Path-independent quantum gates: general formalism and algebraic structure

Ancilla systems are often indispensable to universal control of a nearly isolated central system. However, ancilla systems are typically more vulnerable to environmental noise, limiting the performance of such ancilla-assisted control. To address this challenge, we propose a general class of path-independent (PI) quantum gates [1], which integrate quantum error correction and quantum control and therefore can be resilient to ancilla noise. Furthermore, we reveal the underlying algebraic structure for such PI gates, which we call the PI matrix algebra. The PI matrix algebra is defined on both the ancilla and central systems but isomorphic to the ordinary matrix algebra defined on the ancilla system alone. With such an algebraic structure, we provide a unifying criterion for PI gates against general ancilla errors, with ancilla dephasing and relaxation errors as typical examples.
[1] W. -L. Ma et al., Phys. Rev. Lett. 125, 110503 (2020).