Calibrated Decoders for Experimental Quantum Error Correction

Edward H. Chen, Theodore J. Yoder, Youngseok Kim, Neereja Sundaresan, Srikanth Srinivasan, Muyuan Li, Antonio D. Córcoles, Andrew W. Cross, and Maika Takita

Phys. Rev. Lett. 128, 110504 – Published 17 March 2022

Abstract

Arbitrarily long quantum computations require quantum memories that can be repeatedly measured without being corrupted. Here, we preserve the state of a quantum memory, notably with the additional use of flagged error events. All error events were extracted using fast, midcircuit measurements and resets of the physical qubits. Among the error decoders we considered, we introduce a perfect matching decoder that was calibrated from measurements containing up to size-four correlated events. To compare the decoders, we used a partial postselection scheme shown to retain ten times more data than full postselection. We observed logical errors per round of $2.2 \pm 0.1 \times 10^{- 2}$ (decoded without postselection) and $5.1 \pm 0.7 \times 10^{- 4}$ (full postselection), which was less than the physical measurement error of $7 \times 10^{- 3}$ and therefore surpasses a pseudothreshold for repeated logical measurements.

Received 14 October 2021
Accepted 10 February 2022

DOI:https://doi.org/10.1103/PhysRevLett.128.110504

Physics Subject Headings (PhySH)

Quantum benchmarking Quantum control Quantum correlations in quantum information Quantum error correction Quantum memories Surface code quantum computing Topological quantum computing

Superconducting qubits

Quantum Information, Science & TechnologyGeneral Physics

Authors & Affiliations

Edward H. Chen ^1,*, Theodore J. Yoder ^2,†, Youngseok Kim², Neereja Sundaresan², Srikanth Srinivasan ², Muyuan Li ², Antonio D. Córcoles ², Andrew W. Cross ², and Maika Takita ²

¹IBM Quantum, Almaden Research Center, San Jose, California 95120, USA
²IBM Quantum, T.J. Watson Research Center, Yorktown Heights, New York 10598, USA

^*Cospokesperson. ehchen@ibm.com
^†Cospokesperson. ted.yoder@ibm.com

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Vol. 128, Iss. 11 — 18 March 2022

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Figure 1
(Color) (a) Experiments were performed on ibmq_kolkata, which had 27 qubits connected in a heavy hexagon (HH) topology. The seven qubits used for the $⟦ 4, 1, 2 ⟧$ code are colored yellow, blue, and pink. (b) The code layout indicates a single weight-four, $X$ stabilizer (pink), and two weight-two, $Z$ stabilizers (blue) on the four data qubits (yellow) labeled $d_{i}$ for integers “ $i$ ” from 0 to 3. For the weight-two stabilizers, superscripts “0,2” (“1,3”) indicate the data qubits on which they operate. The reduced connectivity of the graph is addressed by flag qubits (blue) alternating between (i) being used as weight-two stabilizers, and (ii) as intermediary qubits used to detect errors on the center, syndrome qubit (pink). (c) Circuit diagram for the code layout in (b) applied to an initial $| - ⟩_{L}$ logical state with repeated $X$ - (pink), flag (white), and $Z$ -check (blue) stabilizer measurements, together comprising a round, with midcircuit reset operations (“0”) applied between rounds. In this illustration, the final measurement measures the four data qubits in the $X$ basis due to the application of Hadamard gates.
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Figure 2
(Color) (a) Decoder graph for the code layout depicted in Fig. 1. Syndrome measurements from weight-four (2) stabilizers are mapped to the pink (blue) nodes, and the weight-two flag measurements are mapped onto the white nodes. Identical to Fig. 1, “0,2” (“1,3”) denotes the left- (right-) hand side of the code layout, with the colors of the circuits and syndrome measurement matching in both figures. For initial $| - / + ⟩_{L}$ states stabilized by the circuit in Fig. 1, there are three different possible size-four hyperedges within each round, each highlighted in dark blue across three consecutive rounds. The boundary nodes in black have, by definition, edges with weight “0” connecting them; rendering all boundary nodes to be effectively a single node for the purposes of the decoding process. (b) Applying the technique introduced in the main text to the experiment in Fig. 1, we estimated the correlation probabilities for all of the hyperedges shown in (a). The probabilities are sorted from largest to smallest based on the results from a least-squares fit using a six-parameter noise model. Points with darker colors represent hyperedges of greater sizes, as shown in the lower half of the plot. Hyperedges with indices greater than 93 (shaded gray) had no analytical expression, but were still experimentally adjusted to quantify the impact of computational leakage. The result of fitting the six-parameter noise model (fit, pink dash) agreed well with the analytical (red dash) curve generated using noise terms from simultaneous randomized benchmarking [33].
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Figure 3
(Color) (top) Fraction of total results used for the logical states [(a) $| 0 ⟩_{L}$ , (b) $| - ⟩_{L}$ ] as the number of stabilizer rounds were repeated from 0 to 10 times when full (blue squares), none (yellow circles), or partial (gray triangles) postselection analysis was used. For partial postselection, the analytical decoder was used to exclude ambiguous shots. (bottom) The corresponding logical errors versus number of rounds. The dashed red lines indicate the pseudothreshold as determined by the best (average) physical measurement errors of $7 \times 10^{- 3}$ ( $7.7 \times 10^{- 3}$ ).
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Figure 4
(Color) Logical errors per round initially in $| - / + ⟩_{L}$ states under various analysis methods with acceptance probability per round labeled above. Results varied depending on whether the flag events were directly used for decoding (flag) or indirectly used for decoding using a deflagging procedure. (a) Logical error per round for full and none postselection methods. 25.5% of the counts were rejected with each round for the full postselection scheme. (b) Comparison between errors using three decoder graphs on data without postselection. (c) Comparison between errors using three decoder graphs on data with partial postselection. The approximate percentage of counts rejected for each stabilizer round are indicated above each bar.
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