection, depolarizing models, randomized compiling, dynamical decoupling, ZNE, and PEC). The work provides a practical blueprint for executing 2D fermionic dynamics on quantum hardware, highlighting the resource scales and error considerations necessary to access physics beyond classical capabilities and guiding hardware-aware implementation choices. The approach paves a path toward scalable quantum simulation of strongly correlated fermions and informs future experimental and architectural developments in quantum computing for condensed-matter and quantum chemistry applications.

Abstract

The simulation of quantum many-body systems, relevant for quantum chemistry and condensed matter physics, is one of the most promising applications of near-term quantum computers before fault-tolerance. However, since the vast majority of quantum computing technologies are built around qubits and discrete gate-based operations, the translation of the physical problem into this framework is a crucial step. This translation will often be device specific, and a suboptimal implementation will be punished by the exponential compounding of errors on real devices. The importance of an efficient mapping is already revealed for models of spinful fermions in two or three dimensions, which naturally arise when the relevant physics relates to electrons. Using the most direct and well-known mapping, the Jordan-Wigner transformation, leads to a non-local representation of local degrees of freedom, and necessities efficient decompositions of non-local unitary gates into a sequence of hardware accessible local gates. In this paper, we provide a step-by-step recipe for simulating the paradigmatic two-dimensional Fermi-Hubbard model on a quantum computer using only local operations. To provide the ingredients for such a recipe, we briefly review the plethora of different approaches that have emerged recently but focus on the Derby-Klassen compact fermion mapping in order to make our discussion concrete. We provide a detailed recipe for an end-to-end simulation including embedding on a physical device, preparing initial states such as ground states, simulation of unitary time evolution, and measurement of observables and spectral functions. We explicitly compute the resource requirements for simulating a global quantum quench and conclude by discussing the challenges and future directions for simulating strongly-correlated fermionic matter on quantum computers.

Figures (20)

• Figure 1: Spinful fermions are represented by a square lattice of spinless fermions. The blue vertices represent the spin-up fermions and the red vertices represent spin-down fermions. The green oval region is a unit cell with two fermions with opposite spins. The dashed arrows indicate the inter-cell hopping terms in the Fermi-Hubbard model in Eq. \ref{['eq:Hubbard model']}, and the dotted lines indicate the intra-cell interaction terms.
• Figure 2: Examples of the Majorana operators mapped to local qubit operators, see Eqs. \ref{['eq:Compact_Fermoin_E_def']},\ref{['eq:Compact_Fermion_V_def']},\ref{['eq:Compact_Fermoin_F_def']}. We show the Pauli operators that are involved, but for the correct phases please refer to the main text.
• Figure 3: Action of stabilizers $\mathbb{J}$, which have eigenvalue $+1$ in the physical sector. These operators are related to the toric code stabilizers. Primary qubits are shown as grey circles, and secondary qubits are purple. By convention, we can distinguish between plaquette (case 1) and star (case 2) operators based on the rotation direction of the bonds. Equivalently, we can choose to draw a lattice for the secondary qubits, as shown in purple, which makes this artificial distinction clearer.
• Figure 4: Possible embeddings of the spinful fermion lattice from Fig. \ref{['fig:Sq_Spinfull']} on different local geometries: (a) Square, (b) Diamond, (c) Heavy-honeycomb. The grey sites and bonds indicated the physical qubits and connections on the device. The coloured sites and dashed lines indicate the sites and coupling of the desired lattice.
• Figure 5: Example of a half-filled spin density wave configuration for the fermions with spins in a checkerboard pattern, along with the corresponding Pauli operators to create this state on top of the vacuum state.