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Predictive supremacy of informationally-restricted quantum perceptron

Shubhayan Sarkar

Abstract

In the current world, the use of artificial intelligence is penetrating every aspect of human life. The basic element of any artificial intelligence is a digital neuron, called a perceptron, while its quantum analogue is called a quantum perceptron. Here, we introduce a model of perceptron called the informationally-restricted measurement-based perceptron (IMP), where each input is composed of two bits, while at the node, depending on a free input variable, the perceptron decides which bit to evaluate. Additionally, the states transmitted from the input to the node are restricted to a bit (qubit). We establish that under this restriction, the quantum IMP predicts better than a classical IMP. This means that under dimensional restriction of the transmitted states, when both the classical and quantum perceptrons learn the same, the quantum perceptron predicts better than the classical perceptron. For our purpose, we find specific learned values of the perceptron that can display the advantage of a quantum perceptron over its classical counterpart. Restricting to discrete binary inputs, we establish that the observed quantum advantage is universal, that is, for any non-trivial function implementable by both the quantum and classical IMP, one can always find a quantum implementation that outperforms the predictive capability of every classical one. This points to the fact that, given identical learning and resources, a quantum perceptron would predict better than any classical one.

Predictive supremacy of informationally-restricted quantum perceptron

Abstract

In the current world, the use of artificial intelligence is penetrating every aspect of human life. The basic element of any artificial intelligence is a digital neuron, called a perceptron, while its quantum analogue is called a quantum perceptron. Here, we introduce a model of perceptron called the informationally-restricted measurement-based perceptron (IMP), where each input is composed of two bits, while at the node, depending on a free input variable, the perceptron decides which bit to evaluate. Additionally, the states transmitted from the input to the node are restricted to a bit (qubit). We establish that under this restriction, the quantum IMP predicts better than a classical IMP. This means that under dimensional restriction of the transmitted states, when both the classical and quantum perceptrons learn the same, the quantum perceptron predicts better than the classical perceptron. For our purpose, we find specific learned values of the perceptron that can display the advantage of a quantum perceptron over its classical counterpart. Restricting to discrete binary inputs, we establish that the observed quantum advantage is universal, that is, for any non-trivial function implementable by both the quantum and classical IMP, one can always find a quantum implementation that outperforms the predictive capability of every classical one. This points to the fact that, given identical learning and resources, a quantum perceptron would predict better than any classical one.
Paper Structure (5 sections, 2 theorems, 19 equations, 1 figure, 4 tables)

This paper contains 5 sections, 2 theorems, 19 equations, 1 figure, 4 tables.

Table of Contents

  1. All Implementable functions of a two-bit classical perceptron
  2. Witnesses of quantum advantage for every classically implementable perceptron function
  3. $f$: Single-variable type
  4. $f$: And/OR Type
  5. $f$: Assymetric type

Key Result

Lemma 1

The classical bound of psucand is $\beta_C=\frac{13}{40}$.

Figures (1)

  • Figure 1: Perceptrons. (a) McCulloch-Pitts neuron with Rosenblatt's weights and bias. Each input $x_i$ is multiplied by a weight $w_i$, and all weighted inputs are summed along with a bias $b$. The neuron then compares this sum with a fixed threshold depending on the function $f$; if the sum is greater than or equal to the threshold, the neuron outputs $1$, otherwise it outputs $0$. (b) Informationally-restricted quantum perceptron. Similar to the McCulloch-Pitts neuron, but each input encodes two bits $x_{i,0}x_{i,1}$ for $i=1,2$. However, the channel connecting the inputs to the node can transmit only a bit (qubit) of information; that is, the maximal dimension of the transmitted system that encodes the inputs is two. At the node, the input $s=0,1$ decides the bit to evaluate. It is important here that $s$ is chosen after the transmitted system reaches the node.

Theorems & Definitions (4)

  • Lemma 1
  • proof
  • Lemma 2
  • proof