Developing reliable multi-qubit gates in ultracold atomic systems involves challenges at multiple levels. For instance, it is challenging to benchmark a multi-qubit gate for two reasons: (i) complete characterisation, i.e., process tomography of the gate requires an exponentially large number of measurements and therefore not possible and (ii) it is in general classically intractible to compute the expected multi-qubit gate, as it involves exponentiating a very large Hamiltonian matrix. Moreover, it is essential to understand the sources of error in any quantum control and in a multi-qubit operation, there can be a vast number of error processes, which is challenging to model. Therefore, this presents a combination of theoretical and experimental challenges. Below, I describe a roadmap to tackle this problem:
1. Developing a figure of merit
Since one cannot completely characterise (i.e, a process tomography) a multi-qubit gate, the alternative is to identify a small set of properties of the gate, i.e, figures of merit, from which we can learn the most about the errors. In our recent work, we have shown that such figures of merit can be constructed using the reduced Choi matrix of the multi-qubit gate (see this paper and this paper).
2. Developing measurement protocols for the figure of merit
The next challenge would be to develop efficient protocols to measure the reduced Choi matrix. We have shown that entanglement and information scrambling can be used to efficiently measure the reduced Choi matrix (see this paper).
3. Experimental implementation
The following challenge would be to implement the measurement protocol and benchmark an example multi-qubit gate. Our plan is to develop a tweezer array system and implement a plaquette-Rydberg gate and benchmark it.
4. Developing an error model for the experimental system
Once an experimental platform is available, we can use the benchmarking tools developed above to construct a theoretical model to descibe the error processes in the syste, This will enhance our predictive power when we design new multi-qubit gates.
5. Error mitigation
With a known error-model, a natural next step is to us it to mitigate the error, using both theoretical and experimental tools. One of the long-shot objectoives here would be to determine lower bounds on the experimental errors.