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Lecture notes in computer science

A Programming Language Characterizing Quantum Polynomial Time

Emmanuel Hainry F, Romain Péchoux, Mário Silva

Published Jan 1, 2023• 6 citations
Publisher

Abstract

Abstract We introduce a first-order quantum programming language, named foq , whose terminating programs are reversible. We restrict foq to a strict and tractable subset, named pfoq , of terminating programs with bounded width, that provides a first programming language-based characterization of the quantum complexity class fbqp . We finally present a tractable semantics-preserving algorithm compiling a pfoq program to a quantum circuit of size polynomial in the number of input qubits.

DOI
10.1007/978-3-031-30829-1_8

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Abstract We introduce a first-order quantum programming language, named foq , whose terminating programs are reversible. We restrict foq to a strict and tractable subset, named pfoq , of terminating programs with bounded width, that provides a first programming language-based characterization of the quantum complexity class fbqp . We finally present a tractable semantics-preserving algorithm compiling a pfoq program to a quantum circuit of size polynomial in the number of input qubits.
As quantum computing emerges as a promising computational paradigm, quantum programming languages provide the tools that bridge the distance between abstract programming and its hardware implementation. In some cases, restricted programming languages may even provide an avenue for more efficient circuit compilation strategies. In this work, we introduce foq , a first-order quantum programming language which allows for quantum control and recursion, and where a syntactically restricted subset of programs ( pfoq ) is shown to be sound and complete for quantum polytime computation. This is achieved by bounding both the recursion depth and the branching width of programs, which we demonstrate to still be compatible with various interesting applications, such as quantum teleportation and the quantum Fourier transform. pfoq constitutes the first programming-language-based characterization of the quantum complexity class fbqp , and we provide a semantics-preserving compilation algorithm such that any pfoq program can be compiled into a quantum circuit that grows polynomially on its number of input qubits, using an anchoring-and-merging technique to solve the problem of branch sequentialization.
Polylogarithmic time delineates a relevant notion of feasibility on several classical computational models such as Boolean circuits or parallel random access machines. As far as the quantum paradigm is concerned, this notion yields the complexity class FBQPOLYLOG of functions approximable in polylogarithmic time with a quantum random-access Turing machine. We introduce a quantum programming language with first-order recursive procedures, which provides the first programming-language-based characterization of FBQPOLYLOG. Each program computes a function in FBQPOLYLOG (soundness) and, conversely, each function of this complexity class is computed by a program (completeness). We also provide a compilation strategy from programs to uniform families of quantum circuits of polylogarithmic depth and polynomial size, whose set of computed functions is known as QNC, and recover the well-known separation result FBQPOLYLOG ⊊\subsetneq⊊ QNC.
Abstract. We introduce the language QML, a functional language for quantum computations on finite types. Its design is guided by its categorical semantics: QML programs are interpreted by morphisms in the category FQC of finite quantum computations, which provides a constructive semantics of irreversible quantum computations realizable as quantum gates. QML integrates reversible and irreversible quantum computations in one language, using first order strict linear logic to make weakenings explicit. Strict programs are free from decoherence and hence preserve superpositions and entanglement – which is essential for quantum parallelism. 1
Term Rewrite Systems (TRS) is a computational model offering a level of abstraction well-suited towards static analysis, e.g., termination or complexity analyses. In this paper, we introduce Quantum Term Rewrite Systems (QTRS), an extension of TRS to quantum computing, thus allowing to benefit from quantum advantage while being able to certify the complexity. We ensure that QTRS correspond to physically realizable processes and adapt techniques to obtain termination certificates or generic bounds on the reduction length. We delineate a class of terminating QTRS that can be compiled to uniform families of quantum circuits of size bounded by the reduction length. Conversely, this class is universal for quantum circuits. In particular, we show a characterization of the class of functions computable in quantum polynomial time, known as FBQP\mathtt{FBQP}FBQP.