In recent years, artificial intelligence (AI) has become an integral part of daily life, serving as a transformative tool across various professional domains [1] and driving personal applications through advancements in transformer models that power large language models (LLMs) [2]. However, both training and inference of AI models demand substantial computational and energy resources, which are becoming increasingly challenging to access [3, 4]. While server-class GPUs are effective for training, their energy inefficiency [5] and high costs present significant barriers [6]. Additionally, the environmental impact of energy-intensive AI systems has raised critical concerns about their role in exacerbating climate change [4]. Amdahl's law predicts that performance and efficiency gains are best achieved through innovative application-specific accelerator architectures rather than scaling up multi-core general-purpose processors [7]. Consequently, applicationspecific integrated circuits (ASICs), both digital and analog, have emerged as critical solutions for enabling highefficiency training and inference of artificial neural networks [7, 8, 9]. Digital accelerators are widely adopted for training workloads. Notable examples include the Brainwave Neural Processing Unit (NPU) [10], Google's Tensor Processing Unit (TPU) [11], and low-precision inference accelerators such as YodaNN [5], the Unified Neural Processing Unit (UNPU) [12], and BRein Memory [13].

As NAND formula algorithm allow us to evaluate quality Quantum computing is a computation paradigm using of a position in a two-player game tree, we illustrate its properties of quantum mechanics to perform information potential application by using it as a training tool for a processing. Many famous quantum algorithms have been quantum agent in a simple two-player game. With the shown to outperform their equivalent classical algorithm[1], speed-up proposed by this algorithm, we are able to perform [2]. Quantum walk is a way to compose many promising evaluation of deeper trees in equivalent time (twice deeper quantum algorithms. It can be viewed as the quantum exploration for a binary tree). By using quantum algorithm analogues of classical random walks [3]. In several studies, to perform such explorations, we expect agents to achieve it has been shown that it could provide some algorithmic better performances in their learning process.

In this paper, we show a reciprocal implementation of the "softmax" nonlinearity that is usually used to enforce local competition between neurons [Peterson, 1989]. We show that the circuit is passive and incrementally passive, and we explicitly compute its content and co-content functions. This circuit adds a new element to the library of the analog circuit designer that can be combined with reciprocal constraint boxes [Harris, 1988] and nonlinear resistive fuses [Harris, 1989] to form fast, analog VLSI optimization networks.

In this paper, we show a reciprocal implementation of the "softmax" nonlinearity that is usually used to enforce local competition between neurons [Peterson, 1989]. We show that the circuit is passive and incrementally passive, and we explicitly compute its content and co-content functions. This circuit adds a new element to the library of the analog circuit designer that can be combined with reciprocal constraint boxes [Harris, 1988] and nonlinear resistive fuses [Harris, 1989] to form fast, analog VLSI optimization networks.

Reciprocal circuit elements facilitate such an implementation since they 882 The "Softmax" Nonlinearity 883 can be combined with other reciprocal elements to form an analog network having Lyapunov-like functions: the network content or co-content. In this paper, we show a reciprocal implementation of the "softmax" nonlinearity that is usually used to enforce local competition between neurons [Peterson, 1989]. We show that the circuit ispassive and incrementally passive, and we explicitly compute its content and co-content functions. This circuit adds a new element to the library of the analog circuit designer that can be combined with reciprocal constraint boxes [Harris, 1988] and nonlinear resistive fuses [Harris, 1989] to form fast, analog VLSI optimization networks.