Quantum computers are devices which allow the solution of problems unsolvable to their classical counterparts. As an error-corrected quantum computer is still a decade away the quantum computing community has dedicated much attention to developing algorithms for currently available Noisy Intermediate-Scale Quantum computers (NISQ). Thus far, within NISQ, optimization problems are one of the most commonly studied and are exclusively tackled with the Quantum Approximate Optimization Algorithm (QAOA). This algorithm predominantly computes graph partitions with a maximal separation of edges (MaxCut), but can also be modified to calculate other properties of graphs. Here, I present a novel quantum optimization algorithm which uses exponentially less qubits as compared to the QAOA while requiring a significantly reduced number of quantum operations to solve the MaxCut problem. Such an improved performance allowed me to partition a graphs 32 nodes on publicly available 5 qubit gate-based solid state quantum devices without any preprocessing such as division of the graph into smaller subgraphs. This results represent a 40% increase in graph size as compared to state-of-art experiments on solid state devices such as Google Sycamore. The obtained lower bound is 54.9% on the solution for actual hardware benchmarks and 77.6% on ideal quantum simulators. Furthermore, large-scale optimization problems represented by graphs of a 128 nodes are tackled with quantum simulators, again without any pre-division into smaller subproblems and a lower solution bound of 67.9% is achieved.
Quantum computers are devices which allow the solution of problems unsolvable to their classical counterparts. As an error-corrected quantum computer is still a decade away the quantum computing community has dedicated much attention to developing algorithms for currently available Noisy Intermediate-Scale Quantum computers (NISQ). Thus far, within NISQ, optimization problems are one of the most commonly studied and are exclusively tackled with the Quantum Approximate Optimization Algorithm (QAOA). This algorithm predominantly computes graph partitions with a maximal separation of edges (MaxCut), but can also be modified to calculate other properties of graphs. Here, I present a novel quantum optimization algorithm which uses exponentially less qubits as compared to the QAOA while requiring a significantly reduced number of quantum operations to solve the MaxCut problem. Such an improved performance allowed me to partition a graphs 32 nodes on publicly available 5 qubit gate-based solid state quantum devices without any preprocessing such as division of the graph into smaller subgraphs. This results represent a 40% increase in graph size as compared to state-of-art experiments on solid state devices such as Google Sycamore. The obtained lower bound is 54.9% on the solution for actual hardware benchmarks and 77.6% on ideal quantum simulators. Furthermore, large-scale optimization problems represented by graphs of a 128 nodes are tackled with quantum simulators, again without any pre-division into smaller subproblems and a lower solution bound of 67.9% is achieved.
Quantum computers are devices, which allow more efficient solutions of problems as compared to their classical counterparts.As the timeline to developing a quantum-error corrected computer is unclear, the quantum computing community has dedicated much attention to developing algorithms for currently available noisy intermediate-scale quantum computers (NISQ).Thus far, within NISQ, optimization problems are one of the most commonly studied and are quite often tackled with the quantum approximate optimization algorithm (QAOA).This algorithm is best known for computing graph partitions with a maximal separation of edges (MaxCut), but can easily calculate other problems related to graphs.Here, I present a novel quantum optimization algorithm, which uses exponentially less qubits as compared to the QAOA while requiring a significantly reduced number of quantum operations to solve the MaxCut problem.Such an improved performance allowed me to partition graphs with 32 nodes on publicly available 5 qubit gate-based quantum computers without any preprocessing such as division of the graph into smaller subgraphs.These results represent a 40% increase in graph size as compared to state-of-art experiments on gate-based quantum computers such as Google Sycamore.The obtained lower bound is 54.9% on the solution for actual hardware benchmarks and 77.6% on ideal simulators of quantum computers.Furthermore, large-scale optimization problems represented by graphs of a 128 nodes are tackled with simulators of quantum computers, again without any predivision into smaller subproblems and a lower solution bound of 67.9% is achieved.The study presented here paves way to using powerful genetic optimizer in synergy with quantum computers.
Abstract MaxCut is a key NP-hard combinatorial optimization problem. Quantum computing offers methods to solve such problems potentially better than classical counterparts, with the Quantum Approximate Optimization Algorithm (QAOA) being a state-of-the-art example. However, the performance of quantum methods is currently hindered by hardware noise and limited qubit volumes. We present a variational Qubit-Efficient MaxCut (QEMC) algorithm that requires only O(logN) O ( log N ) qubits to tackle graphs of size N , an exponential reduction compared to QAOA. We demonstrate cutting-edge performance for 32-node graph instances (5 qubits) on real superconducting hardware, and for graphs with up to 2048 nodes (11 qubits) via classical simulations. The QEMC algorithm is based on an innovative encoding scheme, with potentially broad applicability, that empowers it with strong noise resilience, but also enables its efficient classical simulation. As such, the QEMC algorithm provides a challenging benchmark for QAOA on noisy devices and offers a novel quantum-inspired approach.
NP-hard problems are not believed to be exactly solvable through general polynomial time algorithms. Hybrid quantum-classical algorithms to address such combinatorial problems have been of great interest in the past few years. Such algorithms are heuristic in nature and aim to obtain an approximate solution. Significant improvements in computational time and/or the ability to treat large problems are some of the principal promises of quantum computing in this regard. The hardware, however, is still in its infancy and the current noisy intermediate-scale quantum (NISQ) computers are not able to optimize industrially relevant problems. Moreover, the storage of qubits and introduction of entanglement require extreme physical conditions. An issue with quantum optimization algorithms such as the quantum approximate optimization algorithm is that they scale linearly with problem size. In this paper, we build upon a proprietary methodology which scales logarithmically with problem size---opening an avenue for treating optimization problems of unprecedented scale on gate-based quantum computers. To test the performance of the algorithm, we first find a way to apply it to a handful of NP-hard problems: Maximum Cut, Minimum Partition, Maximum Clique, Maximum Weighted Independent Set. Subsequently, these algorithms are tested on a quantum simulator with graph sizes of over a hundred nodes and on a real quantum computer up to graph sizes of 256. To our knowledge, these constitute the largest realistic combinatorial optimization problems ever run on a NISQ device, overcoming previous problem sizes by almost tenfold.
Abstract In the noisy intermediate-scale quantum (NISQ) era, quantum approximate optimisation algorithms are important tools for solving NP-hard problems such as combinatorial optimisation. However, most QAOA implementations ignore the inherent noise in quantum circuit environments. This study proposes a noisy quantum approximate optimisation algorithm (NQAOA) and analyses its performance under noisy conditions through numerical simulations. We compare the approximate ratios (AR) of various graphs under different optimisation strategies and noise conditions, and assess the impact of circuit depth, node count, edge count, and noise coefficient on the maximum cut (MaxCut). Experimental results show that noise leads to overfitting, and the approximate ratio ( AR ) exhibits fluctuating changes as the edge count e increases. The AR values under ideal conditions and the expected results are both superior to those under noisy conditions. This study is the first to systematically integrate multiple types of quantum noise into the QAOA framework, quantitatively analysing the synergistic effects of noise type, optimiser performance, and graph structure. Through theoretical derivations and numerical simulations, we established linear/exponential theoretical boundaries for AR decay related to noise coefficients, providing a critical reference for designing noise-resistant quantum optimisation algorithms on NISQ devices.
The design of fast algorithms for combinatorial optimization greatly contributes to a plethora of domains such as logistics, finance, and chemistry. Quantum approximate optimization algorithms (QAOAs), which utilize the power of quantum machines and inherit the spirit of adiabatic evolution, are approaches to tackle combinatorial problems with potential runtime speedups. However, hurdled by the limited quantum resources nowadays, QAOAs are infeasible to manipulate large-scale problems. To address this issue, here we revisit the MaxCut problem via the divide-and-conquer heuristic: seek the solutions of subgraphs in parallel and then merge these solutions to obtain the global solution. Because of the Z2 symmetry in MaxCut, we prove that the merging process can be further cast into a new MaxCut problem and thus be addressed by QAOAs or other MaxCut solvers. In view of this, we propose QAOA-in-QAOA (QAOA2) to solve arbitrary large-scale MaxCut problems using small quantum machines. We also prove that the performance of QAOA2 is lower bounded with respect to the divide-and-conquer process. Experiment results illustrate that, under different graph settings, QAOA2 attains a competitive or even better performance over classical algorithms when the node count is around 2000. Our method can be seamlessly embedded into other advanced strategies to enhance the capability of QAOAs in large-scale combinatorial optimization problems.
Combinatorial optimization lies at the heart of numerous real-world applications. For a broad category of optimization problems, quantum computing is expected to exhibit quantum speed-up over classic computing. Among various quantum algorithms, the Quantum Approximate Optimization Algorithm (QAOA), as one of variational quantum algorithms, shows promise on demonstrating quantum advantage on noisy intermediate-scale quantum (NISQ) hardware. However, with increasing problem size, the circuit depth demanded by original QAOA scales rapidly and quickly surpasses the threshold at which meaningful results can be obtained. To address this challenge, in this work, we propose a variant of QAOA (termed linear chain QAOA) and demonstrate its advantages over original QAOA on paradigmatic MaxCut problems. In original QAOA, each graph edge is encoded with one entangling gate. In our ansatz, we locate a linear chain from the original MaxCut graph and place entangling gates sequentially along this chain. This linear-chain ansatz is featured by shallow quantum circuits and with the low execution time that scales independently of the problem size. Leveraging this ansatz, we demonstrate an approximation ratio of 0.78 (without post-processing) on non-hardware-native random regular MaxCut instances with 100 vertices in a digital quantum processor using 100 qubits. Our findings offer new insights into the design of hardware-efficient ansatz and point toward a promising route for tackling large-scale combinatorial optimization problems on NISQ devices.