Correcting coherent quantum errors by going with the flow

Abstract

The performance of a given quantum error correction (QEC) code depends upon the noise model that is assumed. Independent Pauli noise, applied after each quantum operation, is a simplistic noise model that is easy to simulate and understand in the context of stabilizer codes. Although such a noise model is artificial, it is equivalent to independent, random, unbiased qubit rotations. What about spatially or temporally correlated qubit rotations? Such a noise model is applicable to global operations (e.g., NMR or ESR), common control sources (e.g., lasers), or slow drift (e.g., charge or magnetic noise) in various qubit technologies. In the worst case, such errors can combine constructively and result in a post-correction failure rate that increases with the number of error correction cycles. However, we show that this worst case does not generally arise unless taking active corrective actions while performing QEC. That is, by employing virtual Pauli frame updates ("passive" error correction) rather than physical corrections ("active" error correction), coherent errors do not compound appreciably. Starting in a random Pauli frame is also advantageous. In fact, through perturbation theory arguments and supporting numerical simulations, we show that the logical qubit performance beyond distance 3 for correlated single-qubit Hamiltonian noise models (i.e., global errant qubit rotations), when employing these "lazy" strategies, essentially matches the performance of Pauli noise model with the same process fidelity (fidelity after one application). In a more general circuit model of noise, correlations may add constructively within syndrome extraction rounds but Pauli frame randomization from passive error correction mitigates this effect across multiple rounds.

Figures (5)

• Figure 1: A distance three medial surface code. The qubits are on the vertices and the plaquettes represent stabilizers. Each of the color represents an $X$-type and a $Z$-type stabilizer. The particular choice is arbitrary.
• Figure 2: For a $d=3$ medial surface code, error correction performance due to continuous/discrete $X$ noise with/without codespace randomization and with/without Pauli-$Z$ flips to induce a random walk in the codespace. Where present, the Pauli-$Z$ flips are tracked and perfectly corrected. This is what the angle brackets in the legend are meant to indicate. The sample uncertainty for each curve is comparable to its thickness.
• Figure 3: For a $d=3$ medial surface code, error correction performance due to continuous/discrete $X$ and $Z$ noise with/without codespace randomization. The $X$-noise only analogs from Fig. (\ref{['fig:msc3']}) are re-plotted but multiplied by two for comparison. The error bars indicate sample uncertainty estimates (where it is significant relative to the thickness of the respective curve).
• Figure 4: For a $d=5$ repetition code, error correction performance due to continuous/discrete $X$ noise with/without Pauli-$Z$ flips. Also, for comparison, leading order analytical results of Eq. (\ref{['eq:pfnsigma0']}) (solid gray) and Eq. (\ref{['eq:pf_pLN2']}) (dashed gray) with $\mu_{\epsilon} = \epsilon$, $\sigma_{\epsilon} = 0$ and using $a_5 = {5 \choose 3} = 10$, $b_5 = {5 \choose 3} + {5 \choose 4} + {5 \choose 5} = 16$ for the repetition code. Pauli-$Z$ flips have no effect on the fidelity of states in this non-quantum code, so these do not need to be corrected. They do, however, flip the relative signs of $\left\lvert\bar{\psi}\right\rangle$ and $\bar{X} \left\lvert\bar{\psi}\right\rangle$, providing a random walk effect to suppress the penalty of continuous noise. Error bars indicate sample uncertainty estimates (where it is applicable and significant relative to the thickness of the respective curve).
• Figure 5: For a $d=5$ repetition code, error correction performance due to global/local randomly sampled $X$ noise with/without temporal correlations and with/without Pauli-$Z$ flips. Local noise cases are multiplied by $5!!$ to make a proper comparison with global noise for which a $d!!$ penalty is understood according to Sec. \ref{['sec:passive_gn']} and Sec. \ref{['sec:tcgn']}. The sample uncertainty for each curve is comparable to its thickness.