Warm-starting quantum optimization

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.

Classically-inspired Mixers for QAOA Beat Goemans-Williamson's Max-Cut at Low Circuit Depths

“…The standard QAOA [3] is, therefore, a special case of our custom mixer approach as the standard starting state has each qubit on the x-axis (with Cartesian coordinates (x j , y j , z j ) = (1, 0, 0)) and with the standard mixer H B = n j=1 σ x j , the unitary operator e −iβ k H B corresponds to rotations (by 2β k ) about the x-axis. Additionally, we also recover the results of Egger et al [5] by restricting the initialization to the xz-plane with x > 0.…”

“…These instances, which we refer to as G, have a varied distribution of Max-Cuts, which is important when testing heuristics and algorithms for solving this problem. We consider comparisons with respect to a recent warm-starts approach of Egger et al [5] in Appendix C. We find that QAOA on graph instances with both positive and negative edge-weights has a much higher variance in approximation ratios achieved, and therefore we will present results for positive weight only and mixed weight graphs separately. This is not surprising since mixed-weight graphs are difficult to approximate; in fact, the only classical approximation factor known for mixed weight graphs is an O(1/ log(n)) approximation, when the sum of all the edge-weights is positive [19].…”

“…To give fair comparison against QAOA-Warm [4] (and also for comparisons with [5]), the remainder of this section will assume that we are using rank-2 initializations and vertex-at-top rotations unless otherwise stated, since these were the recommended setting in [4]. 3.…”

“…In recent work, Tate et al [4] and Egger et * Email of the corresponding author: swatig@gatech.edu al. [5] explored using classical algorithms to specify the initial state for QAOA. Following the classical optimization literature ( [6][7][8]), we will refer to these classically-inspired initializations as warm starts.…”

“…The approach by Tate et al [4] considered warm-starting QAOA using rank-2 and rank-3 Burer-Monteiro locally optimal solutions for Max-Cut; however, their method plateaus even at low circuit depths of p = 1 for some initializations and is unable to improve the cut quality for some instances. Egger et al [5] similarly considered warmstarts for quadratic unconstrained binary optimization (QUBO) problems; however, they only focus on rank-1 solutions from classical linear programming relaxations for QUBO. Another method by Egger et al initializes the quantum state via a particular cut in the graph; this QAOA variant can recover the value of the cut but not much is known regarding the convergence properties of this approach as the circuit depth increases.…”