Bravyi Sergey, Lee Dongjin, Li Zhi, Yoshida Beni
IBM T. J. Watson Research Center, IBM Quantum, Yorktown Heights, New York 10598, USA.
Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada.
Phys Rev Lett. 2025 May 30;134(21):210602. doi: 10.1103/PhysRevLett.134.210602.
It is commonly believed that logical states of quantum error-correcting codes have to be highly entangled such that codes capable of correcting more errors require more entanglement to encode a qubit. Here, we show that the validity of this belief depends on the specific code and the choice of entanglement measure. To this end, we characterize a tradeoff between the code distance d quantifying the number of correctable errors, and the geometric entanglement measure of logical states quantifying their maximal overlap with product states or more general "topologically trivial" states. The maximum overlap is shown to be exponentially small in d for three families of codes: (1) low-density parity check codes with commuting check operators, (2) stabilizer codes, and (3) codes with a constant encoding rate. Equivalently, the geometric entanglement of any logical state of these codes grows at least linearly with d. On the opposite side, we also show that this distance-entanglement tradeoff does not hold in general. For any constant d and k (number of logical qubits), we show there exists a family of codes such that the geometric entanglement of some logical states approaches zero in the limit of large code length.
人们普遍认为，量子纠错码的逻辑态必须高度纠缠，以至于能够纠正更多错误的码在编码一个量子比特时需要更多的纠缠。在此，我们表明这种观点的正确性取决于具体的码以及纠缠度量的选择。为此，我们刻画了量化可纠正错误数量的码距(d)与量化逻辑态与积态或更一般的“拓扑平凡”态的最大重叠的逻辑态几何纠缠度量之间的权衡。对于三类码，最大重叠在(d)中呈指数级小：（1）具有可交换校验算子的低密度奇偶校验码，（2）稳定子码，以及（3）具有恒定编码率的码。等效地，这些码的任何逻辑态的几何纠缠至少随(d)线性增长。另一方面，我们也表明这种距离 - 纠缠权衡一般不成立。对于任何常数(d)和(k)（逻辑量子比特数），我们表明存在一族码，使得在大码长的极限下某些逻辑态的几何纠缠趋近于零。
