96 research outputs found
Reducing Binary Quadratic Forms for More Scalable Quantum Annealing
Recent advances in the development of commercial quantum annealers such as the D-Wave 2X allow solving NP-hard optimization problems that can be expressed as quadratic unconstrained binary programs. However, the relatively small number of available qubits (around 1000 for the D-Wave 2X quantum annealer) poses a severe limitation to the range of problems that can be solved. This paper explores the suitability of preprocessing methods for reducing the sizes of the input programs and thereby the number of qubits required for their solution on quantum computers. Such methods allow us to determine the value of certain variables that hold in either any optimal solution (called strong persistencies) or in at least one optimal solution (weak persistencies). We investigate preprocessing methods for two important NP-hard graph problems, the computation of a maximum clique and a maximum cut in a graph. We show that the identification of strong and weak persistencies for those two optimization problems is very instance-specific, but can lead to substantial reductions in the number of variables
Posiform planting: generating QUBO instances for benchmarking
We are interested in benchmarking both quantum annealing and classical algorithms for minimizing quadratic unconstrained binary optimization (QUBO) problems. Such problems are NP-hard in general, implying that the exact minima of randomly generated instances are hard to find and thus typically unknown. While brute forcing smaller instances is possible, such instances are typically not interesting due to being too easy for both quantum and classical algorithms. In this contribution, we propose a novel method, called posiform planting, for generating random QUBO instances of arbitrary size with known optimal solutions, and use those instances to benchmark the sampling quality of four D-Wave quantum annealers utilizing different interconnection structures (Chimera, Pegasus, and Zephyr hardware graphs) and the simulated annealing algorithm. Posiform planting differs from many existing methods in two key ways. It ensures the uniqueness of the planted optimal solution, thus avoiding groundstate degeneracy, and it enables the generation of QUBOs that are tailored to a given hardware connectivity structure, provided that the connectivity is not too sparse. Posiform planted QUBOs are a type of 2-SAT boolean satisfiability combinatorial optimization problems. Our experiments demonstrate the capability of the D-Wave quantum annealers to sample the optimal planted solution of combinatorial optimization problems with up to 5, 627 qubits
Inferring the Dynamics of the State Evolution During Quantum Annealing
To solve an optimization problem using a commercial quantum annealer, one has
to represent the problem of interest as an Ising or a quadratic unconstrained
binary optimization (QUBO) problem and submit its coefficients to the annealer,
which then returns a user-specified number of low-energy solutions. It would be
useful to know what happens in the quantum processor during the anneal process
so that one could design better algorithms or suggest improvements to the
hardware. However, existing quantum annealers are not able to directly extract
such information from the processor. Hence, in this work we propose to use
advanced features of D-Wave 2000Q to indirectly infer information about the
dynamics of the state evolution during the anneal process. Specifically, D-Wave
2000Q allows the user to customize the anneal schedule, that is, the schedule
with which the anneal fraction is changed from the start to the end of the
anneal. Using this feature, we design a set of modified anneal schedules whose
outputs can be used to generate information about the states of the system at
user-defined time points during a standard anneal. With this process, called
"slicing", we obtain approximate distributions of lowest-energy anneal
solutions as the anneal time evolves. We use our technique to obtain a variety
of insights into the annealer, such as the state evolution during annealing,
when individual bits in an evolving solution flip during the anneal process and
when they stabilize, and we introduce a technique to estimate the freeze-out
point of both the system as well as of individual qubits
Initial state encoding via reverse quantum annealing and h-gain features
Quantum annealing is a specialized type of quantum computation that aims to
use quantum fluctuations in order to obtain global minimum solutions of
combinatorial optimization problems. D-Wave Systems, Inc., manufactures quantum
annealers, which are available as cloud computing resources, and allow users to
program the anneal schedules used in the annealing computation. In this paper,
we are interested in improving the quality of the solutions returned by a
quantum annealer by encoding an initial state. We explore two D-Wave features
allowing one to encode such an initial state: the reverse annealing and the
h-gain features. Reverse annealing (RA) aims to refine a known solution
following an anneal path starting with a classical state representing a good
solution, going backwards to a point where a transverse field is present, and
then finishing the annealing process with a forward anneal. The h-gain (HG)
feature allows one to put a time-dependent weighting scheme on linear ()
biases of the Hamiltonian, and we demonstrate that this feature likewise can be
used to bias the annealing to start from an initial state. We also consider a
hybrid method consisting of a backward phase resembling RA, and a forward phase
using the HG initial state encoding. Importantly, we investigate the idea of
iteratively applying RA and HG to a problem, with the goal of monotonically
improving on an initial state that is not optimal. The HG encoding technique is
evaluated on a variety of input problems including the weighted Maximum Cut
problem and the weighted Maximum Clique problem, demonstrating that the HG
technique is a viable alternative to RA for some problems. We also investigate
how the iterative procedures perform for both RA and HG initial state encoding
on random spin glasses with the native connectivity of the D-Wave Chimera and
Pegasus chips.Comment: arXiv admin note: substantial text overlap with arXiv:2009.0500
Advanced anneal paths for improved quantum annealing
Advances in quantum annealing technology make it possible to obtain high
quality approximate solutions of important NP-hard problems. With the newer
generations of the D-Wave annealer, more advanced features are available which
allow the user to have greater control of the anneal process. In this
contribution, we study how such features can help in improving the quality of
the solutions returned by the annealer. Specifically, we focus on two of these
features: reverse annealing and h-gain. Reverse annealing (RA) was designed to
allow refining a known solution by backward annealing from a classical state
representing the solution to a mid-anneal point where a transverse field is
present, followed by an ordinary forward anneal, which is hoped to improve on
the previous solution. The h-gain (HG) feature stands for time-dependent gain
in Hamiltonian linear () biases and was originally developed to help study
freezeout times and phase transitions in spin glasses. Here we apply HG to bias
the quantum state in the beginning of the annealing process towards the known
solution as in the RA case, but using a different apparatus. We also
investigate a hybrid reverse annealing/h-gain schedule, which has a backward
phase resembling an RA step and whose forward phase uses the HG idea. To
optimize the parameters of the schedules, we employ a Bayesian optimization
framework. We test all techniques on a variety of input problems including the
weighted Maximum Cut problem and the weighted Maximum Clique problem. Our
results show that each technique may dominate the others depending on the input
instance, and that the HG technique is a viable alternative to RA for some
problems
Peering into the Anneal Process of a Quantum Annealer
Commercial adiabatic quantum annealers have the potential to solve important
NP-hard optimization problems efficiently. The newest generation of those
machines additionally allows the user to customize the anneal schedule, that
is, the schedule with which the anneal fraction is changed from the start to
the end of the annealing. In this work we use the aforementioned feature of the
D-Wave 2000Q to attempt to monitor how the anneal solution evolves during the
anneal process. This process we call slicing: at each time slice during the
anneal, we are able to obtain an approximate distribution of anneal solutions.
We use our technique to obtain a variety of insights into the D-Wave 2000Q. For
example, we observe when individual bits flip during the anneal process and
when they stabilize, which allows us to determine the freeze-out point for each
qubit individually. We highlight our results using both random QUBO (quadratic
unconstrained binary optimization) instances and, for better visualization,
instances which we specifically optimize (using our own genetic algorithm) to
exhibit a pronounced evolution of its solution during the anneal
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