Tradeoffs for reliable quantum information storage in 2D systems (0909.5200v1)

Abstract: We ask whether there are fundamental limits on storing quantum information reliably in a bounded volume of space. To investigate this question, we study quantum error correcting codes specified by geometrically local commuting constraints on a 2D lattice of finite-dimensional quantum particles. For these 2D systems, we derive a tradeoff between the number of encoded qubits k, the distance of the code d, and the number of particles n. It is shown that kd^2=O(n) where the coefficient in O(n) depends only on the locality of the constraints and dimension of the Hilbert spaces describing individual particles. We show that the analogous tradeoff for the classical information storage is k\sqrt{d} =O(n).

Tradeoffs for Reliable Quantum Information Storage in 2D Systems

This paper presents a rigorous investigation into the constraints of reliable quantum information storage within a two-dimensional spatial framework. The authors focus on quantum error-correcting codes functioning under geometrically local constraints, addressing fundamental limits related to encoding quantum information in 2D systems.

Key Findings

The central result derived in the paper is the tradeoff formula for quantum error-correcting codes in 2D lattices: $kd^2 = O(n)$. Here, $k$ represents the number of logical qubits, $d$ denotes the code distance, and $n$ is the number of physical qubits. This formulation underscores that the product of the number of encoded qubits and the square of the code distance scales linearly with the number of quantum particles involved. The dependency on geometric locality and the dimensionality of the individual particle's Hilbert spaces is noteworthy.

Contrastingly, for classical information storage, the analogous tradeoff is given by $k\sqrt{d} = O(n)$. This highlights distinct differences in scalability and storage efficiency between classical and quantum informational systems.

Detailed Analysis

1. Geometric Locality and Quantum Entanglement:
   • The constraints utilized are geometrically local commuting projectors, which are vital for defining quantum error correcting codes. The commutativity property allows application of error correction algorithms based on syndrome measurements efficiently.
   • The role of quantum entanglement specific to 2D systems is pivotal in establishing these bounds, underlining the dimensional dependency embedded within quantum error correction paradigms.
2. Comparison with Classical Codes:
   • The results are sharp contrasts to classical codes, demonstrating quantum codes' unique behavior in handling entanglement and enhancing stability in informational structures.
   • This tradeoff suggests quantum information's intrinsic constraints due to geometric and entanglement properties, differentiating it markedly from classical information paradigms.
3. Implications for Topological Quantum Computation:
   • Quantum codes can be seen as models for topological quantum order (TQO), influencing potential implementations for topological quantum computation schemes. Examples such as Levin-Wen string-net models, and Kitaev's anyon-based computation, illustrate practical applications aligning with these theoretical results.
4. Holographic Principe Implications:
   • The bounds may suggest connections with the holographic principle, indicating fascinating avenues in linking physical laws with quantum informational limits.

Potential Applications and Future Work

While the tradeoffs provide theoretical constraints, practical applications in quantum computation, specifically in developing robust quantum storage systems with minimal error rates, are ripe for exploration. Future research may involve expanding these findings to systems of higher dimensions ($D > 2$), scrutinizing error correction efficacy across various dimensional scales.

Furthermore, exploring different geometrical configurations—non-Euclidean or irregular lattices—could yield enhanced understanding of quantum locality's role and lead to new code designs leveraging surface codes or other topological strategies.

In summary, the bounds derived are integral in progressing the understanding of quantum information limits within constrained spatial frameworks, paving the way for more nuanced computational structures and refined quantum error correction techniques.