16 research outputs found
Multistart Methods for Quantum Approximate Optimization
Hybrid quantum-classical algorithms such as the quantum approximate
optimization algorithm (QAOA) are considered one of the most promising
approaches for leveraging near-term quantum computers for practical
applications. Such algorithms are often implemented in a variational form,
combining classical optimization methods with a quantum machine to find
parameters to maximize performance. The quality of the QAOA solution depends
heavily on quality of the parameters produced by the classical optimizer.
Moreover, the presence of multiple local optima in the space of parameters
makes it harder for the classical optimizer. In this paper we study the use of
a multistart optimization approach within a QAOA framework to improve the
performance of quantum machines on important graph clustering problems. We also
demonstrate that reusing the optimal parameters from similar problems can
improve the performance of classical optimization methods, expanding on similar
results for MAXCUT
Evaluating Quantum Approximate Optimization Algorithm: A Case Study
Quantum Approximate Optimization Algorithm (QAOA) is one of the most
promising quantum algorithms for the Noisy Intermediate-Scale Quantum (NISQ)
era. Quantifying the performance of QAOA in the near-term regime is of utmost
importance. We perform a large-scale numerical study of the approximation
ratios attainable by QAOA is the low- to medium-depth regime. To find good QAOA
parameters we perform 990 million 10-qubit QAOA circuit evaluations. We find
that the approximation ratio increases only marginally as the depth is
increased, and the gains are offset by the increasing complexity of optimizing
variational parameters. We observe a high variation in approximation ratios
attained by QAOA, including high variations within the same class of problem
instances. We observe that the difference in approximation ratios between
problem instances increases as the similarity between instances decreases. We
find that optimal QAOA parameters concentrate for instances in out benchmark,
confirming the previous findings for a different class of problems
Parameter Transfer for Quantum Approximate Optimization of Weighted MaxCut
Finding high-quality parameters is a central obstacle to using the quantum
approximate optimization algorithm (QAOA). Previous work partially addresses
this issue for QAOA on unweighted MaxCut problems by leveraging similarities in
the objective landscape among different problem instances. However, we show
that the more general weighted MaxCut problem has significantly modified
objective landscapes, with a proliferation of poor local optima. Our main
contribution is a simple rescaling scheme that overcomes these deleterious
effects of weights. We show that for a given QAOA depth, a single "typical"
vector of QAOA parameters can be successfully transferred to weighted MaxCut
instances. This transfer leads to a median decrease in the approximation ratio
of only 2.0 percentage points relative to a considerably more expensive direct
optimization on a dataset of 34,701 instances with up to 20 nodes and multiple
weight distributions. This decrease can be reduced to 1.2 percentage points at
the cost of only 10 additional QAOA circuit evaluations with parameters sampled
from a pretrained metadistribution, or the transferred parameters can be used
as a starting point for a single local optimization run to obtain approximation
ratios equivalent to those achieved by exhaustive optimization in of
our cases
Graph decomposition techniques for solving combinatorial optimization problems with variational quantum algorithms
The quantum approximate optimization algorithm (QAOA) has the potential to
approximately solve complex combinatorial optimization problems in polynomial
time. However, current noisy quantum devices cannot solve large problems due to
hardware constraints. In this work, we develop an algorithm that decomposes the
QAOA input problem graph into a smaller problem and solves MaxCut using QAOA on
the reduced graph. The algorithm requires a subroutine that can be classical or
quantum--in this work, we implement the algorithm twice on each graph. One
implementation uses the classical solver Gurobi in the subroutine and the other
uses QAOA. We solve these reduced problems with QAOA. On average, the reduced
problems require only approximately 1/10 of the number of vertices than the
original MaxCut instances. Furthermore, the average approximation ratio of the
original MaxCut problems is 0.75, while the approximation ratios of the
decomposed graphs are on average of 0.96 for both Gurobi and QAOA. With this
decomposition, we are able to measure optimal solutions for ten 100-vertex
graphs by running single-layer QAOA circuits on the Quantinuum trapped-ion
quantum computer H1-1, sampling each circuit only 500 times. This approach is
best suited for sparse, particularly -regular graphs, as -regular graphs
on vertices can be decomposed into a graph with at most
vertices in polynomial time. Further reductions can be obtained with a
potential trade-off in computational time. While this paper applies the
decomposition method to the MaxCut problem, it can be applied to more general
classes of combinatorial optimization problems
Learning to Optimize Variational Quantum Circuits to Solve Combinatorial Problems
Quantum computing is a computational paradigm with the potential to
outperform classical methods for a variety of problems. Proposed recently, the
Quantum Approximate Optimization Algorithm (QAOA) is considered as one of the
leading candidates for demonstrating quantum advantage in the near term. QAOA
is a variational hybrid quantum-classical algorithm for approximately solving
combinatorial optimization problems. The quality of the solution obtained by
QAOA for a given problem instance depends on the performance of the classical
optimizer used to optimize the variational parameters. In this paper, we
formulate the problem of finding optimal QAOA parameters as a learning task in
which the knowledge gained from solving training instances can be leveraged to
find high-quality solutions for unseen test instances. To this end, we develop
two machine-learning-based approaches. Our first approach adopts a
reinforcement learning (RL) framework to learn a policy network to optimize
QAOA circuits. Our second approach adopts a kernel density estimation (KDE)
technique to learn a generative model of optimal QAOA parameters. In both
approaches, the training procedure is performed on small-sized problem
instances that can be simulated on a classical computer; yet the learned RL
policy and the generative model can be used to efficiently solve larger
problems. Extensive simulations using the IBM Qiskit Aer quantum circuit
simulator demonstrate that our proposed RL- and KDE-based approaches reduce the
optimality gap by factors up to 30.15 when compared with other commonly used
off-the-shelf optimizers.Comment: To appear in the proceedings of the Thirty-Fourth AAAI Conference on
Artificial Intelligence (AAAI), New York, USA, February 202
Quantum annealing initialization of the quantum approximate optimization algorithm
The quantum approximate optimization algorithm (QAOA) is a prospective
near-term quantum algorithm due to its modest circuit depth and promising
benchmarks. However, an external parameter optimization required in QAOA could
become a performance bottleneck. This motivates studies of the optimization
landscape and search for heuristic ways of parameter initialization. In this
work we visualize the optimization landscape of the QAOA applied to the MaxCut
problem on random graphs, demonstrating that random initialization of the QAOA
is prone to converging to local minima with sub-optimal performance. We
introduce the initialization of QAOA parameters based on the Trotterized
quantum annealing (TQA) protocol, parameterized by the Trotter time step. We
find that the TQA initialization allows to circumvent the issue of false minima
for a broad range of time steps, yielding the same performance as the best
result out of an exponentially scaling number of random initializations.
Moreover, we demonstrate that the optimal value of the time step coincides with
the point of proliferation of Trotter errors in quantum annealing. Our results
suggest practical ways of initializing QAOA protocols on near-term quantum
devices and reveals new connections between QAOA and quantum annealing.Comment: 10 pages, 9 figures; typos corrected, references adde
Of Representation Theory and Quantum Approximate Optimization Algorithm
In this paper, the Quantum Approximate Optimization Algorithm (QAOA) is
analyzed by leveraging symmetries inherent in problem Hamiltonians. We focus on
the generalized formulation of optimization problems defined on the sets of
-element -ary strings. Our main contribution encompasses dimension
reductions for the originally proposed QAOA. These reductions retain the same
problem Hamiltonian as the original QAOA but differ in terms of their mixer
Hamiltonian, and initial state. The vast QAOA space has a daunting dimension of
exponential scaling in , where certain reduced QAOA spaces exhibit
dimensions governed by polynomial functions. This phenomenon is illustrated in
this paper, by providing partitions corresponding to polynomial dimensions in
the corresponding subspaces. As a result, each reduced QAOA partition
encapsulates unique classical solutions absent in others, allowing us to
establish a lower bound on the number of solutions to the initial optimization
problem. Our novel approach opens promising practical advantages in
accelerating the class of QAOA approaches, both quantum-based and classical
simulation of circuits, as well as a potential tool to cope with barren
plateaus problem
To quantum or not to quantum: towards algorithm selection in near-term quantum optimization
The Quantum Approximate Optimization Algorithm (QAOA) constitutes one of the
often mentioned candidates expected to yield a quantum boost in the era of
near-term quantum computing. In practice, quantum optimization will have to
compete with cheaper classical heuristic methods, which have the advantage of
decades of empirical domain-specific enhancements. Consequently, to achieve
optimal performance we will face the issue of algorithm selection, well-studied
in practical computing. Here we introduce this problem to the quantum
optimization domain.
Specifically, we study the problem of detecting those problem instances of
where QAOA is most likely to yield an advantage over a conventional algorithm.
As our case study, we compare QAOA against the well-understood approximation
algorithm of Goemans and Williamson (GW) on the Max-Cut problem. As exactly
predicting the performance of algorithms can be intractable, we utilize machine
learning to identify when to resort to the quantum algorithm. We achieve
cross-validated accuracy well over 96\%, which would yield a substantial
practical advantage. In the process, we highlight a number of features of
instances rendering them better suited for QAOA. While we work with simulated
idealised algorithms, the flexibility of ML methods we employed provides
confidence that our methods will be equally applicable to broader classes of
classical heuristics, and to QAOA running on real-world noisy devices