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International Conference on Communications

Feynman Meets Turing: The Infeasibility of Digital Compilers for Gate-Based Quantum Computing

Yannik N. Bock, Holger Boche, Zoe Garcia del Toro, Frank H. P. Fitzek

Published Jun 9, 2024• 4 citations
Publisher

Abstract

We consider the problem of computing gate-circuit approximations of quantum algorithms, i.e., unitary operators, from the perspective of computable (effective) analysis. The scientific community thinks the Solovay-Kitaev theorem a mile-stone in quantum compiling - the task of computing gate-circuit approximations - because it proves the existence of efficient quantum compilers in an analytic sense. However, since we cannot represent unitary operators in a mere analytical way on digital computers, contemporary digital implementations of quantum compiling resort to heuristic numerics and remain below the computational performance engineers hope to realize using the result of Solovay and Kitaev. This paper discusses quantum compiling within a framework of computable analysis, establishing a concept of computable unitary operators for digital computing based on the theory of Turing machines. Particularly, we prove that digital quantum compiling is uncomputable due to the underlying algebraic structure. Finally, we discuss several implications of our findings for heuristic digital implementations of quantum compiling, hinting toward possible research directions to thoroughly understand the relevant bottlenecks.

DOI
10.1109/icc51166.2024.10622486

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We consider the problem of computing gate-circuit approximations of quantum algorithms, i.e., unitary operators, from the perspective of computable (effective) analysis. The scientific community thinks the Solovay-Kitaev theorem a mile-stone in quantum compiling - the task of computing gate-circuit approximations - because it proves the existence of efficient quantum compilers in an analytic sense. However, since we cannot represent unitary operators in a mere analytical way on digital computers, contemporary digital implementations of quantum compiling resort to heuristic numerics and remain below the computational performance engineers hope to realize using the result of Solovay and Kitaev. This paper discusses quantum compiling within a framework of computable analysis, establishing a concept of computable unitary operators for digital computing based on the theory of Turing machines. Particularly, we prove that digital quantum compiling is uncomputable due to the underlying algebraic structure. Finally, we discuss several implications of our findings for heuristic digital implementations of quantum compiling, hinting toward possible research directions to thoroughly understand the relevant bottlenecks.
We consider a formalism ofquantum compiler functions– functions that map unitary matrices to corresponding gate-circuit approximations – and prove the infeasibility of digitally computing such functions. Since the gate-circuit model of quantum computing emerged, much research has been conducted to find algorithmic solutions to thequantum compiler problem. The renownedSolovay-Kitaev theoremproves the existence of quantum compiler functions that provide low-complexity gatecircuit approximations to arbitrary unitary matrices, which is indispensable for the practical feasibility of gate-based quantum computing. However, the mere existence of such functions does not imply theirrealizabilityby means of analgorithm– a constructive procedure executed by aTuring machine. In fact, no algorithm for computing any quantum compiler function is known today. The present article demonstrates that no such algorithm can exist. We prove that no quantum compiler function can satisfyBanach-Mazur computability, which is a formalization of algorithmic feasibility with (mathematically) weak requirements. In consequence, there definitely does not exist a Turing machine that computes any quantum compiler function in the above sense, nor can there exist aconstructive proofof the existence of any such function. Furthermore, we discuss proposed methods of quantum compiling and analyze them in the context of our results.
We consider exact quantum circuit synthesis, quantum gate efficiency, and the spectral gap conjecture from the perspective of computable analysis. Circuit synthesis, in both its exact and its approximate variant, is fundamental to the circuit model of quantum computing. As an engineering problem, however, the practical and theoretical aspects of quantum circuit synthesis are far from being fully understood. Particularly, this concerns explicit methods for gate-agnostic circuit synthesis and questions of gate efficiency. More than 20 years ago, Harrow et al. published their famous spectral gap theorem – given a suitable family of quantum gates, it is possible to approximate any unitary transformation by means of a quantum circuit whose length is proportional to the required accuracy's logarithm. Moreover, Harrow et al. suspected that all universal gate families allow for this type of approximation, a hypothesis that became known as the spectral gap conjecture and remains unproven until today. Being an entirely classical task, quantum circuit synthesis must be considered in the context of digital computing – that is, in the context of Turing computability and computable analysis. Using the relevant mathematical framework, we establish no-go results concerning exact quantum circuit synthesis and quantum big-O analysis. Our findings relate to the theory of approximate t-designs, which has recently received notable attention through the literature. Moreover, as follows from our findings, the existence of an algorithm that computes leading big-O coefficients would prove the spectral gap conjecture true within the computable special unitary group.
The Solovay-Kitaev algorithm is a fundamental result in quantum computation. It gives an algorithm for efficiently compiling arbitrary unitaries using universal gate sets: any unitary can be approximated by short gates sequences, whose length scales merely poly-logarithmically with accuracy. As a consequence, the choice of gate set is typically unimportant in quantum computing. However, the Solovay-Kitaev algorithm requires the gate set to be inverse-closed. It has been a longstanding open question if efficient algorithmic compilation is possible without this condition. In this work, we provide the first inverse-free Solovay-Kitaev algorithm, which makes no assumption on the structure within a gate set beyond universality, answering this problem in the affirmative, and providing an efficient compilation algorithm in the absence of inverses for both SU(d)\text{SU}(d)SU(d) and SL(d,C)\text{SL}(d, \mathbb{C})SL(d,C). The algorithm works by showing that approximate gate implementations of the generalized Pauli group can self-correct their errors.
We consider a formal model of quantum circuit description languages (QCDLs) in which semantically meaningful programs correspond to computable unitary matrices. We show that any semantically universal QCDL -- that is, any QCDL able to describe all computable unitary matrices, which in turn form the set of matrices we can meaningfully represent on digital hardware -- cannot have a semi-decidable set of semantically meaningful descriptions. In particular, no such language admits a compiler that reliably recognizes all valid program descriptions. This result stands in contrast to classical programming languages. While compilation in languages such as C or C++ may itself involve non-terminating computations, the set of semantically meaningful programs remains recursively enumerable, since successful compilation provides a witness of validity. The essential difference lies in the nature of the semantic domains: classical languages describe partial recursive functions, whereas QCDLs describe total unitary operators. Our analysis establishes a fundamental limitation of quantum circuit description languages and highlights a structural distinction between classical and quantum models of computation at the level of formal language theory.
We investigate the feasibility of computing quantum gate-circuit emulation (QGCE)functions and quantum gate-circuit concatenation (QGCC) functions on digital hardware. QGCE functions serve the purpose of rewriting gate-circuits comprised of gates from a varying (possibly universal) input gate-set to gate-circuits comprised of gates from a fixed target gate set. Analogously, QGCC functions serve the purpose of finding an approximation to the concatenation of two arbitrary elements of a varying list of input gate circuits in terms of another element from the same list. Problems of this kind occur regularly in quantum computing and are often considered an easy task for the digital computers controlling the quantum hardware. However, recent results employing a rigorous mathematical theory of computability indicate that this may not be the case. This paper extends the aforementioned theory, providing two relevant insights: Upon applying a rigorous theory of computability, QGCE functions and QGCC functions turn out to be uncomputable on digital hardware. The results remain valid when we restrict the set of feasible inputs for these functions to one-parameter families of fixed gate sets, which is applicable even to standard one-qubit systems. Our insights underline the possibility that several ideas from the theory of quantum computing may require a rethinking in order to become feasible for practical implementation.
We investigate the feasibility of computing quantum gate-circuit emulation (QGCE) and quantum gate-circuit concatenation (QGCC) on digital hardware. QGCE serves the purpose of rewriting gate circuits comprised of gates from a varying input gate set to gate circuits formed of gates from a fixed target gate set. Analogously, QGCC serves the purpose of finding an approximation to the concatenation of two arbitrary elements of a varying list of input gate circuits in terms of another element from the same list. Problems of this kind occur regularly in quantum computing and are often assumed an easy task for the digital computers controlling the quantum hardware. Arguably, this belief is due to analogical reasoning: The classical Boolean equivalents of QGCE and QGCC are natively computable on digital hardware. In the present paper, we present two insights in this regard: Upon applying a rigorous theory of computability, QGCE and QGCC turn out to be uncomputable on digital hardware. The results remain valid when we restrict the set of feasible inputs for the relevant functions to one parameter families of fixed gate sets. Our results underline the possibility that several ideas from quantum-computing theory may require a rethinking to become feasible for practical implementation.