# iterative_mle_state_estimate¶

forest.benchmarking.tomography.iterative_mle_state_estimate(results: List[forest.benchmarking.observable_estimation.ExperimentResult], qubits: List[int], epsilon=0.1, entropy_penalty=0.0, beta=0.0, tol=1e-09, maxiter=10000) → numpy.ndarray

Given tomography data, use one of three iterative algorithms to return an estimate of the state.

“… [The iterative] algorithm is characterized by a very high convergence rate and features a simple adaptive procedure that ensures likelihood increase in every iteration and convergence to the maximum-likelihood state.” [DIMLE1]

There are three options triggered by appropriately setting input parameters:

• MLE only: entropy_penalty=0.0 and beta=0.0
• MLE + maximum entropy: entropy_penalty= (non-zero) and beta=0.0
• MLE + hedging: entropy_penalty=0.0 and beta= (non-zero).

The basic algorithm is due to [DIMLE1], with improvements from [DIMLE2], [HMLE], and [IHMLE].

 [DIMLE1] (1, 2, 3) Diluted maximum-likelihood algorithm for quantum tomography. Řeháček et al. PRA 75, 042108 (2007). https://doi.org/10.1103/PhysRevA.75.042108 https://arxiv.org/abs/quant-ph/0611244
 [DIMLE2] (1, 2) Quantum-State Reconstruction by Maximizing Likelihood and Entropy. Teo et al. PRL 107, 020404 (2011). https://doi.org/10.1103/PhysRevLett.107.020404 https://arxiv.org/abs/1102.2662
 [HMLE] (1, 2) Hedged Maximum Likelihood Quantum State Estimation. Blume-Kohout. PRL, 105, 200504 (2010). https://doi.org/10.1103/PhysRevLett.105.200504 https://arxiv.org/abs/1001.2029
 [IHMLE] Iterative Hedged MLE from Yong Siah Teo’s PhD thesis. see Eqn. 1.5.13 on page 88: Numerical Estimation Schemes for Quantum Tomography. Y. S. Teo. PhD Thesis, from National University of Singapore, (2013). https://arxiv.org/pdf/1302.3399.pdf
Parameters: results – Measured results from a state tomography experiment qubits – All qubits that were tomographized. This specifies the order in which qubits will be kron’ed together; the first qubit in the list is the left-most tensor factor. epsilon – the dilution parameter used in [DIMLE1]. In practice epsilon ~ 1/num_shots entropy_penalty – the entropy penalty parameter from [DIMLE2], i.e. lambda beta – The Hedging parameter from [HMLE], i.e. beta tol – The largest difference in the Frobenious norm between update steps that will cause the algorithm to conclude that it has converged. maxiter – The maximum number of iterations to perform before aborting the procedure. A point estimate of the quantum state rho