Abstract
There is an increasing interest in quantum algorithms for problems of integer programming and combinatorial optimization. Classical solvers for such problems employ relaxations, which replace binary variables with continuous ones, for instance in the form of higher-dimensional matrix-valued problems (semidefinite programming). Under the Unique Games Conjecture, these relaxations often provide the best performance ratios available classically in polynomial time. Here, we discuss how to warm-start quantum optimization with an initial state corresponding to the solution of a relaxation of a combinatorial optimization problem and how to analyze properties of the associated quantum algorithms. In particular, this allows the quantum algorithm to inherit the performance guarantees of the classical algorithm. We illustrate this in the context of portfolio optimization, where our results indicate that warm-starting the Quantum Approximate Optimization Algorithm (QAOA) is particularly beneficial at low depth. Likewise, Recursive QAOA for MAXCUT problems shows a systematic increase in the size of the obtained cut for fully connected graphs with random weights, when Goemans-Williamson randomized rounding is utilized in a warm start. It is straightforward to apply the same ideas to other randomized-rounding schemes and optimization problems.
Highlights
Gate-based quantum computers are expected to help solve problems in quantum chemistry [1,2,3], machine learning [4, 5], financial simulation [6,7,8,9,10,11,12,13] and combinatorial optimization [14, 15]
We introduce a variant of WS-quantum approximate optimization algorithm (QAOA) that utilizes a regularization parameter ε ∈ [0, 0.5] and changes the rotation angle creating the initial state according to θi = 2 arcsin c∗i √
If the Unique Games Conjecture is true, these guarantees cannot be improved upon by classical or quantum algorithms running in polynomial time, unless we can solve NP-Hard problems in polynomial time
Summary
Gate-based quantum computers are expected to help solve problems in quantum chemistry [1,2,3], machine learning [4, 5], financial simulation [6,7,8,9,10,11,12,13] and combinatorial optimization [14, 15]. A classical optimizer seeks the optimal values of β and γ to create a trial state which minimizes the energy of the Hamiltonian HC This algorithm has lacked theoretical guarantees on its performance ratio and for certain problem instances of MAXCUT it cannot, with constant depth, outperform the classical. The best-known continuous relaxations of MAXCUT and many other problems take the form of semidefinite programs [48] These can be solved efficiently both in theoretical models of computation [49], where a real-number arithmetic operation can be performed in unit time, and in practice1 [51]. We explore warm-starting QAOA (WS-QAOA) numerically in Sec. 3 by relaxing Quadratic Unconstrained Binary Optimization problems to continuous ones which provide QAOA with a good initial solution.
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