Reasoning in ZX calculus and quantum natural language processing

量子计算机有潜力解决经典计算机难以解决的任务，从而在各个领域彻底改变科学和技术。然而，量子算法仍然基于量子电路，这是一种相当不直观的低级语言，不支持抽象。一个有希望的替代方案来自牛津大学的量子计算小组。他们的图形语言允许更直观的量子计算方法。在我的博士学位，我将与亚历克斯·基辛格，在牛津大学量子组的联合负责人，和鲍勃Coecke，在牛津Quantinuum的首席科学家，探索量子计算的两个领域：（a）量子电路优化和（B）在量子自然语言处理推理-都利用图形语言。因此，这个哲学博士福尔斯属于EPSRC量子技术研究领域。牛津大学开发的图形语言最初是作为量子力学的分类视图创建的（Coecke & Kissinger，2017）。这种语言，以其直观的视觉表示，简化了复杂的推理过程。一个特别有用的应用是ZX演算-一种由Coecke和邓肯（2008）开发的量子电路的图形语言。它将量子门分解为更基本的组成元素。使用八个简单而完整的拓扑规则（Vilmart 2019），ZX演算是一个强大的，直观的工具来推理量子电路。量子电路优化的目标是根据给定的要求重写量子电路，例如路由条件，或者减少（噪声）门的数量（邓肯等人，2020年）。当前的优化技术通常基于重写规则的子集对贪婪例程进行操作。在我的博士学位中，我将利用我在机器学习方面的经验来建立这项工作。有更精细的方法来应用电路重写可以大大提高性能。Krueger（2022）使用模拟退火和遗传算法的一些早期尝试在相对较小的电路中显示出良好的结果。我将继续这一努力，从其他形式推理任务中已证明的优化技术中汲取灵感，如定理证明（例如，Kaliszyk等人，2018).虽然ZX演算用于低级别的量子电路，但图形语言也用于高级任务。DisCoCirc是量子计算机上的自然语言处理框架（Coecke，2021）。DisCoCirc通过区分单词的句法相互作用和它们的语义表示来对整个文本的意义进行建模。这种分离创造了更多的可理解性和显式推理的潜力。在我的博士学位期间，我将从两个方向探索DisCoCirc中的推理：一方面，更好地理解基于神经网络的DisCoCirc系统如何学习是至关重要的。为此，我将探索推理任务的意义的替代表示。另一方面，我将继续通过探索单个规则及其相互作用来发展对推理的正式理解（如Rodatz等人，2021年）。首先，我将探索合取（由Duneau（2021）提出），否定和后来的其他运算符（如量词）之间的相互作用。使用图形语言，ZX图和自然语言中的量子电路推理变得相关。虽然量子电路更加结构化，但量子电路上的推理可以激发自然语言中的推理，例如，在我们关于会话否定的工作中（Rodatz et al.，2021年）。此外，DisCoCirc框架可以在量子计算机上获得计算加速（Zeng & Coecke，2016）。因此，有效的优化技术使更多种类的自然语言推理实验成为可能。量子计算中两个不同主题的计划组合令人兴奋和充满希望，因为它们共享图形语言的方法。

Quantum computers have the potential to solve tasks intractable on classical computers, thereby revolutionising science and technology in various fields. However, quantum algorithms are still based on quantum circuits, a rather unintuitive low-level language without support for abstraction. One promising alternative comes from the quantum computing group at Oxford. Their diagrammatic languages allow for more intuitive approaches to quantum computing. In my DPhil, I will work with Aleks Kissinger, joint head at the quantum group at Oxford, and Bob Coecke, chief scientist at Quantinuum in Oxford, to explore two areas in quantum computing: (a) quantum circuit optimisation and (b) reasoning in quantum natural language processing - both utilising diagrammatic languages. As such, this DPhil falls within the EPSRC quantum technologies research area.The diagrammatic language developed at Oxford was initially created as a categorical view of quantum mechanics (Coecke & Kissinger, 2017). This language, with its intuitive visual representation, simplifies complex reasoning processes. A particularly useful application is ZX calculus - a diagrammatic language for quantum circuits developed by Coecke and Duncan (2008). It splits quantum gates into more fundamental, compositional elements. Using eight simple yet complete topological rules (Vilmart, 2019), ZX calculus is a potent, intuitive tool to reason over quantum circuits.The goal of quantum circuit optimisation is to rewrite quantum circuits for given requirements, such as routing conditions, or to reduce the number of (noisy) gates (Duncan et al., 2020). Current optimisation techniques often operate on greedy routines based on a subset of rewrite rules. In my DPhil, I will use my experience with machine learning to build on this work. Having more elaborate methods of applying circuit rewrites can improve performance drastically. Some early attempts to use simulated annealing and genetic algorithms by Krueger (2022) show good results for relatively small circuits. I will continue this endeavour by taking inspiration from proven optimisation techniques in other formal reasoning tasks such as theorem proving (e.g., Kaliszyk et al., 2018).While ZX calculus is for low-level quantum circuits, diagrammatic languages are also used for high-level tasks. DisCoCirc is a framework for natural language processing on quantum computers (Coecke, 2021). DisCoCirc models the meaning of entire texts by differentiating the syntactic interaction of words from their semantic representation. This separation creates more understandability and the potential for explicit reasoning.During my DPhil, I will explore reasoning in DisCoCirc from two directions: On the one hand, it is essential to understand better how neural-net-based DisCoCirc systems learn. For this, I will explore alternative representations of meaning for reasoning tasks. On the other hand, I will continue developing a formal understanding of reasoning by exploring individual rules and their interactions (as in Rodatz et al., 2021). Initially, I will explore the interaction between conjunction (proposed by Duneau (2021)), negation and later other operators, such as quantifiers.Using diagrammatic languages, reasoning on quantum circuits in ZX diagrams and in natural language become related. While quantum circuits are more structured, reasoning over quantum circuits can inspire reasoning in natural language, as, for example, in our work on conversational negation (Rodatz et al., 2021). Additionally, the DisCoCirc framework can gain computational speedups on quantum computers (Zeng & Coecke, 2016). Therefore, efficient optimisation techniques enable a larger variety of experiments on reasoning in natural language.The planned combination of two different topics in quantum computing is exciting and promising, given their shared approach of diagrammatic languages.