Tapering off qubits to simulate fermionic Hamiltonians

TL;DR

The paper develops symmetry-exploiting encodings to reduce the qubit count required for simulating fermionic Hamiltonians, balancing first-quantization approaches (good for few particles) with second-quantization LDPC and graph-based schemes (broader applicability and sparsity). It introduces sparse, efficiently simulable encodings, analyzes decoding and spectral-gap properties, and shows how discrete $\mathbb{Z}_2$ symmetries can taper off qubits via stabilizer methods. The contributions include concrete sparsity bounds, practical decoding considerations, and explicit tapering strategies for molecular Hamiltonians, with emphasis on near-term variational quantum algorithms. The work provides a framework for scalable, symmetry-aware quantum simulations and outlines open questions about optimal codes and extending tapering beyond $\nu<1/4$.

Abstract

We discuss encodings of fermionic many-body systems by qubits in the presence of symmetries. Such encodings eliminate redundant degrees of freedom in a way that preserves a simple structure of the system Hamiltonian enabling quantum simulations with fewer qubits. First we consider $U(1)$ symmetry describing the particle number conservation. Using a previously known encoding based on the first quantization method a system of $M$ fermi modes with $N$ particles can be simulated on a quantum computer with $Q=N\log{(M)}$ qubits. We propose a new version of this encoding tailored to variational quantum algorithms. Also we show how to improve sparsity of the simulator Hamiltonian using orthogonal arrays. Next we consider encodings based on the second quantization method. It is shown that encodings with a given filling fraction $ν=N/M$ and a qubit-per-mode ratio $η=Q/M<1$ can be constructed from efficiently decodable classical LDPC codes with the relative distance $2ν$ and the encoding rate $1-η$. A family of codes based on high-girth bipartite graphs is discussed. Graph-based encodings eliminate roughly $M/N$ qubits. Finally we consider discrete symmetries, and show how to eliminate qubits using previously known encodings, illustrating the technique for simple molecular-type Hamiltonians.

Figures (3)

• Figure 1: A lower bound on the number of fermionic modes $M$ that can be simulated for fixed values of $N$ and $Q$ using the graph-based encodings.
• Figure 2: Example of a Young tableau $(\lambda,T)$ and the basis vector $|s(\lambda)\rangle$. Here $N=7$ and $(\lambda_1,\lambda_2,\lambda_3)=(4,2,1)$.
• Figure 3: Example of a girth-$6$ bipartite graph with $12$ vertices and $16$ edges. Each empty or filled circle is a vertex.