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Quantum algorithms for exact and approximate optimization
Barton, Brandon A.
Barton, Brandon A.
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2024
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2025-05-26
Abstract
Following early breakthroughs for quantum algorithms in database search and integer factoring, over recent years, quantum computing hardware technologies and quantum algorithms have undergone rapid developments. Yet, the search for “killer” applications of quantum computing has remained elusive amidst the noisy intermediate scale quantum computing (NISQ) era, where noise from the environment provides a challenge for quantum algorithms requiring a fault-tolerant machine. Among prevalent applications such as quantum simulation and quantum chemistry, hard optimization tasks provide a promising class of problems where quantum algorithms hope to demonstrate advantage beyond quadratic speed-ups. This work investigates the application of heuristic quantum algorithms to combinatorial optimization, where we consider the class of binary constraint satisfaction problems. We introduce two new quantum algorithms for exact and approximate optimization, and demonstrate our methods with extensive quantum simulations that exhaust the capabilities of current supercomputers for classical simulation of quantum dynamics. In particular, we investigate a well-known problem known as MAX-3-XORSAT, in the complexity class, NP-hard. First, we introduce a new method called Spectrally Folded Quantum
Optimization (SFQO) which transforms the energy landscape of the problem, allowing approximate solutions to be readily obtained with guaranteed approximation ratios. Secondly, we introduce a new non-classical steering mechanism called Iterative Symphonic Tunneling for Satisfiability problems
(IST-SAT) which uses macroscopic quantum tunneling effects to guide the sufficiently good approximations towards the true global optima. The first work we present in this thesis, Spectrally Folded Quantum Optimization, investigates the ability of quantum algorithms to find approximate solutions to the MAX-3-XORSAT hypergraph problem class. We identify several distinct physical mechanisms associated to these problems, which make the task of finding exact solutions hard for all previously known classical and quantum algorithms. However, we find the same mechanisms that prevent quantum algorithms to find exact solutions, do not necessarily hold for approximate optimization. Spectrally folded quantum optimization implements a classical deformation of the constraint satisfaction problem energy landscape, which allows quantum algorithms to find constant fractions of the optimal solution with increasing problem size. We provide theoretical performance predictions for the algorithm, and benchmark our methods with extensive quantum simulations. Our results demonstrate that all of our numerical simulations agree with our predictions at the system sizes we can classically simulate using the best super-computing resources for classical simulation of quantum dynamics. This work suggests that quantum algorithms may be more powerful than previously thought for the task of approximate optimization. In the second work in this thesis, Iterative Symphonic Tunneling for Satisfiability problems, we present a non-classical steering mechanism for quantum optimization algorithms based on the use of high frequency oscillating “AC” drives. To demonstrate this mechanism we introduce an iterative quantum algorithm, IST-SAT, which does not require computing gradients or extensive fine-tuning. Using an initial classical or quantum algorithm to approximate the MAX-3-XORSAT problem, IST-SAT sets parameters in single-qubit oscillating drives according to the bits in the initial solution, which induces further macroscopic tunneling effects towards the true ground state(s) of the problem. IST-SAT converges to the ground state in an iterative manner, measured by the number of spin flips away from the ground state(s), also known as Hamming distance. We identify what it means to have a sufficient initial approximation for the IST-SAT, which defines a radius of convergence for the algorithm. The numerical results we obtain demonstrate that IST-SAT monotonically improves in performance, when provided initial states that are closer and closer to the ground state of the problem. When provided with an initial approximation at or above the radius of convergence, our results suggest that IST-SAT converges in polynomial time. Together, the results presented in this work obtain exponential speed-ups for obtaining approximate solutions, and polynomial speed-ups for exact problem solving, over the best known quantum and classical algorithms. While NP-hard optimization problems remain hard to solve exactly, by combining the methods in this work, we show that our algorithm can converge to the true ground state in polynomial time when provided a sufficiently good initial state. The novel mechanisms in this work thus pave new pathways for achieving quantum advantage on well-known hard problems, such as MAX-3-XORSAT. We expect the methods in this work to be amenable for experimental demonstrations on current or near-term quantum hardware, thus providing an exciting opportunity to demonstrate utility of quantum computers in the NISQ era.
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