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Numerous quantum algorithms require the use of quantum error correction to overcome the intrinsic unreliability of physical qubits. However, quantum error correction imposes a unique performance bottleneck, known asT-complexity, that can make an implementation of an algorithm as a quantum program run more slowly than on idealized hardware. In this work, we identify that programming abstractions for control flow, such as the quantum if-statement, can introduce polynomial increases in theT-complexity of a program. If not mitigated, this slowdown can diminish the computational advantage of a quantum algorithm. To enable reasoning about the costs of control flow, we present a cost model that a developer can use to accurately analyze theT-complexity of a program under quantum error correction and pinpoint the sources of slowdown. To enable the mitigation of these costs, we present a set of program-level optimizations that a developer can use to rewrite a program to reduce itsT-complexity, predict theT-complexity of the optimized program using the cost model, and then compile it to an efficient circuit via a straightforward strategy. We implement the program-level optimizations in Spire, an extension of the Tower quantum compiler. Using a set of 11 benchmark programs that use control flow, we empirically show that the cost model is accurate, and that Spire’s optimizations recover programs that are asymptotically efficient, meaning their runtimeT-complexity under error correction is equal to their time complexity on idealized hardware. Our results show that optimizing a program before it is compiled to a circuit can yield better results than compiling the program to an inefficient circuit and then invoking a quantum circuit optimizer found in prior work. For our benchmarks, only 2 of 8 tested quantum circuit optimizers recover circuits with asymptotically efficientT-complexity. Compared to these 2 optimizers, Spire uses 54×–2400× less compile time. 

Quantum algorithms for tasks such as factorization, search, and simulation rely on control flow such as branching and iteration that depends on the value of data in superposition. High-level programming abstractions for control flow, such as switches, loops, higher-order functions, and continuations, are ubiquitous in classical languages. By contrast, many quantum languages do not provide high-level abstractions for control flow in superposition, and instead require the use of hardware-level logic gates to implement such control flow. The reason for this gap is that whereas a classical computer supports control flow abstractions using a program counter that can depend on data, the typical architecture of a quantum computer does not analogously provide a program counter that can depend on data in superposition. As a result, the complete set of control flow abstractions that can be correctly realized on a quantum computer has not yet been established. In this work, we provide a complete characterization of the properties of control flow abstractions that are correctly realizable on a quantum computer. First, we prove that even on a quantum computer whose program counter exists in superposition, one cannot correctly realize control flow in quantum algorithms by lifting the classical conditional jump instruction to work in superposition. This theorem denies the ability to directly lift general abstractions for control flow such as the λ-calculus from classical to quantum programming. In response, we present the necessary and sufficient conditions for control flow to be correctly realizable on a quantum computer. We introduce the quantum control machine, an instruction set architecture featuring a conditional jump that is restricted to satisfy these conditions. We show how this design enables a developer to correctly express control flow in quantum algorithms using a program counter in place of logic gates. 

Computations in physical simulation, computer graphics, and probabilistic inference often require the differentiation of discontinuous processes due to contact, occlusion, and changes at a point in time. Popular differentiable programming languages, such as PyTorch and JAX, ignore discontinuities during differentiation. This is incorrect forparametric discontinuities—conditionals containing at least one real-valued parameter and at least one variable of integration. We introduce Potto, the first differentiable first-order programming language to soundly differentiate parametric discontinuities. We present a denotational semantics for programs and program derivatives and show the two accord. We describe the implementation of Potto, which enables separate compilation of programs. Our prototype implementation overcomes previous compile-time bottlenecks achieving an 88.1x and 441.2x speed up in compile time and a 2.5x and 7.9x speed up in runtime, respectively, on two increasingly large image stylization benchmarks. We showcase Potto by implementing a prototype differentiable renderer with separately compiled shaders. 

Programmers and researchers are increasingly developing surrogates of programs, models of a subset of the observable behavior of a given program, to solve a variety of software development challenges. Programmers train surrogates from measurements of the behavior of a program on a dataset of input examples. A key challenge of surrogate construction is determining what training data to use to train a surrogate of a given program. We present a methodology for sampling datasets to train neural-network-based surrogates of programs. We first characterize the proportion of data to sample from each region of a program's input space (corresponding to different execution paths of the program) based on the complexity of learning a surrogate of the corresponding execution path. We next provide a program analysis to determine the complexity of different paths in a program. We evaluate these results on a range of real-world programs, demonstrating that complexity-guided sampling results in empirical improvements in accuracy. 

A streaming probabilistic program receives a stream of observations and produces a stream of distributions that are conditioned on these observations. Efficient inference is often possible in a streaming context using Rao-Blackwellized particle filters (RBPFs), which exactly solve inference problems when possible and fall back on sampling approximations when necessary. While RBPFs can be implemented by hand to provide efficient inference, the goal of streaming probabilistic programming is to automatically generate such efficient inference implementations given input probabilistic programs. In this work, we propose semi-symbolic inference, a technique for executing probabilistic programs using a runtime inference system that automatically implements Rao-Blackwellized particle filtering. To perform exact and approximate inference together, the semi-symbolic inference system manipulates symbolic distributions to perform exact inference when possible and falls back on approximate sampling when necessary. This approach enables the system to implement the same RBPF a developer would write by hand. To ensure this, we identify closed families of distributions – such as linear-Gaussian and finite discrete models – on which the inference system guarantees exact inference. We have implemented the runtime inference system in the ProbZelus streaming probabilistic programming language. Despite an average 1.6× slowdown compared to the state of the art on existing benchmarks, our evaluation shows that speedups of 3×-87× are obtainable on a new set of challenging benchmarks we have designed to exploit closed families. 

Probabilistic programming languages aid developers performing Bayesian inference. These languages provide programming constructs and tools for probabilistic modeling and automated inference. Prior work introduced a probabilistic programming language, ProbZelus, to extend probabilistic programming functionality to unbounded streams of data. This work demonstrated that the delayed sampling inference algorithm could be extended to work in a streaming context. ProbZelus showed that while delayed sampling could be effectively deployed on some programs, depending on the probabilistic model under consideration, delayed sampling is not guaranteed to use a bounded amount of memory over the course of the execution of the program. In this paper, we the present conditions on a probabilistic program’s execution under which delayed sampling will execute in bounded memory. The two conditions are dataflow properties of the core operations of delayed sampling: the m -consumed property and the unseparated paths property . A program executes in bounded memory under delayed sampling if, and only if, it satisfies the m -consumed and unseparated paths properties. We propose a static analysis that abstracts over these properties to soundly ensure that any program that passes the analysis satisfies these properties, and thus executes in bounded memory under delayed sampling.