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/d1e13811549074d4505fcfee0cc567c58411d0d6f013eda1be5c3affbdfc001e.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(0.018117) ├┤ Ry(0.10105) ├┤ Ry(0.019897) ├┤ Ry(0.08765) ├»
   └──────────────┘└─────────────┘└──────────────┘└─────────────┘»
«   ┌──────────────┐┌──────────────┐┌──────────────┐┌─────────────┐»
«q: ┤ Ry(0.077702) ├┤ Ry(0.026494) ├┤ Ry(0.064612) ├┤ Ry(0.01686) ├»
«   └──────────────┘└──────────────┘└──────────────┘└─────────────┘»
«   ┌──────────────┐┌──────────────┐┌──────────────┐┌──────────────┐»
«q: ┤ Ry(0.062518) ├┤ Ry(0.067858) ├┤ Ry(0.040003) ├┤ Ry(0.084404) ├»
«   └──────────────┘└──────────────┘└──────────────┘└──────────────┘»
«   ┌──────────────┐┌──────────────┐┌──────────────┐┌──────────────┐»
«q: ┤ Ry(0.079587) ├┤ Ry(0.079168) ├┤ Ry(0.088069) ├┤ Ry(0.087022) ├»
«   └──────────────┘└──────────────┘└──────────────┘└──────────────┘»
«   ┌─────────────┐┌──────────────┐┌──────────────┐┌──────────────┐»
«q: ┤ Ry(0.09341) ├┤ Ry(0.081158) ├┤ Ry(0.090059) ├┤ Ry(0.088593) ├»
«   └─────────────┘└──────────────┘└──────────────┘└──────────────┘»
«   ┌──────────────┐┌──────────────┐┌──────────────┐┌──────────────┐»
«q: ┤ Ry(0.091316) ├┤ Ry(0.071942) ├┤ Ry(0.057282) ├┤ Ry(0.053931) ├»
«   └──────────────┘└──────────────┘└──────────────┘└──────────────┘»
«   ┌──────────────┐┌──────────────┐┌──────────────┐┌──────────────┐»
«q: ┤ Ry(0.055501) ├┤ Ry(0.072047) ├┤ Ry(0.098122) ├┤ Ry(0.074456) ├»
«   └──────────────┘└──────────────┘└──────────────┘└──────────────┘»
«   ┌─────────────┐┌──────────────┐
«q: ┤ Ry(0.08964) ├┤ Ry(0.087755) ├
«   └─────────────┘└──────────────┘

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

model.plot_state_mat()
../_images/03ec08986bfadfd8bbb6e3913ca25c4dfa08c0479c8519d44f440905b61acc1b.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": 244722,
    "1": 755278
}

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/91875b4931792ca6209115ed23cf27e8349ec41003d63825220f7e52d5fec5c0.png

References