Distances

In this section, we explore various binary distance measures that are widely used in classical computing. To refresh the nomenclature used:

	1 (Presence in x)	0 (Absence in x)
1 (Presence in y)	\(a = x\cdot y\)	\(b = \overline{x}\cdot y\)
0 (Absence in y)	\(c = x\cdot \overline{y}\)	\(d = \overline{x}\cdot \overline{y}\)

Hamming distance

Hamming distance can be defined as a metric that quantifies the dissimilarity between two binary states of equal length by measuring the number of positions at which the corresponding bits differ. The formula is as follows:

\[d_{Hamming}\left(x,y\right)=b+c\]

• This metric can NOT have NaN values \(\left(\frac{0}{0}\right)\).

To use the Hamming distance metric to generate a quantum superposition, ensure to set the metric parameter to ‘hamming’.

Example: Generation of a Hamming quantum superposition.

from qsimov_cloud_client import QsimovCloudClient

# Initialize QsimovCloudClient with your access token
client = QsimovCloudClient("your_access_token")

# Set parameters for the service
client.set_metric("hamming")
client.set_state(state_bin='0111010')
client.set_range(["0", "3"])
client.can_have_nan(False)
client.set_ancilla_mode("clean")

# Generate a quantum circuit
circuit_superposition = client.generate_circuit()

# Access the result
print("The resulting circuit in qasm is:", circuit_superposition.get_qasm_code())

The presented code facilitates the creation of a quantum superposition within the states that have a distance greater than or equal to \(0\) and less than or equal to \(3\) from the reference state \(0111010\ (58)\). If we simulate the generated QASM code and plot the probabilities of the register where the superposition occurs, obtained from the state vector, we will obtain the following result.

Note

The reference state is highlighted in nova mint color.