Fermionic Resource States via fPEPS

• The paper introduces a framework using fPEPS, where local swap rules absorb fermionic signs to ensure accurate simulation of fermionic statistics.
• The approach employs variational optimization of local tensors, achieving high accuracy and numerical stability for both quadratic and interacting fermionic systems.
• The methodology effectively decouples sign management from local interactions, reducing computational overhead and enabling scalable resource state engineering in 2D systems.

Entangled resource states simulating fermionic statistics are quantum many-body states designed such that their entanglement structure and algebraic properties encode the antisymmetry, exchange rules, and non-local sign structures foundational to fermionic systems. In modern quantum information science and computational physics, such resource states are crucial for simulating strongly correlated electrons, topological phases, and for implementing quantum algorithms sensitive to fermionic behavior.

1. Tensor Network Formulations for Fermionic Resource States

A dominant approach to constructing entangled resource states with fermionic statistics is the fermionic Projected Entangled Pair States (fPEPS) framework. In fPEPS, the global wavefunction is written as a contraction of local site tensors, each incorporating both the physical (on-site) fermionic degree of freedom and several virtual fermionic modes which bind neighboring sites into maximally entangled pairs. The local fPEPS ansatz is:

$$\Psi = W_{\{\alpha,\beta,\gamma,\delta\}} \prod_{i,j} Q_{i,j} \prod_{i,j} H_{i,j} \prod_{i,j} V_{i,j} |0\rangle$$

where

• $$Q_{i,j} = A_{l r u d k}^{[i,j]} c_{i,j}^\dagger{}^{k} \alpha_{i,j}^l \beta_{i,j}^r \gamma_{i,j}^u \delta_{i,j}^d$$ encodes both physical and virtual modes, subject to a well-defined fermionic parity constraint $(l+r+u+d+k)\bmod 2 = P_{i,j}$.
• $$H_{i,j} = 1 + \beta_{i,j}^\dagger \alpha_{i,j+1}^\dagger$$ constructs horizontal entangled virtual bonds; $$V_{i,j} = 1 + \delta_{i,j}^\dagger \gamma_{i+1,j}^\dagger$$ for vertical bonds.

The complete tensor network is then contracted, enforcing the correct fermionic anticommutation relations at both the virtual and physical level.

Crucially, all necessary fermionic signs are absorbed locally using a fermionic swap rule without requiring an additional string-bond or extra index to track non-local sign information during contraction: $W^\dagger A B W = W^\dagger \tilde{B} \tilde{A} W$ with sign structures controlled by the parity of $A$ and $B$ and their tensor indices. This ensures local absorption of nontrivial fermionic phases in the tensor entries, rendering efficient and numerically stable contraction possible—matching the computational scaling and contraction techniques of bosonic PEPS (Pizorn et al., 2010).

2. Variational Methods: Simulation and Optimization

Ground states of fermionic many-body Hamiltonians are simulated by variationally optimizing the local tensors encoding the fPEPS state. The cost function is the energy per site, which can be recast as: $$E = \frac{A \cdot H_{\text{eff}} \cdot A}{A \cdot N_{\text{eff}} \cdot A}$$ where $H_{\text{eff}}$ and $N_{\text{eff}}$ are effective Hamiltonian and norm matrices formed by contracting all other degrees of freedom in the network, with fermionic signs already absorbed. Optimization reduces to solving a generalized eigenvalue problem: $H_{\text{eff}} A = \lambda N_{\text{eff}} A$ with parity built in, naturally splitting the problem into even-even and odd-odd block sectors.

Expectation values of physical observables $\langle\Psi|\mathcal{O}|\Psi\rangle$ are computed via a double-layer tensor network formed by pairing "ket" and "bra" physical indices and absorbing local swap-rule sign factors; contraction proceeds by representing tensor rows or columns as MPS or MPO objects and applying truncations as for spin PEPS (Pizorn et al., 2010).

3. Absorption of Fermionic Statistics: Local Versus Non-Local Sign Structure

Historically, fermionic tensor networks required explicit nonlocal string bonds (e.g., Jordan-Wigner chains or auxiliary indices) to manage the sign structure arising from fermionic antisymmetry. The local swap rule implementation in fPEPS allows for all sign factors—resulting from the anti-commutation of creation and annihilation operators—to be managed locally at each site. This approach produces several advantages:

• Computational efficiency: No increased bond dimension or contraction complexity due to additional sign-tracking indices.
• Numerical stability: All sign bookkeeping is handled at the local tensor level, preventing propagation of sign-related numerical noise.
• Physical fidelity: All nontrivial fermionic phases—essential for accurate physical simulation and resource state preparation—are encoded directly in the structure of the local tensors and their contraction order.

4. Performance on Quadratic and Interacting Fermionic Models

The fPEPS method was tested on both quadratic (free) and interacting lattice fermion systems. For the bilinear model: $$H = \sum_{\langle\mu\nu\rangle} \left[c_\mu^\dagger (c_\nu - \gamma c_\nu^\dagger) + \text{h.c.}\right] - 2\sum_\nu \lambda c_\nu^\dagger c_\nu$$ simulations on $4 \times 4$ and $10 \times 10$ lattices with $D=2$ or $D=4$ virtual bond dimension yielded very low relative ground state energy errors, especially in the gapped regime ($\lambda > 2$). The method is robust even for small $D$ due to strong locality of the sign management and efficient truncation.

For an interacting nearest-neighbor density model: $$H = -\sum_{\langle\mu\nu\rangle} (c^\dagger_\nu c_\mu + \text{h.c.}) + V \sum_{\langle\mu\nu\rangle} n_\nu n_\mu$$ energy convergence, particle number accuracy, and two-point density correlations matched exact results within $1\%$ or better in moderate system sizes after a few variational sweeps and bond-dimension optimization.

5. Implications for Fermionic Quantum Simulation and Resource State Engineering

The fPEPS framework implements the following workflow for simulating fermionic statistics:

• The ground state ansatz is built from local tensors with entangled virtual fermion pairs, enforced parity constraints, and local swap rules.
• All fermionic signs and anti-commutation rules are encoded in the local tensor structure, rendering the resource state suitable for variational optimization and efficient contraction.
• No auxiliary string bonds are added; local contractions suffice.
• The algorithm demonstrates high accuracy, low resource overhead, and numerical stability for both quadratic and interacting theories.

This methodology provides:

• Faithful entanglement structure (including nontrivial sign structure) essential for simulating quantum fermion systems and for using these states as resource states in quantum information.
• A means to systematically increase expressivity (through bond dimension $D$) for arbitrarily accurate approximations, especially efficient in gapped phases.
• A clear separation between sign structures from statistics and from local physical interactions, clarifying resource state engineering for quantum emulation.

6. Limitations and Scaling Behavior

The fPEPS approach, while efficient for gapped systems or systems with only short-range entanglement, encounters increased demands for bond dimension and computational resources in gapless phases (e.g., systems with extended Fermi surfaces or algebraic correlations). The entanglement scaling in such systems implies that capturing long-range correlations or critical features may require higher $D$, with corresponding increases in numerical cost.

In summary, entangled resource states based on locally antisymmetric fPEPS simulate fermionic statistics directly in their tensor structure and contraction rules, enabling efficient, accurate, and scalable modeling of 2D fermionic lattice systems and providing a practical foundation for quantum simulation and quantum information processing tasks involving fermions (Pizorn et al., 2010).