Source code for qmlt.numerical.losses

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
# Copyright 2018 Xanadu Quantum Technologies Inc.

# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at

# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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"""
Losses
========================================================

**Module name:** :mod:qmlt.numerical.losses

.. currentmodule:: qmlt.numerical.losses

.. codeauthor:: Maria Schuld <maria@xanadu.ai>

A collection of loss functions for numpy.

Summary
-------

.. autosummary::
trace_distance
expectation
square_loss
_softmax
cross_entropy_with_softmax

Code details
------------
"""

import numpy as np

[docs]def trace_distance(rho, sigma): r"""Trace distance :math:\frac{1}{2}\tr \{ \sqrt{ (\rho - \sigma)^2} \} between quantum states :math:\rho and :math:\sigma. Args: rho (ndarray or list): 2-dimensional square matrix representing the state :math:\rho. sigma (ndarray or list): 2-dimensional square matrix of the same dimensions and dtype as rho, representing the state :math:\sigma Returns: float: Scalar trace distance. """ rho = np.array(rho) sigma = np.array(sigma) if rho.shape != sigma.shape: raise ValueError("Cannot compute the trace distance if inputs have" " different shapes {} and {}".format(rho.shape, sigma.shape)) if rho.ndim != 2: raise ValueError("Trace distance loss expects 2-d arrays representing density matrices.") diffs = rho - sigma eigvals = np.linalg.eigvals(diffs) return 0.5 * sum(np.absolute(eigvals))
[docs]def expectation(rho, operator): r""" Expectation value :math:\tr\{ \rho O\} of operator :math:O with respect to the quantum state :math:\rho. Args: rho (ndarray or list): 2-dimensional array representing the state :math:\rho. operator (ndarray or list): 2-dimensional array of the same dimensions and dtype as rho, representing the operator :math:O Returns: float: Scalar expectation value. """ rho = np.array(rho) operator = np.array(operator) if rho.shape != operator.shape: raise ValueError("Cannot compute expectation value if rho and operator have" " different shapes {} and {}".format(rho.shape, operator.shape)) if rho.ndim != 2: raise ValueError("Expectation loss expects a 2-d array representing a density matrix.") exp = np.trace(rho@operator) if np.imag(exp) > 1e-5: raise ValueError("Expectation value has a non-negligible imaginary contribution." "Something went wrong.") return exp
[docs]def square_loss(outputs, targets): r"""Mean squared loss :math:0.5 \sum\limits_{m=1}^M |y^m - t^m|^2 between outputs :math:y^m and targets :math:t^m for :math:m = 1,...,M. Args: 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. Returns: float: Scalar mean squared loss. """ outputs = np.array(outputs) targets = np.array(targets) if outputs.shape != targets.shape: raise ValueError("Cannot compute squared loss if outputs and targets have" " different shapes {} and {}".format(outputs.shape, targets.shape)) if outputs.ndim > 2: raise ValueError("Mean squared loss expects 1-d outputs, dimension of current outputs" " is {}.".format(outputs.ndim - 1)) diff = outputs - targets res = 0.5*sum(np.dot(d, d) for d in diff) return res
[docs]def _softmax(logits): r"""Softmax function, turns a vector of real values into a vector of probabilities. Args: logits (ndarray 1-d): Real 1-d vector of model outputs Returns: ndarray: Vector of probabilities """ exps = np.exp(logits - np.max(logits)) return exps / np.sum(exps)
[docs]def cross_entropy_with_softmax(outputs, targets): r""" Cross-entropy loss that measures the probability error in discrete classification tasks (with mutually exclusive classes). Useful for one-hot-encoded vectors. Args: 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. Return: float: Scalar loss. """ probs = [_softmax(outp) for outp in outputs] terms = np.array([y*np.log(p) for p, y in zip(probs, targets)]) loss = (-1)*np.sum(terms) return loss