Direct characterization of quantum dynamics with noisy ancilla

We present methods for the direct characterization of quantum dynamics in which both the principal and ancilla systems undergo noisy processes. Using a concatenated error detection code, we discriminate between located and unlocated errors on the principal system in what amounts to filtering of ancilla noise. The example of composite noise involving amplitude damping and depolarizing channels is used to demonstrate the method, while we find the rate of noise filtering is more generally dependent on code distance. Our results indicate the accuracy of quantum process characterization can be greatly improved while remaining within reach of current experimental capabilities.

Figure 1

Outline of a quantum circuit used to directly characterize the dynamics of a principal system P. The principal system is first entangled with the ancilla system A before being subject to some noisy dynamics. The process matrix is then constructed from an ensemble of stabilizer measurements (for details see Sec. 2a and Fig. 2). (a) illustrates a conventional DCQD circuit, where the P and A are maximally entangled in a Bell state $|{\mathrm{\Phi }}^{+}\rangle$ and characterization assumes a noiseless ancilla. (b) illustrates a concatenated DCQD circuit, with an initial entangled state $|0\rangle$ [Eq. (8)], which supports errors on the ancilla subsystem.

Figure 2

Schematic of the DCQD process. Begin with the state $\rho =|0\rangle \langle 0|$ initialized in the code space of ${\mathcal{S}}_1$. Next, subject $\rho$ to some dynamics resulting in $\mathcal{E}\left(\rho \right)$. A preprocessing operation $O_j=\{\mathbb{I},U_j,P_j\}$ is then applied just prior to the stabilizer generators yielding an error syndrome $e_i$. An ensemble of syndrome measurements given the preprocessing $O_j$ is used to deduce part of the process matrix ${\chi }_{mn}\left(O_j\right)=\{{\chi }_{ii},\text{Im}{\chi }_{Ji},\text{Re}{\chi }_{Ji}\}$ as described in Eqs. (4, 5, 6).

Figure 3

Simulated AD channel process matrices constructed from ensembles of syndrome measurements according to the DCQD procedure, namely Eqs. (4, 5, 6). Probabilities are determined from ${10}^6$ Monte Carlo events using the numerical parameters $\gamma =.4, p=.1$ were used for the amplitude damping and depolarizing channels respectively. The real and imaginary parts for $\chi$ constructed with a noiseless ancilla is given in (a), (b) and its difference from the theoretical value appears in (c). A noisy ancilla reduces the accuracy of the DCQD procedure as seen in (d)–(f) for which a standard $\left[\right[4,0,2\left]\right]$ has been used. As seen in (g)–(i), weight-one ancilla errors are mitigated by utilizing a concatenated $\left[\right[6,0,2\left]\right]$ code. The constructed $\chi$ matrix is characterized by a high degree of fidelity, $F({\mathcal{E}}_{\mathbf{P}}^{AD}\left(\rho \right),{\mathcal{E}}^{\left[\right[6,0,2\left]\right]}\left(\rho \right))=.9884$.

Figure 4

Probabilities of located error syndrome to occur in the presence of a noiseless principal system P in the presence of independent depolarizing channels acting on the ancilla A with probability $p$. The blue dashed line [${\mathcal{P}}_{\mathbb{1}}={(1-p)}^4$] shows the probability for the identity operator to occur while the probability for the identity syndrome to appear in the Monte Carlo simulation is given by the blue circles (denoted by $p_{\mathbb{1}}$). Stabilizer operators therefore occur with probability $\mathrm{\Delta }{p}_{\mathbb{1}}=p_{\mathbb{1}}-{\mathcal{P}}_{\mathbb{1}}$. Orange circles ($p_{00}$) denote the rate at which the remaining located (00) syndromes occur. Green circles (dashed lines) denote the simulated (theoretical) failure rate $p_F=p_{00}+\mathrm{\Delta }{p}_{\mathbb{1}}$ (${\mathcal{P}}_{\mathbb{1}}=p_2/3+2p_3/9+21p_4/81$, see text for derivation) indicating a located error syndrome from Table 1 is caused by an operator different than $E_i$.