  Path-independent (PI) quantum control has recently been proposed to integrate
quantum error correction and quantum control [Phys. Rev. Lett. 125, 110503
(2020)], achieving fault-tolerant quantum gates against ancilla errors. Here we
reveal the underlying algebraic structure of PI quantum control. The PI
Hamiltonians and propagators turn out to lie in an algebra isomorphic to the
ordinary matrix algebra, which we call the PI matrix algebra. The PI matrix
algebra, defined on the Hilbert space of a composite system (including an
ancilla system and a central system), is isomorphic to the matrix algebra
defined on the Hilbert space of the ancilla system. By extending the PI matrix
algebra to the Hilbert-Schmidt space of the composite system, we provide an
exact and unifying condition for PI quantum control against ancilla noise.
