Determining the ability for universal quantum computing: Testing controllability via dimensional expressivity

Fernando Gago-Encinas1, Tobias Hartung2,3, Daniel M. Reich1, Karl Jansen4, and Christiane P. Koch1

1Fachbereich Physik and Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany
2Northeastern University London, Devon House, St Katharine Docks, London, E1W 1LP, United Kingdom
3Khoury College of Computer Sciences, Northeastern University, 440 Huntington Avenue, 202 West Village H Boston, MA 02115, USA
4NIC, DESY Zeuthen, Platanenallee 6, 15738 Zeuthen, Germany

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Operator controllability refers to the ability to implement an arbitrary unitary in SU(N) and is a prerequisite for universal quantum computing. Controllability tests can be used in the design of quantum devices to reduce the number of external controls. Their practical use is hampered, however, by the exponential scaling of their numerical effort with the number of qubits. Here, we devise a hybrid quantum-classical algorithm based on a parametrized quantum circuit. We show that controllability is linked to the number of independent parameters, which can be obtained by dimensional expressivity analysis. We exemplify the application of the algorithm to qubit arrays with nearest-neighbour couplings and local controls. Our work provides a systematic approach to the resource-efficient design of quantum chips.

Controllability tells us whether we can implement every conceivable unitary operation on a quantum system with control fields that we can change as a function of time. This property is important for qubit arrays, since universal quantum computing requires a device that can realize any quantum logic operation. Since every control field takes up physical space, requires calibration and is potentially a source of noise, it becomes essential to find device designs with as few controls and qubit couplings as possible, as quantum devices grow larger. Controllability tests can help us achieve this goal.

Here we present a hybrid quantum-classical test that combines measurements on a quantum device and classical calculations. Our algorithm is based on the concept of parametric quantum circuits, the quantum counterpart of Boolean circuits where some of the logic gates depend on different parameters. We leverage dimensional expressivity analysis to identify all parameters in the circuit that are redundant and can be removed. We show that, for any qubit array, a parametric quantum circuit can be defined such that the number of independent parameters reflects the controllability of the original quantum system.

We hope that this test will provide a useful tool for studying these circuits and for designing controllable quantum devices that can be scaled to larger dimensions.

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