Propagation of generalized Pauli errors in qudit Clifford circuits

It is important for performance studies in quantum technologies to analyze quantum circuits in the presence of noise. We introduce an error probability tensor, a tool to track generalized Pauli error statistics of qudits within quantum circuits composed of qudit Clifford gates. Our framework is compatible with qudit stabilizer quantum error-correcting codes. We show how the error probability tensor can be applied in the most general case, and we demonstrate an error analysis of bipartite qudit repeaters with quantum error correction. We provide an exact analytical solution of the error statistics of the state distributed by such a repeater. For a fixed number of degrees of freedom, we observe that higher dimensional qudits can outperform qubits in terms of distributed entanglement.

Figure 1

A quantum circuit with noise modeled by error channels ${\mathcal{F}}_i$, after ideal unitary gates $U_i$ (top), is mathematically equivalent to an ideal quantum circuit followed by some error channel $\mathcal{E}$ (bottom).

Figure 2

Propagation rules of generalized Pauli errors for the $F$, $M\left(l\right)$, cx, and cz gate. Across the $F$ gate, $X$ propagates into $Z$, and $Z$ into $X^{-1}$. Across two-qudit gates, some single-qudit errors propagate into two-qudit errors; e.g., $X\otimes \mathbb{1}$ propagates into $X\otimes Z$ across the cz gate.

Figure 3

Across a sequence of cx${}^{a_i}$ and cz${}^{b_i}$ gates, the error $X^j{Z}^k\otimes X^{\mathbf{l}}{Z}^{\mathbf{m}}$ propagates into $X^j{Z}^{k+\mathbf{l}·\mathbf{b}-\mathbf{m}·\mathbf{a}}\otimes X^{\mathbf{l}+j\mathbf{a}}{Z}^{\mathbf{m}+j\mathbf{b}}$. This defines the automorphism ${\pi }_{C(\mathbf{a},\mathbf{b})}$ in Eq. (17):

Figure 4

A quantum circuit diagram representation of the qudit repeater line between Alice and Bob. Intermediate repeater stations are introduced to shorten the transmission distance of the qudits. All outcomes $c_i$ of the $X$ measurement at repeater $i$ are transmitted to Bob (who counts as repeater $N$) for the Pauli-frame recovery of $|\mathrm{\Psi }\rangle$.

Figure 5

Error statistics, Eq. (29), of a state distributed with a ququint repeater line encoded with the $[[5,1,3{]]}_5$ quantum polynomial code. The entries of the (reduced) error probability tensor are plotted as functions of the number of repeater stations, $N$, where transmission, measurement, gate, and storage error rates are set to $f_{\mathrm{T}}=0.05$, $f_{\mathrm{M}}=0.01$, $f_{\mathrm{G}}=0.001$, and $f_{\mathrm{S}}=0.0001$, respectively. There are 1 green (circles; no errors), 4 red (triangles up; $X_{\mathrm{B}}^r$ error), 4 blue (triangles down; $Z_{\mathrm{B}}^s$ error), and 16 gray identical curves (squares; $X_{\mathrm{B}}^r{Z}_{\mathrm{B}}^s$ error). The pink curve (dots) shows the difference between red and blue curves.

Figure 6

The fidelity, $F\left(k_{\max }\right)$, of the distributed state (top), and the probability of successfully distributing the state (bottom) as a function of unnoticed $\left(f_{\mathrm{T}}\right)$ and noticed $\left(f_{\mathrm{abs}}\right)$ transmission error rates $f:=f_{\mathrm{T}}=f_{\mathrm{abs}}$ for different abortion strategies. The repeater line has $N=2$ stations, and measurement, gate, and storage error rates are set to $f_{\mathrm{M}}=0.01$, $f_{\mathrm{G}}=0.001$, and $f_{\mathrm{S}}=0.0001$, respectively. The gray (dotted) curve shows the performance of an unencoded qudit with $D=13$. The other curves show the performance of the $[[13,1,7{]]}_{13}$ quantum polynomial code for different abortion conditions $k_{\max }$. If at any of the repeater stations the number of qudits marked as “?” is greater than $k_{\max }$, the protocol is aborted.

Figure 7

The logarithmic negativity ${\mathcal{E}}_{\mathcal{N}}\left(\rho \right)$ of the state $\rho$ distributed via a repeater line with $N=50$ stations and encoded with a $[[2d-1,1,d{]]}_D$ code for varying qudit dimension $D$ and code distance $d$. The ambient Hilbert space of a logical qudit is $\mathcal{H}$; i.e., if for example $D=100$ and $dim\mathcal{H}={10}^{70}$, one logical qudit is encoded into 35 physical qudits. The transmission, measurement, gate, and storage error rates are set to $f_{\mathrm{T}}=0.05$, $f_{\mathrm{M}}=0.01$, $f_{\mathrm{G}}=0.001$, and $f_{\mathrm{S}}=0.0001$, respectively. The dots represent parameters for which quantum polynomial codes exist, namely $1\le d\le (D+1)/2$ for every fixed prime $D$. The crosses represent the codes listed in Table 1. The white curve exemplary shows codes with constant code distance $d=15$.

Figure 8

Adapted from Fig. 3 of Ref. [15]. Sources of undetected errors on the measurement outcome at repeater $i\in \{2,...,N\}$. If an $X$ error occurs on qudit $i-1$ (one gate, one transmission), it induces a $Z$ error on qudit $i$ across the cz gate. If this happens, or $Z$ errors directly occur on qudit $i$ (two gates, one transmission, one measurement), the measurement outcome might be erroneous.