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Algorithms

Quantum Circuit-Width Reduction through Parameterisation and Specialisation

Youssef Moawad, Wim A. Vanderbauwhede, R. Steijl

Published May 5, 2023• 0 citations
Publisher

Abstract

As quantum computing technology continues to develop, the need for research into novel quantum algorithms is growing. However, such algorithms cannot yet be reliably tested on actual quantum hardware, which is still limited in several ways, including qubit coherence times, connectivity, and available qubits. To facilitate the development of novel algorithms despite this, simulators on classical computing systems are used to verify the correctness of an algorithm, and study its behaviour under different error models. In general, this involves operating on a memory space that grows exponentially with the number of qubits. In this work, we introduce quantum circuit transformations that allow for the construction of parameterised circuits for quantum algorithms. The parameterised circuits are in an ideal form to be processed by quantum compilation tools, such that the circuit can be partially evaluated prior to simulation, and a smaller specialised circuit can be constructed by eliminating fixed input qubits. We show significant reduction in the number of qubits for various quantum arithmetic circuits. Divide-by-n-bits quantum integer dividers are used as an example demonstration. It is shown that the complexity reduces from 4n+2 to 3n+2 qubits in the specialised versions. For quantum algorithms involving divide-by-8 arithmetic operations, a reduction by 28=256 in required memory is achieved for classical simulation, reducing the memory required from 137 GB to 0.53 GB.

DOI
10.3390/a16050241

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As quantum computing technology continues to develop, the need for research into novel quantum algorithms is growing. However, such algorithms cannot yet be reliably tested on actual quantum hardware, which is still limited in several ways, including qubit coherence times, connectivity, and available qubits. To facilitate the development of novel algorithms despite this, simulators on classical computing systems are used to verify the correctness of an algorithm, and study its behaviour under different error models. In general, this involves operating on a memory space that grows exponentially with the number of qubits. In this work, we introduce quantum circuit transformations that allow for the construction of parameterised circuits for quantum algorithms. The parameterised circuits are in an ideal form to be processed by quantum compilation tools, such that the circuit can be partially evaluated prior to simulation, and a smaller specialised circuit can be constructed by eliminating fixed input qubits. We show significant reduction in the number of qubits for various quantum arithmetic circuits. Divide-by-n-bits quantum integer dividers are used as an example demonstration. It is shown that the complexity reduces from 4n+2 to 3n+2 qubits in the specialised versions. For quantum algorithms involving divide-by-8 arithmetic operations, a reduction by 28=256 in required memory is achieved for classical simulation, reducing the memory required from 137 GB to 0.53 GB.
Quantum computing is an emerging technology in which quantum mechanical properties are suitably utilized to perform certain compute-intensive operations faster than classical computers. Quantum algorithms are designed as a combination of quantum circuits that each require a large number of quantum gates, which is a challenge considering the limited number of qubit resources available in quantum computing systems. Our work proposes a technique to optimize quantum arithmetic algorithms by reducing the hardware resources and the number of qubits based on ZX calculus. We have utilised ZX calculus rewrite rules for the optimization of fault-tolerant quantum multiplier circuits where we are able to achieve a significant reduction in the number of ancilla bits and T-gates as compared to the originally required numbers to achieve fault-tolerance. Our work is the first step in the series of arithmetic circuit optimization using graphical rewrite tools and it paves the way for advancing the optimization of various complex quantum circuits and establishing the potential for new applications of the same.
Quantum computing is an emerging technology in which quantum mechanical properties are suitably utilized to perform certain compute-intensive operations faster than classical computers. Quantum algorithms are designed as a combination of quantum circuits that each require a large number of quantum gates, which is a challenge considering the limited number of qubit resources available in quantum computing systems. Our work proposes a technique to optimize quantum arithmetic algorithms by reducing the hardware resources and the number of qubits based on ZX calculus. We have utilized ZX calculus rewrite rules for the optimization of fault-tolerant quantum multiplier circuits where we are able to achieve a significant reduction in the number of ancilla bits and T-gates as compared to the originally required numbers to achieve fault-tolerance. Our work is the first step in the series of arithmetic circuit optimization using graphicalrewrite tools and it pavesthe way for advancing the optimization of various complex quantum circuits and establishing the potential for new applications of the same.
Practical realizations of quantum computers are poised to deliver outstanding computational capabilities far beyond the reach of any classical supercomputer. While classical systems are reliably implemented using CMOS technology, the fabrication of quantum hardware has not yet reached a commercially viable level of maturity. Nowadays, many different technologies are being developed as competing candidates to build the first error-corrected quantum computer. Among the software resources required to operate such a system, quantum compilers translate a high-level description of a quantum algorithm into low-level, technology-dependent quantum instructions. Many quantum algorithms, including Shor's and Grover's algorithms, require to compute some classical logic functions on the quantum computer. The first part of this thesis deals with the problem of compiling quantum circuits that perform such classical Boolean functions, called oracles, into a fault-tolerant set of instructions. Oracle circuits often demand many resources in terms of the number of qubits and gates. The focus is on compiling quantum circuits starting from multi-level logic networks representing large Boolean functions while minimizing the resource footprint of the obtained quantum circuits. At the same time, the trade-off between memory resources and the number of operations typical of such a compilation process is explored. As part of the effort in reducing the resources required to perform a given quantum program, I also present some quantum circuit optimization techniques. The researched algorithms leverage data structures and techniques borrowed and adapted from classical logic synthesis, e.g., SAT solvers, LUT mapping, and multi-level logic networks. I implemented all the compilation algorithms presented in this thesis as part of open-source projects. In particular, I develop and maintain caterpillar---a C++ header-only library dedicated to the quantum compilation of oracles and quantum memory management. The second part of this thesis describes how to equip quantum programming language compilers with automatic accuracy management. Despite the availability of many of such languages, resource estimation of quantum algorithms does not yet support taking approximation errors into account. This general methodology is applicable to any programming language and I demonstrate its integration into the Q# compiler. The technique consists of providing language support to ease the accuracy-aware programming task. The user can define accuracy parameters that will be automatically optimized according to a constraint and a cost function directly generated from the source code. In the presented practical evaluations, the constraint function is the overall allowed accuracy, while the cost function is application-dependent and related to the number of gates. During the optimization process, such functions are evaluated hundreds of times. For this reason, we extract them as (near-)symbolic expressions, whose evaluation time does not depend on the quantum algorithm size. The algorithms and the methodologies presented in this thesis are part of a widespread effort of the research community to build a complete and efficient software stack to program and control the first practical universal quantum computer.
Quantum-mechanical phenomena are playing an increasing role in information processing as transistor sizes approach the nanometer level, while the securest forms of communication rely on quantum data encoding. When they involve a finite number of basis states, these phenomena can be modeled as quantum circuits, the quantum analogue of conventional or classical logic circuits. Simulation of quantum circuits can therefore be used as a tool to evaluate issues in the design of quantum information processors. Unfortunately, simulating such phenomena efficiently is exceedingly difficult. The matrices representing quantum operators (gates) and vectors modeling quantum states grow exponentially with the number of quantum bits. The information represented by quantum states and operators often exhibits structure that can be exploited when simulating certain classes of quantum circuits. We study the development of simulation methods that run on classical computers and take advantage of such repetitions and redundancies. In particular, we define a new data structure for simulating quantum circuits called the quantum information decision diagram (QuIDD). A QuIDD is a compressed graph representation of a vector or matrix and permits computations to be performed directly on the compressed data. We develop a comprehensive set of algorithms for operating on QuIDDs in both the state-vector and density-matrix formats, and evaluate their complexity. These algorithms have been implemented in a general-purpose simulator program for quantum-mechanical applications called QuIDDPro. Through extensive experiments conducted on representative quantum simulation applications, including Grover's search algorithm, error characterization, and reversible circuits, we demonstrate that QuIDDPro is faster than other existing quantum-mechanical simulators such as the National Institute of Standards and Technology's QCSim program, and is far more memory-efficient. Using QuIDDPro, we explore the advantages of quantum computation over classical computation, simulate quantum errors and error correction, and study the impact of numerical precision on the fidelity of simulations. We also develop several novel algorithms for testing quantum circuit equivalence and compare them empirically. The QuIDDPro software is equipped with a user-friendly interface and is distributed with numerous example scripts. It has been used as a laboratory supplement for quantum computing courses at several universities.
Abstract Gate-level quantum circuits are often derived manually from higher level algorithms. While this suffices for small implementations and demonstrations, ultimately automatic circuit design will be required to realise complex algorithms using hardware-specific operations and connectivity. Therefore, ab initio creation of circuits within a machine, either a classical computer or a hybrid quantum–classical device, is of key importance. We explore a range of established and novel techniques for the synthesis of new circuit structures, the optimisation of parameterised circuits, and the efficient removal of low-value gates via the quantum geometric tensor. Using these techniques we tackle the tasks of automatic encoding of unitary processes and translation (recompilation) of a circuit from one form to another. Using emulated quantum computers with various noise-free gate sets we provide simple examples involving up to 10 qubits, corresponding to 20 qubits in the augmented space we use. Further applications of specific relevance to chemistry modelling are considered in a sister paper, ‘Exploiting subspace constraints and ab initio variational methods for quantum chemistry’. The emulation environments used were QuEST, QuESTlink and pyQuEST. All resources will be made openly accessible and are currently available upon request.
Parameterised quantum circuits (PQCs) hold great promise for demonstrating quantum advantages in practical applications of quantum computation. Examples of successful applications include the variational quantum eigensolver, the quantum approximate optimisation algorithm, and quantum machine learning. However, before executing PQCs on real quantum devices, they undergo compilation and optimisation procedures. Given the inherent error-proneness of these processes, it becomes crucial to verify the equivalence between the original PQC and its compiled or optimised version. Unfortunately, most existing quantum circuit verifiers cannot directly handle parameterised quantum circuits; instead, they require parameter substitution to perform verification. In this paper, we address the critical challenge of equivalence checking for PQCs. We propose a novel compact representation for PQCs based on tensor decision diagrams. Leveraging this representation, we present an algorithm for verifying PQC equivalence without the need for instantiation. Our approach ensures both effectiveness and efficiency, as confirmed by experimental evaluations. The decision-diagram representations offer a powerful tool for analysing and verifying parameterised quantum circuits, bridging the gap between theoretical models and practical implementations.