Numerical Certificates

estimatemse(circ, pstr::PauliString, n_mcsamples::Integer, thetas=π; stateoverlapfunc=overlapwithzero, circuit_is_reversed=false, customtruncfunc=nothing)

Function to estimate the mean square error of a truncated circuit simulation using Monte Carlo sampling. Returns the mean squared error of the truncated Pauli propagation simulation averaged over the thetas∈ [theta, theta] of the angle theta of each PauliRotation. Currently, the function only supports circuits with PauliRotation and CliffordGate gates.

The length the thetas vector should be equal to the number of parametrized gates in the circuit. Alternatively, thetas can be a single real number applicable for all parametrized gates. The default thetas=π or any other non-array values assume that the circuit consists only of PauliRotation -CliffordGate. For PauliRotation, the value should be the integration range of the parameters around zero.

An initial state overlap function stateoverlapfunc can be provided to calculate the overlap of the backpropagated Pauli strings with the initial state. Importantly, the kwargs can be used to set the truncation parameters of the Pauli propagation. Currently supported are max_weight, max_freq, and max_sins. Note that min_abs_coeff is not supported here, as we consider errors integrated over the angles. max_freq effectively truncates small coefficients below (1/2)^max_freq on average over thetas ∈ [-π, π]. A custom truncation function can be passed as customtruncfunc with the signature customtruncfunc(pstr::PauliStringType, coefficient)::Bool.

estimatemse!(
circ, pstr::PauliString, error_array::AbstractVector, thetas, split_probabilities; 
stateoverlapfunc=overlapwithzero, circuit_is_reversed=false, max_weight=Inf, max_freq=Inf, max_sins=Inf, customtruncfunc=nothing
)

In-place version of estimatemse. This function takes an array error_array of length n_mcsamples as an argument and modifies it in-place. It further assumes that the thetas and split_probabilities are already correctly calculated and provided as arguments. In general they will be vectors, but they can also be real numbers. A custom truncation function can be passed as customtruncfunc with the signature customtruncfunc(pstr::PauliStringType, coefficient)::Bool.