The eligibility of various advanced quantum algorithms will be questioned if they can not guarantee privacy. To fill this knowledge gap, here we devise an efficient quantum differentially private (QDP) Lasso estimator to solve sparse regression tasks. Concretely, given N d-dimensional data points with N ≪ d, we first prove that the optimal classical and quantum non-private Lasso requires Ω(N + d) and Ω(√N + √d) runtime, respectively. We next prove that the runtime cost of QDP Lasso is dimension independent, i.e., O(N5/2), which implies that the QDP Lasso can be faster than both the optimal classical and quantum non-private Lasso. Last, we exhibit that the QDP Lasso attains a near-optimal utility bound Õ(N-2/3) with privacy guarantees and discuss the chance to realize it on near-term quantum chips with advantages.