Quantum variational learning for quantum error-correcting codes
Chenfeng Cao, Chao Zhang, Zipeng Wu, Markus Grassl, Bei Zeng
Abstract
Quantum error correction is believed to be a necessity for large-scale fault-tolerant quantum computation. In the past two decades, various constructions of quantum error-correcting codes (QECCs) have been developed, leading to many good code families. However, the majority of these codes are not suitable for near-term quantum devices. Here we present VarQEC, a noise-resilient variational quantum algorithm to search for quantum codes with a hardware-efficient encoding circuit. The cost functions are inspired by the most general and fundamental requirements of a QECC, the Knill-Laflamme conditions. Given the target noise channel (or the target code parameters) and the hardware connectivity graph, we optimize a shallow variational quantum circuit to prepare the basis states of an eligible code. In principle, VarQEC can find quantum codes for any error model, whether additive or non-additive, degenerate or non-degenerate, pure or impure. We have verified its effectiveness by (re)discovering some symmetric and asymmetric codes, e.g., <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:msup><mml:mn>2</mml:mn><mml:mrow class="MJX-TeXAtom-ORD"><mml:mi>n</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:msup><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mn>2</mml:mn></mml:msub></mml:math> for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>n</mml:mi></mml:math> from 7 to 14. We also found new <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>6</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mn>2</mml:mn></mml:msub></mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>7</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mn>2</mml:mn></mml:msub></mml:math> codes that are not equivalent to any stabilizer code, and extensive numerical evidence with VarQEC suggests that a <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo stretchy="false">(</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>7</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo>,</mml:mo><mml:mn>3</mml:mn><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mo stretchy="false">)</mml:mo><mml:mn>2</mml:mn></mml:msub></mml:math> code does not exist. Furthermore, we found many new channel-adaptive codes for error models involving nearest-neighbor correlated errors. Our work sheds new light on the understanding of QECC in general, which may also help to enhance near-term device performance with channel-adaptive error-correcting codes.