Monodimensional Angular Model

Research provenance

The model introduced here is presented in A Preliminary Study for a Quantum-like Robot Perception Model (2020), and later incorporated into Quantum-like Modeling of Cognitive Architectures for Robotics.

import numpy as np
import matplotlib.pyplot as plt
from qiskit.visualization import plot_histogram
from qrobot.models import AngularModel

In this notebook we present a 1-dimensional (\(n=1\)) demo for the AngularModel class. The model integrates a sequence of normalized sensor events in one qubit and produces a binary outcome when that qubit is measured.

n = 1

Here, we considered a time window of \(\tau > 1\).

tau = 30

For each event \(x_t\in[0,1]\), the model applies a fractional rotation \(R_y(\pi x_t/\tau)\). Since every rotation uses the same axis, the final angle is

\[ \theta=\frac{\pi}{\tau}\sum_t x_t=\pi\bar{x}, \]

where \(\bar{x}\) is the mean input in the temporal window. Consequently, \(P(0)=\cos^2(\theta/2)\) and \(P(1)=\sin^2(\theta/2)\).

Input definition

We start by defining an arbitrary continuous input sequence. Its first half spans the full normalized interval, while the second half is biased toward larger readings:

sequence = list()

# Balanced events (between 0 and 1)
for i in range(0, int(tau / 2)):
    sequence.append(np.random.randint(0, 1000) / 1000)

# Unbalanced events (balanced between .5 and 1)
for i in range(int(tau / 2), tau):
    sequence.append(np.random.randint(500, 1000) / 1000)

plt.figure()
plt.stem(sequence, linefmt="C0-", markerfmt="C0o", basefmt="C0-")
plt.xlabel("Time index t")
plt.ylabel("Input value x")
plt.title("Input sequence")
plt.show()
../_images/aee0bbda7f6b80a26a3743704a46c399c26ff8fd4b04cd01f669b3adec465604.png

Encode the input in the model

We initialize the model by instantiating an object with \(n\) and \(\tau\)

model = AngularModel(n, tau)

Using the encode method, we can encode each event’s data in the model (for multidimensional inputs, a second loop is needed in order to loop through the \(n\) dimensions of the input).

model.clear()  # Keep this cell repeatable by discarding any earlier encoding.

for t in range(0, model.tau):  # loop throug the event sequence
    model.encode(sequence[t], dim=0)

The model is implemented by a Qiskit quantum circuit:

model.print_circuit()

   ┌──────────────┐┌───────────────┐┌──────────────┐┌─────────────┐»
q: ┤ Ry(0.062937) ├┤ Ry(0.0075398) ├┤ Ry(0.079482) ├┤ Ry(0.06168) ├»
   └──────────────┘└───────────────┘└──────────────┘└─────────────┘»
«   ┌──────────────┐┌──────────────┐┌──────────────┐┌──────────────┐»
«q: ┤ Ry(0.068906) ├┤ Ry(0.010472) ├┤ Ry(0.021153) ├┤ Ry(0.064717) ├»
«   └──────────────┘└──────────────┘└──────────────┘└──────────────┘»
«   ┌───────────────┐┌──────────────┐┌─────────────┐┌──────────────┐»
«q: ┤ Ry(0.0071209) ├┤ Ry(0.026913) ├┤ Ry(0.10179) ├┤ Ry(0.056339) ├»
«   └───────────────┘└──────────────┘└─────────────┘└──────────────┘»
«   ┌──────────────┐┌──────────────┐┌──────────────┐┌──────────────┐»
«q: ┤ Ry(0.077178) ├┤ Ry(0.079273) ├┤ Ry(0.032987) ├┤ Ry(0.098646) ├»
«   └──────────────┘└──────────────┘└──────────────┘└──────────────┘»
«   ┌──────────────┐┌─────────────┐┌──────────────┐┌──────────────┐»
«q: ┤ Ry(0.080425) ├┤ Ry(0.10231) ├┤ Ry(0.064822) ├┤ Ry(0.055397) ├»
«   └──────────────┘└─────────────┘└──────────────┘└──────────────┘»
«   ┌──────────────┐┌──────────────┐┌──────────────┐┌──────────────┐»
«q: ┤ Ry(0.077283) ├┤ Ry(0.074037) ├┤ Ry(0.061261) ├┤ Ry(0.073932) ├»
«   └──────────────┘└──────────────┘└──────────────┘└──────────────┘»
«   ┌──────────────┐┌──────────────┐┌──────────────┐┌─────────────┐»
«q: ┤ Ry(0.076445) ├┤ Ry(0.057386) ├┤ Ry(0.086603) ├┤ Ry(0.10043) ├»
«   └──────────────┘└──────────────┘└──────────────┘└─────────────┘»
«   ┌─────────────┐┌──────────────┐
«q: ┤ Ry(0.10367) ├┤ Ry(0.087755) ├
«   └─────────────┘└──────────────┘

Given the input we defined above, the model is in the following state:

model.plot_state_mat()
../_images/ed7fc4a967abadcf28c306c039ffe0c5e1a382b021812af7d1aa962647a2f69e.png

Density matrix (see Wikipedia): for a finite-dimensional state space, the most general density operator is of the form

\[\rho =\sum _{j}p_{j}|\psi _{j}\rangle \langle \psi _{j}|\]

where the coefficients \(p_{j}\) are non-negative and add up to one, and \(|\psi _{j}\rangle \langle \psi _{j}|\) is an outer product written in bra-ket notation. This represents a mixed state, with probability \( p_{j}\) that the system is in the pure state \(|\psi _{j}\rangle \).

Measurement simulation

A single call to model.decode() performs one measurement and can be interpreted as one stochastic decision:

print(f"One-shot decision: |{model.decode()}⟩")

One-shot decision: |1⟩

Now we instead simulate many shots to expose the probability distribution for the two possible basis-state outcomes \(\lvert 0 \rangle\) and \(\lvert 1 \rangle\) and validate the information encoded by the temporal window:

shots = 1000000
counts = model.measure(shots)

Raw counts for each possible outcome:

import json

print("Aggregated binary outcomes of the circuit:")
print(json.dumps(counts, sort_keys=True, indent=4))

Aggregated binary outcomes of the circuit:
{
    "0": 310830,
    "1": 689170
}

From the raw counts we can obtain the relative frequencies (aka the probabilities) and compare them with the input sequence shape:

plt.figure(figsize=(15, 4), dpi=150)

ax1 = plt.subplot(1, 2, 1)
ax1.stem(sequence, linefmt="C0-", markerfmt="C0o", basefmt="C0-")
ax1.set_xlabel("Time index t")
ax1.set_ylabel("Input value x")
ax1.set_title("Input sequence")

ax2 = plt.subplot(1, 2, 2)
plot_histogram(counts, ax=ax2)
ax2.set_ylabel("")
ax2.set_title("Probabilities")

plt.show()
../_images/4692b0dcc0f1722a916acb8132562d085e5ed16fba5748cec917337b99e7eb06.png

References