Path-Independent Quantum Gates with Noisy Ancilla

Ancilla systems are often indispensable to universal control of a nearly isolated quantum system. However, ancilla systems are typically more vulnerable to environmental noise, which limits the performance of such ancilla-assisted quantum control. To address this challenge of ancilla-induced decoherence, we propose a general framework that integrates quantum control and quantum error correction, so that we can achieve robust quantum gates resilient to ancilla noise. We introduce the path independence criterion for fault-tolerant quantum gates against ancilla errors. As an example, a path-independent gate is provided for superconducting circuits with a hardware-efficient design.

Figure 1

(a) Schematic of a central system with good coherence coupled to an ancilla system with poor coherence. The ancilla dephasing rate $\kappa$ and relaxation rate $\gamma$ of are much larger than the decoherence rate $\zeta$ of the central system. (b),(c) For PI control, the central system undergoes a unitary evolution $U_{fi}$ for the ancilla starting from state $|i⟩$ and finally measured in $|f⟩$, regardless of the ancilla paths induced by the control and ancilla error events.

Figure 2

Schematic of ancilla evolution paths with different kinds of ancilla errors. (a) In the paths with blue line, the ancilla goes from $|i⟩$ to $|f⟩$ without any ancilla error and the central system gate is $U_{fi}$. In the paths with green lines, the ancilla suffers two dephasing errors $|a⟩⟨a|$ and at $|c⟩⟨c|$, while the central system gate is still $U_{fc}{U}_{ca}{U}_{ai}=U_{fi}$, independent of the dephasing error times. (b) In the paths with red lines, the ancilla suffers a relaxation error $|b⟩⟨a|$ (dashed red arrow lines) in the NAS spanned by $\{|a⟩,|b⟩\}$ (purple-shaded region) with the unitary gate as $U_{eb}{U}_{ai}$. In the paths with yellow line, the ancilla suffers two additional dephasing errors $|a⟩⟨a|$ and $|c⟩⟨c|$ but with the same unitary gate as that for a single relaxation error. The solid (hollow) circles represent the initial and final (intermediate) ancilla states, and the red dashed arrows represent the ancilla relaxation errors. Here we adopt the interaction picture associated with $H_0^{\prime }(t)$.