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arXiv:2208.13679 · IEEE/ACM International Symposium on Microarchitecture (MICRO)

Qubit Mapping and Routing via MaxSAT

Abtin Molavi, Amanda Xu, Martin Diges, Lauren Pick, Swamit Tannu, Aws Albarghouthi

Published Oct 1, 2022• 54 citations
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Abstract

Near-term quantum computers will operate in a noisy environment, without error correction. A critical problem for near-term quantum computing is laying out a logical circuit onto a physical device with limited connectivity between qubits. This is known as the qubit mapping and routing (QMR) problem, an intractable combinatorial problem. It is important to solve QMR as optimally as possible to reduce the amount of added noise, which may render a quantum computation useless. In this paper, we present a novel approach for optimally solving the QMR problem via a reduction to maximum satisfiability (MAXSAT). Additionally, we present two novel relaxation ideas that shrink the size of the MAXSAT constraints by exploiting the structure of a quantum circuit. Our thorough empirical evaluation demonstrates (1) the scalability of our approach compared to state-of-the-art optimal QMR techniques (solves more than 3x benchmarks with 40x speedup), (2) the significant cost reduction compared to state-of-the-art heuristic approaches (an average of ~5x swap reduction), and (3) the power of our proposed constraint relaxations.

DOI
10.1109/micro56248.2022.00077

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Near-term quantum computers will operate in a noisy environment, without error correction. A critical problem for near-term quantum computing is laying out a logical circuit onto a physical device with limited connectivity between qubits. This is known as the qubit mapping and routing (QMR) problem, an intractable combinatorial problem. It is important to solve QMR as optimally as possible to reduce the amount of added noise, which may render a quantum computation useless. In this paper, we present a novel approach for optimally solving the QMR problem via a reduction to maximum satisfiability (MAXSAT). Additionally, we present two novel relaxation ideas that shrink the size of the MAXSAT constraints by exploiting the structure of a quantum circuit. Our thorough empirical evaluation demonstrates (1) the scalability of our approach compared to state-of-the-art optimal QMR techniques (solves more than 3x benchmarks with 40x speedup), (2) the significant cost reduction compared to state-of-the-art heuristic approaches (an average of ~5x swap reduction), and (3) the power of our proposed constraint relaxations.
To evaluate a quantum circuit on a quantum processor, one must find a mapping from circuit qubits to processor qubits and plan the instruction execution while satisfying the processor’s constraints. This is known as the qubit mapping and routing ( qmr ) problem. High-quality qmr solutions are key to maximizing the utility of scarce quantum resources and minimizing the probability of logical errors affecting computation. The challenge is that the landscape of quantum processors is incredibly diverse and fast-evolving. Given this diversity, dozens of papers have addressed the qmr problem for different qubit hardware, connectivity constraints, and quantum error correction schemes by a developing a new algorithm for a particular context. We present an alternative approach: automatically generating qubit mapping and routing compilers for arbitrary quantum processors. Though each qmr problem is different, we identify a common core structure— device state machine —that we use to formulate an abstract qmr problem . Our formulation naturally leads to a compact domain-specific language for specifying qmr problems and a powerful parametric algorithm that can be instantiated for any qmr specification. Our thorough evaluation on case studies of important qmr problems shows that generated compilers are competitive with handwritten, specialized compilers in terms of runtime and solution quality.
Abstract In this work we propose a high-quality decomposition approach for qubit routing by swap insertion. This optimization problem arises in the context of compiling quantum algorithms formulated in the circuit model of computation onto specific quantum hardware. Our approach decomposes the routing problem into an allocation subproblem and a set of token swapping problems. This allows us to tackle the allocation part and the token swapping part separately. Extracting the allocation part from the qubit routing model of Nannicini et al. (Optimal qubit assignment and routing via integer programming, 2021, http://arxiv.org/abs/2106.06446 ), we formulate the allocation subproblem as a binary linear program. Herein, we employ a cost function that is a lower bound on the overall routing problem objective. We strengthen the linear relaxation by novel valid inequalities. For the token swapping part we develop an exact branch-and-bound algorithm. In this context, we improve upon known lower bounds on the token swapping problem. Furthermore, we enhance an existing approximation algorithm which runs much faster than the exact approach and typically is able to determine solutions close to the optimum. We present numerical results for the fully integrated allocation and token swapping problem. Obtained solutions may not be globally optimal due to the decomposition and the usage of an approximation algorithm. However, the solutions are obtained fast and are typically close to optimal. In addition, there is a significant reduction in the number of artificial gates and output circuit depth when compared to various state-of-the-art heuristics. Reducing these figures is crucial for minimizing noise when running quantum algorithms on near-term hardware. As a consequence, using the novel decomposition approach leads to compiled algorithms with improved quality. Indeed, when compiled with the novel routing procedure and executed on real hardware, our experimental results for quantum approximate optimization algorithms show an significant increase in solution quality in comparison to standard routing methods.
Qubits are powerful computational resources because each additional qubit doubles the size of the accessible state space. This expressive power comes at a cost: adding qubits also increases circuit size, constraint overhead, and exposure to hardware noise. These limitations are especially critical for near-term quantum computers, making the number of qubits required to formulate a problem a central practical concern. In this work, we present a method for efficiently mapping logistics and routing problems onto qubits through structure-aware encoding reduction. We demonstrate the method on a benchmark problem of latency-resilient routing under dual-link failures. For this benchmark, the reduced formulation substantially lowers both the qubit count and the compiled circuit complexity while preserving the original optimization task. By running the reduced model on the same trapped-ion quantum computer as the benchmark, we obtain a 100X improvement in the success probability. This finding shows how careful problem formulation can make current quantum hardware substantially more effective for practical optimization.
Quantum computing is gaining momentum in revolutionizing the way we approach complex problem-solving. However, the practical implementation of quantum algorithms remains a significant challenge due to the error-prone and hardware limits of near-term quantum devices. For instance, physical qubit connections are limited, which necessitates the use of quantum SWAP gates to dynamically transform the logical topology during execution. In addition, to optimize fidelity, it is essential to ensure that 1) the allocated hardware has a low error rate and 2) the number of SWAP gates injected into the circuit is minimized. To address these challenges, we propose a suite of algorithms: the Fidelity-aware Graph Extraction Algorithm (FGEA) is used to identify the hardware region with the lowest probability of error, the Frequency-based Mapping Algorithm (FMA) allocates logical-physical qubits that reduce the potential distance of topological transformation, and the Heuristic Routing Algorithm (HRA) searches for an optimal swapping injection strategy. We evaluate the proposed algorithms on the IBM-provided Noisy Intermediate-Scale Quantum (NISQ) computer, using a dataset consisting of 17 different quantum circuits of various sizes. The circuits are executed on the IBM Toronto Falcon processor. The three proposed algorithms outperform the existing SABRE algorithm in reducing the number of SWAP gates required. Therefore, our proposed algorithms hold significant promise in enhancing the fidelity and reducing the number of SWAP gates required in implementing Quantum algorithms.
In the era of noisy intermediate-scale quantum (NISQ) computing, the limited connectivity between qubits is one of the common physical limitations faced by current quantum computing devices. Quantum circuit mapping methods transform quantum circuits into equivalent circuits that satisfy physical connectivity constraints by remapping logical qubits, making them executable. The optimization problem of quantum circuit mapping has NP-hard computational complexity, and existing heuristic mapping algorithms still have significant potential for optimization in terms of the number of quantum gates generated. To reduce the number of SWAP gates inserted during mapping, the solution space of the mapping problem is represented as a tree structure, and the mapping process is equivalent to traversing this tree structure. To effectively and efficiently complete the search process, a beam search framework (BSF) is proposed for solving quantum circuit mapping. By iteratively selecting, expanding, and making decisions, high-quality target circuits are generated. Experimental results show that this method can significantly reduce the number of inserted SWAP gates on medium to large circuits, achieving an average reduction of 44% compared to baseline methods, and is applicable to circuits of various sizes and complexities.