Fault-tolerant simulation of Lattice Gauge Theories with gauge covariant codes

TL;DR

The paper addresses the challenge of simulating lattice gauge theories on quantum hardware under noise by constructing gauge-covariant quantum error-correcting codes from Gauss' law, enabling fault-tolerant dynamics. It develops a general framework to express lattice-Hamiltonians in terms of logical operations of the Gauss' law code and maps matter to hardcore bosons, preserving locality. The main contributions include generalizing Gauss' law codes to arbitrary dimensions, providing explicit logical Hamiltonians and two concatenation schemes to achieve a distance-3 CSS-QLDPC code, and proposing universal, fault-tolerant time-evolution strategies via Quantum Signal Processing and Trottterization with minimal ancilla overhead. The work offers a path toward end-to-end dynamical simulations of gauge theories with reduced qubit resources, leveraging the interplay between quantum error correction and gauge symmetries to enable scalable quantum simulations of the Standard Model and related Abelian theories.

Abstract

We show in this paper that a strong and easy connection exists between quantum error correction and Lattice Gauge Theories (LGT) by using the Gauge symmetry to construct an efficient error-correcting code for Abelian LGTs. We identify the logical operations on this gauge covariant code and show that the corresponding Hamiltonian can be expressed in terms of these logical operations while preserving the locality of the interactions. Furthermore, we demonstrate that these substitutions actually give a new way of writing the LGT as an equivalent hardcore boson model. Finally we demonstrate a method to perform fault-tolerant time evolution of the Hamiltonian within the gauge covariant code using both product formulas and qubitization approaches. This opens up the possibility of inexpensive end to end dynamical simulations that save physical qubits by blurring the lines between simulation algorithms and quantum error correcting codes.

Figures (10)

• Figure 1: Labelling convention for links in a plaquette.
• Figure 2: Convention for positive and negative links around a site on the left-hand side, and labelling convention on the right-hand side.
• Figure 3: Representation of a 1-dimensional lattice, showing the labelling convention for sites and links.
• Figure 4: Graphical representation of the additional qubits needed for the concatenation. This figure shows how we can associate every site and link to $3$ qubits now, instead of being one qubit each as before.
• Figure 5: Graphical representation of the quantum error-correcting code. The green circle represents $X$ stabilizers, which are all of weight $2$. The orange area instead represents the weight $9$ Gauss' law stabilizer. On top, the 1-dimensional system is represented to show explicitly that every link and site is in reality made of the $3$ physical qubits below it.

Theorems & Definitions (14)

• Theorem 3.1
• proof
• Lemma 3.2
• Lemma 3.3
• proof
• Theorem 4.1
• Corollary 4.2
• proof
• Theorem 5.1
• proof