Scalable Simulation of Fermionic Encoding Performance on Noisy Quantum Computers

TL;DR

Using a high-performance stabilizer simulator, this work classically simulate the performance of a local encoding known as the Derby-Klassen encoding and compares its performance with the Jordan-Wigner encoding, the ternary tree encoding and the ternary tree encoding.

Abstract

A compelling application of quantum computers with thousands of qubits is quantum simulation. Simulating fermionic systems is both a problem with clear real-world applications and a computationally challenging task. In order to simulate a system of fermions on a quantum computer, one has to first map the fermionic Hamiltonian to a qubit Hamiltonian. The most popular such mapping is the Jordan-Wigner encoding, which suffers from inefficiencies caused by the high weight of some encoded operators. As a result, alternative local encodings have been proposed that solve this problem at the expense of a constant factor increase in the number of qubits required. Some such encodings possess local stabilizers, i.e., Pauli operators that act as the logical identity on the encoded fermionic modes. A natural error mitigation approach in these cases is to measure the stabilizers and discard any run where a measurement returns a -1 outcome. Using a high-performance stabilizer simulator, we classically simulate the performance of a local encoding known as the Derby-Klassen encoding and compare its performance with the Jordan-Wigner encoding and the ternary tree encoding. Our simulations use more complex error models and significantly larger system sizes (up to $18\times18$) than in previous work. We find that the high sampling requirements of postselection methods with the Derby-Klassen encoding pose a limitation to its applicability in near-term devices and call for more encoding-specific circuit optimizations.

Figures (18)

• Figure 1: Ternary Tree Encoding operators are defined by assigning qubit indices to tree nodes that are not leaves and Pauli operators to the edges. This example tree creates 9 anticommuting Pauli operators on 4 qubits, with any subset of 8 of them giving a set of 8 encoded Majorana operators for 4 fermionic modes.
• Figure 2: DK encoding lattice of size $4 \times 4$ with periodic boundaries. Each hollow vertex is identified with the corresponding filled vertex on the opposite boundary. We highlight a stabilizer in red, a vertex operator in blue, and two edge operators in green.
• Figure 3: Circuit for measuring a stabilizer generator of the DK encoding. The top eight qubits are the data qubits. The auxiliary qubit $|+\rangle_a$ is read out to give the measurement result and the auxiliary qubit $|+\rangle_f$ is a flag qubit that is used to detect high-weight errors.
• Figure 4: Efficient DK measurement schedule. For each stabilizer in the lattice, we can measure the hopping term operators corresponding to the same color simultaneously. Each highlighted edge corresponds to two operators in the hopping term: $a_i^{\dagger} a_j$ and $a_j^{\dagger} a_i$; see \ref{['eq:FH-CE-hop']}.
• Figure 5: Clifford and non-Clifford logical rotations.