Abstract
Ensuring the correct functioning of quantum error correction (QEC) circuits is crucial to achieve fault tolerance in realistic quantum processors subjected to noise. The first checkpoint for a fully operational QEC circuit is to create genuine multipartite entanglement across all subsystems of physical qubits. We introduce a conditional witnessing technique to certify genuine multipartite entanglement (GME) that is efficient in the number of subsystems and, importantly, robust against experimental noise and imperfections. Specifically, we prove that the detection of entanglement in a linear number of bipartitions by a number of measurements that also scales linearly, suffices to certify GME. Moreover, our method goes beyond the standard procedure of separating the state from the convex hull of biseparable states, yielding an improved finesse and robustness compared to previous techniques. We apply our method to the noisy readout of stabilizer operators of the distance-three topological color code and its flag-based fault-tolerant version. In particular, we subject the circuits to combinations of three types of noise, namely, uniform depolarizing noise, two-qubit gate depolarizing noise, and bit-flip measurement noise. We numerically compare our method with the standard, yet generally inefficient, fidelity test and to a pair of efficient witnesses, verifying the increased robustness of our method. Last but not least, we provide the full translation of our analysis to a trapped-ion native gate set that makes it suitable for experimental applications.
Highlights
The continued effort to control quantum systems with ever increasing accuracy has led to various quantum processors [1], comprising tens of qubits and, more recently, to the demonstration of quantum supremacy [2]
We consider noisy nonFT and flag-based fault tolerance (FT) plaquette measurement circuits and witness the resulting entangled state using imperfect measurements in three different ways: (i) standard witnessing with a GHZ projector requiring an exponential number of measurements, that is, by measuring fidelity to the ideal output state, (ii) standard-linear witnessing (SL) with a single witness that relies on only a linear number of measurements using the proposed witnesses of Tóth and Gühne [29], and (iii) our proposed conditional genuine multipartite entanglement (GME) witnessing requiring a linear number of witnesses and measurements
We introduce conditional entanglement witnessing as a robust and efficient technique to test GME in multipartite quantum systems
Summary
The continued effort to control quantum systems with ever increasing accuracy has led to various quantum processors [1], comprising tens of qubits and, more recently, to the demonstration of quantum supremacy [2]. As discussed in detail below, the standard method of GME certification using a single witness [26,29], reduces the exponential growth of the required number of bipartitions and measurements to a linear one by trading the individual bipartitions for the convex hull of all biseparable states, and detecting those states outside this convex hull. This advantage comes at a price, namely, a significant reduction in the robustness of the witness against experimental imperfections. VII we provide the conclusions and outlook of our work
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