Quantum model averaging

Abstract

Standard tomographic analyses ignore model uncertainty. It is assumed that a given model generated the data and the task is to estimate the quantum state, or a subset of parameters within that model. Here we apply a model averaging technique to mitigate the risk of overconfident estimates of model parameters in two examples: (1) selecting the rank of the state in tomography and (2) selecting the model for the fidelity decay curve in randomized benchmarking.Parameter estimation is an integral part of physics. Accurate estimates of physical parameters in quantum mechanical models allow for precision quantum control [1,2] which enables practical goals such as quantum computation [3] and quantum metrology [4] which in turn can provide probes of fundamental physics such as gravity wave detection [5].Quantum parameter estimation shares many similarities with its classical counterpart. But there are many subtle and peculiar differences. Even in single parameter estimation, quantum metrology [4,7] shows that we can obtain advantages from quantum resources such as squeezed states of light [9] and entanglement [10]. With such rich structure, many new subtleties [11], considerations [12-14] and generalities [15][16][17] can arise, including entirely new approaches to estimation [18][19][20][21] and verification [22][23][24].

Crystallizing highly-likely subspaces that contain an unknown quantum state of light

“…These strategies depend on the parametric models selected (which in our case corresponds to some pre-chosen reconstruction subspace) to optimize the signal locations, the accuracy of which depend heavily on the correctness of the chosen models. An average of these models may be carried out over the quantum state space to mitigate possible model inaccuracies29. Other methods of choosing reconstruction subspaces include the utilization of other prior knowledge about the source and assigning a partial dependence of the subspace dimension D rec on the number of measurement settings or groups of outcomes30.…”

Photodetector figures of merit in terms of POVMs

“…Model selection [31] is then a nice statistical technique that allows one to rank different models, based on how well the models fit the data and how many fitting parameters they use. This technique is especially useful for reducing the number of parameters if the relevant Hilbert space is large [32,33], as it indeed is for the photo-detection problem.…”

Practical Bayesian tomography

“…Bayesian techniques in the context of quantum tomography were first suggested by Jones [16], Slater [17], Derka et al [18], Bužek et al [19], and Schack et al [20]. In addition to the inclusion of prior information, Bayesian estimation also naturally includes several other experimental advantages, such as optimality [21][22][23], adaptive experimental design [1,24], robust region estimates [25] and model selection criteria [2,26]. These advantages arise from the fact that Bayesian methods provide a complete characterization of the current state of an experimentalist's knowledge after each datum.…”

“…For our application (quantum tomography), Huszár and Houlsby [1] and by Ferrie [2] suggested SMC as numerical tool for implementing (5). Of particular interest is Ferrie's recent work on tomographic region estimators with SMC [25].…”