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Nearest Neighbor Compliance in Quatum Circuit Design. Masterthesis Applied Mathematics TU Delfdt

Jesse Mulderij

Published Jan 1, 2019• 0 citations

Abstract

Over the course of the last decade, an interest has emerged in nearest neighbor constraints for quantum circuit design. The challenge herein is to bothminimize the running time of a circuit and tomodify it such that quantum gates only act on adjacent qubits while leaving the desired computational operation intact. These modifications involve the insertion of SWAP gates, which introduces computational overhead. Several exact and heuristic methods have been developed to determine the minimal number of required SWAP gates. Until now, the model sizes of exact methods scale superexponentionally in the number of qubits of the quantum circuit. The main research goal of this project was to develop an integer linear program that instead scales polynomially both in the number of qubits and the number of quantum gates in the quantumcircuit. First, qubits located on a linear array were considered. The resulting integer linear program proposed requires O(n2m) variables and constraints, where n denotes the number of qubits and m denotes the number of quantum gates of the quantum circuit under consideration. Second, qubits located on a two- or threedimensional grid were considered. The programs both require O(n4m) variables and O(n3m) constraints. Subsequently we prove that, for a fixed quantum circuit, the optimal objective value of the one-dimensional problem is an upper bound for the optimal objective value of the corresponding two- and three-dimensional problems. Using CPLEX’s Branch & Bound method, 131 benchmark instances of various circuit sizes were evaluated. The largest of which contained either eighteen qubits and sixteen quantum gates, or five qubits and 112 quantumgates. Furthermore, we identify problems on the interface of nearest neighbor compliant quantum circuit design and distributed quantum computing. Integer linear program formulations are provided for these additional problems.

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Over the course of the last decade, an interest has emerged in nearest neighbor constraints for quantum circuit design. The challenge herein is to bothminimize the running time of a circuit and tomodify it such that quantum gates only act on adjacent qubits while leaving the desired computational operation intact. These modifications involve the insertion of SWAP gates, which introduces computational overhead. Several exact and heuristic methods have been developed to determine the minimal number of required SWAP gates. Until now, the model sizes of exact methods scale superexponentionally in the number of qubits of the quantum circuit. The main research goal of this project was to develop an integer linear program that instead scales polynomially both in the number of qubits and the number of quantum gates in the quantumcircuit. First, qubits located on a linear array were considered. The resulting integer linear program proposed requires O(n2m) variables and constraints, where n denotes the number of qubits and m denotes the number of quantum gates of the quantum circuit under consideration. Second, qubits located on a two- or threedimensional grid were considered. The programs both require O(n4m) variables and O(n3m) constraints. Subsequently we prove that, for a fixed quantum circuit, the optimal objective value of the one-dimensional problem is an upper bound for the optimal objective value of the corresponding two- and three-dimensional problems. Using CPLEX’s Branch & Bound method, 131 benchmark instances of various circuit sizes were evaluated. The largest of which contained either eighteen qubits and sixteen quantum gates, or five qubits and 112 quantumgates. Furthermore, we identify problems on the interface of nearest neighbor compliant quantum circuit design and distributed quantum computing. Integer linear program formulations are provided for these additional problems.

Synthesis and optimization of quantum circuits have received significant attention from researchers in recent years. Developments in the physical realization of qubits in quantum computing have led to new physical constraints to be addressed. One of the most important constraints that is considered by many researchers is the nearest neighbor constraint which limits the interaction distance between qubits for quantum gate operations. Various works have been reported in the literature that deal with nearest neighbor compliance in multi-dimensional (mostly 1D and 2D) qubit arrangements. This is normally achieved by inserting SWAP gates in the gate netlist to bring the interacting qubits closer together. The main objective function to minimize here is the number of SWAP gates. The present paper proposes an efficient qubit placement strategy in a three-dimensional (3D) grid that considers not only qubit interactions but also the relative positions of the gates in the circuit. Experimental evaluation on a number of benchmark circuits show that the proposed method reduces the number of SWAP gates by 16.2% to 47.0% on the average as compared to recently published works.

Motivated by its promising applications e.g. for database search or factorization, significant progress has been made in the development of automated design methods for quantum circuits. But in order to keep up with recent physical developments in this domain, new technological constraints have to be considered. Limited interaction distance between gate qubits is one of the most common of these constraints. This led to the development of several strategies aiming at making a given quantum circuit nearest neighbor-compliant by inserting SWAP gates into the existing structure. Usually these strategies are of heuristic nature. In this work, we present an exact approach that enables nearest neighbor-compliance by inserting a minimal number of SWAP gates. Experiments demonstrate the applicability of the approach which enabled a comparison of results obtained by heuristic methods to the actual optimum.

A New Heuristic for

N

-Dimensional Nearest Neighbor Realization of a Quantum Circuit

One of the main challenges in quantum computing is to ensure error-free operation of the basic quantum gates. There are various implementation technologies of quantum gates for which the distance between interacting qubits must be kept within a limit for reliable operation. This leads to the so-called requirement of neighborhood arrangements of the interacting qubits, often referred to as nearest neighbor (NN) constraint. This is typically achieved by inserting SWAP gates in the quantum circuits, where a SWAP gate between two qubits exchanges their states. Minimizing the number of SWAP gates to provide NN compliance is an important problem to solve. A number of approaches have been proposed in this regard, based on local and global ordering techniques. In this paper, a generalized approach for combined local and global ordering of qubits have been proposed that is based on an improved heuristic for cost estimation and is also scalable. The approach can be extended to N -dimensional arrangement of qubits, for any arbitrary values of N . Practical constraints, however, restrict the maximum value of N to 3. Extensive experiments on benchmark functions have been carried out to evaluate the performance in terms of SWAP gate requirements. 3-D organization of qubits shows average reductions of 6.7% and 37.4%, respectively, in the number of SWAP gates over 2-D and 1-D organizations. Also compared to the best 2-D and 1-D results reported in the literature, on the average 8.7% and 8.4% reductions, respectively, are observed.

Design and synthesis of quantum circuits have gained huge momentum during the last few years. With various physical constraint sone of the major issues is in making quantum circuits nearest neighbor complaint. The existing one and two dimensional architectures for the placement of qubits does not scale well for larger circuits. The situation thus demands for a faster approach to solve the problem. To make a quantum circuit nearest neighbor compliant, SWAP gates are inserted in the quantum gate circuit to bring the interacting qubits of a gate adjacent. We have proposed a faster method with a simple yet effective algorithm to minimize the number of SWAP gates for adjacent interactions. The time required for processing does not exceed one second even for larger benchmark circuits. The proposed method is found to reduce the number of SWAP gates on an average by 9.5% compared to the best results in previous works on 2D architectures, and by 45.7% compared to the best results on 1D architecture.

The linear nearest neighbor (LNN) restriction, present in several current implementations of 1D and 2D quantum circuits, limits the interaction of qubits to those which are adjacent to each other. While there have been several proposals to optimize the process of achieving LNN compliance in 1D circuits, there are few proposals for 2D circuits. Here, we present a new perspective on this problem for 2D quantum circuits. We propose to see this as a multiobjective optimization problem with two objectives: minimizing the size of the 2D grid in the circuit, and minimizing the number of SWAP gates required to achieve LNN compliance. We present some preliminary results which show that these are two contradictory objectives. Since this is common in multiobjective problems, these results indicate that a multiobjective algorithm might be a suitable way to address this problem, since it would make considerations which currently available methods do not make.

A physical limitation in quantum circuit design is the fact that gates in a quantum system can only act on qubits that are physically adjacent in the architecture. To overcome this problem, SWAP gates need to be inserted to make the circuit physically realizable. The nearest neighbour compliance problem (NNCP) asks for an optimal embedding of qubits in a given architecture such that the total number of SWAP gates to be inserted is minimized. In this paper we study the NNCP on general quantum architectures. Building upon an existing linear programming formulation, we show how the model can be reduced by exploiting the symmetries of the graph underlying the formulation. The resulting model is equivalent to a generalized network flow problem and follows from an in-depth analysis of the automorphism group of specific Cayley graphs. As a byproduct of our approach, we show that the NNCP is polynomial time solvable for several classes of symmetric quantum architectures. Numerical tests on various architectures indicate that the reduction in the number of variables and constraints is on average at least 90%. In particular, NNCP instances on the star architecture can be solved for quantum circuits up to 100 qubits and more than 1000 quantum gates within a very short computation time. These results are far beyond the computational capacity when solving the instances without the exploitation of symmetries.