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International SoC Design Conference

Mixed-Precision Neural Network Quantization with Quantum-Inspired Annealers

Cing-Yuan Lai, Chin-Fu Nien, Lien-Po Yu, Lai Chao-Sung

Published Oct 15, 2025• 0 citations
Publisher

Abstract

Mixed-precision quantization (MPQ) aims to assign different bit-widths to different neural network layers, balancing accuracy and efficiency. Prior studies have formulated the MPQ search problem as an integer linear programming (ILP) prob-lem, relying on layer-wise importance indicators derived from quantization-aware training. This work proposes an alternative approach that reformulates the MPQ as a Quadratic Uncon-strained Binary Optimization (QUBO) problem, which can be solved effectively using the quantum-inspired annealers. This method leverages learned importance indicators while enabling flexible incorporation of additional constraints via soft penalty terms. Experimental results on the Imagenette dataset demon-strate that our QUBO-based approach achieves quantization policies with comparable performance to the ILP-based method, validating its effectiveness.

DOI
10.1109/isocc66390.2025.11329857

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Mixed-precision quantization (MPQ) aims to assign different bit-widths to different neural network layers, balancing accuracy and efficiency. Prior studies have formulated the MPQ search problem as an integer linear programming (ILP) prob-lem, relying on layer-wise importance indicators derived from quantization-aware training. This work proposes an alternative approach that reformulates the MPQ as a Quadratic Uncon-strained Binary Optimization (QUBO) problem, which can be solved effectively using the quantum-inspired annealers. This method leverages learned importance indicators while enabling flexible incorporation of additional constraints via soft penalty terms. Experimental results on the Imagenette dataset demon-strate that our QUBO-based approach achieves quantization policies with comparable performance to the ILP-based method, validating its effectiveness.
Quantum annealers are a class of quantum technologies that have attracted increasing attention due to their natural suitability for solving combinatorial optimization problems.Recent works reformulated the linear regression as a Quadratic Unconstrained Binary Optimization (QUBO) problem, enabling its implementation on quantum annealers and offering potential speed-ups for large datasets.However, this QUBO-based formulation requires the definition of a precision vector to represent the real-valued regression coefficients as integers variables, which introduces limitations related to the quantization accuracy.In this work, we investigate several numerical representations strategies for defining the precision vector.By performing a set of numerical experiments with synthetic datasets, we analyze the performance of these strategies in different configurations.Our results that the strategy usually known as conventional binary representation provides the best trade-off between performance and resource efficiency for quantum-assisted linear regression.
Quantum annealers such as the commercially available D-Wave machines are designed to natively solve quadratic unconstrained binary optimization (QUBO) problems. While most of the well-known NP-hard optimization problems can easily be formulated as quadratic binary problems, such formulations also contain constraints, which commonly are added to the objective function in the form of penalties to obtain a QUBO version. However, the standard method for defining such penalties leads to QUBOs that are dense and therefore take too much of the resources of the quantum annealer. In this paper, we describe an alternative approach to the constraint embedding problem that uses mixed-integer linear programming (MILP) and is scalable to problems of arbitrary number of variables.
Mixed-precision quantization requires discrete allocation of weight and activation bit-widths, followed by recovery of the selected network. We develop a task-aware quadratic unconstrained binary optimization (QUBO) surrogate with separate weight and activation profiles, a bit-operation (BOP) cost, and selected structural priors. QUBO provides a network-wide allocation that can be refined through direct validation-based PROTES search. On a compact NAFBlock-based denoiser, the refined route achieves 37.192 dB after LSQ+ at 4.035\% routed-layer BOPs, versus 37.092 dB at 4.101\% for a HAWQ-style baseline. The repeated-search primary experiment shows that LSQ+ largely closes the quality gap between QUBO allocation and expensive direct refinement. An additional restoration architecture retains a larger recovered gain, indicating that refinement's value depends on architecture and recovery. We evaluate quality, achieved cost, routing stability and optimization expense together. Deployment measurements characterize a fake-quantized floating-point implementation; BOP reductions describe analytical allocation savings.
Variational quantum circuits for image classification suffer from barren plateaus, while quantum kernel methods scale quadratically with dataset size. We propose an iterative framework based on Quadratic Unconstrained Binary Optimization (QUBO) for training the classifier head of convolutional neural networks (CNNs) via quantum annealing, entirely avoiding gradient-based circuit optimization. Following the Extreme Learning Machine paradigm, convolutional filters are randomly initialized and frozen, and only the fully connected layer is optimized. At each iteration, a convex quadratic surrogate derived from the feature Gram matrix replaces the non-quadratic cross-entropy loss, yielding an iteration-stable curvature proxy. A per-output decomposition splits the CCC-class problem into CCC independent QUBOs, each with (d+1)K(d+1)K(d+1)K binary variables, where ddd is the feature dimension and KKK is the bit precision, so that problem size depends on the image resolution and bit precision, not on the number of training samples. We evaluate the method on six image-classification benchmarks (sklearn digits, MNIST, Fashion-MNIST, CIFAR-10, EMNIST, KMNIST). A precision study shows that accuracy improves monotonically with bit resolution, with 10 bits representing a practical minimum for effective optimization; the 15-bit formulation remains within the qubit and coupler limits of current D-Wave Advantage hardware. The 20-bit formulation matches or exceeds classical stochastic gradient descent on MNIST, Fashion-MNIST, and EMNIST, while remaining competitive on CIFAR-10 and KMNIST. All experiments use simulated annealing, establishing a baseline for direct deployment on quantum annealing hardware.
This work introduces a post-training quantization (PTQ) method for dense neural networks via a novel ADAROUND-based QUBO formulation. Using the Frobenius distance between the theoretical output and the dequantized output (before the activation function) as the objective, an explicit QUBO whose binary variables represent the rounding choice for each weight and bias is obtained. Additionally, by exploiting the structure of the coefficient QUBO matrix, the global problem can be exactly decomposed into nnn independent subproblems of size f+1f+1f+1, which can be efficiently solved using some heuristics such as simulated annealing. The approach is evaluated on MNIST, Fashion-MNIST, EMNIST, and CIFAR-10 across integer precisions from int8 to int1 and compared with a round-to-nearest traditional quantization methodology.
Quantum annealing is a model for quantum computing that is aimed at solving hard optimization problems by representing them as quadratic unconstrained binary optimization (QUBO) problems. Although many NP-hard problems can easily be formulated as binary-variable problems with a quadratic objective function, such formulations typically also include constraints, which are not allowed in a QUBO. Hence, such constraints are usually incorporated in the objective function as additive penalty terms. While there is substantial previous work on implementing linear equality constraints, the case of inequality constraints has not much been studied. In this paper, we propose a new approach for formulating and embedding inequality constraints as penalties and describe early implementation results.