A no-go theorem for a two-dimensional self-correcting quantum memory based on stabilizer codes

Abstract: We study properties of stabilizer codes that permit a local description on a regular D-dimensional lattice. Specifically, we assume that the stabilizer group of a code (the gauge group for subsystem codes) can be generated by local Pauli operators such that the support of any generator is bounded by a hypercube of constant size. Our first result concerns the optimal scaling of the distance $d$ with the linear size of the lattice $L$. We prove an upper bound $d=O(L^{D-1})$ which is tight for D=1,2. This bound applies to both subspace and subsystem stabilizer codes. Secondly, we analyze the suitability of stabilizer codes for building a self-correcting quantum memory. Any stabilizer code with geometrically local generators can be naturally transformed to a local Hamiltonian penalizing states that violate the stabilizer condition. A degenerate ground-state of this Hamiltonian corresponds to the logical subspace of the code. We prove that for D=1,2 the height of the energy barrier separating different logical states is upper bounded by a constant independent of the lattice size L. The same result holds if there are unused logical qubits that are treated as "gauge qubits". It demonstrates that a self-correcting quantum memory cannot be built using stabilizer codes in dimensions D=1,2. This result is in sharp contrast with the existence of a classical self-correcting memory in the form of a two-dimensional ferromagnet. Our results leave open the possibility for a self-correcting quantum memory based on 2D subsystem codes or on 3D subspace or subsystem codes.

Evaluation of a No-Go Theorem for Self-Correcting Quantum Memories

The paper by Sergey Bravyi and Barbara M. Terhal addresses a significant question in the field of quantum information theory: is it possible to realize a self-correcting quantum memory (SCQM) within a two-dimensional (2D) framework utilizing stabilizer codes with local generators? While SCQMs, akin to classical counterparts like the 2D Ising ferromagnet, would offer substantial advancements for quantum computing, the paper presents evidence against such feasibility in 2D settings using existing models.

Overview of Stabilizer Codes and Quantum Memory

The authors explore stabilizer codes within a lattice framework, focusing on the spatial limitations and interaction range permissible for quantum error correction. Key parameters include the code distance ( d ) and the energy barrier ( d^\ddag ), which reflect the robustness of quantum memory against local errors and logical errors, respectively. The core results derive fundamental limits on the achievable code distance and energy barrier in dimensions ( D = 1 ) and ( D = 2 ).

Upper Bounds and the No-Go Theorem

Code Distance

Bravyi and Terhal establish that the code distance ( d ) of any stabilizer code on a 2D lattice is upper-bounded by ( d = O(L) ), where ( L ) denotes the linear size of the lattice. This implies that while linear scaling of the distance with system size is possible, achieving superior scaling akin to ( L^2 ) is unfeasible in two dimensions using stabilizer codes. This is derived using a "cleaning lemma" that allows local elimination of logical operator components from subsets of the lattice, thus affording an effective constraint on the code distance.

Energy Barrier

The analysis of the energy barrier reveals that ( d^\ddag ) is limited to be constant in two dimensions. This is a critical insight into the unsuitability of 2D stabilizer codes for SCQMs, as it suggests that the energy landscape for implementing logical operations does not impose a significant threshold that scales with system size, making error accumulation likely in the presence of environmental noise.

Implications and Future Directions

The presented results sharply contrast with classical systems that support self-correction in two dimensions, underscoring a fundamental distinction between classical and quantum error correction mechanisms. The exploration of 2D stabilizer codes yields that they inherently face structural barriers against supporting SCQM characteristics.

Future avenues may include investigating three-dimensional (3D) frameworks or subsystem codes, which exhibit more complex structures and potentially beneficial logical properties that evade the no-go constraints. In particular, studying 3D stabilizer codes or more sophisticated Hamiltonian models could open pathways to realizing SCQMs if they demonstrate macroscopic energy barriers and enhanced code distances. This line of investigation may align with proposals such as the Bacon-Shor code, which offer a 3D model without the restrictions posed by the paper's theorem.

The paper further motivates a detailed characterization of subsystem codes, where gauge degrees of freedom could potentially be harnessed to overcome limitations found in traditional stabilizer codes. Such explorations could redefine the landscape for practically implementing viable quantum memories.

In summary, the paper presents a rigorous verification of limitations within stabilizer codes when applied to SCQMs in 2D, advancing our understanding of the constraints and possibilities in quantum information sciences.