II. THEORY Here, we first describe our target application, namely, determination of the ground-state wavefunctions for one-dimensional Schrödinger equations with bound PESs. We then explain how algorithms are represented in our linear PS approach, before describing how the performance of the computer-generated code is optimized using a SA protocol. We also give details of the input, output, and internal function sets that are used in our implementation to form solutions to Eq. (1). A. Problem representation As a benchmark problem, we will seek to identify codes that can produce the ground-state solution ψ(x) for the 1D time-independent Schrödinger equation, −ℏ22𝑚𝑑2𝜓(𝑥)𝑑𝑥2+𝑉(𝑥)𝜓(𝑥)=𝐸0𝜓(𝑥). [as-pasted] Here, m is the particle mass and E0 is the total energy of the eigenstate ψ(x); in what follows, we adopt atomic units such that ℏ = 1, and we will only consider systems for which m = 1. As emphasized below and in contrast to previous research,50 we are aiming to generate entire computer programs that can solve the eigenproblem of Eq. (1) to give ψ(x) for more general V(x). In all the following discussion, we will aim to generate algorithms for PESs V(x) that are of a general polynomial form given by 𝑉(𝑥)=∑𝑘=1𝑁𝑝𝑎𝑘𝑥𝑘, [as-pasted] ascii transcription: V(x) = Sum[k=1...N_p, a_k * x^k], where Np is the polynomial order and a is a set of random coefficients chosen in the range [−5,+5]. To encourage generation of algorithms that are transferrable across different polynomial orders, the order Np is randomly selected Np ∈ [2, 6] whenever a PES is required as input during our PS strategy. Furthermore, we require that any V(x) is bound such that it is greater than a sufficiently large value [chosen to be V(x) = 5 here] at the boundaries of the coordinate range-of-interest (see below). This setup for V(x) is chosen because, by choosing different random coefficients a, it allows us to generate arbitrary numbers of different V(x) examples that can be used as target inputs to verify the accuracy of codes generated by PS. Based on the above definitions, our goal is to now automatically synthesize algorithms that output ψ(x) for a given V(x). However, to make progress, we must first define how the mathematical problem captured by Eq. (1) will be represented in our computer-generated codes; here, we adopt a computationally simple and practical approach, representing both V(x) and ψ(x) as real-valued numbers on a uniform grid x. The vectors denoting these properties on the grid-points will be labeled V and ψ, respectively; in all calculations reported below, the uniform grid comprises ng = 101 grid-points uniformly distributed on the range 𝑥𝑖∈[−5,+5]. This grid-based approach is well-known in the field of quantum simulations, as in the discrete variable representation (DVR52–55) that forms the basis of common strategies for solving Eq. (1); the connection between DVR methods and the algorithms generated here is discussed further below. Alternative representations of the input V(x) and output ψ(x) can clearly be explored in the future, whereas the focus of the current article is in providing an initial proof-of-concept. B. Program representation Previous work on PS has employed a variety of different structures for representing computer programs, such as tree structures, directed acyclic graphs, and fixed-size grids.[43-45,49,50] In this article, we choose to use a program representation that is analogous to that used in CGP, namely, using a grid of functional nodes that operate on input vectors, constants, and matrices.[44,56,57] Although we anticipate that a tree structure, as commonly employed in GP, can be used to solve the same problem, a grid-like code representation is selected here for its simplicity, flexibility, and ease of interpretation. Of course, there is no guarantee that the chosen program representation used here is optimal in any way, and the impact of the program structure will be a subject of interest in future work. ........................ this doesn't look like synthesis of a computer program to me, more synthesis of an equation. i may be wrong. it looks like it turned up as a result for a websearch on 'julia' because it was published in 'july'. it doens't mention julia.