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Efficient Simulation of Sparse, Non-Local Fermion Models

Reinis Irmejs, J. Ignacio Cirac

TL;DR

The paper tackles the challenge of simulating sparse, non-local fermionic models on all-to-all connected qubit hardware by introducing an auxiliary-fermion encoding that eliminates Jordan–Wigner strings. Stabilizers built from auxiliary Majorana modes render interaction terms into local Pauli operators with constant weight, while initializing the auxiliary sector in a mutual +1 eigenstate preserves the original physics during dynamics. A graph-coloring-based grouping of stabilizers minimizes ancilla requirements to $\nu=\lceil \chi/2 \rceil$ per site, and an ordered-state preparation enables efficient initialization in $\mathcal{O}(\log\nu)$ depth. The results show that long-time dynamics can be implemented with asymptotically optimal circuit depth, matching the performance of fermionic hardware up to constants and offering a practical route for simulating sparse fermionic systems on qubit-based quantum computers. The approach reduces the previous $O(\log N)$ multiplicative overhead to an additive overhead, with total ancilla scaling as $O(dN)$, thereby closing a key gap in the quantum simulation of sparse fermionic models.

Abstract

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 $O(\log N)$ 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.

Efficient Simulation of Sparse, Non-Local Fermion Models

TL;DR

The paper tackles the challenge of simulating sparse, non-local fermionic models on all-to-all connected qubit hardware by introducing an auxiliary-fermion encoding that eliminates Jordan–Wigner strings. Stabilizers built from auxiliary Majorana modes render interaction terms into local Pauli operators with constant weight, while initializing the auxiliary sector in a mutual +1 eigenstate preserves the original physics during dynamics. A graph-coloring-based grouping of stabilizers minimizes ancilla requirements to per site, and an ordered-state preparation enables efficient initialization in depth. The results show that long-time dynamics can be implemented with asymptotically optimal circuit depth, matching the performance of fermionic hardware up to constants and offering a practical route for simulating sparse fermionic systems on qubit-based quantum computers. The approach reduces the previous multiplicative overhead to an additive overhead, with total ancilla scaling as , thereby closing a key gap in the quantum simulation of sparse fermionic models.

Abstract

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 fermionic modes in which each participates in at most a constant number 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 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 ancillary qubits.

Paper Structure

This paper contains 38 sections, 2 theorems, 90 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Method
  3. Sparse, non-local Hamiltonian
  4. Setup
  5. Constructing the encoding
  6. Constructing the stabilizers.
  7. Stabilizer initial state.
  8. Preparing the full auxiliary state
  9. Grouping the stabilizers.
  10. Full auxiliary state.
  11. Permutation to an ordered state.
  12. Ordered state preparation.
  13. Summary of the Time Dynamics Simulation
  14. Depth per Trotter step.
  15. Trotter Error.
  16. ...and 23 more sections

Key Result

Lemma 1

Fix a color $\gamma$ and an auxiliary index $\ell$. Let be the set of operators defined as above from the edges of $E_{\gamma}$ and an arbitrary orientation. Then the operators in $\mathcal{P}_{\gamma}^{(\ell)}$ mutually commute.

Figures (2)

  • Figure 1: Physical modes $a_i$ (blue) and auxiliary modes $b_i^{(\ell)}$ (olive) together with the JW ordering path (dashed). The rectangular selection encompasses one whole auxiliary mode family.
  • Figure 2: (a) Proper edge coloring of the interaction graph with $N=8$ vertices. (b) Subgraph of green and blue edges. Arrows denote the assigned edge orientations. With this ordering, all depicted stabilizers $P_{ij}$ commute and can be encoded in the same auxiliary register $\ell$. (c) Operator preparing the initial state on auxiliary modes $\ell=1$, associated with the blue and green stabilizers. (d) Permutations relevant for stabilizers associated with the blue and green colors, bringing them into an ordered form.

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Lemma 2
  • proof