# Losses¶

Module name: qmlt.numerical.losses

A collection of loss functions for numpy.

## Summary¶

 trace_distance(rho, sigma) Trace distance $$\frac{1}{2}\tr \{ \sqrt{ (\rho - \sigma)^2} \}$$ between quantum states $$\rho$$ and $$\sigma$$. expectation(rho, operator) Expectation value $$\tr\{ \rho O\}$$ of operator $$O$$ with respect to the quantum state $$\rho$$. square_loss(outputs, targets) Mean squared loss $$0.5 \sum\limits_{m=1}^M |y^m - t^m|^2$$ between outputs $$y^m$$ and targets $$t^m$$ for $$m = 1,...,M$$. _softmax(logits) Softmax function, turns a vector of real values into a vector of probabilities. cross_entropy_with_softmax(outputs, targets) Cross-entropy loss that measures the probability error in discrete classification tasks (with mutually exclusive classes).

## Code details¶

qmlt.numerical.losses.trace_distance(rho, sigma)[source]

Trace distance $$\frac{1}{2}\tr \{ \sqrt{ (\rho - \sigma)^2} \}$$ between quantum states $$\rho$$ and $$\sigma$$.

Parameters: rho (ndarray or list) – 2-dimensional square matrix representing the state $$\rho$$. sigma (ndarray or list) – 2-dimensional square matrix of the same dimensions and dtype as rho, representing the state $$\sigma$$ Scalar trace distance. float
qmlt.numerical.losses.expectation(rho, operator)[source]

Expectation value $$\tr\{ \rho O\}$$ of operator $$O$$ with respect to the quantum state $$\rho$$.

Parameters: rho (ndarray or list) – 2-dimensional array representing the state $$\rho$$. operator (ndarray or list) – 2-dimensional array of the same dimensions and dtype as rho, representing the operator $$O$$ Scalar expectation value. float
qmlt.numerical.losses.square_loss(outputs, targets)[source]

Mean squared loss $$0.5 \sum\limits_{m=1}^M |y^m - t^m|^2$$ between outputs $$y^m$$ and targets $$t^m$$ for $$m = 1,...,M$$.

Parameters: outputs (ndarray or list) – array of dimension M x 1 containing the 1-dimensional outputs. targets (ndarray or list) – array of the same dimension and type as outputs, containing the targets. Scalar mean squared loss. float
qmlt.numerical.losses._softmax(logits)[source]

Softmax function, turns a vector of real values into a vector of probabilities.

Parameters: logits (ndarray 1-d) – Real 1-d vector of model outputs Vector of probabilities ndarray
qmlt.numerical.losses.cross_entropy_with_softmax(outputs, targets)[source]

Cross-entropy loss that measures the probability error in discrete classification tasks (with mutually exclusive classes). Useful for one-hot-encoded vectors.

Parameters: outputs (ndarray) – Real 2-dim array representing a batch of model outputs. Also called logits. targets (ndarray) – Real 2-dim array representing a batch of target outputs. Scalar loss. float