Hierarchical memories: Simulating quantum LDPC codes with local gates
Christopher A. Pattison, Anirudh Krishna, John Preskill
Abstract
Constant-rate low-density parity-check (LDPC) codes are promising candidates for constructing efficient fault-tolerant quantum memories. However, if physical gates are subject to geometric-locality constraints, it becomes challenging to realize these codes. In this paper, we construct a new family of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">[</mml:mo><mml:mo stretchy="false">[</mml:mo><mml:mi>N</mml:mi><mml:mo>,</mml:mo><mml:mi>K</mml:mi><mml:mo>,</mml:mo><mml:mi>D</mml:mi><mml:mo stretchy="false">]</mml:mo><mml:mo stretchy="false">]</mml:mo></mml:math> codes, referred to as hierarchical codes, that encode a number of logical qubits <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>K</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant="normal">&#x03A9;</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>N</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mo>/</mml:mo></mml:mrow><mml:mi>log</mml:mi><mml:mo>&#x2061;</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mi>N</mml:mi><mml:msup><mml:mo stretchy="false">)</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mo stretchy="false">)</mml:mo></mml:math>. The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>N</mml:mi><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>t</mml:mi><mml:mi>h</mml:mi></mml:mrow></mml:msup></mml:math> element of this code family is obtained by concatenating a constant-rate quantum LDPC code with a surface code; nearest-neighbor gates in two dimensions are sufficient to implement the corresponding syndrome-extraction circuit and achieve a threshold. Below threshold the logical failure rate vanishes superpolynomially as a function of the distance <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>D</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>N</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>. We present a bilayer architecture for implementing the syndrome-extraction circuit, and estimate the logical failure rate for this architecture. Under conservative assumptions, we find that the hierarchical code outperforms the basic encoding where all logical qubits are encoded in the surface code.