From d6284576c7cfb5356eb0195b80904f589fc3f8a3 Mon Sep 17 00:00:00 2001 From: Supreeth Mysore Venkatesh Date: Mon, 7 Apr 2025 18:59:14 +0200 Subject: [PATCH 1/7] feat: add demo on qubit-efficient VQAs for image segmentation --- .../qubit_efficient_encoding/ABE_ACE.png | Bin 0 -> 1702380 bytes .../qubit_efficient_encoding/scalability.png | Bin 0 -> 33223 bytes .../vqa-segmentation.png | Bin 0 -> 1361558 bytes ...ial_qubit_efficient_encoding.metadata.json | 77 ++ .../tutorial_qubit_efficient_encoding.py | 891 ++++++++++++++++++ 5 files changed, 968 insertions(+) create mode 100644 _static/demonstration_assets/qubit_efficient_encoding/ABE_ACE.png create mode 100644 _static/demonstration_assets/qubit_efficient_encoding/scalability.png create mode 100644 _static/demonstration_assets/qubit_efficient_encoding/vqa-segmentation.png create mode 100644 demonstrations/tutorial_qubit_efficient_encoding.metadata.json create mode 100644 demonstrations/tutorial_qubit_efficient_encoding.py diff --git 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a/demonstrations/tutorial_qubit_efficient_encoding.metadata.json b/demonstrations/tutorial_qubit_efficient_encoding.metadata.json new file mode 100644 index 0000000000..ed770bf736 --- /dev/null +++ b/demonstrations/tutorial_qubit_efficient_encoding.metadata.json @@ -0,0 +1,77 @@ +{ + "title": "Qubit-Efficient Encoding Techniques for Solving QUBO Problems", + "authors": [ + { + "username": "supreethmv" + } + ], + "dateOfPublication": "2024-11-15T00:00:00+00:00", + "dateOfLastModification": "2025-04-07T00:00:00+00:00", + "categories": ["Optimization"], + "tags": ["qubit-efficient encoding", "QUBO", "VQA", "image segmentation"], + "previewImages": [ + { + "type": "thumbnail", + "uri": "/_static/demonstration_assets/qubit_efficient_encoding/vqa-segmentation.png" + } + ], + "seoDescription": "Explore scalable qubit-efficient variational quantum algorithms (VQAs) for solving QUBO problems, with applications in image segmentation and beyond.", + "doi": "10.1109/QCE60285.2024.00059", + "references": [ + { + "id": "smv-nisq-seg", + "type": "article", + "title": "Qubit-Efficient Variational Quantum Algorithms for Image Segmentation", + "authors": "S. M. Venkatesh, A. Macaluso, M. Nuske, M. Klusch, and A. Dengel", + "year": "2024", + "journal": "2024 IEEE International Conference on Quantum Computing and Engineering (QCE)", + "doi": "10.1109/QCE60285.2024.00059", + "url": "https://doi.org/10.1109/QCE60285.2024.00059", + "pages": "450–456" + }, + { + "id": "smv-qseg", + "type": "article", + "title": "Q-Seg: Quantum Annealing-Based Unsupervised Image Segmentation", + "authors": "S. M. Venkatesh, A. Macaluso, M. Nuske, M. Klusch, and A. Dengel", + "year": "2024", + "journal": "IEEE Computer Graphics and Applications", + "doi": "10.1109/MCG.2024.3455012", + "url": "https://doi.org/10.1109/MCG.2024.3455012", + "pages": "27–39" + }, + { + "id": "qaoa", + "type": "preprint", + "title": "A quantum approximate optimization algorithm", + "authors": "E. Farhi, J. Goldstone, and S. Gutmann", + "year": "2014", + "url": "https://arxiv.org/abs/1411.4028" + }, + { + "id": "pge", + "type": "article", + "title": "Noisy intermediate-scale quantum computing algorithm for solving an n-vertex MaxCut problem with log(n) qubits", + "authors": "M. J. Rančić", + "year": "2023", + "journal": "Physical Review Research", + "doi": "10.1103/PhysRevResearch.5.L012021", + "url": "https://doi.org/10.1103/PhysRevResearch.5.L012021" + }, + { + "id": "abe", + "type": "article", + "title": "Qubit-efficient encoding schemes for binary optimisation problems", + "authors": "B. Tan, M.-A. Lemonde, S. Thanasilp, J. Tangpanitanon, and D. G. Angelakis", + "year": "2021", + "journal": "Quantum", + "doi": "10.22331/q-2021-05-04-454", + "url": "https://doi.org/10.22331/q-2021-05-04-454" + } + ], + "basedOnPapers": ["10.1109/QCE60285.2024.00059"], + "referencedByPapers": [], + "relatedContent": [], + "hardware": [] + } + \ No newline at end of file diff --git a/demonstrations/tutorial_qubit_efficient_encoding.py b/demonstrations/tutorial_qubit_efficient_encoding.py new file mode 100644 index 0000000000..108b8a3d5e --- /dev/null +++ b/demonstrations/tutorial_qubit_efficient_encoding.py @@ -0,0 +1,891 @@ +r"""Qubit-Efficient Encoding Techniques for Solving QUBO Problems + +=================================== + +""" + +###################################################################### +# 🧠 Introduction + +# --------------- + +# + +# Classical algorithms are already incredibly effective at solving many optimization problems — + +# especially when the number of variables is in the low thousands. However, when scaling up to + +# problems involving **tens or hundreds of thousands** of binary variables, classical solvers begin to + +# falter due to the **combinatorial explosion** in the search space. + +# + +# Quantum computing offers a compelling alternative — not by brute force, but by **exploring + +# exponentially large solution spaces in parallel** using quantum superposition and entanglement. One + +# popular quantum algorithm for such problems is the **Quantum Approximate Optimization Algorithm + +# (QAOA)** [#qaoa]_, a textbook example of a variational quantum algorithm (VQA). However, QAOA requires + +# **one qubit per binary variable**, which becomes **prohibitively resource-intensive** for + +# large-scale problems. + +# + +# For instance, solving a QUBO (Quadratic Unconstrained Binary Optimization) problem with 10,000 + +# variables using QAOA would require 10,000 logical qubits — a scale **far beyond current quantum + +# hardware capabilities**. + +# + +###################################################################### +# 🔍 Why Qubit-Efficient Methods? + +# ------------------------------- + +# + +# This demo explores **qubit-efficient alternatives** to QAOA that are not only suitable for today’s + +# NISQ (Noisy Intermediate-Scale Quantum) devices, but also **generalizable across combinatorial + +# optimization problems**. + +# + +# We specifically demonstrate these methods in the context of **unsupervised image segmentation**, by: + +# + +# 1. Formulating segmentation as a **min-cut problem** over a graph derived from the image. + +# 2. Reformulating the min-cut as a **QUBO problem**. + +# 3. Solving the QUBO using **qubit-efficient VQAs**, namely [#smv-nisq-seg]_: + +# + +# - **Parametric Gate Encoding (PGE)** + +# - **Ancilla Basis Encoding (ABE)** + +# - **Adaptive Cost Encoding (ACE)** + +# + +# These methods only require a **logarithmic number of qubits** with respect to the problem size + +# (i.e., number of binary variables). This makes them **scalable and implementable** even on near-term + +# quantum hardware. Moreover, the exact graph-based image segmentation technique have been explored + +# also using quantum annealers [#smv-qseg]_. + +# + +# While this demo walks through an image segmentation example, the encoding schemes presented here can + +# be applied to **any binary combinatorial optimization problem**. + +# .. figure:: ../_static/demonstration_assets/qubit_efficient_encoding/vqa-segmentation.png +# :scale: 75% +# :align: center +# :alt: Architecture for segmenting an image by transforming it into a graph and solving the corresponding minimum cut as a QUBO problem using variational quantum circuits [#smv-nisq-seg]_. +# +# Architecture for segmenting an image by transforming it into a graph and solving the corresponding minimum cut as a QUBO problem using variational quantum circuits [#smv-nisq-seg]_. + +# -------------- + +# + +###################################################################### +# 🧱 Import dependencies + +# ---------------------- + +# + +# We begin by importing the standard Python libraries for graph construction and visualization, and + +# PennyLane modules for quantum circuit construction and simulation. This setup ensures a seamless + +# interface for hybrid quantum-classical processing. + +# + +import numpy as np +import networkx as nx +import pennylane as qml +from scipy.optimize import minimize, differential_evolution +import matplotlib.pyplot as plt + +from math import ceil, log2 +import time + +###################################################################### +# 🖼️ Generate an Example Image + +# ---------------------------- + +# + +# To keep things intuitive, we generate a toy grayscale image. This synthetic image will be segmented + +# using quantum algorithms. + +# + +# We use a simple 4×4 grid for clarity. The pixel intensities will act as a proxy for similarity when + +# we later construct a graph. Each pixel becomes a vertex, and edges represent similarity (e.g., + +# inverse of intensity difference). + +# + +height, width = 4, 4 +image = np.array( + [ + [0.97, 0.87, 0.09, 0.92], + [0.93, 0.10, 0.12, 0.93], + [0.94, 0.15, 0.88, 0.94], + [0.92, 0.17, 0.10, 0.92], + ] +) + + +plt.imshow(image, interpolation="nearest", cmap=plt.cm.gray, vmin=0, vmax=1) +plt.axis("off") +plt.show() + +###################################################################### +# 🔗 Representing Image as a Graph + +# -------------------------------- + +# + +# | Each image is modeled as an undirected weighted grid graph: + +# | - Each pixel becomes a node. - An edge exists between neighboring pixels. - The weight reflects + +# similarity (e.g., inverse absolute difference in grayscale values). + +# + +# This transforms the segmentation task into a **minimum cut problem**, where the goal is to partition + +# the graph to minimize the sum of cut edge weights. + +# + + +def image_to_grid_graph(gray_img, sigma=0.5): + # Convert image to grayscale + # gray_img = np.dot(img, [0.2989, 0.5870, 0.1140]) + h, w = gray_img.shape + # Initialize graph nodes and edges + nodes = np.zeros((h * w, 1)) + edges = [] + nx_elist = [] + # Compute node potentials and edge weights + min_weight = 1 + max_weight = 0 + for i in range(h * w): + x, y = i // w, i % w + nodes[i] = gray_img[x, y] + if x > 0: + j = (x - 1) * w + y + # weight = np.exp(-dist.euclidean([gray_img[x,y]], [gray_img[x-1,y]])**2 / (2*sigma**2)) + weight = 1 - np.exp(-((gray_img[x, y] - gray_img[x - 1, y]) ** 2) / (2 * sigma**2)) + edges.append((i, j, weight)) + nx_elist.append(((x, y), (x - 1, y), np.round(weight, 2))) + if min_weight > weight: + min_weight = weight + if max_weight < weight: + max_weight = weight + if y > 0: + j = x * w + y - 1 + # weight = np.exp(-dist.euclidean([gray_img[x,y]], [gray_img[x,y-1]])**2 / (2*sigma**2)) + weight = 1 - np.exp(-((gray_img[x, y] - gray_img[x, y - 1]) ** 2) / (2 * sigma**2)) + # print('weight',weight) + edges.append((i, j, weight)) + nx_elist.append(((x, y), (x, y - 1), np.round(weight, 2))) + if min_weight > weight: + min_weight = weight + if max_weight < weight: + max_weight = weight + a = -1 + b = 1 + if max_weight - min_weight: + normalized_edges = [ + ( + node1, + node2, + -1 + * np.round( + ((b - a) * ((edge_weight - min_weight) / (max_weight - min_weight))) + a, 2 + ), + ) + for node1, node2, edge_weight in edges + ] + normalized_nx_elist = [ + ( + node1, + node2, + -1 + * np.round( + ((b - a) * ((edge_weight - min_weight) / (max_weight - min_weight))) + a, 2 + ), + ) + for node1, node2, edge_weight in nx_elist + ] + else: + normalized_edges = [ + (node1, node2, -1 * np.round(edge_weight, 2)) for node1, node2, edge_weight in edges + ] + normalized_nx_elist = [ + (node1, node2, -1 * np.round(edge_weight, 2)) for node1, node2, edge_weight in nx_elist + ] + return nodes, edges, nx_elist, normalized_edges, normalized_nx_elist + + +pixel_values, elist, nx_elist, normalized_elist, normalized_nx_elist = image_to_grid_graph(image) + + +pixel_values, elist, nx_elist, normalized_elist, normalized_nx_elist + +G = nx.grid_2d_graph(image.shape[0], image.shape[1]) +G.add_weighted_edges_from(normalized_nx_elist) + + +def draw(G): + plt.figure(figsize=(6, 6)) + default_axes = plt.axes(frameon=True) + pos = {(x, y): (y, -x) for x, y in G.nodes()} + nx.draw_networkx(G, pos) + nodes = nx.draw_networkx_nodes( + G, pos, node_color=1 - pixel_values, node_size=3000, cmap=plt.cm.Greys, vmin=0, vmax=1 + ) + nodes.set_edgecolor("k") + edge_labels = nx.get_edge_attributes(G, "weight") + nx.draw_networkx_edge_labels( + G, pos=pos, edge_labels=edge_labels, font_size=10, font_color="tab:blue" + ) + plt.axis("off") + + +# plt.savefig("2x2_graph.jpg", dpi=600, bbox_inches='tight', pad_inches=0) +draw(G) +plt.show() + +###################################################################### +# 🔁 Parametric Gate Encoding (PGE) + +# --------------------------------- + +# + +# In PGE, we directly encode the binary segmentation mask in the rotation parameters of a diagonal + +# gate [#pge]_. + +# + +# Only :math:`\log_2(n)` qubits are required, which is a **huge gain** over methods like QAOA that + +# need :math:`O(n)`. + +# + +# Quantum Circuit: + +# ~~~~~~~~~~~~~~~~ + +# + +# - Apply Hadamard gates to initialize a uniform superposition. + +# - Apply a diagonal gate :math:`U(\vec{\theta})` whose entries encode binary values using a + +# thresholding function :math:`f(\theta_i)`. + +# - Evaluate the cost function: + +# + +# .. math:: + +# + +# + +# C(\vec{\theta}) = \langle \psi(\vec{\theta}) | L_G | \psi(\vec{\theta}) \rangle + +# + +# where :math:`L_G` is the graph Laplacian. + +# + +# The result is optimized using a classical optimizer, and :math:`\theta_i` are mapped to bits + +# depending on whether :math:`\theta_i < \pi` or not. + +# + +n = G.number_of_nodes() +n_qubits = int(np.ceil(np.log2(n))) +n_qubits + + +def R(theta): + if abs(theta) > 2 * np.pi or abs(theta) < 0: + theta = abs(theta) - (np.floor(abs(theta) / (2 * np.pi)) * (2 * np.pi)) + return 0 if 0 <= theta < np.pi else 1 + + +# Define a PennyLane quantum device (using default.qubit simulator) +dev = qml.device("default.qubit", wires=n_qubits) + + +# Quantum circuit as a QNode +@qml.qnode(dev) +def circuit(params, H): + # N = len(params) + N = int(np.log2(len(params))) + for i in range(N): + qml.Hadamard(wires=i) + + diagonal_elements = [np.exp(1j * np.pi * R(param)) for param in params] + # qml.DiagonalQubitUnitary(np.diag(diagonal_elements), wires=list(range(N))) + qml.DiagonalQubitUnitary(diagonal_elements, wires=list(range(N))) + + return qml.expval(qml.Hermitian(H, wires=list(range(N)))) + + +def cost_fn(params, H): + global optimization_iteration_count + optimization_iteration_count += 1 + cost = circuit(params, H) + print(f"@ Iteration {optimization_iteration_count} Cost :", cost) + return cost + + +def decode(optimal_params): + return [R(param) for param in optimal_params] + + +def objective_value(x, w): + return np.dot(x, np.dot(w, x)) + + +def new_nisq_algo_solver(G, optimizer_method="Powell", initial_params_seed=123): + global optimization_iteration_count + optimization_iteration_count = 0 + n = G.number_of_nodes() + w = nx.adjacency_matrix(G).todense() + D_G = np.diag(list(dict(G.degree()).values())) + A_G = w + L_G = D_G - A_G + n_padding = 2 ** int(np.ceil(np.log2(n))) - n + L_G = np.pad(L_G, [(0, n_padding), (0, n_padding)], mode="constant") + H = L_G + + np.random.seed(seed=initial_params_seed) + initial_params = np.random.uniform(low=0.5, high=2 * np.pi, size=(n + n_padding)) + + result = minimize( + fun=cost_fn, + x0=initial_params, + args=(H,), + method=optimizer_method, + ) + + optimal_params, expectation_value = result.x, result.fun + x = np.real(decode(optimal_params)) + x = x[:n] + cut_value = objective_value(x, w) + + return x, expectation_value, cut_value + + +binary_solution, expectation_value, cut_value = new_nisq_algo_solver(G, optimizer_method="Powell") +binary_solution, expectation_value, cut_value + + +def decode_binary_string(x, height, width): + mask = np.zeros([height, width]) + for index, segment in enumerate(x): + mask[index // width, index % width] = segment + return mask + + +mask = decode_binary_string(x, height, width) +plt.imshow(mask, interpolation="nearest", cmap=plt.cm.gray, vmin=0, vmax=1) +plt.axis("off") +plt.show() + + +###################################################################### +# 🧪 Ancilla Basis Encoding (ABE) + +# ------------------------------- + +# + +# | ABE uses :math:`\log_2(n) + 1` qubits: + +# | - :math:`\log_2(n)` **register qubits** represent binary states. + +# | - One **ancilla qubit** helps encode the binary decision. + +# + +# Each computational basis state of the register corresponds to one binary variable. The ancilla + +# qubit’s amplitude is used to decide the bit value [#abe]_: + +# + +# .. math:: + +# + +# + +# x_{v_i} = \begin{cases} 0 & \text{if } |a_i|^2 > |b_i|^2 \\ 1 & \text{otherwise} \end{cases} + +# + +# The cost function is parameterized over **expectation values** of basis state projectors: + +# + +# .. math:: + +# + +# + +# C(\vec{\theta}) = \sum_{i,j} Q_{ij} \frac{\langle \hat{P}_i^1 \rangle \langle \hat{P}_j^1 \rangle}{\langle \hat{P}_i \rangle \langle \hat{P}_j \rangle} (1 - \delta_{ij}) + \sum_i Q_{ii} \frac{\langle \hat{P}_i^1 \rangle}{\langle \hat{P}_i \rangle} + +# .. figure:: ../_static/demonstration_assets/qubit_efficient_encoding/ABE_ACE.png +# :scale: 75% +# :align: center +# :alt: Quantum circuit of ABE/ACE for solving QUBO of size 4 using 3 qubits (2 register + 1 ancilla). Basis state amplitudes are used to decode the binary solution [#smv-nisq-seg]_. +# +# Quantum circuit of ABE/ACE for solving QUBO of size 4 using 3 qubits (2 register + 1 ancilla). Basis state amplitudes are used to decode the binary solution [#smv-nisq-seg]_. + + +# + + +def get_qubo_matrix(W): + """Computes the QUBO matrix for the Minimum Cut problem given a weight matrix W.""" + n = W.shape[0] # Number of nodes + Q = np.zeros((n, n)) # Initialize QUBO matrix + + for i in range(n): + Q[i, i] = np.sum(W[i]) # Diagonal terms (degree of node) + for j in range(n): + if i != j: + Q[i, j] = -W[i, j] # Off-diagonal terms (negative adjacency) + + return Q + + +def get_circuit(nq, n_layers): + def circuit(params): + for i in range(nq): + qml.Hadamard(wires=i) + for layer_i in range(n_layers): + for qubit_i in range(nq - 1): + qml.CNOT(wires=[qubit_i, (qubit_i + 1) % nq]) + for qubit_i in range(nq): + qml.RY(params[(nq * layer_i) + qubit_i], wires=qubit_i) + return qml.probs(wires=range(nq)) + + return circuit + + +def get_projectors(probabilities, n_c): + P, P1 = [0] * n_c, [0] * n_c + for k, v in probabilities.items(): + index = int(k[1:], 2) + P[index] += v + if k[0] == "1": + P1[index] += v + return P, P1 + + +def objective_value(x, w): + return sum(w[i, j] * x[i] * (1 - x[j]) for i in range(len(x)) for j in range(len(x))) + + +def decode_probabilities(probabilities): + binary_solution, n_r = [], len(list(probabilities.keys())[0]) - 1 + for i in range(2**n_r): + zero_state, one_state = "0" + format(i, "b").zfill(n_r), "1" + format(i, "b").zfill(n_r) + if zero_state in probabilities and one_state in probabilities: + binary_solution.append(0 if probabilities[zero_state] > probabilities[one_state] else 1) + elif zero_state in probabilities: + binary_solution.append(0) + elif one_state in probabilities: + binary_solution.append(1) + return binary_solution + + +def evaluate_cost(params, circuit, qubo_matrix): + global valid_probabilities, optimization_iteration_count, best_cost, best_params + optimization_iteration_count += 1 + raw_probs = circuit(params) # NumPy array + + # Convert probability array to dictionary with bitstrings + num_qubits = int(np.log2(len(raw_probs))) + bitstrings = [format(i, f"0{num_qubits}b") for i in range(len(raw_probs))] + probabilities = {bitstrings[i]: raw_probs[i] for i in range(len(raw_probs))} + + n_c = 2 ** (len(bitstrings[0]) - 1) + P, P1 = get_projectors(probabilities, n_c) + if 0 in P: + return float("inf") + + valid_probabilities = probabilities.copy() + cost = sum( + ( + qubo_matrix[i][j] * P1[i] / P[i] + if i == j + else (qubo_matrix[i][j] * P1[i] * P1[j]) / (P[i] * P[j]) + ) + for i in range(len(qubo_matrix)) + for j in range(len(qubo_matrix)) + ) + + # Track the best cost and parameters manually + if cost < best_cost: + best_cost = cost + best_params = params.copy() # Store the best parameter set + + print(f"@ Iteration {optimization_iteration_count} Cost :", cost) + return cost + + +def ancilla_basis_encoding( + G, initial_params, n_layers=1, optimizer_method="differential_evolution" +): + w = nx.adjacency_matrix(G).todense() + Q = get_qubo_matrix(W=w) + nc, nq = len(G.nodes()), ceil(log2(len(G.nodes()))) + 1 + dev = qml.device("default.qubit", wires=nq, shots=None) + qnode = qml.QNode(get_circuit(nq, n_layers), dev) + global valid_probabilities, optimization_iteration_count, best_cost, best_params # Track the best parameters + valid_probabilities, optimization_iteration_count = [], 0 + best_cost = float("inf") # Initialize best cost + best_params = None # Initialize best parameters + if optimizer_method == "differential_evolution": + optimization_result = differential_evolution( + func=evaluate_cost, + bounds=[(-np.pi, np.pi)] * len(initial_params), + args=(qnode, Q), + strategy="best1bin", + maxiter=100, + popsize=10, + ) + else: + optimization_result = minimize( + fun=evaluate_cost, + x0=initial_params, + args=(qnode, Q), + method=optimizer_method, + bounds=[(-np.pi, np.pi)] * len(initial_params), + ) + if not valid_probabilities: + return False, [0] * nc, 0, 0, [0] * len(initial_params) + best_raw_probs = qnode(best_params) + bitstrings = [format(i, f"0{nq}b") for i in range(len(best_raw_probs))] + best_probabilities = {bitstrings[i]: best_raw_probs[i] for i in range(len(best_raw_probs))} + binary_solution = decode_probabilities(best_probabilities)[:nc] + print("binary_solution", binary_solution) + return binary_solution, best_cost, objective_value(np.array(binary_solution), w), best_params + + +###################################################################### +# Although the code can handle various methods available in ``scipy.minimize``, in this demo we will + +# be using a more robust genetic algorithm ``differential_evolution`` as the classical optimizer. + +# + +initial_params_seed, scipy_optimizer_method, n_layers = 42, "differential_evolution", 4 +nq = ceil(log2(len(G.nodes()))) + 1 +np.random.seed(initial_params_seed) +initial_params = np.random.uniform(low=-np.pi, high=np.pi, size=nq * n_layers) +print(f"Executing QC with {n_layers} layers and {scipy_optimizer_method} optimizer.") +start_time = time.time() +solution, value, cut_value, optimal_params = ancilla_basis_encoding( + G, initial_params, n_layers, scipy_optimizer_method +) +end_time = time.time() +print(solution, value, cut_value, optimal_params) + +mask = decode_binary_string(solution, height, width) +plt.imshow(mask, interpolation="nearest", cmap=plt.cm.gray, vmin=0, vmax=1) +plt.axis("off") +plt.show() + +###################################################################### +# 🔁 Adaptive Cost Encoding (ACE) + +# ------------------------------- + +# + +# ACE modifies ABE by skipping the QUBO formulation entirely during evaluation. + +# + +# Instead of optimizing over probabilities, we decode a binary vector :math:`x` from the measured + +# output and evaluate the **original min-cut cost function** directly [#smv-nisq-seg]_: + +# + +# .. math:: + +# + +# + +# C(x) = \sum_{i < j} x_i (1 - x_j) w(v_i, v_j) + +# + +# This: - Removes the need for a QUBO matrix, which can have loss of generality for constrained + +# optimization problems when the penalty coefficient is not set appropriately. - Improves convergence + +# (the cost function is discrete and combinatorial). - Aligns optimization directly with the true task + +# objective. + +# + + +def evaluate_cost_mincut(params, circuit, w): + global valid_probabilities, optimization_iteration_count, best_cost, best_params + optimization_iteration_count += 1 + raw_probs = circuit(params) # NumPy array + + # Convert probability array to dictionary with bitstrings + num_qubits = int(np.log2(len(raw_probs))) + bitstrings = [format(i, f"0{num_qubits}b") for i in range(len(raw_probs))] + probabilities = {bitstrings[i]: raw_probs[i] for i in range(len(raw_probs))} + + n_c = 2 ** (len(bitstrings[0]) - 1) + P, P1 = get_projectors(probabilities, n_c) + if 0 in P: + return float("inf") + valid_probabilities = probabilities.copy() + binary_solution = decode_probabilities(valid_probabilities)[: len(w)] + cost = objective_value(np.array(binary_solution), w) + + # Track the best cost and parameters manually + if cost < best_cost: + best_cost = cost + best_params = params.copy() # Store the best parameter set + + print(f"@ Iteration {optimization_iteration_count} Cost :", cost) + return cost + + +def adaptive_cost_encoding_de( + G, initial_params, n_layers=1, optimizer_method="differential_evolution" +): + w = nx.adjacency_matrix(G).todense() + nc, nq = len(G.nodes()), ceil(log2(len(G.nodes()))) + 1 + dev = qml.device("default.qubit", wires=nq, shots=None) + qnode = qml.QNode(get_circuit(nq, n_layers), dev) + global valid_probabilities, optimization_iteration_count, best_cost, best_params # Track the best parameters + valid_probabilities, optimization_iteration_count = [], 0 + best_cost = float("inf") # Initialize best cost + best_params = None # Initialize best parameters + + if optimizer_method == "differential_evolution": + optimization_result = differential_evolution( + func=evaluate_cost_mincut, + bounds=[(-np.pi, np.pi)] * len(initial_params), + args=(qnode, w), + strategy="best1bin", + maxiter=100, + popsize=10, + ) + else: + optimization_result = minimize( + fun=evaluate_cost_mincut, + x0=initial_params, + args=(qnode, w), + method=optimizer_method, + bounds=[(-np.pi, np.pi)] * len(initial_params), + ) + + if not valid_probabilities: + return False, [0] * nc, 0, 0, [0] * len(initial_params) + + best_raw_probs = qnode(best_params) + bitstrings = [format(i, f"0{nq}b") for i in range(len(best_raw_probs))] + best_probabilities = {bitstrings[i]: best_raw_probs[i] for i in range(len(best_raw_probs))} + binary_solution = decode_probabilities(best_probabilities)[:nc] + print("binary_solution", binary_solution) + return binary_solution, best_cost, objective_value(np.array(binary_solution), w), best_params + + +initial_params_seed, scipy_optimizer_method, n_layers = 42, "differential_evolution", 4 +nq = ceil(log2(len(G.nodes()))) + 1 +np.random.seed(initial_params_seed) +initial_params = np.random.uniform(low=-np.pi, high=np.pi, size=nq * n_layers) +print(f"Executing QC with {n_layers} layers and Differential Evolution optimizer.") +start_time = time.time() +solution, value, cut_value, optimal_params = adaptive_cost_encoding_de( + G, initial_params, n_layers, optimizer_method=scipy_optimizer_method +) +end_time = time.time() +print(solution, value, cut_value, optimal_params) + +mask = decode_binary_string(solution, height, width) +plt.imshow(mask, interpolation="nearest", cmap=plt.cm.gray, vmin=0, vmax=1) +plt.axis("off") +plt.show() + +###################################################################### +# 📈 Evaluation & Observations + +# ---------------------------- + +# + +# - PGE has a compact circuit but many parameters. + +# - ABE introduces a more efficient encoding at the cost of probabilistic decoding. + +# - ACE improves convergence by using the real problem cost and avoiding indirect probability-based + +# objectives. + +# + +# Theoretical scaling advantages: - Qubit requirement is :math:`O(\log n)`. - Parametric gates and + +# depth grow modestly with problem size. + +# + +# Given L as the number of ansatz layers, n is the number of QUBO variables, here’s the complexity + +# comparison of QAOA, PGE, and ABE/ACE in terms of quantum resources for solving QUBO problems. + + +# .. figure:: ../_static/demonstration_assets/qubit_efficient_encoding/scalability.png +# :scale: 75% +# :align: center +# :alt: Scalability analysis of VQAs for solving QUBO problems in terms of quantum resources [#smv-nisq-seg]_. +# +# Scalability analysis of VQAs for solving QUBO problems in terms of quantum resources [#smv-nisq-seg]_. + +# + +###################################################################### +# 📎 Final Notes + +# -------------- + +# + +# This demo showcases a scalable, qubit-efficient strategy for quantum-enhanced image segmentation. + +# + +# By formulating image segmentation as a graph cut, reducing it to a QUBO, and then solving with + +# advanced encodings like PGE, ABE, and ACE, we show that near-term quantum devices can tackle + +# meaningful vision tasks. + +# + +###################################################################### +# -------------- + +# + +# 📄 References + +# ------------- + +# + +# .. [#smv-nisq-seg] + +# S. M. Venkatesh, A. Macaluso, M. Nuske, M. Klusch, and A. Dengel, “Qubit-Efficient Variational + +# Quantum Algorithms for Image Segmentation,” 2024 IEEE International Conference on Quantum Computing + +# and Engineering (QCE), Montreal, QC, Canada, 2024, pp. 450–456. DOI: + +# `10.1109/QCE60285.2024.00059 `__, + +# `arXiv:2405.14405 `__ + +# + +# .. [#smv-qseg] + +# S. M. Venkatesh, A. Macaluso, M. Nuske, M. Klusch and A. Dengel, “Q-Seg: Quantum Annealing-Based + +# Unsupervised Image Segmentation,” in IEEE Computer Graphics and Applications, vol. 44, no. 5, + +# pp. 27-39, Sept.-Oct. 2024, doi: + +# `10.1109/MCG.2024.3455012 `__, + +# `arXiv:2311.12912 `__. + +# + +# .. [#qaoa] + +# Farhi, E., Goldstone, J., & Gutmann, S. (2014). A quantum approximate optimization algorithm. + +# arXiv preprint `arXiv:1411.4028 `__. + +# + +# .. [#pge] + +# Rančić, M. J. (2023). Noisy intermediate-scale quantum computing algorithm for solving an + +# n-vertex MaxCut problem with log (n) qubits. Physical Review Research, 5(1), L012021. DOI: + +# `10.1103/PhysRevResearch.5.L012021 `__ + +# + +# .. [#abe] + +# Tan, B., Lemonde, M. A., Thanasilp, S., Tangpanitanon, J., & Angelakis, D. G. (2021). + +# Qubit-efficient encoding schemes for binary optimisation problems. Quantum, 5, 454. DOI: + +# `10.22331/q-2021-05-04-454 `__ + +# + +###################################################################### +# About the author +# ---------------- +# From ca36351d92a3b7533c8d117cf8f95c928d569142 Mon Sep 17 00:00:00 2001 From: Supreeth Mysore Venkatesh Date: Mon, 7 Apr 2025 19:26:20 +0200 Subject: [PATCH 2/7] debug dot product --- demonstrations/tutorial_qubit_efficient_encoding.py | 4 +++- 1 file changed, 3 insertions(+), 1 deletion(-) diff --git a/demonstrations/tutorial_qubit_efficient_encoding.py b/demonstrations/tutorial_qubit_efficient_encoding.py index 108b8a3d5e..0a6015596d 100644 --- a/demonstrations/tutorial_qubit_efficient_encoding.py +++ b/demonstrations/tutorial_qubit_efficient_encoding.py @@ -382,7 +382,9 @@ def decode(optimal_params): def objective_value(x, w): - return np.dot(x, np.dot(w, x)) + x = np.asarray(x).flatten() + w = np.asarray(w) + return float(x @ w @ x) def new_nisq_algo_solver(G, optimizer_method="Powell", initial_params_seed=123): From a10a9a362d12468cd908fd4890a732925d796b37 Mon Sep 17 00:00:00 2001 From: Supreeth Mysore Venkatesh Date: Mon, 7 Apr 2025 19:41:15 +0200 Subject: [PATCH 3/7] fix variable names --- demonstrations/tutorial_qubit_efficient_encoding.py | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/demonstrations/tutorial_qubit_efficient_encoding.py b/demonstrations/tutorial_qubit_efficient_encoding.py index 0a6015596d..bf63242bf7 100644 --- a/demonstrations/tutorial_qubit_efficient_encoding.py +++ b/demonstrations/tutorial_qubit_efficient_encoding.py @@ -428,7 +428,7 @@ def decode_binary_string(x, height, width): return mask -mask = decode_binary_string(x, height, width) +mask = decode_binary_string(binary_solution, height, width) plt.imshow(mask, interpolation="nearest", cmap=plt.cm.gray, vmin=0, vmax=1) plt.axis("off") plt.show() From f2dc9acf8fc4d12e5f567eaab00765551c04b215 Mon Sep 17 00:00:00 2001 From: Supreeth Mysore Venkatesh Date: Tue, 8 Apr 2025 12:02:14 +0200 Subject: [PATCH 4/7] styling, captions and seed fix --- .../tutorial_qubit_efficient_encoding.py | 372 +++--------------- 1 file changed, 55 insertions(+), 317 deletions(-) diff --git a/demonstrations/tutorial_qubit_efficient_encoding.py b/demonstrations/tutorial_qubit_efficient_encoding.py index bf63242bf7..a8741337fd 100644 --- a/demonstrations/tutorial_qubit_efficient_encoding.py +++ b/demonstrations/tutorial_qubit_efficient_encoding.py @@ -1,190 +1,114 @@ -r"""Qubit-Efficient Encoding Techniques for Solving QUBO Problems +r""" -=================================== +.. _qubit_efficient_encoding: + +Qubit-Efficient Encoding Techniques for Solving QUBO Problems +============================================================= """ ###################################################################### -# 🧠 Introduction - -# --------------- - -# +# Introduction +# ------------ # Classical algorithms are already incredibly effective at solving many optimization problems — - # especially when the number of variables is in the low thousands. However, when scaling up to - # problems involving **tens or hundreds of thousands** of binary variables, classical solvers begin to - # falter due to the **combinatorial explosion** in the search space. -# - # Quantum computing offers a compelling alternative — not by brute force, but by **exploring - # exponentially large solution spaces in parallel** using quantum superposition and entanglement. One - # popular quantum algorithm for such problems is the **Quantum Approximate Optimization Algorithm - # (QAOA)** [#qaoa]_, a textbook example of a variational quantum algorithm (VQA). However, QAOA requires - # **one qubit per binary variable**, which becomes **prohibitively resource-intensive** for - # large-scale problems. -# - # For instance, solving a QUBO (Quadratic Unconstrained Binary Optimization) problem with 10,000 - # variables using QAOA would require 10,000 logical qubits — a scale **far beyond current quantum - # hardware capabilities**. -# ###################################################################### -# 🔍 Why Qubit-Efficient Methods? - -# ------------------------------- - -# +# Why Qubit-Efficient Methods? +# ---------------------------- # This demo explores **qubit-efficient alternatives** to QAOA that are not only suitable for today’s - # NISQ (Noisy Intermediate-Scale Quantum) devices, but also **generalizable across combinatorial - # optimization problems**. -# - # We specifically demonstrate these methods in the context of **unsupervised image segmentation**, by: - -# - # 1. Formulating segmentation as a **min-cut problem** over a graph derived from the image. - # 2. Reformulating the min-cut as a **QUBO problem**. - # 3. Solving the QUBO using **qubit-efficient VQAs**, namely [#smv-nisq-seg]_: - -# - # - **Parametric Gate Encoding (PGE)** - # - **Ancilla Basis Encoding (ABE)** - # - **Adaptive Cost Encoding (ACE)** -# - # These methods only require a **logarithmic number of qubits** with respect to the problem size - # (i.e., number of binary variables). This makes them **scalable and implementable** even on near-term - -# quantum hardware. Moreover, the exact graph-based image segmentation technique have been explored - +# quantum hardware. Moreover, the exact graph-based image segmentation technique has been explored # also using quantum annealers [#smv-qseg]_. -# - # While this demo walks through an image segmentation example, the encoding schemes presented here can - # be applied to **any binary combinatorial optimization problem**. - # .. figure:: ../_static/demonstration_assets/qubit_efficient_encoding/vqa-segmentation.png # :scale: 75% # :align: center -# :alt: Architecture for segmenting an image by transforming it into a graph and solving the corresponding minimum cut as a QUBO problem using variational quantum circuits [#smv-nisq-seg]_. -# -# Architecture for segmenting an image by transforming it into a graph and solving the corresponding minimum cut as a QUBO problem using variational quantum circuits [#smv-nisq-seg]_. - -# -------------- - -# +# :alt: Architecture for segmenting an image by transforming it into a graph and solving the corresponding minimum cut as a QUBO problem using variational quantum circuits (Fig. 1 in [#smv-nisq-seg]_). +# Architecture for segmenting an image by transforming it into a graph and solving the corresponding minimum cut as a QUBO problem using variational quantum circuits (Fig. 1 in [#smv-nisq-seg]_). ###################################################################### -# 🧱 Import dependencies - -# ---------------------- - +# Import dependencies +# ------------------- # # We begin by importing the standard Python libraries for graph construction and visualization, and - # PennyLane modules for quantum circuit construction and simulation. This setup ensures a seamless - # interface for hybrid quantum-classical processing. -# - import numpy as np import networkx as nx import pennylane as qml from scipy.optimize import minimize, differential_evolution import matplotlib.pyplot as plt - from math import ceil, log2 import time ###################################################################### -# 🖼️ Generate an Example Image - -# ---------------------------- - -# - +# Generate an Example Image +# ------------------------- # To keep things intuitive, we generate a toy grayscale image. This synthetic image will be segmented - # using quantum algorithms. -# - # We use a simple 4×4 grid for clarity. The pixel intensities will act as a proxy for similarity when - # we later construct a graph. Each pixel becomes a vertex, and edges represent similarity (e.g., - # inverse of intensity difference). -# - -height, width = 4, 4 -image = np.array( - [ - [0.97, 0.87, 0.09, 0.92], - [0.93, 0.10, 0.12, 0.93], - [0.94, 0.15, 0.88, 0.94], - [0.92, 0.17, 0.10, 0.92], - ] -) +height,width = 4,4 +image = np.array([ + [0.97, 0.92, 0.05, 0.92], + [0.93, 0.10, 0.12, 0.93], + [0.94, 0.15, 0.88, 0.94], + [0.98, 0.17, 0.10, 0.92] + ]) -plt.imshow(image, interpolation="nearest", cmap=plt.cm.gray, vmin=0, vmax=1) -plt.axis("off") +plt.imshow(image, interpolation='nearest', cmap=plt.cm.gray, vmin=0, vmax=1) +plt.xticks([]) +plt.yticks([]) plt.show() ###################################################################### -# 🔗 Representing Image as a Graph - -# -------------------------------- - -# - +# Representing Image as a Graph +# ----------------------------- # | Each image is modeled as an undirected weighted grid graph: - -# | - Each pixel becomes a node. - An edge exists between neighboring pixels. - The weight reflects - -# similarity (e.g., inverse absolute difference in grayscale values). - -# +# | - Each pixel becomes a node. +# | - An edge exists between neighboring pixels. +# | - The weight reflect similarity (e.g., inverse absolute difference in grayscale values). # This transforms the segmentation task into a **minimum cut problem**, where the goal is to partition - # the graph to minimize the sum of cut edge weights. -# - - def image_to_grid_graph(gray_img, sigma=0.5): # Convert image to grayscale # gray_img = np.dot(img, [0.2989, 0.5870, 0.1140]) @@ -257,13 +181,9 @@ def image_to_grid_graph(gray_img, sigma=0.5): pixel_values, elist, nx_elist, normalized_elist, normalized_nx_elist = image_to_grid_graph(image) - -pixel_values, elist, nx_elist, normalized_elist, normalized_nx_elist - G = nx.grid_2d_graph(image.shape[0], image.shape[1]) G.add_weighted_edges_from(normalized_nx_elist) - def draw(G): plt.figure(figsize=(6, 6)) default_axes = plt.axes(frameon=True) @@ -279,66 +199,28 @@ def draw(G): ) plt.axis("off") - -# plt.savefig("2x2_graph.jpg", dpi=600, bbox_inches='tight', pad_inches=0) draw(G) plt.show() ###################################################################### -# 🔁 Parametric Gate Encoding (PGE) - -# --------------------------------- - -# - +# Parametric Gate Encoding (PGE) +# ------------------------------ # In PGE, we directly encode the binary segmentation mask in the rotation parameters of a diagonal - # gate [#pge]_. - -# - # Only :math:`\log_2(n)` qubits are required, which is a **huge gain** over methods like QAOA that - # need :math:`O(n)`. - -# - # Quantum Circuit: - # ~~~~~~~~~~~~~~~~ - -# - # - Apply Hadamard gates to initialize a uniform superposition. - # - Apply a diagonal gate :math:`U(\vec{\theta})` whose entries encode binary values using a - # thresholding function :math:`f(\theta_i)`. - # - Evaluate the cost function: - -# - # .. math:: - -# - -# - # C(\vec{\theta}) = \langle \psi(\vec{\theta}) | L_G | \psi(\vec{\theta}) \rangle - -# - # where :math:`L_G` is the graph Laplacian. - -# - # The result is optimized using a classical optimizer, and :math:`\theta_i` are mapped to bits - # depending on whether :math:`\theta_i < \pi` or not. -# - n = G.number_of_nodes() n_qubits = int(np.ceil(np.log2(n))) n_qubits @@ -353,17 +235,14 @@ def R(theta): # Define a PennyLane quantum device (using default.qubit simulator) dev = qml.device("default.qubit", wires=n_qubits) - # Quantum circuit as a QNode @qml.qnode(dev) def circuit(params, H): - # N = len(params) N = int(np.log2(len(params))) for i in range(N): qml.Hadamard(wires=i) diagonal_elements = [np.exp(1j * np.pi * R(param)) for param in params] - # qml.DiagonalQubitUnitary(np.diag(diagonal_elements), wires=list(range(N))) qml.DiagonalQubitUnitary(diagonal_elements, wires=list(range(N))) return qml.expval(qml.Hermitian(H, wires=list(range(N)))) @@ -430,63 +309,30 @@ def decode_binary_string(x, height, width): mask = decode_binary_string(binary_solution, height, width) plt.imshow(mask, interpolation="nearest", cmap=plt.cm.gray, vmin=0, vmax=1) -plt.axis("off") +plt.xticks([]) +plt.yticks([]) plt.show() - ###################################################################### -# 🧪 Ancilla Basis Encoding (ABE) - -# ------------------------------- - -# - +# Ancilla Basis Encoding (ABE) +# ---------------------------- # | ABE uses :math:`\log_2(n) + 1` qubits: - # | - :math:`\log_2(n)` **register qubits** represent binary states. - # | - One **ancilla qubit** helps encode the binary decision. -# - # Each computational basis state of the register corresponds to one binary variable. The ancilla - # qubit’s amplitude is used to decide the bit value [#abe]_: -# - # .. math:: - -# - -# - # x_{v_i} = \begin{cases} 0 & \text{if } |a_i|^2 > |b_i|^2 \\ 1 & \text{otherwise} \end{cases} - -# - # The cost function is parameterized over **expectation values** of basis state projectors: - -# - # .. math:: - -# - -# - # C(\vec{\theta}) = \sum_{i,j} Q_{ij} \frac{\langle \hat{P}_i^1 \rangle \langle \hat{P}_j^1 \rangle}{\langle \hat{P}_i \rangle \langle \hat{P}_j \rangle} (1 - \delta_{ij}) + \sum_i Q_{ii} \frac{\langle \hat{P}_i^1 \rangle}{\langle \hat{P}_i \rangle} - # .. figure:: ../_static/demonstration_assets/qubit_efficient_encoding/ABE_ACE.png # :scale: 75% # :align: center -# :alt: Quantum circuit of ABE/ACE for solving QUBO of size 4 using 3 qubits (2 register + 1 ancilla). Basis state amplitudes are used to decode the binary solution [#smv-nisq-seg]_. -# -# Quantum circuit of ABE/ACE for solving QUBO of size 4 using 3 qubits (2 register + 1 ancilla). Basis state amplitudes are used to decode the binary solution [#smv-nisq-seg]_. - - -# - +# :alt: Quantum circuit of ABE/ACE for solving QUBO of size 4 using 3 qubits (2 register + 1 ancilla). Basis state amplitudes are used to decode the binary solution (Fig. 3 in [#smv-nisq-seg]_). +# Quantum circuit of ABE/ACE for solving QUBO of size 4 using 3 qubits (2 register + 1 ancilla). Basis state amplitudes are used to decode the binary solution (Fig. 3 in [#smv-nisq-seg]_). def get_qubo_matrix(W): """Computes the QUBO matrix for the Minimum Cut problem given a weight matrix W.""" @@ -498,10 +344,8 @@ def get_qubo_matrix(W): for j in range(n): if i != j: Q[i, j] = -W[i, j] # Off-diagonal terms (negative adjacency) - return Q - def get_circuit(nq, n_layers): def circuit(params): for i in range(nq): @@ -512,10 +356,8 @@ def circuit(params): for qubit_i in range(nq): qml.RY(params[(nq * layer_i) + qubit_i], wires=qubit_i) return qml.probs(wires=range(nq)) - return circuit - def get_projectors(probabilities, n_c): P, P1 = [0] * n_c, [0] * n_c for k, v in probabilities.items(): @@ -525,11 +367,9 @@ def get_projectors(probabilities, n_c): P1[index] += v return P, P1 - def objective_value(x, w): return sum(w[i, j] * x[i] * (1 - x[j]) for i in range(len(x)) for j in range(len(x))) - def decode_probabilities(probabilities): binary_solution, n_r = [], len(list(probabilities.keys())[0]) - 1 for i in range(2**n_r): @@ -542,7 +382,6 @@ def decode_probabilities(probabilities): binary_solution.append(1) return binary_solution - def evaluate_cost(params, circuit, qubo_matrix): global valid_probabilities, optimization_iteration_count, best_cost, best_params optimization_iteration_count += 1 @@ -577,7 +416,6 @@ def evaluate_cost(params, circuit, qubo_matrix): print(f"@ Iteration {optimization_iteration_count} Cost :", cost) return cost - def ancilla_basis_encoding( G, initial_params, n_layers=1, optimizer_method="differential_evolution" ): @@ -616,15 +454,11 @@ def ancilla_basis_encoding( print("binary_solution", binary_solution) return binary_solution, best_cost, objective_value(np.array(binary_solution), w), best_params - ###################################################################### # Although the code can handle various methods available in ``scipy.minimize``, in this demo we will - # be using a more robust genetic algorithm ``differential_evolution`` as the classical optimizer. -# - -initial_params_seed, scipy_optimizer_method, n_layers = 42, "differential_evolution", 4 +initial_params_seed, scipy_optimizer_method, n_layers = 111, "differential_evolution", 4 nq = ceil(log2(len(G.nodes()))) + 1 np.random.seed(initial_params_seed) initial_params = np.random.uniform(low=-np.pi, high=np.pi, size=nq * n_layers) @@ -638,47 +472,23 @@ def ancilla_basis_encoding( mask = decode_binary_string(solution, height, width) plt.imshow(mask, interpolation="nearest", cmap=plt.cm.gray, vmin=0, vmax=1) -plt.axis("off") +plt.xticks([]) +plt.yticks([]) plt.show() ###################################################################### -# 🔁 Adaptive Cost Encoding (ACE) - -# ------------------------------- - -# - +# Adaptive Cost Encoding (ACE) +# ---------------------------- # ACE modifies ABE by skipping the QUBO formulation entirely during evaluation. - -# - # Instead of optimizing over probabilities, we decode a binary vector :math:`x` from the measured - # output and evaluate the **original min-cut cost function** directly [#smv-nisq-seg]_: - -# - # .. math:: - -# - -# - # C(x) = \sum_{i < j} x_i (1 - x_j) w(v_i, v_j) - -# - # This: - Removes the need for a QUBO matrix, which can have loss of generality for constrained - # optimization problems when the penalty coefficient is not set appropriately. - Improves convergence - # (the cost function is discrete and combinatorial). - Aligns optimization directly with the true task - # objective. -# - - def evaluate_cost_mincut(params, circuit, w): global valid_probabilities, optimization_iteration_count, best_cost, best_params optimization_iteration_count += 1 @@ -705,8 +515,7 @@ def evaluate_cost_mincut(params, circuit, w): print(f"@ Iteration {optimization_iteration_count} Cost :", cost) return cost - -def adaptive_cost_encoding_de( +def adaptive_cost_encoding( G, initial_params, n_layers=1, optimizer_method="differential_evolution" ): w = nx.adjacency_matrix(G).todense() @@ -746,14 +555,13 @@ def adaptive_cost_encoding_de( print("binary_solution", binary_solution) return binary_solution, best_cost, objective_value(np.array(binary_solution), w), best_params - -initial_params_seed, scipy_optimizer_method, n_layers = 42, "differential_evolution", 4 +initial_params_seed, scipy_optimizer_method, n_layers = 111, "differential_evolution", 4 nq = ceil(log2(len(G.nodes()))) + 1 np.random.seed(initial_params_seed) initial_params = np.random.uniform(low=-np.pi, high=np.pi, size=nq * n_layers) print(f"Executing QC with {n_layers} layers and Differential Evolution optimizer.") start_time = time.time() -solution, value, cut_value, optimal_params = adaptive_cost_encoding_de( +solution, value, cut_value, optimal_params = adaptive_cost_encoding( G, initial_params, n_layers, optimizer_method=scipy_optimizer_method ) end_time = time.time() @@ -761,133 +569,63 @@ def adaptive_cost_encoding_de( mask = decode_binary_string(solution, height, width) plt.imshow(mask, interpolation="nearest", cmap=plt.cm.gray, vmin=0, vmax=1) -plt.axis("off") +plt.xticks([]) +plt.yticks([]) plt.show() ###################################################################### -# 📈 Evaluation & Observations - +# Evaluation & Observations # ---------------------------- - -# - # - PGE has a compact circuit but many parameters. - # - ABE introduces a more efficient encoding at the cost of probabilistic decoding. - # - ACE improves convergence by using the real problem cost and avoiding indirect probability-based - # objectives. - -# - # Theoretical scaling advantages: - Qubit requirement is :math:`O(\log n)`. - Parametric gates and - # depth grow modestly with problem size. - -# - # Given L as the number of ansatz layers, n is the number of QUBO variables, here’s the complexity - # comparison of QAOA, PGE, and ABE/ACE in terms of quantum resources for solving QUBO problems. - - # .. figure:: ../_static/demonstration_assets/qubit_efficient_encoding/scalability.png # :scale: 75% # :align: center -# :alt: Scalability analysis of VQAs for solving QUBO problems in terms of quantum resources [#smv-nisq-seg]_. -# -# Scalability analysis of VQAs for solving QUBO problems in terms of quantum resources [#smv-nisq-seg]_. - -# +# :alt: Scalability analysis of VQAs for solving QUBO problems in terms of quantum resources (Table 2 in [#smv-nisq-seg]_). +# Scalability analysis of VQAs for solving QUBO problems in terms of quantum resources (Table 2 in [#smv-nisq-seg]_). ###################################################################### -# 📎 Final Notes - +# Final Notes # -------------- - -# - # This demo showcases a scalable, qubit-efficient strategy for quantum-enhanced image segmentation. - -# - # By formulating image segmentation as a graph cut, reducing it to a QUBO, and then solving with - # advanced encodings like PGE, ABE, and ACE, we show that near-term quantum devices can tackle - # meaningful vision tasks. -# - ###################################################################### -# -------------- - -# - -# 📄 References - +# References # ------------- - -# - # .. [#smv-nisq-seg] - # S. M. Venkatesh, A. Macaluso, M. Nuske, M. Klusch, and A. Dengel, “Qubit-Efficient Variational - # Quantum Algorithms for Image Segmentation,” 2024 IEEE International Conference on Quantum Computing - # and Engineering (QCE), Montreal, QC, Canada, 2024, pp. 450–456. DOI: - # `10.1109/QCE60285.2024.00059 `__, - # `arXiv:2405.14405 `__ - -# - # .. [#smv-qseg] - # S. M. Venkatesh, A. Macaluso, M. Nuske, M. Klusch and A. Dengel, “Q-Seg: Quantum Annealing-Based - # Unsupervised Image Segmentation,” in IEEE Computer Graphics and Applications, vol. 44, no. 5, - # pp. 27-39, Sept.-Oct. 2024, doi: - # `10.1109/MCG.2024.3455012 `__, - # `arXiv:2311.12912 `__. - -# - # .. [#qaoa] - # Farhi, E., Goldstone, J., & Gutmann, S. (2014). A quantum approximate optimization algorithm. - # arXiv preprint `arXiv:1411.4028 `__. - -# - # .. [#pge] - # Rančić, M. J. (2023). Noisy intermediate-scale quantum computing algorithm for solving an - # n-vertex MaxCut problem with log (n) qubits. Physical Review Research, 5(1), L012021. DOI: - # `10.1103/PhysRevResearch.5.L012021 `__ - -# - # .. [#abe] - # Tan, B., Lemonde, M. A., Thanasilp, S., Tangpanitanon, J., & Angelakis, D. G. (2021). - # Qubit-efficient encoding schemes for binary optimisation problems. Quantum, 5, 454. DOI: - # `10.22331/q-2021-05-04-454 `__ -# - ###################################################################### # About the author # ---------------- -# +# \ No newline at end of file From fa88959eea3d5f8c1af0a0732b68964175390466 Mon Sep 17 00:00:00 2001 From: Supreeth Mysore Venkatesh Date: Tue, 8 Apr 2025 12:36:08 +0200 Subject: [PATCH 5/7] minor styling fix --- demonstrations/tutorial_qubit_efficient_encoding.py | 9 ++------- 1 file changed, 2 insertions(+), 7 deletions(-) diff --git a/demonstrations/tutorial_qubit_efficient_encoding.py b/demonstrations/tutorial_qubit_efficient_encoding.py index a8741337fd..2c90cb0bd8 100644 --- a/demonstrations/tutorial_qubit_efficient_encoding.py +++ b/demonstrations/tutorial_qubit_efficient_encoding.py @@ -1,5 +1,4 @@ r""" - .. _qubit_efficient_encoding: Qubit-Efficient Encoding Techniques for Solving QUBO Problems @@ -10,7 +9,6 @@ ###################################################################### # Introduction # ------------ - # Classical algorithms are already incredibly effective at solving many optimization problems — # especially when the number of variables is in the low thousands. However, when scaling up to # problems involving **tens or hundreds of thousands** of binary variables, classical solvers begin to @@ -31,7 +29,6 @@ ###################################################################### # Why Qubit-Efficient Methods? # ---------------------------- - # This demo explores **qubit-efficient alternatives** to QAOA that are not only suitable for today’s # NISQ (Noisy Intermediate-Scale Quantum) devices, but also **generalizable across combinatorial # optimization problems**. @@ -60,8 +57,6 @@ ###################################################################### # Import dependencies # ------------------- -# - # We begin by importing the standard Python libraries for graph construction and visualization, and # PennyLane modules for quantum circuit construction and simulation. This setup ensures a seamless # interface for hybrid quantum-classical processing. @@ -592,7 +587,7 @@ def adaptive_cost_encoding( ###################################################################### # Final Notes -# -------------- +# ----------- # This demo showcases a scalable, qubit-efficient strategy for quantum-enhanced image segmentation. # By formulating image segmentation as a graph cut, reducing it to a QUBO, and then solving with # advanced encodings like PGE, ABE, and ACE, we show that near-term quantum devices can tackle @@ -600,7 +595,7 @@ def adaptive_cost_encoding( ###################################################################### # References -# ------------- +# ---------- # .. [#smv-nisq-seg] # S. M. Venkatesh, A. Macaluso, M. Nuske, M. Klusch, and A. Dengel, “Qubit-Efficient Variational # Quantum Algorithms for Image Segmentation,” 2024 IEEE International Conference on Quantum Computing From 897f467497e90662c486dd3ea2b2abaa3c7eea45 Mon Sep 17 00:00:00 2001 From: Supreeth Mysore Venkatesh Date: Thu, 10 Apr 2025 17:16:35 +0200 Subject: [PATCH 6/7] Style and output fix --- .../tutorial_qubit_efficient_encoding.py | 41 +++++++++---------- 1 file changed, 20 insertions(+), 21 deletions(-) diff --git a/demonstrations/tutorial_qubit_efficient_encoding.py b/demonstrations/tutorial_qubit_efficient_encoding.py index 2c90cb0bd8..4df84f8948 100644 --- a/demonstrations/tutorial_qubit_efficient_encoding.py +++ b/demonstrations/tutorial_qubit_efficient_encoding.py @@ -13,14 +13,14 @@ # especially when the number of variables is in the low thousands. However, when scaling up to # problems involving **tens or hundreds of thousands** of binary variables, classical solvers begin to # falter due to the **combinatorial explosion** in the search space. - +# # Quantum computing offers a compelling alternative — not by brute force, but by **exploring # exponentially large solution spaces in parallel** using quantum superposition and entanglement. One # popular quantum algorithm for such problems is the **Quantum Approximate Optimization Algorithm # (QAOA)** [#qaoa]_, a textbook example of a variational quantum algorithm (VQA). However, QAOA requires # **one qubit per binary variable**, which becomes **prohibitively resource-intensive** for # large-scale problems. - +# # For instance, solving a QUBO (Quadratic Unconstrained Binary Optimization) problem with 10,000 # variables using QAOA would require 10,000 logical qubits — a scale **far beyond current quantum # hardware capabilities**. @@ -32,7 +32,7 @@ # This demo explores **qubit-efficient alternatives** to QAOA that are not only suitable for today’s # NISQ (Noisy Intermediate-Scale Quantum) devices, but also **generalizable across combinatorial # optimization problems**. - +# # We specifically demonstrate these methods in the context of **unsupervised image segmentation**, by: # 1. Formulating segmentation as a **min-cut problem** over a graph derived from the image. # 2. Reformulating the min-cut as a **QUBO problem**. @@ -40,12 +40,12 @@ # - **Parametric Gate Encoding (PGE)** # - **Ancilla Basis Encoding (ABE)** # - **Adaptive Cost Encoding (ACE)** - +# # These methods only require a **logarithmic number of qubits** with respect to the problem size # (i.e., number of binary variables). This makes them **scalable and implementable** even on near-term # quantum hardware. Moreover, the exact graph-based image segmentation technique has been explored # also using quantum annealers [#smv-qseg]_. - +# # While this demo walks through an image segmentation example, the encoding schemes presented here can # be applied to **any binary combinatorial optimization problem**. # .. figure:: ../_static/demonstration_assets/qubit_efficient_encoding/vqa-segmentation.png @@ -74,7 +74,7 @@ # ------------------------- # To keep things intuitive, we generate a toy grayscale image. This synthetic image will be segmented # using quantum algorithms. - +# # We use a simple 4×4 grid for clarity. The pixel intensities will act as a proxy for similarity when # we later construct a graph. Each pixel becomes a vertex, and edges represent similarity (e.g., # inverse of intensity difference). @@ -100,7 +100,7 @@ # | - Each pixel becomes a node. # | - An edge exists between neighboring pixels. # | - The weight reflect similarity (e.g., inverse absolute difference in grayscale values). - +# # This transforms the segmentation task into a **minimum cut problem**, where the goal is to partition # the graph to minimize the sum of cut edge weights. @@ -204,16 +204,16 @@ def draw(G): # gate [#pge]_. # Only :math:`\log_2(n)` qubits are required, which is a **huge gain** over methods like QAOA that # need :math:`O(n)`. -# Quantum Circuit: +# +# **Quantum Circuit:** # ~~~~~~~~~~~~~~~~ # - Apply Hadamard gates to initialize a uniform superposition. -# - Apply a diagonal gate :math:`U(\vec{\theta})` whose entries encode binary values using a -# thresholding function :math:`f(\theta_i)`. +# - Apply a diagonal gate :math:`U(\vec{\theta})` whose entries encode binary values using a thresholding function :math:`f(\theta_i)`. # - Evaluate the cost function: # .. math:: # C(\vec{\theta}) = \langle \psi(\vec{\theta}) | L_G | \psi(\vec{\theta}) \rangle # where :math:`L_G` is the graph Laplacian. -# The result is optimized using a classical optimizer, and :math:`\theta_i` are mapped to bits +# The result is optimized using a classical optimizer, and :math:`\theta_i` are mapped to binary values # depending on whether :math:`\theta_i < \pi` or not. n = G.number_of_nodes() @@ -292,8 +292,6 @@ def new_nisq_algo_solver(G, optimizer_method="Powell", initial_params_seed=123): binary_solution, expectation_value, cut_value = new_nisq_algo_solver(G, optimizer_method="Powell") -binary_solution, expectation_value, cut_value - def decode_binary_string(x, height, width): mask = np.zeros([height, width]) @@ -314,10 +312,10 @@ def decode_binary_string(x, height, width): # | ABE uses :math:`\log_2(n) + 1` qubits: # | - :math:`\log_2(n)` **register qubits** represent binary states. # | - One **ancilla qubit** helps encode the binary decision. - +# # Each computational basis state of the register corresponds to one binary variable. The ancilla # qubit’s amplitude is used to decide the bit value [#abe]_: - +# # .. math:: # x_{v_i} = \begin{cases} 0 & \text{if } |a_i|^2 > |b_i|^2 \\ 1 & \text{otherwise} \end{cases} # The cost function is parameterized over **expectation values** of basis state projectors: @@ -326,8 +324,8 @@ def decode_binary_string(x, height, width): # .. figure:: ../_static/demonstration_assets/qubit_efficient_encoding/ABE_ACE.png # :scale: 75% # :align: center -# :alt: Quantum circuit of ABE/ACE for solving QUBO of size 4 using 3 qubits (2 register + 1 ancilla). Basis state amplitudes are used to decode the binary solution (Fig. 3 in [#smv-nisq-seg]_). -# Quantum circuit of ABE/ACE for solving QUBO of size 4 using 3 qubits (2 register + 1 ancilla). Basis state amplitudes are used to decode the binary solution (Fig. 3 in [#smv-nisq-seg]_). +# :alt: Example of a quantum circuit of ABE/ACE for solving QUBO of size 4 using 3 qubits (2 register + 1 ancilla), whereas in this demo we are solving a QUBO of size 16, so we are simulating 5 qubits. Basis state amplitudes are used to decode the binary solution (Fig. 3 in [#smv-nisq-seg]_). +# Example of a quantum circuit of ABE/ACE for solving QUBO of size 4 using 3 qubits (2 register + 1 ancilla), whereas in this demo we are solving a QUBO of size 16, so we are simulating 5 qubits. Basis state amplitudes are used to decode the binary solution (Fig. 3 in [#smv-nisq-seg]_). def get_qubo_matrix(W): """Computes the QUBO matrix for the Minimum Cut problem given a weight matrix W.""" @@ -430,7 +428,7 @@ def ancilla_basis_encoding( args=(qnode, Q), strategy="best1bin", maxiter=100, - popsize=10, + popsize=15, ) else: optimization_result = minimize( @@ -452,8 +450,9 @@ def ancilla_basis_encoding( ###################################################################### # Although the code can handle various methods available in ``scipy.minimize``, in this demo we will # be using a more robust genetic algorithm ``differential_evolution`` as the classical optimizer. +# You can vary the population size and/or the number of iteration for larger problems in order to find the optimal solution. -initial_params_seed, scipy_optimizer_method, n_layers = 111, "differential_evolution", 4 +initial_params_seed, scipy_optimizer_method, n_layers = 1234, "differential_evolution", 5 nq = ceil(log2(len(G.nodes()))) + 1 np.random.seed(initial_params_seed) initial_params = np.random.uniform(low=-np.pi, high=np.pi, size=nq * n_layers) @@ -529,7 +528,7 @@ def adaptive_cost_encoding( args=(qnode, w), strategy="best1bin", maxiter=100, - popsize=10, + popsize=15, ) else: optimization_result = minimize( @@ -550,7 +549,7 @@ def adaptive_cost_encoding( print("binary_solution", binary_solution) return binary_solution, best_cost, objective_value(np.array(binary_solution), w), best_params -initial_params_seed, scipy_optimizer_method, n_layers = 111, "differential_evolution", 4 +initial_params_seed, scipy_optimizer_method, n_layers = 1234, "differential_evolution", 5 nq = ceil(log2(len(G.nodes()))) + 1 np.random.seed(initial_params_seed) initial_params = np.random.uniform(low=-np.pi, high=np.pi, size=nq * n_layers) From 5e151bbc5d42ae82b65360c2bf2bbdba3c35f8f7 Mon Sep 17 00:00:00 2001 From: Supreeth Mysore Venkatesh Date: Fri, 11 Apr 2025 13:48:24 +0200 Subject: [PATCH 7/7] update math and figure --- .../tutorial_qubit_efficient_encoding.py | 14 ++++++++++---- 1 file changed, 10 insertions(+), 4 deletions(-) diff --git a/demonstrations/tutorial_qubit_efficient_encoding.py b/demonstrations/tutorial_qubit_efficient_encoding.py index 4df84f8948..e0dbe7ccc5 100644 --- a/demonstrations/tutorial_qubit_efficient_encoding.py +++ b/demonstrations/tutorial_qubit_efficient_encoding.py @@ -34,9 +34,9 @@ # optimization problems**. # # We specifically demonstrate these methods in the context of **unsupervised image segmentation**, by: -# 1. Formulating segmentation as a **min-cut problem** over a graph derived from the image. -# 2. Reformulating the min-cut as a **QUBO problem**. -# 3. Solving the QUBO using **qubit-efficient VQAs**, namely [#smv-nisq-seg]_: +# - Formulating segmentation as a **min-cut problem** over a graph derived from the image. +# - Reformulating the min-cut as a **QUBO problem**. +# - Solving the QUBO using **qubit-efficient VQAs**, namely [#smv-nisq-seg]_: # - **Parametric Gate Encoding (PGE)** # - **Ancilla Basis Encoding (ABE)** # - **Adaptive Cost Encoding (ACE)** @@ -48,6 +48,7 @@ # # While this demo walks through an image segmentation example, the encoding schemes presented here can # be applied to **any binary combinatorial optimization problem**. +# # .. figure:: ../_static/demonstration_assets/qubit_efficient_encoding/vqa-segmentation.png # :scale: 75% # :align: center @@ -210,7 +211,8 @@ def draw(G): # - Apply Hadamard gates to initialize a uniform superposition. # - Apply a diagonal gate :math:`U(\vec{\theta})` whose entries encode binary values using a thresholding function :math:`f(\theta_i)`. # - Evaluate the cost function: -# .. math:: +# +# .. math:: # C(\vec{\theta}) = \langle \psi(\vec{\theta}) | L_G | \psi(\vec{\theta}) \rangle # where :math:`L_G` is the graph Laplacian. # The result is optimized using a classical optimizer, and :math:`\theta_i` are mapped to binary values @@ -319,8 +321,10 @@ def decode_binary_string(x, height, width): # .. math:: # x_{v_i} = \begin{cases} 0 & \text{if } |a_i|^2 > |b_i|^2 \\ 1 & \text{otherwise} \end{cases} # The cost function is parameterized over **expectation values** of basis state projectors: +# # .. math:: # C(\vec{\theta}) = \sum_{i,j} Q_{ij} \frac{\langle \hat{P}_i^1 \rangle \langle \hat{P}_j^1 \rangle}{\langle \hat{P}_i \rangle \langle \hat{P}_j \rangle} (1 - \delta_{ij}) + \sum_i Q_{ii} \frac{\langle \hat{P}_i^1 \rangle}{\langle \hat{P}_i \rangle} +# # .. figure:: ../_static/demonstration_assets/qubit_efficient_encoding/ABE_ACE.png # :scale: 75% # :align: center @@ -476,6 +480,7 @@ def ancilla_basis_encoding( # ACE modifies ABE by skipping the QUBO formulation entirely during evaluation. # Instead of optimizing over probabilities, we decode a binary vector :math:`x` from the measured # output and evaluate the **original min-cut cost function** directly [#smv-nisq-seg]_: +# # .. math:: # C(x) = \sum_{i < j} x_i (1 - x_j) w(v_i, v_j) # This: - Removes the need for a QUBO matrix, which can have loss of generality for constrained @@ -578,6 +583,7 @@ def adaptive_cost_encoding( # depth grow modestly with problem size. # Given L as the number of ansatz layers, n is the number of QUBO variables, here’s the complexity # comparison of QAOA, PGE, and ABE/ACE in terms of quantum resources for solving QUBO problems. +# # .. figure:: ../_static/demonstration_assets/qubit_efficient_encoding/scalability.png # :scale: 75% # :align: center