Quantum Divide and Conquer for Combinatorial Optimization and Distributed Computing Zain H. Saleem, 1, ∗ Teague Tomesh, 2, 3, † Michael A. Perlin, 4, ‡ Pranav Gokhale, 3, § and Martin Suchara 1, ¶ 1 Argonne National Laboratory, 9700 S. Cass Ave., Lemont, IL 60439, USA. 2 Department of Computer Science, Princeton University, Princeton, NJ 08540, USA. 3 Super.tech, Chicago, IL, USA. 4 JILA, NIST and University of Colorado, 440 UCB, Boulder, Colorado 80309, USA. (Dated: November 23, 2021) We introduce a quantum divide and conquer algorithm that enables the use of distributed com- puting for constrained combinatorial optimization problems. The algorithm consists of three major components: classical partitioning of a target graph into multiple subgraphs, variational optimiza- tion over these subgraphs, and a quantum circuit cutting procedure that allows the optimization to take place independently on separate quantum processors. We simulate the execution of the quantum divide and conquer algorithm to find approximate solutions to instances of the Maximum Independent Set problem which have nearly twice as many nodes than the number of qubits available on a single quantum processor. I. INTRODUCTION Building quantum chips with a sufficiently large num- ber of qubits suitable for solving practically sized prob- lems has remained an elusive goal. Distributed quantum computing [1, 2] that uses multiple smaller systems con- nected by quantum communication channels promises to offer a path to scalability. Suitable quantum connections can be offered by quantum networks such as the quan- tum internet [3–5], with proof-of-concept experimental realizations under construction by several teams [6–9]. However, distributed quantum computing will require reliable, low-latency quantum connectivity between the quantum systems, and the amount of quantum commu- nication therefore needs to be minimized. This work ad- dresses this challenge by designing a quantum optimiza- tion algorithm that minimizes communication between subproblems. In the absence of a working physical quan- tum communication network, we use the Quantum Di- vide and Conquer (QDC) approach that allows connect- ing quantum systems by using only classical communica- tion. The first component of the QDC algorithm takes an in- put graph and classically partitions it into multiple sub- graphs. Graph partitioning approaches are commonly used for tackling combinatorial optimization problems in the classical algorithms literature [10]. The idea is to break down a large problem into more manageable sub- problems. In this work, we use the Kernighan-Lin al- gorithm [11] to partition graphs into approximately bal- anced subgraphs. The second component of QDC consists of a quantum variational optimization on the partitioned subgraphs. * zsaleem@anl.gov; † ttomesh@princeton.edu ‡ mika.perlin@gmail.com § pranav@super.tech ¶ msuchara@anl.gov We use the dynamic quantum variational ansatz (DQVA) to perform this task [12]. The DQVA algorithm was de- signed for tackling constrained combinatorial optimiza- tion problems, such as Maximum Independent Set (MIS). It is based on an ansatz which dynamically changes its structure to make more efficient use of a fixed allocation of quantum resources. However, DQVA is not the only approach to solving constrained optimization problems. We could have also used quantum local search which was recently demonstrated to efficiently find approximate so- lutions to the MIS problem on large graph instances [13]. Finally, the third component of QDC is the circuit cut- ting technique which divides large quantum circuits into smaller fragments [14–17]. These smaller fragments can then be independently executed on separate quantum de- vices and their outputs can be recombined using classical post-processing to obtain the output of the larger, uncut circuit. While circuit cutting is computationally expen- sive, with a cost that grows exponentially with the num- ber of cuts required to partition a circuit, in our approach we carefully structure the ansatz in such a way that the number of cuts required is tunable to the available post- processing resources. In this work, we combine these three components: (1) graph partitioning, (2) quantum variational optimiza- tion, and (3) circuit cutting to approximate the Maxi- mum Independent Set (MIS) problem for large graphs that would otherwise require more qubits than are cur- rently available on an individual noisy intermediate-scale quantum (NISQ) device [18]. Our key contributions are as follows. First, we provide specifications for building the variational ansatz, choosing the cut locations, and the dynamic ansatz update that enables the execution of QDC with manageable quantum and classical costs. Second, we provide the first demonstration of quantum circuit cutting for a useful, practical application. Using this technique we are able to find approximate solutions to MIS on graphs containing up to 26 nodes while only requiring the simulation of up to 15-qubit quantum cir- arXiv:2107.07532v1 [quant-ph] 15 Jul 2021