Bacon-Shor code with continuous measurement of noncommuting operators

We analyze the operation of a four-qubit Bacon-Shor code with simultaneous continuous measurement of noncommuting gauge operators. The error syndrome in this case is monitored via time-averaged cross-correlators of the output signals. We find the logical error rate for several models of decoherence, and also find the termination rate for this quantum error detecting code. The code operation is comparable to that based on projective measurements when the collapse time scale due to continuous measurements is an order of magnitude less than the time period between the projective measurements. An advantage of the continuous-measurement implementation is the absence of time dependence in the code operation, with passive continuous monitoring of the error syndrome.

Figure 1

The four-qubit Bacon-Shor code contains four physical qubits (shown by circles) and is based on measurement of four operators (dashed lines): $X_1{X}_2, X_3{X}_4, Z_1{Z}_3$, and $Z_2{Z}_4$, called gauge operators. In the conventional code operation they are measured in two steps, thus separating noncommuting pairs (Fig. 2), while in this paper we also analyze the case when all four operators are measured at the same time continuously.

Figure 2

One cycle of the code operation with projective measurement of gauge operators: at step 1 the operators $Z_1{Z}_3$ and $Z_2{Z}_4$ are measured instantaneously, and at step 2 the operators $X_1{X}_2$ and $X_3{X}_4$ are measured (also instantaneously). The duration of each time step is $\mathrm{\Delta }t$, so the cycle duration is $2\mathrm{\Delta }t$. In the absence of errors, product of two measurement results ($\pm 1$) at each step is $+1$.

Figure 3

Encoding circuit, which produces the state $|z+\rangle$ from the state ${|\psi \rangle }_{\mathrm{L}}|0\rangle |0\rangle |0\rangle$. Decoding can be done by running the same circuit backwards after step-1 measurement with the outcome “$++$.”

Figure 4

State evolution and errors with continuous measurement. Thick-line circle illustrates diffusive evolution of the legitimate state [Eq. (39)] within two-dimensional subspace of the gauge qubit due to measurement. The 12 single-qubit errors move the legitimate state to eight other two-dimensional subspaces in ${\mathcal{Q}}_X, {\mathcal{Q}}_Y$, and ${\mathcal{Q}}_Z$ with the mapping described by Eq. (65) (see operators written inside the circles, the corresponding gauge qubit operations are in parentheses). If the second error occurs before the first error is detected (i.e., within the response time $T_{\mathrm{R}}$), it may move the state back to ${\mathcal{Q}}_0$, so that both errors remain undetected (dotted line separates ${\mathcal{Q}}_0$ from error subspaces). Depending on a combination of the two errors, the logical qubit ${|\alpha ,\beta \rangle }_{\mathrm{L}}$ may be affected by logical errors $X_{\mathrm{L}}, Y_{\mathrm{L}}$, or $Z_{\mathrm{L}}$. The solid-line arrows illustrate a scenario of errors $X_1$ and $X_3$, leading to logical error $X_{\mathrm{L}}$; the dashed-line arrows correspond to errors $Z_4$ and $Z_1$, leading to $Z_{\mathrm{L}}$.

Figure 5

The false alarm rate ${\gamma }_{\mathrm{f}\mathrm{al}}$ as a function of the response time $T_{\mathrm{R}}^{\mathrm{r},\mathrm{e}}$ (normalization involves the collapse time ${\tau }_{\mathrm{m}}$). Numerical results for the rectangular and exponential correlator integrations are shown by red squares and blue crosses, respectively. The (upper) red and (lower) blue dashed lines represent the analytics [Eqs. (99) and (98)]. The solid lines include correction factors in the analytical formulas (see main text). The inset shows the termination rate ${\gamma }_{\mathrm{term}}$ as a function of $T_{\mathrm{R}}^{\mathrm{r},\mathrm{e}}$ (the same horizontal axis as in the main panel) for the case when qubits 1 and 2 are subject to pure dephasing with ${\mathrm{\Gamma }}_{\phi ,1}={\mathrm{\Gamma }}_{\phi ,2}={10}^{-3}{\mathrm{\Gamma }}_{\mathrm{m}}$ (no decoherence in the main panel); the horizontal dashed line corresponds to single-qubit errors ${\gamma }_{\mathrm{term}}=({\mathrm{\Gamma }}_{\phi ,1}+{\mathrm{\Gamma }}_{\phi ,2})/2$. We used $\mathrm{\Theta }=1, \eta =1$, and ${\tau }_{\mathrm{c}}={\tau }_{\mathrm{c},\mathrm{opt}}$.

Figure 6

The rate ${\gamma }_Z$ of the logical $Z$ error as a function of the response time $T_{\mathrm{R}}^{\mathrm{e}}$ for the case when qubits 1 and 2 are subject to pure dephasing with ${\mathrm{\Gamma }}_{\phi ,1}={\mathrm{\Gamma }}_{\phi ,2}={10}^{-3}{\mathrm{\Gamma }}_{\mathrm{m}}$. The crosses show numerical results, the solid line is the analytical result (72). The inset shows the diagonal elements of the process matrix (${\chi }_{XX}, {\chi }_{YY}$, and ${\chi }_{ZZ}$) as functions of time $T$ for $T_{\mathrm{R}}^{\mathrm{e}}=20.8{\tau }_{\mathrm{m}}$. The rate ${\gamma }_Z$ is calculated from the slope of ${\chi }_{ZZ}\left(T\right)$. We used $\mathrm{\Theta }=1, \eta =1, {\tau }_{\mathrm{c}}={\tau }_{\mathrm{c},\mathrm{opt}}$, and the exponential kernel for the correlators.

Figure 7

The $\chi$-matrix elements and logical error rate for the case of qubit energy relaxation (amplitude damping). In panels (a) and (b) the energy relaxation affects only qubits 1 and 3, ${\mu }_1={\mu }_3={10}^{-3}{\mathrm{\Gamma }}_{\mathrm{m}}$; in panels (c) and (d) it affects only qubits 1 and 2, ${\mu }_1={\mu }_2={10}^{-3}{\mathrm{\Gamma }}_{\mathrm{m}}$. Panels (a) and (c) show numerically calculated components of the $\chi$ matrix as functions of time $T$; the diagonal elements are depicted by solid lines. The only nonzero off-diagonal elements are ${\chi }_{IZ}$ (real) and ${\chi }_{XY}$ (imaginary) in panel (c), shown by (almost coinciding) dashed green lines. Panels (b) and (d) show, respectively, the logical error rates ${\gamma }_X$ and ${\gamma }_Z$ as functions of the response time $T_{\mathrm{R}}^{\mathrm{e}}$. The crosses represent numerical results and the solid lines represent analytical formulas (74) and (76). We used $\mathrm{\Theta }=1, \eta =1, {\tau }_{\mathrm{c}}={\tau }_{\mathrm{c},\mathrm{opt}}$, and the exponential kernel for the correlators.

Figure 8

Tradeoff between the logical error rate and termination rate for the Bacon-Shor code with continuous measurements. Thick red line: ratio ${\gamma }_{\mathrm{L},\mathrm{cont}}/{\gamma }_{\mathrm{L},\mathrm{proj}}$ of the logical error rates for continuous and projective measurements [Eq. (113)], as a function of the correlator integration time $T_{\mathrm{c}}^{\mathrm{e}}$ normalized by the projective-measurement half-cycle time $\mathrm{\Delta }t$. Thin blue lines: ratio ${\gamma }_{\mathrm{term},\mathrm{cont}}/{\gamma }_{\mathrm{term},\mathrm{proj}}$ of the termination rates for the continuous and projective measurements [Eq. (114)] for several values of the collapse (“measurement”) time ${\tau }_{\mathrm{m}}$: ${\tau }_{\mathrm{m}}/\mathrm{\Delta }t=1$, 0.3, 0.1, and 0.03 (from right to left) and quantum efficiency $\eta =1$ (solid lines) or $\eta =0.5$ (dashed lines). We assume ${\mathrm{\Gamma }}_{\mathrm{d}}={10}^{-4}/\mathrm{\Delta }t$ and $\mathrm{\Theta }=1$.