We employ scaling arguments and optimal fluctuation theory to establish a
general relation between quantum Griffiths singularities and the Harris
criterion for quantum phase transitions in disordered systems. If a clean
critical point violates the Harris criterion, it is destabilized by weak
disorder. At the same time, the Griffiths dynamical exponent z′ diverges upon
approaching the transition, suggesting unconventional critical behavior. In
contrast, if the Harris criterion is fulfilled, power-law Griffiths
singularities can coexist with clean critical behavior but z′ saturates at a
finite value. We present applications of our theory to a variety of systems
including quantum spin chains, classical reaction-diffusion systems and
metallic magnets; and we discuss modifications for transitions above the upper
critical dimension. Based on these results we propose a unified classification
of phase transitions in disordered systems.Comment: 4.5 pages, 1 eps figure, final version as publishe