Quantum Error Correcting Codes in Eigenstates of Translation-Invariant Spin Chains

Area of Science:

• Quantum Information Science
• Condensed Matter Physics
• Quantum Computation

Background:

• Quantum error correction (QECC) is crucial for fault-tolerant quantum computation.
• Topological order systems naturally realize QECCs.
• Holographic duals in CFTs suggest eigenstates may form QECCs.

Purpose of the Study:

• Investigate the general conditions under which many-body model eigenstates form quantum codes.
• Establish connections between quantum chaos, translation invariance, and approximate quantum error-correcting codes (AQECC).

Main Methods:

• Analyze eigenstates of quantum chaotic systems obeying the eigenstate thermalization hypothesis.
• Develop probabilistic methods to derive AQECC from translation-invariant energy eigenstates of spin chains.
• Utilize local symmetries to construct parent Hamiltonians for 1D quantum spin chains.

Main Results:

• Demonstrate that quantum chaotic systems' eigenstates form AQECC.
• Show that AQECC can be probabilistically obtained from translation-invariant spin chains, including integrable models.
• Construct explicit local AQECC in the ground space of 1D ferromagnetic Heisenberg and Motzkin spin chains.

Conclusions:

• Eigenstates of many-body systems, particularly quantum chaotic ones, naturally exhibit properties of approximate quantum error-correcting codes.
• This work provides a pathway to construct non-stabilizer AQECC in physically relevant 1D gapless models.
• The findings bridge quantum chaos, many-body physics, and quantum error correction theory.