28 research outputs found
Comparing Three Generations of D-Wave Quantum Annealers for Minor Embedded Combinatorial Optimization Problems
Quantum annealing is a novel type of analog computation that aims to use
quantum mechanical fluctuations to search for optimal solutions of Ising
problems. Quantum annealing in the Transverse Ising model, implemented on
D-Wave QPUs, are available as cloud computing resources. In this article we
report concise benchmarks across three generations of D-Wave quantum annealers,
consisting of four different devices, for the NP-Hard combinatorial
optimization problems unweighted maximum clique and unweighted maximum cut on
random graphs. The Ising, or equivalently QUBO, formulation of these problems
do not require auxiliary variables for order reduction, and their overall
structure and weights are not highly complex, which makes these problems simple
test cases to understand the sampling capability of current D-Wave quantum
annealers. All-to-all minor embeddings of size , with relatively uniform
chain lengths, are used for a direct comparison across the Chimera, Pegasus,
and Zephyr device topologies. A grid search over annealing times and the minor
embedding chain strengths is performed in order to determine the level of
reasonable performance for each device and problem type. Experiment metrics
that are reported are approximation ratios for non-broken chain samples and
chain break proportions. How fairly the quantum annealers sample optimal
maximum cliques, for instances which contain multiple maximum cliques, is also
quantified using entropy of the measured ground state distributions. The newest
generation of quantum annealing hardware, which has a Zephyr hardware
connectivity, performed the best overall with respect to approximation ratios
and chain break frequencies
4-clique network minor embedding for quantum annealers
Quantum annealing is a proposed algorithm for computing solutions to
combinatorial optimization problems. Current quantum annealing hardware is
relatively sparse and therefore requires graph minor embedding in order to map
an arbitrarily structured problem onto the sparse, and relatively small,
quantum annealing processor. This paper proposes a new minor embedding method
called 4-clique minor embedding. This is in contrast to the standard minor
embedding technique of using a path of linearly connected qubits in order to
represent a logical variable state. The 4-clique minor embedding is possible
because of Pegasus graph connectivity, which is the native hardware graph for
some of the current D-Wave quantum annealers. The Pegasus hardware graph has
many 4-cliques, and it is possible to form a graph composed entirely of paths
of connected 4-cliques, on which a problem can be minor embedded. The 4-clique
chains come at the cost of additional qubit usage on the hardware graph, but
they allow for stronger coupling within each chain thereby increasing chain
integrity and reducing chain breaks. This 4-clique minor embedding technique is
described in detail, and is compared against the standard linear path minor
embedding with some experiments on two D-Wave quantum annealing processors with
Pegasus hardware graphs. 4-clique minor embeddings can use weak chain strengths
while successfully carrying out the computation of minimizing random all-to-all
spin glass problem instances, in contrast to the linear path minor embeddings
which have high chain break frequencies for weak chain strengths. This work
shows that non standard minor embedding methods could be useful. For future
quantum annealing architectures, distributing minor embeddings over more
densely connected regions of hardware instead of linear paths may provide more
robust computations for minor embedding problems
Analysis of a Programmable Quantum Annealer as a Random Number Generator
Quantum devices offer a highly useful function - that is generating random numbers in a non-deterministic way since the measurement of a quantum state is not deterministic. This means that quantum devices can be constructed that generate qubits in a uniform superposition and then measure the state of those qubits. If the preparation of the qubits in a uniform superposition is unbiased, then quantum computers can be used to create high entropy, secure random numbers. Typically, preparing and measuring such quantum systems requires more time compared to classical pseudo random number generators (PRNGs) which are inherently deterministic algorithms. Therefore, the typical use of quantum random number generators (QRNGs) is to provide high entropy secure seeds for PRNGs. Quantum annealing (QA) is a type of analog quantum computation that is a relaxed form of adiabatic quantum computation and uses quantum fluctuations in order to search for ground state solutions of a programmable Ising model. Here we present extensive experimental random number results from a D-Wave 2000Q quantum annealer, totaling over 20 billion bits of QA measurements, which is significantly larger than previous D-Wave QA random number generator studies. Current quantum annealers are susceptible to noise from environmental sources and calibration errors, and are not in general unbiased samplers. Therefore, it is of interest to quantify whether noisy quantum annealers can effectively function as an unbiased QRNG. The amount of data that was collected from the quantum annealer allows a comprehensive analysis of the random bits to be performed using the NIST SP 800-22 Rev 1a testsuite, as well as min-entropy estimates from NIST SP 800-90B. The randomness tests show that the generated random bits from the D-Wave 2000Q are biased, and not unpredictable random bit sequences. With no server-side sampling post-processing, the microsecond annealing time measurements had a min-entropy of
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
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
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 unembedding techniques for quantum annealers
The D-Wave quantum annealers make it possible to obtain high quality
solutions of NP-hard problems by mapping a problem in a QUBO (quadratic
unconstrained binary optimization) or Ising form to the physical qubit
connectivity structure on the D-Wave chip. However, the latter is restricted in
that only a fraction of all pairwise couplers between physical qubits exists.
Modeling the connectivity structure of a given problem instance thus
necessitates the computation of a minor embedding of the variables in the
problem specification onto the logical qubits, which consist of several
physical qubits "chained" together to act as a logical one. After annealing, it
is however not guaranteed that all chained qubits get the same value (-1 or +1
for an Ising model, and 0 or 1 for a QUBO), and several approaches exist to
assign a final value to each logical qubit (a process called "unembedding"). In
this work, we present tailored unembedding techniques for four important
NP-hard problems: the Maximum Clique, Maximum Cut, Minimum Vertex Cover, and
Graph Partitioning problems. Our techniques are simple and yet make use of
structural properties of the problem being solved. Using Erd\H{o}s-R\'enyi
random graphs as inputs, we compare our unembedding techniques to three popular
ones (majority vote, random weighting, and minimize energy). We demonstrate
that our proposed algorithms outperform the currently available ones in that
they yield solutions of better quality, while being computationally equally
efficient
Parallel Quantum Annealing
Quantum annealers of D-Wave Systems, Inc., offer an efficient way to compute
high quality solutions of NP-hard problems. This is done by mapping a problem
onto the physical qubits of the quantum chip, from which a solution is obtained
after quantum annealing. However, since the connectivity of the physical qubits
on the chip is limited, a minor embedding of the problem structure onto the
chip is required. In this process, and especially for smaller problems, many
qubits will stay unused. We propose a novel method, called parallel quantum
annealing, to make better use of available qubits, wherein either the same or
several independent problems are solved in the same annealing cycle of a
quantum annealer, assuming enough physical qubits are available to embed more
than one problem. Although the individual solution quality may be slightly
decreased when solving several problems in parallel (as opposed to solving each
problem separately), we demonstrate that our method may give dramatic speed-ups
in terms of Time-to-Solution (TTS) for solving instances of the Maximum Clique
problem when compared to solving each problem sequentially on the quantum
annealer. Additionally, we show that solving a single Maximum Clique problem
using parallel quantum annealing reduces the TTS significantly.Comment: 13 pages. v4: format improvement