Quantum process tomography of unitary and near-unitary maps

We study quantum process tomography given the prior information that the map is a unitary or close to a unitary process. We show that a unitary map on a $d$-level system is completely characterized by a minimal set of $d^2+d$ elements associated with a collection of POVMs, in contrast to the $d^4-d^2$ elements required for a general completely positive trace-preserving map. To achieve this lower bound, one must probe the map with particular sets of $d$ pure states. We further compare the performance of different estimators inspired by compressed sensing, to reconstruct a near-unitary process from such data. We find that when we have accurate prior information, an appropriate compressed sensing method reduces the required data needed for high-fidelity estimation, and different estimators applied to the same data are sensitive to different types of noise. Convex optimization inspired by the techniques of compressed sensing can therefore be used both as indicators of error models and to validate the use of the prior assumptions.

Figure 1

Fidelity between a unitary map on a $\left(d=5\right)$-dimensional Hilbert space and the LS estimate of the process matrix (averaged over 50 Haar-random unitary maps) as a function of the number of input states in the absence (a) and in the presence (b) of statistical noise. The error bars represent the standard deviation. The data represented in the dotted (red) line are obtained from an informationally complete measurement on the states produced by applying the unitary map to all $d^2$ input states in the order specified in Eq. (9). In the solid (blue) line the first five input states are those of Eq. (20), and the remaining states are chosen from Eq. (9) in an arbitrary order. In the dashed (green) line the first five input states are those of Eq. (19) and the remaining state from Eq. (10). Unit fidelity is obtained after $d=5$ input states in a UIC set. In the presence of statistical error, as seen in (b), the main features of the fidelity remain qualitatively the same—by choosing the correct set of input states we obtain a reliable reconstruction after $d$ input states, up to statistical error. The statistical fluctuations in the data are modeled by Eq. (23), a normal distribution, with $\sigma ={10}^{-4}$.

Figure 2

The fidelity between the estimate and the applied map as a function of the fidelity between the target map and the applied map for the case of coherent (a) and incoherent (b) errors. The error bars represent the standard deviation. The estimates are obtained with data from only the five UIC input states of Eq. (20). Each data point in the plots is obtained by an average over 50 random target unitary maps each with a random error map. (a) Coherent errors: The applied map is given by $U_{\mathrm{a}}=U_{\mathrm{err}}{U}_{\mathrm{t}}$, where $U_{\mathrm{t}}$ is a Haar-random unitary map, $U_{\mathrm{err}}=e^{i\eta H},\eta \in [0,3]$, and $H$ is a random Hermitian matrix selected by the Hilbert-Schmidt measure, normalized to $\mathrm{Tr}H=1$. While the fidelities of the estimator based on the ${\mathrm{CS}}_{\mathrm{Tr}}$ and the LS minimizations remain more or less constant, the fidelity of the estimator based on the ${\mathrm{CS}}_{{\ell }_1}$ minimization decreases as the fidelity between the applied and the target maps decreases. This is an indicator of the sensitivity of ${\mathrm{CS}}_{{\ell }_1}$ to coherent errors. (b) Incoherent errors: The applied map is given by Eq. (26). The numerical simulation was done by choosing at random values of $\xi$ from a uniform distribution on [0,0.6]. For each value we generated Haar-random unitary target maps $U_{\mathrm{t}}$ and 25 randomly selected Kraus operators from the Hilbert-Schmidt measure as prescribed in Ref. [34]. The estimate based on the ${\mathrm{CS}}_{{\ell }_1}$ minimization performs better than the estimate based on the ${\mathrm{CS}}_{\mathrm{Tr}}$ and the LS minimization procedures, and thus, the ${\mathrm{CS}}_{{\ell }_1}$ estimator is more robust to incoherent errors of the form of Eq. (26) than either the ${\mathrm{CS}}_{\mathrm{Tr}}$ or the LS estimator.

Figure 3

The fidelity between the estimate and the applied map, $F(\widehat{\chi },{\chi }_{\mathrm{a}})$ (a), and the estimate and the target map, $F(\widehat{\chi },{\chi }_{\mathrm{t}})$, inset (b), as a function of the number of input states, averaged over 50 applied processes, using different estimators, and under different error models for the applied map. Error bars represent the standard deviation. Each column corresponds to a different magnitude of implementation error, represented by the fidelity between the applied and target maps $F({\chi }_{\mathrm{t}},{\chi }_{\mathrm{a}})$. Top row: Coherent errors as in Fig. 2. For a low error magnitude$F({\chi }_{\mathrm{t}},{\chi }_{\mathrm{a}})=0.97\pm 0.005$, the three estimators return a high-fidelity estimate with the applied and the target maps. The ${\mathrm{CS}}_{{\ell }_1}$ estimator returns a reliable estimate with the information obtained from a single input state, while the ${\mathrm{CS}}_{\mathrm{Tr}}$ and the LS minimization procedures reliably estimate the map after five UIC input states. The ${\mathrm{CS}}_{\mathrm{Tr}}$ and the LS estimators are robust to coherent error and perform essentially the same as the error level increases; here, $F({\chi }_{\mathrm{t}},{\chi }_{\mathrm{a}})=0.90\pm 0.005$ and $F({\chi }_{\mathrm{t}},{\chi }_{\mathrm{a}})=0.83\pm 0.005$. A noticeable slope in the fidelity as a function of the number of input states for the ${\mathrm{CS}}_{{\ell }_1}$ estimator, and a sharp kink in this curve for the LS and ${\mathrm{CS}}_{\mathrm{Tr}}$ estimators, are indicators of the coherent-error type. Bottom row: Incoherent errors as in Fig. 2. The ${\mathrm{CS}}_{{\ell }_1}$ estimator returns a reliable estimate with information for higher error magnitudes; here, $F({\chi }_{\mathrm{t}},{\chi }_{\mathrm{a}})=0.90\pm 0.005$ and $F({\chi }_{\mathrm{t}},{\chi }_{\mathrm{a}})=0.83\pm 0.005$ with the fidelity relatively constant after a single input state. In contrast, the ${\mathrm{CS}}_{\mathrm{Tr}}$ and the LS estimators are more sensitive to an increase in the magnitude of incoherent errors. As the error level increases, the sharp transition in the fidelity as a function of the number of input states becomes smoother as the noise increases. These features of the fidelity curves for the three estimators are indicators of the incoherent-error type.