Feedforward suppression of readout-induced faults in quantum error correction

Abstract

Qubit measurements in quantum devices involve various types of errors, including erroneous state determination, correlated preparation errors and measurement-induced leakage from the computational states. We propose a feedforward protocol to reduce readout-induced faults, applicable for qubits with errors biased between the different states, in settings like quantum error correction with repeated measurement cycles. The method consists of an adaptive readout sequence conditioned on each check qubit's readout result from the previous cycle, which is optimized for the expected measured state. Focusing on a simple realization of conditionally flipping (by an X gate) the state of check qubits before their measurement, we investigate the effect of such state-dependent errors using simulations in the setup of a low-density parity check code. We show that the suggested protocol can reduce both logical errors and decoding time, two important aspects of fault-tolerant quantum computations.

Figures (7)

• Figure 1: A schematic depiction of the adaptive readout sequence, based on the outcome of each qubit's previous measurement. (a) As an example focused on two check qubits of a stabilizer-based error correction code, following the measurement step shown on the left and in the absence of any further error, the first and second check qubits are again expected to be measured in the states $\ket{1}$ and $\ket{0}$ respectively (at the end of the syndrome cycle "SC"). The measurement type (indicated by color) applied to each qubit is conditioned and adapted to its expected state. (b) A possible realization of the adapted measurement is a conditional flip (an X gate) applied to each qubit that is expected to be in $\ket{1}$; in this example, setting the first qubit back to $\ket{0}$ while the second one is left untouched. A classical NOT on the first qubit's measured bit would recover the expected syndrome at the end.
• Figure 2: Results from decoding of simulated syndrome measurements as a function of the backaction error probability (corresponding to leaked check qubits inducing independent Z and X errors on the coupled data qubits), with the data comparing the flip protocol (denoted as "w/ flip") vs. its absence (denoted as "w/o flip"), for error rates as in Eqs. \ref{['eq:fixed_rates']}-\ref{['eq:flip_rates']}. (a) The mean number of BP iterations in each decoding execution. The decoding time is significantly reduced with the flip protocol due to fewer BP iterations being needed. (b) The logical error rate per syndrome cycle, showing a significant reduction to a nearly constant level. See the text for a detailed discussion.
• Figure 3: Decoding of simulated syndrome measurement circuits as a function of the leakage rate, comparing the flip protocol vs. its absence, with error rates in Eqs. \ref{['eq:fixed_rates']}-\ref{['eq:leak_rates']}. (a) The fraction of BP attempts that converged. The increased convergence seen with the flip protocol dramatically reduces the decoding time by eliminating the need for secondary decoder calls. (b) The logical error rate per syndrome cycle, showing again a significant reduction with the flip.
• Figure 4: Decoding results when varying the seepage rate (between 0.2 and 0.7 in increments of magnitude 0.1) simultaneously with the error in the long-idle gate (between 0.005 and 0.01 in steps of 0.001), with the other error rates as in Eqs. \ref{['eq:fixed_rates']}-\ref{['eq:readout_rates']}. (a) The mean number of BP iterations. (b) The logical error per syndrome cycle. Modeling a tradeoff between the efficacy of a leakage removal process and its cost in terms of the prolonged idling of data qubits, the non-monotonic behavior of the decoding quantities without the flip emphasizes the importance of having a variety of tools at hand for optimizing complementing approaches.
• Figure 5: The dependence of decoding characteristics on preparation errors occurring with the probability $p_{\rm corr}$ conditioned on a readout error having occurred in the preceding readout step. The error parameters are as in Eqs. \ref{['eq:fixed_rates2']}-\ref{['eq:readout_rates2']} and $p_{\rm id,l}=0.005$. (a) The mean number of BP iterations per decoding window. (b) The fraction of BP algorithm runs that converged. (c) The mean population of check qubits in the ground state just prior to the syndrome cycles' readout step. (d) The logical error rate per cycle. A significant increase in decoding errors is observed for even small residual preparation errors. See the text for a discussion.