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

Davide Vodola1,2, Manuel Rispler3, Seyong Kim4, and Markus Müller5,6

1Dipartimento di Fisica e Astronomia ``Augusto Righi'' dell'Università di Bologna, I-40127 Bologna, Italy
2INFN, Sezione di Bologna, I-40127 Bologna, Italy
3QuTech, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands
4Department of Physics, Sejong University, 05006 Seoul, Republic of Korea
5Institute for Theoretical Nanoelectronics (PGI-2), Forschungszentrum Jülich, 52428 Jülich, Germany
6Institute for Quantum Information, RWTH Aachen University, 52056 Aachen, Germany

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Mapping the decoding of quantum error correcting (QEC) codes to classical disordered statistical mechanics models allows one to determine critical error thresholds of QEC codes under phenomenological noise models. Here, we extend this mapping to admit realistic, multi-parameter noise models of faulty QEC circuits, derive the associated strongly correlated classical spin models, and illustrate this approach for a quantum repetition code with faulty stabilizer readout circuits. We use Monte-Carlo simulations to study the resulting phase diagram and benchmark our results against a minimum-weight perfect matching decoder. The presented method provides an avenue to assess fundamental thresholds of QEC circuits, independent of specific decoding strategies, and can thereby help guiding the development of near-term QEC hardware.

One of the main obstacles for the realization of a fully functioning quantum computer is the noise affecting the qubits. To protect quantum information against such disturbances, scientists have developed techniques for quantum error correction (QEC). Here, one uses quantum codes that redundantly encode the fragile quantum information in a collection of data qubits. Then quantum circuits can be periodically applied, to detect where in the qubit register errors have occurred. This is done via indirect measurements of correlations among the data qubits via coupling and measurement of auxiliary qubits. The information accumulated from these measurements can then be used to find a suitable correction, which removes the detected errors with a high success probability.

It has been proven that the process of finding the most suitable correction for the quantum codes is equivalent to studying the phase diagram of a classical disordered statistical mechanics model. This connection between quantum and classical physics allows one to harness the knowledge of phase diagram of the classical models to gain insight into the fundamental robustness limits of the quantum codes against noise. For example, locating the phase transitions between ordered and disordered phases in the classical models reveals the parameter regimes for which QEC succeeds or fails, respectively.

So far, however, these mappings have been largely limited to QEC codes with simplified noise models where data qubits can suffer errors and the measurement of the auxiliary qubits is faulty with a certain probability. These simplified descriptions, however, ignore the possibility that errors can occur while executing the quantum circuits used for the QEC procedures, and that such errors can propagate and proliferate through the quantum hardware.

In this work, we extend the approach of mapping QEC codes to classical statistical mechanics models to the more realistic scenario ,in which errors in the quantum circuits and propagation of errors is taken into account explicitly. We illustrate this mapping for a one-dimensional quantum repetition code. In realistic quantum hardware, different circuit elements such as single or two-qubit gates typically have different failure probabilities. In our study, we model this by a multi-parameter microscopic noise model for the QEC circuits, and derive the associated classical statistical physics model. A numerical study of its phase diagram finally allows us to determine the largest possible parameter region, in which successful QEC is feasible.

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