Quantum Hadamard Edge Detection for Image Processing
Based on the paper: "Edge Detection Quantumized: A Novel Quantum Algorithm for Image Processing" https://arxiv.org/html/2404.06889v1
This notebook demonstrates:
-
QPIE (Quantum Probability Image Encoding) encoding
-
QHED (Quantum Hadamard Edge Detection) algorithm
The encoding was implemented based on this paper: https://arxiv.org/pdf/1801.01465
import math
from typing import List
import matplotlib.pyplot as plt
import numpy as np
from classiq import *
photo = []
photo = plt.imread("Marina Bay Sands 128.png")
plt.imshow(photo)
segments = [i / 255 for i in [0, 30, 60, 90, 120, 150, 180, 210, 255]]
image = np.array(photo)
print(segments)
print(image)
for i in range(1, len(image)):
for j in range(len(image[i])):
pixel = image[i][j][0]
for s in segments:
if pixel >= s:
image[i][j][0] = s
image[i][j][1] = s
image[i][j][2] = s
print("AFTER THE UPDATE")
print(image)
plt.imshow(image)
image = np.array(photo)
[0.0, 0.11764705882352941, 0.23529411764705882, 0.35294117647058826, 0.47058823529411764, 0.5882352941176471, 0.7058823529411765, 0.8235294117647058, 1.0]
[[[0.03921569 0.03921569 0.03921569]
[0.03921569 0.03921569 0.03921569]
[0.03921569 0.03921569 0.03921569]
...
[0.04313726 0.04313726 0.04313726]
[0.04705882 0.04705882 0.04705882]
[0.04705882 0.04705882 0.04705882]]
[[0.03921569 0.03921569 0.03921569]
[0.03921569 0.03921569 0.03921569]
[0.03921569 0.03921569 0.03921569]
...
[0.04705882 0.04705882 0.04705882]
[0.04313726 0.04313726 0.04313726]
[0.04705882 0.04705882 0.04705882]]
[[0.03921569 0.03921569 0.03921569]
[0.03921569 0.03921569 0.03921569]
[0.03921569 0.03921569 0.03921569]
...
[0.04705882 0.04705882 0.04705882]
[0.04705882 0.04705882 0.04705882]
[0.04705882 0.04705882 0.04705882]]
...
[[0.03529412 0.03529412 0.02745098]
[0.02352941 0.02352941 0.01960784]
[0.02352941 0.02352941 0.01568628]
...
[0.2784314 0.11764706 0.17254902]
[0.42745098 0.14509805 0.28627452]
[0.44313726 0.14117648 0.30588236]]
[[0.03137255 0.03137255 0.02352941]
[0.03137255 0.03137255 0.02352941]
[0.03137255 0.03137255 0.02352941]
...
[0.24313726 0.11764706 0.14901961]
[0.30980393 0.11764706 0.20392157]
[0.3137255 0.10980392 0.20784314]]
[[0.02745098 0.02745098 0.01960784]
[0.02745098 0.02745098 0.01960784]
[0.02745098 0.02745098 0.01960784]
...
[0.09411765 0.05882353 0.05098039]
[0.09019608 0.05490196 0.04705882]
[0.07843138 0.05490196 0.04313726]]]
AFTER THE UPDATE
[[[0.03921569 0.03921569 0.03921569]
[0.03921569 0.03921569 0.03921569]
[0.03921569 0.03921569 0.03921569]
...
[0.04313726 0.04313726 0.04313726]
[0.04705882 0.04705882 0.04705882]
[0.04705882 0.04705882 0.04705882]]
[[0. 0. 0. ]
[0. 0. 0. ]
[0. 0. 0. ]
...
[0. 0. 0. ]
[0. 0. 0. ]
[0. 0. 0. ]]
[[0. 0. 0. ]
[0. 0. 0. ]
[0. 0. 0. ]
...
[0. 0. 0. ]
[0. 0. 0. ]
[0. 0. 0. ]]
...
[[0. 0. 0. ]
[0. 0. 0. ]
[0. 0. 0. ]
...
[0.23529412 0.23529412 0.23529412]
[0.3529412 0.3529412 0.3529412 ]
[0.3529412 0.3529412 0.3529412 ]]
[[0. 0. 0. ]
[0. 0. 0. ]
[0. 0. 0. ]
...
[0.23529412 0.23529412 0.23529412]
[0.23529412 0.23529412 0.23529412]
[0.23529412 0.23529412 0.23529412]]
[[0. 0. 0. ]
[0. 0. 0. ]
[0. 0. 0. ]
...
[0. 0. 0. ]
[0. 0. 0. ]
[0. 0. 0. ]]]
QPIE Encoding Implementation
Now, convert an image into valid QPIE probability amplitudes. The image is converted to grayscale if needed, flattened to a 1D vector, made non-negative, and L2-normalized so the sum of squared values equals 1.
If the image is all zeros, a uniform normalized vector is returned to avoid division by zero.
def normalize_image(image: np.ndarray) -> np.ndarray:
"""
Normalize image pixels to create valid probability amplitudes.
For QPIE, the image is represented as:
|img⟩ = Σ(x,y) (I_xy / √(Σ I_xy²)) |xy⟩
Args:
image: Input image as numpy array
Returns:
Normalized probability amplitudes
"""
# Flatten the image
flat_image = image.flatten()
# Normalize to get probability amplitudes
# Ensure all values are non-negative
flat_image = np.abs(flat_image)
# Calculate normalization factor
norm_factor = np.sqrt(np.sum(flat_image**2))
if norm_factor == 0:
# Handle edge case of all-zero image
normalized = np.ones_like(flat_image) / np.sqrt(len(flat_image))
else:
normalized = flat_image / norm_factor
return normalized
def image_to_qpie_amplitudes(image: np.ndarray) -> List[float]:
"""
Convert an image to QPIE probability amplitudes.
Args:
image: Input image (grayscale or color)
Returns:
List of normalized probability amplitudes
"""
if len(image.shape) == 3: # Color image
# Convert to grayscale
image = np.mean(image, axis=2)
# Normalize the image
amplitudes = normalize_image(image)
return amplitudes.tolist()
@qfunc
def qpie_encoding(image_qubits: Output[QArray[QBit]]):
"""
QPIE (Quantum Probability Image Encoding) implementation.
Encodes image pixels as probability amplitudes:
|img⟩ = Σ c_i |i⟩
where c_i are normalized pixel values.
Args:
amplitudes: Normalized pixel values as probability amplitudes
image_qubits: Output qubits storing the encoded image
"""
# Calculate number of qubits needed
num_pixels = len(AMPLITUDES)
num_qubits = math.ceil(math.log2(num_pixels))
# Allocate qubits for image encoding
allocate(num_qubits, image_qubits)
# Use inplace_prepare_amplitudes to encode the amplitudes
inplace_prepare_amplitudes(AMPLITUDES, 0, image_qubits)
amplitudes = image_to_qpie_amplitudes(image)
print(f"{amplitudes=}")
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len(amplitudes)
16384
Modified QHED Algorithm
The QHED algorithm detects edges by:
-
Adding an auxiliary qubit in \(|+\rangle\) state
-
Applying amplitude permutation to shift pixels
-
Applying Hadamard to compute differences
-
Measuring to get edge information
from classiq.qmod.symbolic import logical_or
@qfunc
def quantum_edge_detection_scalable(
# n_qubits: CInt, # Number of position qubits
edge_aux: Output[QBit],
position: Output[QNum],
):
# We use the QPIE encoding to encode the image in the quantum state
qpie_encoding(image_qubits=position)
# Create the vertical and horizontal correlation checks
horizontal_edge = QBit()
vertical_edge = QBit()
allocate(horizontal_edge)
allocate(vertical_edge)
# QHED edge detection
within_apply(
within=lambda: H(horizontal_edge),
apply=lambda: (
control(
horizontal_edge, lambda: inplace_add(-1, position)
), # Cyclic shift one left
),
)
within_apply(
within=lambda: H(vertical_edge),
apply=lambda: (
control(
vertical_edge, lambda: inplace_add(-n, position)
), # Cyclic shift one up
),
)
edge_aux |= logical_or(horizontal_edge, vertical_edge)
drop(horizontal_edge)
drop(vertical_edge)
# Create and synthesize the 2x2 model
AMPLITUDES = amplitudes
n = int(np.sqrt(len(AMPLITUDES)))
print(f"Creating {n}x{n} image edge detection model...")
Creating 128x128 image edge detection model...
Synthesize and Analyze the Quantum Circuit
The model is synthesized with a 20-qubit width limit and a long timeout, and finally exported as quantum_image_edge_detection with 15-digit numeric precision.
@qfunc
def main(
pos: Output[QNum],
edge_aux: Output[QNum],
):
quantum_edge_detection_scalable(edge_aux=edge_aux, position=pos)
qmod = create_model(
main,
constraints=Constraints(max_width=20),
preferences=Preferences(timeout_seconds=14400),
)
write_qmod(
qmod,
"quantum_image_edge_detection",
decimal_precision=15,
)
qprog = synthesize(
main,
constraints=Constraints(max_width=20),
preferences=Preferences(timeout_seconds=14400),
)
print(f"\n{n}x{n} Image Circuit Statistics:")
print(f" - Number of qubits: {qprog.data.width}")
print(f" - Circuit depth: {qprog.transpiled_circuit.depth}")
print(
f" - Number of gates: {qprog.transpiled_circuit.count_ops if hasattr(qprog.transpiled_circuit, 'count_ops') else 'N/A'}"
)
128x128 Image Circuit Statistics:
- Number of qubits: 17
- Circuit depth: 32995
- Number of gates: {'u': 16903, 'cx': 16878}
show(qprog)
Quantum program link: https://platform.classiq.io/circuit/36qAK2Al0RP6zgLHxrbafMRglxX
qprog = set_quantum_program_execution_preferences(
qprog, preferences=ExecutionPreferences(num_shots=200000)
)
res = execute(qprog).result()[0].value
res.dataframe
| pos | edge_aux | count | probability | bitstring | |
|---|---|---|---|---|---|
| 0 | 11710 | 0 | 173 | 0.000865 | 010110110111110 |
| 1 | 11839 | 0 | 169 | 0.000845 | 010111000111111 |
| 2 | 11316 | 0 | 162 | 0.000810 | 010110000110100 |
| 3 | 11712 | 0 | 162 | 0.000810 | 010110111000000 |
| 4 | 11711 | 0 | 159 | 0.000795 | 010110110111111 |
| ... | ... | ... | ... | ... | ... |
| 17285 | 16371 | 1 | 1 | 0.000005 | 111111111110011 |
| 17286 | 16372 | 1 | 1 | 0.000005 | 111111111110100 |
| 17287 | 16375 | 1 | 1 | 0.000005 | 111111111110111 |
| 17288 | 16376 | 1 | 1 | 0.000005 | 111111111111000 |
| 17289 | 16377 | 1 | 1 | 0.000005 | 111111111111001 |
17290 rows × 5 columns
Create Edge Image From Measurement Results
If ["edge_aux"] == 1 then it is marked as an edge pixel. The new amplitude is calculated based on the number of shots measured for that pixel, normalized by the total number of shots.
new_amplitudes = dict()
for pos in range(len(AMPLITUDES)):
new_amplitudes[pos] = 0
num_shots = sum([pc.shots for pc in res.parsed_counts])
# if marked as edge, set new amplitude normalized
for pc in res.parsed_counts:
if pc.state["edge_aux"] == 1:
new_amplitudes[pc.state["pos"]] = float(np.sqrt(pc.shots / num_shots))
# set amplitude in image of edges
edge_image = np.zeros((n, n))
for i in range(n):
for j in range(n):
edge_image[i, j] = new_amplitudes[n * i + j]
Analyze amplitude distributions
plt.hist(new_amplitudes.values())
(array([1.2809e+04, 0.0000e+00, 2.0090e+03, 4.9500e+02, 5.3900e+02,
3.1900e+02, 1.4400e+02, 5.1000e+01, 1.6000e+01, 2.0000e+00]),
array([0. , 0.00109545, 0.00219089, 0.00328634, 0.00438178,
0.00547723, 0.00657267, 0.00766812, 0.00876356, 0.00985901,
0.01095445]),
<BarContainer object of 10 artists>)
Some amplitudes are extremely small and can be treated as noise; discarding them yields a cleaner edge image.
edge_image = np.zeros((n, n))
for i in range(n):
for j in range(n):
edge_image[i, j] = (
new_amplitudes[n * i + j] if new_amplitudes[n * i + j] > 0.003 else 0
)
plt.hist(AMPLITUDES)
(array([7454., 4363., 1308., 942., 786., 677., 426., 252., 118.,
58.]),
array([0.00054111, 0.00337295, 0.00620478, 0.00903661, 0.01186845,
0.01470028, 0.01753211, 0.02036395, 0.02319578, 0.02602761,
0.02885945]),
<BarContainer object of 10 artists>)
The resulting edge-detected image
plt.imshow(edge_image, cmap="gray") # , cmap="gray")
<matplotlib.image.AxesImage at 0x162e09e20>
In comparison, the original image is:
plt.imshow(image)
<matplotlib.image.AxesImage at 0x162e2d8b0>
