Quality of two norm solution in basis pursuit

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Suppose I want to solve

$$\min\; \|x\|_0$$ $$\text{st}\; Mx=b$$

with solution $x^*$. What are the known error bounds between $x^*$ and the min two norm solution $\hat x= \arg\min_x\{\|x\|_2: Mx=b\}$? Are there any obvious ones?

Thank you!