Exponential suppression of bit or phase errors with cyclic error correction

Abstract

Realizing the potential of quantum computing requires sufficiently low logical error rates1. Many applications call for error rates as low as 10−15 (refs. 2–9), but state-of-the-art quantum platforms typically have physical error rates near 10−3 (refs. 10–14). Quantum error correction15–17 promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analysing error correlations with high precision, allowing us to characterize error locality while performing quantum error correction. Finally, we perform error detection with a small logical qubit using the 2D surface code on the same device18,19 and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. These experimental demonstrations provide a foundation for building a scalable fault-tolerant quantum computer with superconducting qubits.

Fundamental thresholds of realistic quantum error correction circuits from classical spin models

“…First, p and q both are typically dominated by similar error processes that for many qubit platforms are of comparable error rates, hence we set p = q for simplicity, and additionally, we dial up the strength of r. The known case r = 0 reduces to the phenomenological case without correlated errors, which would correspond to perfect two-qubit gates (p 2 = 0). Increasing r to r = p/2 = q/2 corresponds to p 2 = 5p 1 + O(p 2 1 ), which is roughly compatible with the two-qubit gate being an order of magnitude worse than single-qubit operations (error rate or infidelity) as observed in many experimental realizations [36,40,55,63]. Considering r = p = q corresponds to p 2 being dominant and all other error sources negligible (p sp = p id = p 1 = p m = 0).…”

“…Interestingly, in a recent experimental realization of the phase-flip code [55] circuit errors are reported to be well described by Pauli errors. Casting the experimentally obtained error rates into Eq.…”

“…We illustrate the mapping of circuit-level QEC codes to statistical mechanics models for a 1D quantum repetition code with faulty circuitry described by a multi-parameter microscopic noise model accounting for a number of single-and two-qubit gate noise processes. Whereas this code does not enable correction of arbitrary errors, current experimental efforts focus on achieving repetitive QEC with this code in the regime of beneficial error suppression for increasing code sizes [52][53][54][55]. We show that in contrast to oversimplified phenomenological noise models, new effective correlated noise processes arise, which we systematically derive and quantitatively relate to error rates of the underlying microscopic noise model.…”

Introduction to Semi-Classical Analysis for Digital Errors of Qubit in Quantum Processor

“…Then it has been clarified that nonlinear errors give serious degradations of the capability of quantum computer, by the recurrence effect due to quantum correlation and also by collective decoherence . In order to cope with the quantum errors described in this paper, or to avoid this situation, one method is to further develop the conventional quantum error correction theory based on quantum noise analysis, or to establish a new way to physically suppress such errors [ 32 , 33 , 34 ]. Recently, a number of previously unknown and extremely difficult challenges in the development of an error correctable quantum computer have been reported [ 35 , 36 , 37 , 38 ].…”

LILLIPUT: a lightweight low-latency lookup-table decoder for near-term Quantum error correction

“…In recent years, several preliminary QEC experiments involving repetition codes and Bacon-Shor codes have been successfully demonstrated [6,17,18,27,48,51,65,68,70,84,86,87,89]. However, reaching low logical error rates requires implementation of more Figure 1: (a) In QEC, a logical qubit is encoded using a set of data and parity qubits.…”