Dynamical quantum error correction of unitary operations with bounded controls

Dynamically corrected gates were recently introduced [K. Khodjasteh and L. Viola, Phys. Rev. Lett. 102, 080501 (2009)] as a tool to achieve decoherence-protected quantum gates based on open-loop Hamiltonian engineering. Here, we further expand the framework of dynamical quantum error correction, with emphasis on elucidating under what conditions decoherence suppression can be ensured while performing a generic target quantum gate, using only available bounded-strength control resources. Explicit constructions for physically relevant error models are detailed, including arbitrary linear decoherence and pure dephasing on qubits. The effectiveness of dynamically corrected gates in an illustrative non-Markovian spin-bath setting is investigated numerically, confirming the expected fidelity performance in a wide parameter range. Robustness against a class of systematic control errors is automatically incorporated in the perturbative error regime.

Figure 1

(Color online) The Cayley graph $G\left(\mathcal{G},\Gamma \right)$ with $\mathcal{G}={\mathbb{Z}}_2\otimes {\mathbb{Z}}_2={\left\{g_i\right\}}_{i=1}^4=\left\{\left(0,0\right),\left(0,1\right),\left(1,0\right),\left(1,1\right)\right\}$ and $\Gamma =\left\{h_1,h_2\right\}=\left\{g_2,g_3\right\}=\left\{\left(0,1\right),\left(1,0\right)\right\}$, with an Eulerian path of length $L=8$ depicted. This group with the Cayley graph and the Eulerian cycle depicted here will be used in Sec. 4C to implement DCG constructions for the linear decoherence model.

Figure 2

(Color online) Control profiles for implementing two gates $I_Q=Q^{\prime }Q$ and $Q_{1/2}$ that correspond to a target identity gate and a target gate $Q$, with equal first-order error.

Figure 3

(Color online) Left: Cayley graphs for ${\mathbb{Z}}_2$ (top) and ${\mathbb{Z}}_2\otimes {\mathbb{Z}}_2$ (bottom), relevant to pure dephasing and arbitrary linear decoherence, respectively. Representative Eulerian cycles and the corresponding EDD sequences are depicted. Right: Modified Cayley graphs for DCG constructions based on ${\mathbb{Z}}_2$ (top) and ${\mathbb{Z}}_2\otimes {\mathbb{Z}}_2$ (bottom), respectively. Relevant Eulerian paths and the corresponding DCG sequences are depicted.

Figure 4

(Color online) Improvement ratio as a function of (minimum) gating interval $\tau$ for the two-qubit target gate $W_{\pi /4}^{\left(1,2\right)}$. The “no-improvement” (horizontal) line $r=1$ is included for reference in this and all other figures that use $r$ as a metric for performance. The internal-bath Hamiltonian is kept fixed at $\Gamma =1$, whereas $A/\Gamma$ takes the value ${10}^2$ (brown diamonds), ${10}^1$ (green squares), and 1 (blue circles). Inset: Infidelity error for the corresponding DCG implementation, $1-f_{W_{\text{DCG}}}$, as a function of $\tau$, for the same parameters used in the main panel.

Figure 5

(Color online) Isofidelity contour plots for the implementation of the single-qubit target gate $R_{\pi /4}^{\left(1\right)}$ for fixed (minimum) gating time $\tau =1$. Top: Primitive uncorrected implementation. Bottom: DCG implementation. In both cases, the integers labeling each curve give (the logarithm of) the corresponding infidelity error, with darker colors corresponding to higher fidelity values.

Figure 6

(Color online) Infidelity, $1-f$, as a function of intrabath–system-bath coupling scales for implementing the single-qubit gate $R_{\pi /4}^{\left(1\right)}$ at fixed $A=1$. Primitive vs DCG implementations are contrasted for two different gating times $\tau =0.1$ (top) and $\tau =0.01$ (bottom).

Figure 7

(Color online) Effect of systematic over-rotation errors in the DCG improvement ratio for $Q=R_{\pi /4}^{\left(1\right)}$ as a function of gating interval $\tau$. In both cases, the control error strength is parametrized by $\epsilon ∊\left[0,1\right]$ and we have set $A=1$, $\Gamma =0$.