[Paper Review] Solving various NP-Hard problems using exponentially fewer qubits on a Quantum Computer

Yagnik Chatterjee, Eric Bourreau+1 more|arXiv (Cornell University)|Jan 17, 2023

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces a logarithmic qubit encoding method that enables solving various NP-hard combinatorial optimization problems—such as Maximum Cut, Minimum Partition, Maximum Clique, and Maximum Weighted Independent Set—on gate-based quantum computers using exponentially fewer qubits than classically required. By reformulating problems into a QUBO framework compatible with a logarithmic encoding scheme, the authors achieve the largest real-world quantum runs to date, demonstrating successful execution on a 256-node Maximum Cut problem and a 64-object Minimum Partition problem using real quantum hardware.

ABSTRACT

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 QAOA 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. In order 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.

Motivation & Objective

• To overcome the linear qubit scaling limitation of existing quantum optimization algorithms like QAOA, which restricts the size of solvable problems on current NISQ devices.
• To develop a general-purpose quantum algorithm that can encode and solve diverse NP-hard combinatorial optimization problems using only O(log n) qubits, regardless of problem size.
• To demonstrate the feasibility of running large-scale, industrially relevant optimization problems on current noisy intermediate-scale quantum (NISQ) hardware by leveraging a proprietary logarithmic encoding technique.
• To extend the applicability of the logarithmic encoding method beyond Maximum Cut to other NP-hard problems via reductions and QUBO reformulation.
• To validate the method experimentally through simulations and real quantum hardware runs, achieving unprecedented problem sizes (up to 256 nodes) on a universal gate-based quantum processor.

Proposed method

• The core method uses a logarithmic qubit encoding where each node in a graph is represented by a binary label encoded in log₂n qubits, drastically reducing the number of required qubits from n to ⌈log₂n⌉.
• The algorithm maps NP-hard problems to the Maximum Cut problem via problem-specific reductions, enabling the use of the same logarithmic encoding framework across multiple problem types.
• For general Quadratic Unconstrained Binary Optimization (QUBO) problems, the method introduces an sQUBO representation that allows direct integration with the logarithmic encoding by transforming the QUBO matrix into a form compatible with the Hamiltonian structure of the algorithm.
• The quantum circuit is implemented using a variational quantum approach with a classical optimizer (e.g., COBYLA or genetic algorithm) to iteratively adjust parameters and minimize the expectation value of the problem Hamiltonian.
• The expectation value of the observable is computed via Pauli string decomposition, where the Hamiltonian is expressed as a sum of tensor products of Pauli matrices, enabling measurement-based energy estimation on NISQ devices.
• The method is tested on both quantum simulators (for graphs up to 100+ nodes) and real quantum hardware (IBM Osprey, up to 256 nodes), with results compared across different optimizers and problem instances.

Experimental results

Research questions

• RQ1 Can NP-hard optimization problems be solved using exponentially fewer qubits than their classical counterparts by exploiting logarithmic encoding?
• RQ2 To what extent can the logarithmic encoding method be generalized beyond Maximum Cut to other NP-hard problems such as Minimum Partition and Maximum Clique?
• RQ3 What is the maximum problem size that can be practically solved on current NISQ devices using this logarithmic encoding approach?
• RQ4 How does the performance of the algorithm compare between quantum simulators and real quantum hardware across different problem instances and optimizers?
• RQ5 Can the sQUBO representation enable seamless integration of arbitrary QUBO problems into the logarithmic encoding framework without loss of generality or scalability?

Key findings

• The authors successfully solved a 256-node Maximum Cut problem on real quantum hardware, representing the largest realistic combinatorial optimization problem solved on a universal gate-based quantum computer to date.
• A 64-object Minimum Partition problem was solved on real hardware, marking a significant advancement over previous problem sizes, which were limited to around 25-30 objects.
• The algorithm achieved stable convergence on quantum simulators for problem instances with over 100 nodes, demonstrating scalability and robustness in the presence of noise.
• The use of classical optimizers like COBYLA and the genetic algorithm led to consistent and measurable improvements in solution quality across all tested problem sizes.
• The method reduced the required number of qubits from n to ⌈log₂n⌉, enabling the encoding of problems with up to 256 nodes using only 8 qubits, a 32-fold reduction in qubit count.
• The results show that smaller, higher-fidelity quantum devices can outperform larger, noisier ones when using this logarithmic encoding, due to reduced circuit depth and fewer SWAP gates.