Existing SNP-heritability estimation methods that leverage GWAS summary statistics produce estimators that are less efficient than the restricted maximum likelihood (REML) estimator using individual-level data under linear mixed models (LMMs). Increasing the precision of a heritability estimator is particularly important for regional analyses, as local genetic variances tend to be small. We introduce a new estimator for local heritability, "HEELS", which attains comparable statistical efficiency as REML (\emph{i.e.} relative efficiency greater than 92%) but only requires summary-level statistics -- Z-scores from the marginal association tests plus the empirical LD matrix. HEELS significantly improves the statistical efficiency of the existing summary-statistics-based heritability estimators-- for instance, HEELS produces heritability estimates that are more than 3-fold and 7-times less variable than GRE and LDSC, respectively. Moreover, we introduce a unified framework to evaluate and compare the performance of different LD approximation strategies. We propose representing the empirical LD as the sum of a low-rank matrix and a banded matrix. This approximation not only reduces the storage and memory cost of using the LD matrix, but also improves the computational efficiency of the HEELS estimation. We demonstrate the statistical efficiency of HEELS and the advantages of our proposed LD approximation strategies both in simulations and through empirical analyses of the UK Biobank data.