Local stabilizer codes in three dimensions without string logical operators (1101.1962v2)

Abstract: We suggest concrete models for self-correcting quantum memory by reporting examples of local stabilizer codes in 3D that have no string logical operators. Previously known local stabilizer codes in 3D all have string-like logical operators, which make the codes non-self-correcting. We introduce a notion of "logical string segments" to avoid difficulties in defining one dimensional objects in discrete lattices. We prove that every string-like logical operator of our code can be deformed to a disjoint union of short segments, and each segment is in the stabilizer group. The code has surface-like logical operators whose partial implementation has unsatisfied stabilizers along its boundary.

• The paper introduces the concept of logical string segments to reform string-like logical operators into short stabilizer group segments.
• It classifies CSS stabilizer codes in 3D, demonstrating that a subset exists without string logical operators and with a macroscopic, linear code distance.
• The study provides empirical formulas for predicting logical qubits and shows that high energy barriers confine thermal excitations to enhance quantum memory stability.

Local Stabilizer Codes in Three Dimensions Without String Logical Operators

The research conducted in this paper explores the potential for developing self-correcting quantum memories utilizing local stabilizer codes in three dimensions (3D) that are devoid of string logical operators. Previous work in 3D stabilizer codes often encountered the problem of string logical operators, which hinder the codes from achieving self-correction. This paper introduces the concept of "logical string segments" and proposes models for local stabilizer codes in 3D that do not harbor such operators. The key contribution is proving that any string-like logical operator within these codes can be deformed into a union of short segments, each a member of the stabilizer group. These insights could provide a path toward practical implementations of quantum memory with topological order that remain stable at finite temperatures.

Key Contributions and Findings

1. Logical String Segments: The introduction of logical string segments serves as a mechanism to analyze string-like logical operators in discrete lattice systems. These segments provide a method to ensure that string logical operators do not inadvertently disrupt encoded quantum information due to interactions with a thermal environment.
2. Classification of Cubic Codes: The paper methodically classifies all conceivable CSS stabilizer codes with two types of generators and proves that there exists a subset of these that are devoid of string logical operators. The classification involves identifying commutation relations of corner operators and analyzing them with respect to translation invariance and singlet properties.
3. Code Distance: Through rigorous analysis, the paper establishes that these codes maintain a macroscopic code distance, exhibiting linear growth with the lattice size. This outcome is an essential trait for ensuring robust error correction in quantum memories.
4. Empirical Formulae for Logical Qubits: The paper provides empirical formulae to predict the number of logical qubits as a function of the system size for each type of code. These formulae capture the intricate dependencies of the logical qubits on the lattice dimensions, providing insights useful for code design and system realization.
5. Thermal Robustness: The paper explores the Hamiltonian model representing the sum of local generators to simulate adverse effects stemming from thermal interactions. It argues that certain configurations confine excitations to boundaries, suggesting that the energy barriers associated with transitioning between ground states are particularly high.

Implications

These findings have significant implications for both quantum information theory and practical quantum computing. The codes' design, which circumvents the issue of string logical operators, indicates a potential pathway towards achieving self-correcting quantum memories. Furthermore, the detailed empirical and theoretical analyses of code properties provide a foundation for future research on quantum codes' stability and robustness under thermal influences.

Future Directions

The work opens the door for further exploration of stabilizer codes in higher dimensions, aiming to extend these ideas to more complex topologies and interactions. Another critical direction is developing efficient decoding algorithms that could exploit the unique structure of these codes to enhance fault-tolerance in quantum information systems.

Moreover, investigating the relationship between different dimensions and topological properties in stabilizer codes might yield new insights into quantum error correction mechanisms applicable across various quantum computing architectures.

In summary, this paper makes substantial contributions to our understanding of stabilizer codes in three dimensions, particularly in their application to self-correcting quantum memories, and lays groundwork for future advancements in quantum memory technologies.