We establish a symmetry classification for a general class of quantum feedback control. For successive feedback control with a nonadaptive sequence of bare measurements (i.e., with positive Kraus operators), we prove that the symmetry classification collapses to the ten-fold ${\mathrm{AZ}}^{†}$ classes, specifying the allowed topology of CPTP maps associated with feedback control. We demonstrate that a chiral Maxwell’s demon with Gaussian measurement errors exhibits quantized winding numbers. Moreover, for general (nonbare) measurements, we explicitly construct a protocol that falls outside the ten-fold classification. These results broaden and clarify the principles in engineering topological aspects of quantum control robust against disorder and imperfections.

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• Open quantum systems
• Quantum control
• Quantum feedback
• Topological phases of matter
• Non-Hermitian systems
• Topology