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On the Complexity of Quantum Circuit Compilation
Abstract
Quantum circuit compilation (QCC) is an important problem in the emerging field of quantum computing. The problem has a strong relevance to combinatorial search, as solving approaches recently published include constraint programming and temporal planning. In this paper, we focus on a complexity analysis of quantum circuit compilation. We formulate a makespan optimization problem based on QCC, and prove that the problem is NP-complete. To the best of our knowledge, this is the first study on the theoretical complexity of QCC.
... On the flip side, by linking distributed quantum processors, several new challenges arise [12,15,23,31,52,79,90]. Here we consider the compilation problem, which is generally tough to solve, even on single processor, and for which an NP-hardness proof is available [10]. An ever growing literature arises with a variety of proposals for local computing [9,11,35,46,47,50,62,68,69,71,75,83,91,93,98] and for distributed computing [8, 24-26, 36, 40, 77, 81, 96, 97]. ...
... Therefore, finding a mapping with minimum depth overhead is an optimization problem. We refer to it as the quantum circuit compilation problem (QCC), which is proved to be NP-hard [10]. Its version on distributed architectures, which we refer to as the distributed quantum circuit compilation problem (DQCC), is likely to be at least as hard as QCC. ...
... 9 Assigning logical qubits to physical ones -i.e., qubit mapping -is another critical step for compilation and it deserves dedicated analysis [5,27], out of the scope of this work. 10 The design of a distributed quantum architecture can easily adapt to satisfy requirements coming from assumptions on classical technologies, since these are very advanced. To not further weigh down the formalism, we remodel the instance, by considering as main nodes, the processors, corresponding to an enumeration for the partition , i.e., = { 1 , 2 , . . . ...
Practical distributed quantum computing requires the development of efficient compilers, able to make quantum circuits compatible with some given hardware constraints. This problem is known to be tough, even for local computing. Here, we address it on distributed architectures. As generally assumed in this scenario, telegates represent the fundamental remote (inter-processor) operations. Each telegate consists of several tasks: i) entanglement generation and distribution, ii) local operations, and iii) classical communications. Entanglement generations and distribution is an expensive resource, as it is time-consuming. To mitigate its impact, we model an optimization problem that combines running-time minimization with the usage of distributed entangled states. Specifically, we formulated the distributed compilation problem as a dynamic network flow. To enhance the solution space, we extend the formulation, by introducing a predicate that manipulates the circuit given in input and parallelizes telegate tasks.
To evaluate our framework, we split the problem into three sub-problems, and solve it by means of an approximation routine. Experiments demonstrate that the run-time is resistant to the problem size scaling. Moreover, we apply the proposed algorithm to compile circuits under different topologies, showing that topologies with a higher ratio between edges and nodes give rise to shallower circuits.
... Typically, a quantum circuit compilation (or transpilation) procedure is required to adapt a given circuit to be able to be compatible with the capabilities of the devices (e.g. converting gates to native gates, applying SWAP gates to connect qubits which are not physically connected) [7]. ...
... Here, we run the algorithm two times and take the best results using optimization level 3, and sabre-sabre layout and routing methods. Although, It is possible to obtain better gate counts with more runs or different transpilation algorithms, the best values obtained wouldn't change our conclusions.7 This value is chosen after a survey of devices listed on IBM Quantum Cloud. ...
Barren plateaus appear to be a major obstacle for using variational quantum algorithms to simulate large-scale quantum systems or to replace traditional machine learning algorithms. They can be caused by multiple factors such as the expressivity of the ansatz, excessive entanglement, the locality of observables under consideration, or even hardware noise. We propose classical splitting of parametric ansatz circuits to avoid barren plateaus. Classical splitting is realized by subdividing an N qubit ansatz into multiple ansätze that consist of O(logN)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(\log N)$$\end{document} qubits. We show that such an approach allows for avoiding barren plateaus and carry out numerical experiments, and perform binary classification on classical and quantum datasets. Moreover, we propose an extension of the ansatz that is compatible with variational quantum simulations. Finally, we discuss a speed-up for gradient-based optimization and hardware implementation, robustness against noise and parallelization, making classical splitting an ideal tool for noisy intermediate scale quantum (NISQ) applications.
... In order to ensure reliable execution of the resulting circuit, it is crucial to keep the overhead introduced through routing as small as possible. Unfortunately, determining optimal mappings of quantum circuits is an NP-hard problem [7], [15]-mainly due to the involved search space growing exponentially in the number of considered qubits. This is exacerbated by the fact that quantum computers are not available in arbitrary sizes. ...
... This leads to circuits with many SWAP gates-an undesirable property on NISQ devices due to the relatively high error rate of two-qubit gates. However, finding good solutions to the mapping problem is a challenging task in general and has even been shown to be NP-hard [7], [15], [19], [20]. Although these works make slightly different assumptions about the qubit mapping problem, the problem's complexity stays the same overall. ...
Compiling a high-level quantum circuit down to a low-level description that can be executed on state-of-the-art quantum computers is a crucial part of the software stack for quantum computing. One step in compiling a quantum circuit to some device is quantum circuit mapping, where the circuit is transformed such that it complies with the architecture’s limited qubit connectivity. Because the search space in quantum circuit mapping grows exponentially in the number of qubits, it is desirable to consider as few of the device’s physical qubits as possible in the process. Previous work conjectured that it suffices to consider only subarchitectures of a quantum computer composed of as many qubits as used in the circuit. In this work, we refute this conjecture and establish criteria for judging whether considering larger parts of the architecture might yield better solutions to the mapping problem. We show that determining subarchitectures that are of minimal size, i.e., from which no physical qubit can be removed without losing the optimal mapping solution for some quantum circuit, is a very hard problem. Based on a relaxation of the criteria for optimality, we introduce a relaxed consideration that still maintains optimality for practically relevant quantum circuits. Eventually, this results in two methods for computing near-optimal sets of subarchitectures—providing the basis for efficient quantum circuit mapping solutions. We demonstrate the benefits of this novel method for state-of-the-art quantum computers by IBM, Google and Rigetti.
... In order to ensure reliable execution of the resulting circuit, it is crucial to keep the overhead introduced through routing as small as possible. Unfortunately, determining optimal mappings of quantum circuits is an NP-hard problem [4], [13]-mainly due to the involved search space growing exponentially in the number of considered qubits. This is exacerbated by the fact that quantum computers are not available in arbitrary sizes. ...
... This leads to circuits with many SWAP gates-an undesirable property on NISQ devices due to relatively high error rate of two-qubit gates. However, finding good solutions to the mapping problem is a challenging task in general and has even been shown to be NP-hard [4], [13]. ...
Compiling a high-level quantum circuit down to a low-level description that can be executed on state-of-the-art quantum computers is a crucial part of the software stack for quantum computing. One step in compiling a quantum circuit to some device is quantum circuit mapping, where the circuit is transformed such that it complies with the architecture's limited qubit connectivity. Because the search space in quantum circuit mapping grows exponentially in the number of qubits, it is desirable to consider as few of the device's physical qubits as possible in the process. Previous work conjectured that it suffices to consider only subarchitectures of a quantum computer composed of as many qubits as used in the circuit. In this work, we refute this conjecture and establish criteria for judging whether considering larger parts of the architecture might yield better solutions to the mapping problem. Through rigorous analysis, we show that determining subarchitectures that are of minimal size, i.e., of which no physical qubit can be removed without losing the optimal mapping solution for some quantum circuit, is a very hard problem. Based on a relaxation of the criteria for optimality, we introduce a relaxed consideration that still maintains optimality for practically relevant quantum circuits. Eventually, this results in two methods for computing near-optimal sets of subarchitectures$\unicode{x2014}$providing the basis for efficient quantum circuit mapping solutions. We demonstrate the benefits of this novel method for state-of-the-art quantum computers by IBM, Google and Rigetti.
... Many arXiv:2203.00698v1 [quant-ph] 1 Mar 2022 of these problems have high worst-case complexity-some have even been proven to be NP-complete [6] or QMA-complete 1 [25]. Hence, efficient methods to tackle practically relevant instances are needed. ...
... A product state is a quantum state where the state of the whole system is described as the product of the states of the individual qubits, i.e., |ϕ = |ϕ n−1 ⊗ · · · ⊗ |ϕ 0 with each |ϕ i being an arbitrary single qubit state for i from 0 to n − 1. An important subclass of quantum circuits possessing this property are reversible circuits 6 [18], which effectively represent bijective Boolean functions and form a dedicated research area on their own. However, while restricting the potential quantum states to product states allows to efficiently analyze the respective circuits, this restriction prohibits a fundamental quantum mechanical phenomenon to be employed: entanglement. ...
Satisfiability Testing (SAT) techniques are well-established in classical computing where they are used to solve a broad variety of problems, e.g., in the design of classical circuits and systems. Analogous to the classical realm, quantum algorithms are usually modelled as circuits and similar design tasks need to be tackled. Thus, it is natural to pose the question whether these design tasks in the quantum realm can also be approached using SAT techniques. To the best of our knowledge, no SAT formulation for arbitrary quantum circuits exists and it is unknown whether such an approach is feasible at all. In this work, we define a propositional SAT encoding that, in principle, can be applied to arbitrary quantum circuits. However, we show that due to the inherent complexity of representing quantum states, constructing such an encoding is not feasible in general. Therefore, we establish general criteria for determining the feasibility of the proposed encoding and identify classes of quantum circuits fulfilling these criteria. We explicitly demonstrate how the proposed encoding can be applied to the class of Clifford circuits as a representative. Finally, we empirically demonstrate the applicability and efficiency of the proposed encoding for Clifford circuits. With these results, we lay the foundation for continuing the ongoing success of SAT in classical circuit and systems design for quantum circuits.
... As a result, reducing the number of quantum gates through the optimization proposed in this article would directly influence the mapping of quantum circuits. This mapping is known to be NP-Hard [6,38]. ...
This paper presents novel methods for optimizing multi-controlled quantum gates, which naturally arise in high-level quantum programming. Our primary approach involves rewriting $U(2)$ gates as $SU(2)$ gates, utilizing one auxiliary qubit for phase correction. This reduces the number of CNOT gates required to decompose any multi-controlled quantum gate from $O(n^2)$ to at most $32n$. Additionally, we can reduce the number of CNOTs for multi-controlled Pauli gates from $16n$ to $12n$ and propose an optimization to reduce the number of controlled gates in high-level quantum programming. We have implemented these optimizations in the Ket quantum programming platform and demonstrated significant reductions in the number of gates. For instance, for a Grover's algorithm layer with 114 qubits, we achieved a reduction in the number of CNOTs from 101,245 to 2,684. This reduction in the number of gates significantly impacts the execution time of quantum algorithms, thereby enhancing the feasibility of executing them on NISQ computers.
... However, since any additional gate increases the error rate and, hence, reduces the accuracy of the computation, it is vital to keep the number of additionally added gates as low as possible. It has been shown that even this small part in the compilation flow is an NP-complete problem [76]. ...
Quantum computers are becoming a reality and numerous quantum computing applications with a near-term perspective (e.g., for finance, chemistry, machine learning, and optimization) and with a long-term perspective (e.g., for cryptography or unstructured search) are currently being investigated. However, designing and realizing potential applications for these devices in a scalable fashion requires automated, efficient, and user-friendly software tools that cater to the needs of end users, engineers, and physicists at every level of the entire quantum software stack. Many of the problems to be tackled in that regard are similar to design problems from the classical realm for which sophisticated design automation tools have been developed in the previous decades. The Munich Quantum Toolkit (MQT) is a collection of software tools for quantum computing developed by the Chair for Design Automation at the Technical University of Munich which explicitly utilizes this design automation expertise. Our overarching objective is to provide solutions for design tasks across the entire quantum software stack. This entails high-level support for end users in realizing their applications, efficient methods for the classical simulation, compilation, and verification of quantum circuits, tools for quantum error correction, support for physical design, and more. These methods are supported by corresponding data structures (such as decision diagrams) and core methods (such as SAT encodings/solvers). All of the developed tools are available as open-source implementations and are hosted on https://github.com/cda-tum.
... Of course, one can also try to maximize algorithm-specific performance, e.g., the probability of success. The problem of optimizing a circuit with respect to a particular objective is generally very difficult [77,22,7], such that there is no efficient algorithm to find the global minimum except for small circuits. Some approaches are based on meet-in-the-middle [8] or satisfiability (SAT) solvers [144]. ...
The rapid advancements in quantum computing necessitate a scientific and rigorous approach to the construction of a corresponding software ecosystem, a topic underexplored and primed for systematic investigation. This chapter takes an important step in this direction: It presents scientific considerations essential for building a quantum software ecosystem that makes quantum computing available for scientific and industrial problem solving. Central to this discourse is the concept of hardware-software co-design, which fosters a bidirectional feedback loop from the application layer at the top of the software stack down to the hardware. This approach begins with compilers and low-level software that are specifically designed to align with the unique specifications and constraints of the quantum processor, proceeds with algorithms developed with a clear understanding of underlying hardware and computational model features, and extends to applications that effectively leverage the capabilities to achieve a quantum advantage. We analyze the ecosystem from two critical perspectives: the conceptual view, focusing on theoretical foundations, and the technical infrastructure, addressing practical implementations around real quantum devices necessary for a functional ecosystem. This approach ensures that the focus is towards promising applications with optimized algorithm-circuit synergy, while ensuring a user-friendly design, an effective data management and an overall orchestration. Our chapter thus offers a guide to the essential concepts and practical strategies necessary for developing a scientifically grounded quantum software ecosystem.
... In the current NISQ era, the primary objective of quantum compilation is to optimize quantum circuits by reducing the circuit depth and the number of two-qubit gates, while considering the topology and features of the underlying hardware structure. Although it has been demonstrated that finding the optimal quantum circuit is typically an NP-hard problem, [41][42][43] in practice, our focus is on searching for sub-optimal quantum circuits within a feasible timeframe. For certain circuits with special structures, such as the QAOA circuit, particular compilation strategies can outperform general-purpose compilers. ...
We introduce Quafu-Qcover, an open-source cloud-based software package developed for solving combinatorial optimization problems using quantum simulators and hardware backends. Quafu-Qcover provides a standardized and comprehensive workflow that utilizes the quantum approximate optimization algorithm (QAOA). It facilitates the automatic conversion of the original problem into a quadratic unconstrained binary optimization (QUBO) model and its corresponding Ising model, which can be subsequently transformed into a weight graph. The core of Qcover relies on a graph decomposition-based classical algorithm, which efficiently derives the optimal parameters for the shallow QAOA circuit. Quafu-Qcover incorporates a dedicated compiler capable of translating QAOA circuits into physical quantum circuits that can be executed on Quafu cloud quantum computers. Compared to a general-purpose compiler, our compiler demonstrates the ability to generate shorter circuit depths, while also exhibiting superior speed performance. Additionally, the Qcover compiler has the capability to dynamically create a library of qubits coupling substructures in real-time, utilizing the most recent calibration data from the superconducting quantum devices. This ensures that computational tasks can be assigned to connected physical qubits with the highest fidelity. The Quafu-Qcover allows us to retrieve quantum computing sampling results using a task ID at any time, enabling asynchronous processing. Moreover, it incorporates modules for results preprocessing and visualization, facilitating an intuitive display of solutions for combinatorial optimization problems. We hope that Quafu-Qcover can serve as an instructive illustration for how to explore application problems on the Quafu cloud quantum computers.
... The minimum mapping problem, aiming to insert the fewest swaps, is NP-Complete [Botea et al. 2018], requiring heuristic or approximate algorithms for nearoptimal solutions. Researchers explored reinforcement learning in [Pozzi et al. 2022]. ...
The rapid development of Quantum Computing (QC) as a promising computing paradigm has garnered significant attention for its ability to harness quantum mechanical properties for computation. With classical computing facing limitations outlined by Moore's Law, QC emerges as a potential alternative for tackling complex computational problems. Yet, to unlock its full potential, robust and optimized compilers are pivotal, especially in addressing challenges posed by circuits with numerous qubits. In this systematic literature review, we analyze 18 articles to identify proposed optimizations for quantum compilers , exploring their applications, performance impacts, and emerging trends. Our primary goal is to offer valuable insights into the recent advancements and challenges in QC compiler optimizations. This will be achieved through the clustering of optimization groups, ultimately facilitating further progress in the development of highly optimized quantum algorithms and circuits.
... In the present NISQ era, the primary objective of quantum compilation is to optimize quantum circuits by reducing the circuit depth and the number of two-qubit gates, while considering the topology and characteristics of the underlying hardware structure. Although it has been shown that finding the optimal quantum circuit is usually an NP-hard problem [41][42][43], in practice, we only need to search for sub-optimal quantum circuits within a reasonable time. For some circuits with special structures, such as the QAOA circuit, specific compiling strategies can outperform general-purpose compilers [44][45][46][47][48]. Inspired by previous works and in consideration of the topology and characteristics of the Quafu quantum processor, we design a specific compiler for Quafu-Qcover that exhibits superior performance compared to general-purpose compilers like Qiskit. ...
We present Quafu-Qcover, an open-source cloud-based software package designed for combinato-rial optimization problems that support both quantum simulators and hardware backends. Quafu-Qcover provides a standardized and complete workflow for solving combinatorial optimization problems using the Quantum Approximate Optimization Algorithm (QAOA). It enables the automatic modeling of the original problem as a quadratic unconstrained binary optimization (QUBO) model and corresponding Ising model, which can be further transformed into a weight graph. The core of Qcover relies on a graph decomposition-based classical algorithm, which enables obtaining the optimal parameters for the shallow QAOA circuit more efficiently. Quafu-Qcover includes a specialized compiler that translates QAOA circuits into physical quantum circuits capable of execution on Quafu cloud quantum computers. Compared to a general-purpose compiler, ours generates shorter circuit depths while also possessing better speed performance. The Qcover compiler can establish a library of qubits coupling substructures in real time based on the updated calibration data of the superconducting quantum devices, ensuring that the task is executed on physical qubits with higher fidelity. The Quafu-Qcover allows us to retrieve quantum computer sampling result information at any time using task ID, enabling asynchronous processing. Besides, it includes modules for result preprocessing and visualization, allowing for an intuitive display of the solution to combinatorial optimization problems. We hope that Quafu-Qcover can serve as a guiding example for how to explore application problems on the Quafu cloud quantum computers.
... In the present NISQ era, the primary objective of quantum compilation is to optimize quantum circuits by reducing the circuit depth and the number of two-qubit gates, while considering the topology and characteristics of the underlying hardware structure. Although it has been shown that finding the optimal quantum circuit is usually an NP-hard problem [41][42][43], in practice, we only need to search for sub-optimal quantum circuits within a reasonable time. For some circuits with special structures, such as the QAOA circuit, specific compiling strategies can outperform general-purpose compilers [44][45][46][47][48]. Inspired by previous works and in consideration of the topology and characteristics of the Quafu quantum processor, we design a specific compiler for Quafu-Qcover that exhibits superior performance compared to general-purpose compilers like Qiskit. ...
We present Quafu-Qcover, an open-source cloud-based software package designed for combinatorial optimization problems that support both quantum simulators and hardware backends. Quafu-Qcover provides a standardized and complete workflow for solving combinatorial optimization problems using the Quantum Approximate Optimization Algorithm (QAOA). It enables the automatic modeling of the original problem as a quadratic unconstrained binary optimization (QUBO) model and corresponding Ising model, which can be further transformed into a weight graph. The core of Qcover relies on a graph decomposition-based classical algorithm, which enables obtaining the optimal parameters for the shallow QAOA circuit more efficiently. Quafu-Qcover includes a specialized compiler that translates QAOA circuits into physical quantum circuits capable of execution on Quafu cloud quantum computers. Compared to a general-purpose compiler, ours generates shorter circuit depths while also possessing better speed performance. The Qcover compiler can establish a library of qubits coupling substructures in real time based on the updated calibration data of the superconducting quantum devices, ensuring that the task is executed on physical qubits with higher fidelity. The Quafu-Qcover allows us to retrieve quantum computer sampling result information at any time using task ID, enabling asynchronous processing. Besides, it includes modules for result preprocessing and visualization, allowing for an intuitive display of the solution to combinatorial optimization problems. We hope that Quafu-Qcover can serve as a guiding example for how to explore application problems on the Quafu cloud quantum computers
... Its extension to restricted topologies is also difficult, often leading to a large multiplicative overhead [40]. Note that being able to find truly optimal decompositions of arbitrary unitaries would amount to determining their gate complexities, which is an NP-complete problem [41]. It is therefore natural to use numeric optimization and heuristic methods in the search for efficient decompositions. ...
We consider the problem of the variational quantum circuit synthesis into a gate set consisting of the CNOT gate and arbitrary single-qubit (1q) gates with the primary target being the minimization of the CNOT count. First we note that along with the discrete architecture search suffering from the combinatorial explosion of complexity, optimization over 1q gates can also be a crucial roadblock due to the omnipresence of local minimums (well known in the context of variational quantum algorithms but apparently underappreciated in the context of the variational compiling). Taking the issue seriously, we make an extensive search over the initial conditions an essential part of our approach. Another key idea we propose is to use parametrized two-qubit (2q) controlled phase gates, which can interpolate between the identity gate and the CNOT gate, and allow a continuous relaxation of the discrete architecture search, which can be executed jointly with the optimization over 1q gates. This coherent optimization of the architecture together with 1q gates appears to work surprisingly well in practice, sometimes even outperforming optimization over 1q gates alone (for fixed optimal architectures). As illustrative examples and applications we derive 8 CNOT and T depth 3 decomposition of the 3q Toffoli gate on the nearest-neighbor topology, rediscover known best decompositions of the 4q Toffoli gate on all 4q topologies including a 1 CNOT gate improvement on the star-shaped topology, and propose decomposition of the 5q Toffoli gate on the nearest-neighbor topology with 48 CNOT gates. We also benchmark the performance of our approach on a number of 5q quantum circuits from the ibm_qx_mapping database showing that it is highly competitive with the existing software. The algorithm developed in this work is available as a Python package CPFlow.
... Keeping the number of additionally introduced gates as small as possible during quantum circuit mapping is key for ensuring the successful execution of the quantum circuit. Determining an optimal mapping for a quantum circuit is an NP-hard problem [24]. ...
... Based on the general setting introduced above, the compilation of entangling operations into native gate sets is of central importance for efficient QIP. Since the general problem is NP-hard already for qubits [28], quantum computers critically depend on efficient, if not optimal, compilation methods. Given the more complicated entanglement structure of qudits, compared to their binary counterpart, the compilation problem becomes much more challenging in high-dimensional spaces. ...
Most quantum computing architectures to date natively supportmulti-malued logic, albeit being typically operated in a binary fash-ion. Multi-valued, or qudit, quantum processors have access tomuch richer forms of quantum entanglement, which promise tosignificantly boost the performance and usefulness of quantum de-vices. However, much of the theory as well as corresponding designmethods required for exploiting such hardware remain insufficientand generalizations from qubits are not straightforward. A partic-ular challenge is the compilation of quantum circuits into sets ofnative qudit gates supported by state-of-the-art quantum hardware.In this work, we address this challenge by introducing a completeworkflow for compiling any two-qudit unitary into an arbitrarynative gate set. Case studies demonstrate the feasibility of both, theproposed approach as well as the corresponding implementation(which is freely available at github.com/cda-tum/qudit-entanglement-compilation).
... Based on the general setting introduced above, the compilation of entangling operations into native gate sets is of central importance for efficient QIP. Since the general problem is NP-hard already for qubits [28], quantum computers critically depend on efficient, if not optimal, compilation methods. Given the more complicated entanglement structure of qudits, compared to their binary counterpart, the compilation problem becomes much more challenging in high-dimensional spaces. ...
Most quantum computing architectures to date natively support multi-valued logic, albeit being typically operated in a binary fashion. Multi-valued, or qudit, quantum processors have access to much richer forms of quantum entanglement, which promise to significantly boost the performance and usefulness of quantum devices. However, much of the theory as well as corresponding design methods required for exploiting such hardware remain insufficient and generalizations from qubits are not straightforward. A particular challenge is the compilation of quantum circuits into sets of native qudit gates supported by state-of-the-art quantum hardware. In this work, we address this challenge by introducing a complete workflow for compiling any two-qudit unitary into an arbitrary native gate set. Case studies demonstrate the feasibility of both, the proposed approach as well as the corresponding implementation (which is freely available at https://github.com/cda-tum/qudit-entanglement-compilation).
... In addition to lower costs, it offers detailed insights on the quantum state during the execution of a quantum circuit that is physically unavailable when running the circuit on an actual quantum computer [8]- [13]. • Compilation: Similar to classical circuits and systems, quantum circuits are initially described at a rather high abstraction level and need to be compiled to a representation that adheres to all the constraints imposed by the target device (e.g., limited gate-set and/or limited connectivity) [14]- [18]. • Verification: Since compilation significantly changes the structure of quantum circuits, it is crucial to ensure that the resulting circuits still realize the originally intended functionality. ...
Quantum computers promise to efficiently solve important problems classical computers never will. However, in order to capitalize on these prospects, a fully automated quantum software stack needs to be developed. This involves a multitude of complex tasks from the classical simulation of quantum circuits, over their compilation to specific devices, to the verification of the circuits to be executed as well as the obtained results. All of these tasks are highly non-trivial and necessitate efficient data structures to tackle the inherent complexity. Starting from rather straight-forward arrays over decision diagrams (inspired by the design automation community) to tensor networks and the ZX-calculus, various complementary approaches have been proposed. This work provides a look "under the hood" of today's tools and showcases how these means are utilized in them, e.g., for simulation, compilation, and verification of quantum circuits.
... • Quantum Circuit Compilation (QCC) [29][30][31]. QCC is a key technique to transform the nonconforming quantum circuit to an executable circuit on the target quantum platform according to its constraints, including the native gateset, connectivity and so on. ...
Quantum computing is a game-changing technology for global academia, research centers and industries including computational science, mathematics, finance, pharmaceutical, materials science, chemistry and cryptography. Although it has seen a major boost in the last decade, we are still a long way from reaching the maturity of a full-fledged quantum computer. That said, we will be in the Noisy-Intermediate Scale Quantum (NISQ) era for a long time, working on dozens or even thousands of qubits quantum computing systems. An outstanding challenge, then, is to come up with an application that can reliably carry out a nontrivial task of interest on the near-term quantum devices with non-negligible quantum noise. To address this challenge, several near-term quantum computing techniques, including variational quantum algorithms, error mitigation, quantum circuit compilation and benchmarking protocols, have been proposed to characterize and mitigate errors, and to implement algorithms with a certain resistance to noise, so as to enhance the capabilities of near-term quantum devices and explore the boundaries of their ability to realize useful applications. Besides, the development of near-term quantum devices is inseparable from the efficient classical simulation, which plays a vital role in quantum algorithm design and verification, error-tolerant verification and other applications. This review will provide a thorough introduction of these near-term quantum computing techniques, report on their progress, and finally discuss the future prospect of these techniques, which we hope will motivate researchers to undertake additional studies in this field.
... The QCM problem is, given a quantum circuit, a quantum architecture (denoted by a coupling graph in the paper), and a number k, asking whether k SWAP gates is enough to transform the quantum circuit into an equivalent one that is compliant with the quantum architecture's constraint. Deciding the minimal number of swap actions is an NP-complete problem [2,22], which implies that it is unlikely to find a polynomial time algorithm. There are many algorithms for QCM based on heuristic and approximation methods [5,14,25,26,29], which provide efficient solutions by taking advantage of the inner structural features of the quantum circuits and quantum devices. ...
A quantum circuit must be preprocessed before implementing on NISQ devices due to the connectivity constraint. Quantum circuit mapping (QCM) transforms the circuit into an equivalent one that is compliant with the NISQ device’s architecture constraint by adding SWAP gates. The QCM problem asks the minimal number of auxiliary SWAP gates, and is NP-complete. The complexity of QCM with fixed parameters is studied in the paper. We give an exact algorithm for QCM, and show that the algorithm runs in polynomial time if the NISQ device’s architecture is fixed. If the number of qubits of the quantum circuit is fixed, we show that the QCM problem is NL-complete by a reduction from the undirected shortest path problem. Moreover, the fixed-parameter complexity of QCM is W[1]-hard when parameterized by the number of qubits of the quantum circuit. We prove the result by a reduction from the clique problem. If taking the depth of the quantum circuits and the coupling graphs as parameters, we show that the QCM problem is still NP-complete over shallow quantum circuits, and planar, bipartite and degree bounded coupling graphs.
... After quantum circuits started being analyzed as reversible circuits formed from Toffoli gates, many exact methods and heuristics were proposed -a not so recent but complete review is provided by Saeedi and Markov [30]. The complexity class of QCL was discussed first in [19], which has been used as a foundation for proving the complexity of different QCL variations, such as in [7,32,34]. ...
The quantum circuit layout (QCL) problem is to map a quantum circuit such that the constraints of the device are satisfied. We introduce a quantum circuit mapping heuristic, QXX, and its machine learning version, QXX-MLP. The latter infers automatically the optimal QXX parameter values such that the layed out circuit has a reduced depth. In order to speed up circuit compilation, before laying the circuits out, we are using a Gaussian function to estimate the depth of the compiled circuits. This Gaussian also informs the compiler about the circuit region that influences most the resulting circuit’s depth. We present empiric evidence for the feasibility of learning the layout method using approximation. QXX and QXX-MLP open the path to feasible large scale QCL methods.
... In addition to lower costs, it offers detailed insights on the quantum state during the execution of a quantum circuit that is physically unavailable when running the circuit on an actual quantum computer [8]- [13]. • Compilation: Similar to classical circuits and systems, quantum circuits are initially described at a rather high abstraction level and need to be compiled to a representation that adheres to all the constraints imposed by the target device (e.g., limited gate-set and/or limited connectivity) [14]- [18]. • Verification: Since compilation significantly changes the structure of quantum circuits, it is crucial to ensure that the resulting circuits still realize the originally intended functionality. ...
Quantum computers promise to efficiently solve important problems classical computers never will. However, in order to capitalize on these prospects, a fully automated quantum software stack needs to be developed. This involves a multitude of complex tasks from the classical simulation of quantum circuits, over their compilation to specific devices, to the verification of the circuits to be executed as well as the obtained results. All of these tasks are highly non-trivial and necessitate efficient data structures to tackle the inherent complexity. Starting from rather straightforward arrays over decision diagrams (inspired by the design automation community) to tensor networks and the ZX-calculus, various complementary approaches have been proposed. This work provides a look “under the hood” of today’s tools and showcases how these means are utilized in them, e.g., for simulation, compilation, and verification of quantum circuits.
... While many (heuristic) techniques have been proposed in the past that allow to determine suitable mappings, e.g., [11]- [19], determining truly optimal solutions (with as little overhead as possible) revealed to be a challenging problem. In fact, the mapping problem has been shown to be NPcomplete [21], [27]. ...
... While many (heuristic) techniques have been proposed in the past that allow to determine suitable mappings, e.g., [11]- [19], determining truly optimal solutions (with as little overhead as possible) revealed to be a challenging problem. In fact, the mapping problem has been shown to be NPcomplete [21], [27]. ...
Executing quantum circuits on currently available quantum computers requires compiling them to a representation that conforms to all restrictions imposed by the targeted architecture. Due to the limited connectivity of the devices' physical qubits, an important step in the compilation process is to map the circuit in such a way that all its gates are executable on the hardware. Existing solutions delivering optimal solutions to this task are severely challenged by the exponential complexity of the problem. In this paper, we show that the search space of the mapping problem can be limited drastically while still preserving optimality. The proposed strategies are generic, architecture-independent, and can be adapted to various mapping methodologies. The findings are backed by both, theoretical considerations and experimental evaluations. Results confirm that, by limiting the search space, optimal solutions can be determined for instances that timeouted before or speed-ups of up to three orders of magnitude can be achieved.
... Thus, the compiler has to "move" the required qubits together on the coupling graph, usually via SWAP gates. Although various settings of the mapping and SWAP insertion problem have been proved NP-hard [12], [14]- [16], given the limited QC resource in the NISQ era, we strive for optimal mapping solutions, as the high error rates and short coherence limit the circuit size and depth. ...
Before quantum error correction (QEC) is achieved, quantum computers focus on noisy intermediate-scale quantum (NISQ) applications. Compared to the well-known quantum algorithms requiring QEC, like Shor's or Grover's algorithm, NISQ applications have different structures and properties to exploit in compilation. A key step in compilation is mapping the qubits in the program to physical qubits on a given quantum computer, which has been shown to be an NP-hard problem. In this paper, we present OLSQ-GA, an optimal qubit mapper with a key feature of simultaneous SWAP gate absorption during qubit mapping, which we show to be a very effective optimization technique for NISQ applications. For the class of quantum approximate optimization algorithm (QAOA), an important NISQ application, OLSQ-GA reduces depth by up to 50.0% and SWAP count by 100% compared to other state-of-the-art methods, which translates to 55.9% fidelity improvement. The solution optimality of OLSQ-GA is achieved by the exact SMT formulation. For better scalability, we augment our approach with additional constraints in the form of initial mapping or alternating matching, which speeds up OLSQ-GA by up to 272X with no or little loss of optimality.
... A good quantum compiler must translate an input program into the most efficient equivalent of itself [13], getting the most out of the available hardware. In general, the quantum compilation problem is NP-Hard [14,15]. On noisy devices, quantum compilation encompasses the following tasks: gate synthesis [16], which is the decomposition of an arbitrary unitary operation into a quantum circuit made of single-qubit and two-qubit gates from a universal gate set; compliance with the hardware architecture, starting from an initial mapping of the virtual qubits to the physical ones, and moving through subsequent mappings by means of a clever swapping strategy; and noise awareness. ...
Current quantum processors are noisy and have limited coherence and imperfect gate implementations. On such hardware, only algorithms that are shorter than the overall coherence time can be implemented and executed successfully. A good quantum compiler must translate an input program into the most efficient equivalent of itself, getting the most out of the available hardware. In this work, we present novel deterministic algorithms for compiling recurrent quantum circuit patterns in polynomial time. In particular, such patterns appear in quantum circuits that are used to compute the ground-state properties of molecular systems using the variational quantum eigensolver method together with the RyRz heuristic wavefunction Ansätz. We show that our pattern-oriented compiling algorithms, combined with an efficient swapping strategy, produces—in general—output programs that are comparable to those obtained with state-of-the-art compilers, in terms of CNOT count and CNOT depth. In particular, our solution produces unmatched results on RyRz circuits.
... In particular, f ð cÞ ¼ fðq C 0 ; q 1 Þ; ðq C 1 ; q 0 Þ; ðq C 2 ; q 5 Þ; ðq C 3 ; q 6 Þ; ðq C 4 ; q 7 Þg. The circuit mapping problem has been proved to be NPcomplete in [53], and, as a consequence, the computation of its exact solution could be not suitable to deal with future generation of quantum processors characterized by thousands or millions of qubits 10 . Thus, there is a strong need of approximate algorithms capable of efficiently computing sub-optimal solutions for this problem and pave the way towards the next generation of compilers for quantum computers. ...
Quantum computers have become reality thanks to the effort of some majors in developing innovative technologies that enable the usage of quantum effects in computation, so as to pave the way towards the design of efficient quantum algorithms to use in different applications domains, from finance and chemistry to artificial and computational intelligence. However, there are still some technological limitations that do not allow a correct design of quantum algorithms, compromising the achievement of the so-called quantum advantage. Specifically, a major limitation in the design of a quantum algorithm is related to its proper mapping to a specific quantum processor so that the underlying physical constraints are satisfied. This hard problem, known as circuit mapping, is a critical task to face in quantum world, and it needs to be efficiently addressed to allow quantum computers to work correctly and productively. In order to bridge above gap, this paper introduces a very first circuit mapping approach based on deep neural networks, which opens a completely new scenario in which the correct execution of quantum algorithms is supported by classical machine learning techniques. As shown in experimental section, the proposed approach speeds up current state-of-the-art mapping algorithms when used on 5-qubits IBM Q processors, maintaining suitable mapping accuracy.
Dynamically field-programmable qubit arrays (DPQA) have recently emerged as a promising platform for quantum information processing. In DPQA, atomic qubits are selectively loaded into arrays of optical traps that can be reconfigured during the computation itself. Leveraging qubit transport and parallel, entangling quantum operations, different pairs of qubits, even those initially far away, can be entangled at different stages of the quantum program execution. Such reconfigurability and non-local connectivity present new challenges for compilation, especially in the layout synthesis step which places and routes the qubits and schedules the gates. In this paper, we consider a DPQA architecture that contains multiple arrays and supports 2D array movements, representing cutting-edge experimental platforms. Within this architecture, we discretize the state space and formulate layout synthesis as a satisfiability modulo theories problem, which can be solved by existing solvers optimally in terms of circuit depth. For a set of benchmark circuits generated by random graphs with complex connectivities, our compiler OLSQ-DPQA reduces the number of two-qubit entangling gates on small problem instances by 1.7x compared to optimal compilation results on a fixed planar architecture. To further improve scalability and practicality of the method, we introduce a greedy heuristic inspired by the iterative peeling approach in classical integrated circuit routing. Using a hybrid approach that combined the greedy and optimal methods, we demonstrate that our DPQA-based compiled circuits feature reduced scaling overhead compared to a grid fixed architecture, resulting in 5.1X less two-qubit gates for 90 qubit quantum circuits. These methods enable programmable, complex quantum circuits with neutral atom quantum computers, as well as informing both future compilers and future hardware choices.
Kuantum hesaplama, geleneksel bilgisayarların yapamayacağı kadar karmaşık hesaplamaları çok daha hızlı ve daha verimli gerçekleştirmeye olanak tanıyan bir teknolojidir. Ancak kuantum bilgisayarların çalıştırılması için özel olarak tasarlanmış kuantum algoritmalara ihtiyaç duyulmaktadır. Bu algoritmaların kuantum bilgisayarlarda verimli bir şekilde çalıştırabilmek için uygun derleyici ve kuantum bilgisayar seçimi kritik öneme sahiptir. Bu çalışmada kauntum programlama ve derleyicileri hakkında bilgiler verilerek, literatürdeki kuantum derleyicilerin karşılaştırmaları gerçekleştirilmiştir. Örnek bir soyut kuantum devre 5 kübtlik ibmq_belem, ibmq_quito ve ibmq_manila kuantum bilgisayarlarında çalıştırılarak, kuantum devrelerin çalışma mantığı uygulamalı olarak açıklanmıştır. Yapılan analizlerler sonucu L tipi kübit bağlantısına sahip ibmq_manila bilgisayarının ortalama %86 ile daha başarılı sonuçlar ürettiği gözlemlenmiştir. Diğer taraftan T tipi kübit bağlantılarına sahip ibmq_quito ve ibmq_belem bilgisayarlarının ürettikleri sonuçların başarısı ortaalama %82 ve %48 ile sınırlı kalmaktadır. Aynı kübit bağlantısına sahip bu bilgisayarların başarımları arasındaki gözle görülür farkın sebebi kübit ve bağlantılardaki hata oranlarının olduğu sonucuna varılmıştır.
The qubit routing problem, also known as the swap minimization problem, is a (classical) combinatorial optimization problem that arises in the design of compilers of quantum programs. We study the qubit routing problem from the viewpoint of theoretical computer science, while most of the existing studies investigated the practical aspects. We concentrate on the linear nearest neighbor (LNN) architectures of quantum computers, in which the graph topology is a path. Our results are three-fold. (1) We prove that the qubit routing problem is NP-hard. (2) We give a fixed-parameter algorithm when the number of two-qubit gates is a parameter. (3) We give a polynomial-time algorithm when each qubit is involved in at most one two-qubit gate.KeywordsQubit routingQubit allocationFixed-parameter tractability
The capabilities of quantum computers, such as the number of supported qubits and maximum circuit depth, have grown exponentially in recent years. Commercially relevant applications that take advantage of quantum computing are expected to be available soon. In this paper, we shed light on the possibilities of accelerating database tasks using quantum computing with examples of optimizing queries and transaction schedules and present some open challenges for future studies in the field.
Quantum circuit mapping is an essential process required by executing quantum circuits using a noisy intermediate-scale quantum (NISQ) device. Since qubits and quantum gates of a NISQ device are error-prone and variable in quality, it is crucial to choose qubits or quantum gates in a variation-aware manner to maximize the success rate of executing circuits. To this end, this paper proposes a variation-aware method for quantum circuit mapping through the cooperation of multiple agents. Each agent in the proposed method can gradually construct a physical circuit that respects the device's connectivity constraints by inserting a SWAP gate at each step. Moreover, at each step, the circuit information of each agent is shared within the agent population through a communication mechanism that combines global and local information exchange, so that agents with poor fitness can get an opportunity to improve their physical circuits. The experimental results on extensive benchmark circuits confirm that the proposed method can effectively and consistently improve the overall circuit fidelity compared with the state-of-the-art methods.
We propose a method for performing software pipelining on quantum for-loop programs to exploit parallelism in and across iterations. We redefined concepts useful in program optimization, including array aliasing, instruction dependency, and resource conflict required in optimizing quantum programs. Using these concepts, we present a software pipelining framework exploiting instruction-level parallelism in quantum loop programs. This method is further enhanced with several improvements to reduce total gate count and program depth. The optimization method is then evaluated on some popular quantum algorithms like Grover and QAOA, and compared under different configurations and with several baseline compilers. The evaluation results show that our approach can schedule loop programs with depth close to the depth of the entire loop unrolling while generating smaller code sizes and consuming much less time. This is the first step towards optimization of a quantum program with such loop control flow, as far as we know.
The most challenging stage in compilation for near-term quantum computing is layout synthesis, also called qubit mapping, where qubits in quantum programs are mapped to physical qubits. In order to understand the quality of existing solutions, we apply the measure-improve methodology, which has been successful in classical circuit placement, to this problem. We construct quantum mapping examples with known optimal, QUEKO, to measure the optimality gaps of leading heuristic compilers. On the revelation of large gaps, we set out to close them with optimal layout synthesis for quantum computing, OLSQ, a more efficient formulation of the layout synthesis problem into mathematical programming. We accelerate OLSQ with the transition mode and expand its solution space with domain-specific knowledge on applications like quantum approximate optimization algorithm, QAOA.KeywordsQubit mappingQubit allocationQubit placementQuantum computingQuantum schedulingSWAP minimization
The Quantum Approximate Optimization Algorithm (QAOA) is an algorithmic framework for finding approximate solutions to combinatorial optimization problems, derived from an approximation to the Quantum Adiabatic Algorithm (QAA). In solving combinatorial optimization problems with constraints in the context of QAOA, one needs to find a way to encode problem constraints into the scheme. In this paper, we propose and discuss several QAOA-based algorithms to solve combinatorial optimization problems with equality and/or inequality constraints. We formalize the encoding method of different types of constraints, and demonstrate the effectiveness and efficiency of the proposed scheme by providing examples and results for some well-known NP optimization problems. Compared to previous constraint-encoding methods, we argue our work leads to a more generalized framework for finding, in the context of QAOA, higher-quality approximate solutions to combinatorial problems with various types of constraints.
Variational quantum algorithms have been introduced as a promising class of quantum-classical hybrid algorithms that can already be used with the noisy quantum computing hardware available today by employing parameterized quantum circuits. Considering the non-trivial nature of quantum circuit compilation and the subtleties of quantum computing, it is essential to verify that these parameterized circuits have been compiled correctly. Established equivalence checking procedures that handle parameter-free circuits already exist. However, no methodology capable of handling circuits with parameters has been proposed yet. This work fills this gap by showing that verifying the equivalence of parameterized circuits can be achieved in a purely symbolic fashion using an equivalence checking approach based on the ZX-calculus. At the same time, proofs of inequality can be efficiently obtained with conventional methods by taking advantage of the degrees of freedom inherent to parameterized circuits. We implemented the corresponding methods and proved that the resulting methodology is complete. Experimental evaluations (using the entire parametric ansatz circuit library provided by Qiskit as benchmarks) demonstrate the efficacy of the proposed approach. The implementation is open source and publicly available as part of the equivalence checking tool QCEC (https://github.com/cda-tum/qcec) which is part of the Munich Quantum Toolkit (MQT).
Dynamic simulation of materials is a promising application for near-term quantum computers. Current algorithms for Hamiltonian simulation, however, produce circuits that grow in depth with increasing simulation time, limiting feasible simulations to short-time dynamics. Here, we present a method for generating circuits that are constant in depth with increasing simulation time for a specific subset of one-dimensional (1D) materials Hamiltonians, thereby enabling simulations out to arbitrarily long times. Furthermore, by removing the effective limit on the number of feasibly simulatable time-steps, the constant-depth circuits enable Trotter error to be made negligibly small by allowing simulations to be broken into arbitrarily many time-steps. For an N -spin system, the constant-depth circuit contains only $\mathcal {O}(N^{2})$ O ( N 2 ) CNOT gates. Such compact circuits enable us to successfully execute long-time dynamic simulation of ubiquitous models, such as the transverse field Ising and XY models, on current quantum hardware for systems of up to 5 qubits without the need for complex error mitigation techniques. Aside from enabling long-time dynamic simulations with minimal Trotter error for a specific subset of 1D Hamiltonians, our constant-depth circuits can advance materials simulations on quantum computers more broadly in a number of indirect ways.
In order to make the most of the increasing computational power of recently developed quantum computers, it is crucial to perform an efficient mapping of a given quantum circuit that realizes the desired quantum algorithm to the targeted quantum computer (so-called technology mapping). In most cases, the limitations of the targeted quantum hardware have not been taken into account when generating these quantum circuits in the first place. Thus, the technology mapping is likely to induce a considerable overhead for such circuits in order prepare them for the execution on the actual device. In this work, we consider the realization of reversible circuits consisting of multiple-controlled Toffoli gates on IBM quantum computers. Using templates for the realization of the reversible/quantum gates allows to perform a topology-aware decomposition of MCT gates that exhibits the potential of significant reductions of the technology mapping overhead.
The Quantum Circuit Compilation Problem (QCCP) is challenging to the Artificial Intelligence community. It was already tackled with temporal planning, constraint programming, greedy heuristics and other techniques. In this paper, QCCP is formulated as a scheduling problem and solved by a genetic algorithm. We focus on QCCP for Quantum Approximation Optimization Algorithms (QAOA) applied to the MaxCut problem and consider Noisy Intermediate Scale Quantum (NISQ) hardware architectures. Based on the fact that these algorithms apply a set of basic quantum operations repeatedly over a number of rounds, we propose a genetic algorithm approach, termed Decomposition Based Genetic Algorithm (DBGA), that in each round extends the partial solutions obtained for the previous ones. DBGA is compared to the state of the art across a set of conventional instances. The results of the experimental study provided interesting insight in the problem structure and showed that DBGA is quite competitive with the state of the art. In particular, DBGA outperformed the best current method on the largest instances and provided new best solutions to most of them.
Before quantum error correction (QEC) is achieved, quantum computers focus on noisy intermediate-scale quantum (NISQ) applications.
Compared to the well-known quantum algorithms requiring QEC, like Shor’s or Grover’s algorithm, NISQ applications have different structures and properties to exploit in compilation.
A key step in compilation is mapping the qubits in the program to physical qubits on a given quantum computer, which has been shown to be an NP-hard problem.
In this paper, we present OLSQ-GA, an optimal qubit mapper with a key feature of simultaneous SWAP gate absorption during qubit mapping, which we show to be a very effective optimization technique for NISQ applications.
For the class of quantum approximate optimization algorithm (QAOA), an important NISQ application, OLSQ-GA reduces depth by up to 50.0% and SWAP count by 100% compared to other state-of-the-art methods, which translates to 55.9% fidelity improvement.
The solution optimality of OLSQ-GA is achieved by the exact SMT formulation.
For better scalability, we augment our approach with additional constraints in the form of initial mapping or alternating matching, which speeds up OLSQ-GA by up to 272X with no or little loss of optimality.
The qubit mapping approach serves to transform a quantum logical circuit (LC) into a physical one that satisfies the connectivity constraints imposed by the noisy intermediate-scale quantum (NISQ) devices. The quality of the physical circuit generated by a mapping approach depends largely on the initial mapping, which specifies the correspondence between the qubits in the LC and the qubits on the NISQ device. There are a total of
$n!$
different initial mappings for a qubit mapping problem with
$n$
qubits, and among them, there is at least one initial mapping corresponding to the smallest physical circuit that this mapping approach can output. Finding such an initial mapping is very important for reliable computations on the NISQ device. To this end, we propose an iterated local search framework as well as a heuristic circuit mapper. In this framework, we perform multiple local searches on the space of initial mappings, and during each local search, several promising neighborhoods of the current initial mapping are generated and evaluated by invoking the circuit mapper in a forward or a backward manner. This framework provides a way for the qubit mapping approach to find the best physical circuit that it can produce, allowing it to trade time for circuit quality, which is necessary in the NISQ era. The experimental results demonstrate the stability, scalability, and effectiveness of this approach in reducing the number of additional gates. Moreover, although this approach is a multipass circuit mapping process, it can generate a good-quality physical circuit within half an hour, even for the circuit with more than 10 000 gates.
Quantum computer architectures place restrictions on the availability of quantum gates. While single-qubit gates are usually available on every qubit, multi-qubit gates like the CNOT gate can only be applied to a subset of all pairs of qubits. Thus, a given quantum circuit usually needs to be transformed prior to its execution in order to satisfy these restrictions. Existing transformation approaches mainly focus on using SWAP gates to enable the realization of CNOT gates that are not natively available in the architecture. As the SWAP gate is a composition of CNOT and single-qubit Hadamard gates, such methods may not yield a minimal solution. In this work, we propose a method to find an optimal implementation of non-native CNOTs, i.e. using the minimal number of native CNOT and Hadamard gates, by using a formulation as a Boolean Satisfiability (SAT) problem. While straightforward representations of quantum states, gates and circuits require an exponential number of complex-valued variables, the approach makes use of a dedicated representation that requires only a quadratic number of variables, all of which are Boolean. As confirmed by experimental results, the resulting problem formulation scales considerably well—despite the exponential complexity of the SAT problem—and enables us to determine significantly improved realizations of non-native CNOT gates for the 16-qubit IBM QX5 architecture.
The world of computing is going to shift towards new paradigms able to provide better performance in solving hard problems than classical computation. In this scenario, quantum computing is assuming a key role thanks to the recent technological enhancements achieved by several big companies in developing computational devices based on fundamental principles of quantum mechanics: superposition, entanglement and interference. These computers will be able to yield performance never seen before in several application domains, and the area of artificial intelligence may be the one most affected by this revolution. Indeed, on the one hand, the intrinsic parallelism provided by quantum computers could support the design of efficient algorithms for artificial intelligence such as, for example, the training algorithms of machine learning models, and bio-inspired optimization algorithms; on the other hand, artificial intelligence techniques could be used to reduce the effect of quantum decoherence in quantum computing, and make this type of computation more reliable. This position paper aims at introducing the readers with this new research area and pave the way towards the design of innovative computing infrastructure where both quantum computing and artificial intelligence take a key role in overcoming the performance of conventional approaches.
Recently, the makespan-minimization problem of compiling a general class of quantum algorithms into near-term quantum processors has been introduced to the AI community. The research demonstrated that temporal planning is a strong approach for a class of quantum circuit compilation (QCC) problems. In this paper, we explore the use of constraint programming (CP) as an alternative and complementary approach to temporal planning. We extend previous work by introducing two new problem variations that incorporate important characteristics identified by the quantum computing community. We apply temporal planning and CP to the baseline and extended QCC problems as both stand-alone and hybrid approaches. Our hybrid methods use solutions found by temporal planning to warm start CP, leveraging the ability of the former to find satisficing solutions to problems with a high degree of task optionality, an area that CP typically struggles with. The CP model, benefiting from inferred bounds on planning horizon length and task counts provided by the warm start, is then used to find higher quality solutions. Our empirical evaluation indicates that while stand-alone CP is only competitive for the smallest problems, CP in our hybridization with temporal planning out-performs stand-alone temporal planning in the majority of problem classes.
An optimization variant of a problem of path planning for multiple robots is addressed in this work. The task is to find spatial-temporal path for each robot of a group of robots such that each robot can reach its destination by navigating through these paths. In the optimization variant of the problem, there is an additional requirement that the makespan of the solution must be as small as possible. A proof of the claim that optimal path planning for multiple robots is WP-complete is sketched in this short paper. Copyright © 2010, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.
We investigate the application of temporal planners to the problem of compiling quantum circuits to emerging quantum hardware. While our approach is general, we focus our initial experiments on Quantum Approximate Optimization Algorithm (QAOA) circuits that have few ordering constraints and thus allow highly parallel plans. We report on experiments using several temporal planners to compile circuits of various sizes to a realistic hardware architecture. This early empirical evaluation suggests that temporal planning is a viable approach to quantum circuit compilation.
We study the problem of optimal multi-robot path planning on graphs (MPP)
over four distinct minimization objectives: the total arrival time, the
makespan (last arrival time), the total distance, and the maximum (single-robot
traveled) distance. On the structure side, we show that each pair of these four
objectives induces a Pareto front and cannot always be optimized
simultaneously. Then, through reductions from 3-SAT, we further establish that
computation over each objective is an NP-hard task, providing evidence that
solving MPP optimally is generally intractable. Nevertheless, in a related
paper, we design complete algorithms and efficient heuristics for optimizing
all four objectives, capable of solving MPP optimally or near-optimally for
hundreds of robots in challenging setups.
The authors consider the following generalization of the familiar 15-puzzle, which arises from issues in memory management in distributed systems: Let G be a graph with n vertices with k greater than n pebbles numbered 1,. . . . ,k on distinct vertices. A move consists of transferring a pebble to an adjacent unoccupied vertex. Is one arrangement of the pebbles reachable from another? They present a P-time decision algorithm and prove matching O(n**3 ) upper and lower bounds on the number of moves required. The authors also consider the question of permutation group diameter and obtain the following subexponential bound for certain unbounded cycles: If G (on n letters) is generated by cycles, one of which has prime length p greater than 2n/3, and G is primitive, then G equals A//n or S//n and has diameter greater than 2**6 ROOT **p** plus **4 n**8 .
Cooperative Pathfinding is a multi-agent path planning prob- lem where agents must find non-colliding routes to separate destinations, given full information about the routes of other agents. This paper presents three new algorithms for effi- ciently solving this problem, suitable for use in Real-Time Strategy games and other real-time environments. The algo- rithms are decoupled approaches that break down the prob- lem into a series of single-agent searches. Cooperative A* (CA*) searches space-time for a non-colliding route. Hierar- chical Cooperative A* (HCA*) uses an abstract heuristic to boost performance. Finally, Windowed Hierarchical Coop- erative A* (WHCA*) limits the space-time search depth to a dynamic window, spreading computation over the duration of the route. The algorithms are applied to a series of challeng- ing, maze-like environments, and compared to A* with Local Repair (the current video-games industry standard). The re- sults show that the new algorithms, especially WHCA*, are robust and efficient solutions to the Cooperative Pathfinding problem, finding more successful routes and following better paths than Local Repair A*.
Part I. Fundamental Concepts: 1. Introduction and overview; 2.
Introduction to quantum mechanics; 3. Introduction to computer science;
Part II. Quantum Computation: 4. Quantum circuits; 5. The quantum
Fourier transform and its application; 6. Quantum search algorithms; 7.
Quantum computers: physical realization; Part III. Quantum Information:
8. Quantum noise and quantum operations; 9. Distance measures for
quantum information; 10. Quantum error-correction; 11. Entropy and
information; 12. Quantum information theory; Appendices; References;
Index.
Characterizing quantum supremacy in nearterm devices
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A quantum approximate optimization algorithm
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Quantum computing: a gentle introduction
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A practical quantum instruction set architecture
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Projectq: an open source software framework for quantum computing
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