Relaxation times do not capture logical qubit dynamics

Amit Kumar Pal1,2,3, Philipp Schindler4, Alexander Erhard4, Ángel Rivas5,6, Miguel-Angel Martin-Delgado5,6, Rainer Blatt4,7, Thomas Monz4,8, and Markus Müller2,9,10

1Department of Physics, Indian Institute of Technology Palakkad, Palakkad 678557, India
2Department of Physics, College of Science, Swansea University, Singleton Park, Swansea - SA2 8PP, United Kingdom
3Faculty of Physics, University of Warsaw, ul. Pasteura 5, PL-02-093 Warszawa, Poland
4Institut für Experimentalphysik, Universität Innsbruck, Technikerstr. 25, A-6020 Innsbruck, Austria
5Departamento de Física Teórica, Facultad de Ciencias Físicas, Universidad Complutense, 28040 Madrid, Spain
6CCS -Center for Computational Simulation, Campus de Montegancedo UPM, 28660 Boadilla del Monte, Madrid, Spain.
7Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, Otto-Hittmair-Platz 1, A-6020 Innsbruck, Austria
8Alpine Quantum Technologies GmbH, 6020 Innsbruck, Austria
9Institute for Quantum Information, RWTH Aachen University, D-52056 Aachen, Germany
10Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich, D-52425 Jülich, Germany

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Quantum error correction procedures have the potential to enable faithful operation of large-scale quantum computers. They protect information from environmental decoherence by storing it in logical qubits, built from ensembles of entangled physical qubits according to suitably tailored quantum error correcting encodings. To date, no generally accepted framework to characterise the behaviour of logical qubits as quantum memories has been developed. In this work, we show that generalisations of well-established figures of merit of physical qubits, such as relaxation times, to logical qubits fail and do not capture dynamics of logical qubits. We experimentally illustrate that, in particular, spatial noise correlations can give rise to rich and counter-intuitive dynamical behavior of logical qubits. We show that a suitable set of observables, formed by code space population and logical operators within the code space, allows one to track and characterize the dynamical behaviour of logical qubits. Awareness of these effects and the efficient characterisation tools used in this work will help to guide and benchmark experimental implementations of logical qubits.

Relaxation describes processes that bring a system into equilibrium. A perfect qubit is a closed system that is not able to relax. In practice we do not have a perfect system and the qubit can couple to its environment which can be described by relaxation due to coupling of a perfect qubit to a classical bath. There are various experimental noise processes that couple the qubit to the environment, where in equilibrium all quantum correlations are lost in the bath and the state can be described by classical physics. Under the action of a noise process, the value of an observable that measures quantum correlations will therefore decay with time. Typically, the relaxation of physical qubits is expressed as an exponential decay, only characterized by the relaxation time. This simple picture does not necessarily hold for multiple qubits in experiments where noise processes are spatially correlated. We demonstrate that for small quantum error correction codes, spatially correlated noise can yield complex relaxation behavior that cannot be described by a single relaxation time.

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