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Hamiltonian simulation is a central application of quantum computing, with significant potential in modeling physical systems and solving complex optimization problems. Existing compilers for such simulations typically focus on low-level representations based on Pauli operators, limiting programmability and offering no formal guarantees of correctness across the compilation pipeline. We introduce QBlue, a high-level, formally verified framework for compiling Hamiltonian simulations. QBlue is based on the formalism of second quantization, which provides a natural and expressive way to describe quantum particle systems using creation and annihilation operators. To ensure safety and correctness, QBlue includes a type system that tracks particle types and enforces Hermitian structure. The framework supports compilation to both digital and analog quantum circuits and captures multiple layers of semantics, from static constraints to dynamic evolution. All components of QBlue, including its language design, type system, and compilation correctness, are fully mechanized in the Rocq proof framework, making it the first end-to-end verified compiler for second-quantized Hamiltonian simulation. 

In physics and chemistry, quantum systems are typically modeled using energy constraints formulated as Hamiltonians. Investigations into such systems often focus on the evolution of the Hamiltonians under various initial conditions, an approach summarized as Adiabatic Quantum Computing (AQC). Although this perspective may initially seem foreign to functional programmers, we demonstrate that conventional functional programming abstractions—specifically, the Traversable and Monad type classes—naturally capture the essence of AQC. To illustrate this connection, we introduce EnQ, a functional programming library designed to express diverse optimization problems as energy constraint computations (ECC). The library comprises three core components: generating the solution space, associating energy costs with potential solutions, and searching for optimal or near-optimal solutions. Because EnQ is implemented using standard Haskell, it can be executed directly through conventional classical Haskell compilers. More interestingly, we develop and implement a process to compile EnQ programs into circuits executable on quantum hardware. We validate EnQ’s effectiveness through a number of case studies, demonstrating its capacity to express and solve classical optimization problems on quantum hardware, including search problems, type inference, number partitioning, clique finding, and graph coloring. 

We describe a general formalism for quantum dynamics and show how this formalism subsumes several quantum algorithms including the Deutsch, Deutsch-Jozsa, Bernstein-Vazirani, Simon, and Shor algorithms as well as the conventional approach to quantum dynamics based on tensor networks. The common framework exposes similarities among quantum algorithms and natural quantum phenomena: we illustrate this connection by showing how the correlated behavior of protons in water wire systems that are common in many biological and materials systems parallels the structure of Shor's algorithm. 

Compact closed categories include objects representing higher-order functions and are well-established as models of linear logic, concurrency, and quantum computing. We show that it is possible to construct such compact closed categories for conventional sum and product types by defining a dual to sum types, a negative type, and a dual to product types, a fractional type. Inspired by the categorical semantics, we define a sound operational semantics for negative and fractional types in which a negative type represents a computational effect that ``reverses execution flow'' and a fractional type represents a computational effect that ``garbage collects'' particular values or throws exceptions. Specifically, we extend a first-order reversible language of type isomorphisms with negative and fractional types, specify an operational semantics for each extension, and prove that each extension forms a compact closed category. We furthermore show that both operational semantics can be merged using the standard combination of backtracking and exceptions resulting in a smooth interoperability of negative and fractional types. We illustrate the expressiveness of this combination by writing a reversible SAT solver that uses backtracking search along freshly allocated and de-allocated locations. The operational semantics, most of its meta-theoretic properties, and all examples are formalized in a supplementary Agda package.