I recently stumbled upon a press release from Xanadu.ai stating that

Under the hood, PennyLane's core feature is that it implements a version of the backpropagation algorithm - the workhorse for training deep learning models - that is naturally compatible with quantum devices.

As far as I know, not many algorithms are known to profit from the features of a quantum computer. So is quantum backpropagation one of these algorithms that is theoretically faster than classical backpropagation? Or what advantages should one get from running backpropagation on a quantum computer?

  • $\begingroup$ Are you asking in general or are you asking in the context of PennyLane itself? $\endgroup$
    – cnada
    Nov 14, 2018 at 12:39
  • $\begingroup$ I'm more interested about quantum backpropagation in general. $\endgroup$
    – asmaier
    Nov 14, 2018 at 15:40
  • $\begingroup$ Well at the moment it is a bit investigated but people are mostly looking first at a quantum version of gradient descent, where we try to find a complexity that is better than the classical version. $\endgroup$
    – cnada
    Nov 14, 2018 at 15:46
  • $\begingroup$ Maybe an interesting follow-up on cnada's comment: quantum improvements of gradient descent algorithms are for example covered in this paper: arxiv.org/abs/1711.00465 $\endgroup$
    – arriopolis
    Nov 14, 2018 at 19:57

2 Answers 2


In general, the efficiency of Quantum Machine Learning Techniques will be calibrated and measured more in terms of the energy efficiency, ability to handle complex computational problems, NP-hard problems and the ability to ensemble different domain algorithms than the speed and learning rate. However, there could be exceptionally faster quantum algorithms for a specific set of computational problems.

Quantum Backpropagation can be more energy efficient and faster if the right combination of quantum algorithms is used. It depends on how efficiently the output neuron states converge the quantum correlations employed in the feedforward network and what kind of control system algorithms are used for backpropagation. An efficient control NOT gate with lesser decoherence and optimized topology will be useful for this purpose.

One interesting approach to quantum backpropagation is by implementing a form of quantum adaptive error correction, in the sense that, for a feedforward network, the input layer is conditionally transformed so that it exhibits the firing patterns that solve a given computational problem.

In this approach quantum backpropagation dynamics is integrated in a two-stage neural cognition scheme: there is a feedforward learning stage such that the output neurons’ states, initially separable from the input neurons’ states, that converge during a neural processing time to correlated states with the input layer, and then there is a backpropagation stage, where the output neurons act as a control system that triggers different quantum circuits that are implemented on the input neurons, conditionally transforming their state in such a way that a given computational problem is solved.

The following research paper has a deep analysis on the Quantum Back Propagation dynamics through the application of a Hamiltonian framework. It introduces a Hamiltonian framework for quantum neural machine learning with basic feedforward neural networks integrating quantum measurement theory and dividing the quantum neural dynamics in the learning stage and the backpropagation stage and then apply the framework to two example problems:

  1. The firing pattern selection problem, where the neural network places the input layer in a specific well-defined firing configuration, from an initially arbitrary superposition of neural firing patterns

  2. The n-to-m Boolean functions’ representation problem, where the goal for the network is to correct the input layer so that it represents an arbitrary n-to-m Boolean function.

There is another experimental implementation of Quantum Back Propagation algorithm for a Multi Layer Perceptron based Artificial Neural Network as outlined here.

  • $\begingroup$ The Quantum Back Propagation algorithm for a Multi Layer Perceptron based Artificial Neural Network is a quantum inspired algorithm. Does that not mean it is more a classical algorithm than a quantum one? $\endgroup$
    – cnada
    Nov 15, 2018 at 3:49

In my mind, quantum backpropagation wouldn't involve a linear amount (sparse set) of input output pairs to train it, it would train all possible (full set) input output pairs, before it adjusts a synapse weight.

That would give you the best possible compression for a perceptron that size!!!! (which is amazing) - for backpropagation anyway.... but u wont be able to use ordinary training data, it would have to be procedurally generated because there's an exponential amount of it.


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