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The Quantum Approach to NP-Hard Optimization Problems
Name: Sam Prager
Bio: BSEE Undergraduate. I worked at USC’s Information Sciences Institute (ISI) on the
D-Wave quantum computing research team during summer 2013. During my time at ISI,
I investigated the encoding of NP-Hard problems onto the D-Wave, specifically working
on dense graph partitioning, and was fortunate enough to work with the researchers who
first demonstrated the quantum nature of the D-Wave (ref. [10]).
Keywords: D-Wave, Quantum Computing, Optimization, NP-Hard, Qubit, Adiabatic
Quantum Computing, Quantum Annealing, Ising Model, computing.
Suggested Multimedia: D-Wave website:
[http://www.dwavesys.com/en/dw_homepage.html]
Google’s Quantum AI Lab promo video:
[http://www.youtube.com/watch?v=CMdHDHEuOUE]
Abstract:
Quantum computing is a new and exciting class of technology with the potential
to fundamentally change the way we live. Although still in the early stages of
infancy, quantum computers are already showing a great deal of promise as
game-changing tools, particularly in the study of combinatorial optimization – a
broad and extremely important class of applied mathematics with applications in
just about every field imaginable. Researchers are only beginning to uncover the
capabilities of quantum computers and with recent purchases by some of the
world’s premier technology groups, the technology is poised to open many new
and exciting doors in the coming years.
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Introduction
For years scientists have hypothesized a new class of computer so impossibly
powerful that just one could solve problems that would take all of the world’s
computational power the length of the universe to solve. Quantum computers, so called
because they rely on the strange properties of quantum physics, have the potential to
outclass today’s computers based on classical physics by unimaginable amounts. At the
forefront of quantum computer development is a company called D-Wave Systems. Selfdescribed as “The Quantum Computing Company,” D-Wave is the first and only
company to make commercially available quantum computers [1]. Although there is an
ongoing debate surrounding the extent to which quantum phenomena come into play in
the computer, the latest evidence strongly suggests that they are at the very least present
and customers seem to be convinced. To date D-Wave has built and sold two computers:
The 128 qubit D-Wave One, sold in 2011 to Lockheed Martin and located at USC’s
Information Sciences Institute (ISI) in Marina Del Ray, CA and The 512 qubit D-Wave
Two, purchased in May 2013 by a collaboration of NASA, Google, and the Universities
Space Research Association (Figure 1) [2]. Researchers at these institutions aim to
discover ways in which the D-Wave machines can be used to solve a notoriously difficult
class of problems with far-reaching applications too challenging for classical computers:
discrete combinatorial optimization problems [1]. This hugely important class of
problems has applications in a diverse number of fields ranging from network computing,
machine learning, and artificial intelligence to neuroscience, zoology, and airline travel.
Finding successful solutions to these problems would not only indicate the dawn of a new
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era in computer science and technology; it would revolutionize a vast range of disciplines
and solve a number of today’s grand challenges.
D-Wave Systems
Figure 1: (Left) The D-Wave One and D-Wave Two. (Right) The 512 qubit quantum annealing processor
at the heart of the D-Wave Two
Quantum Computing and the D-Wave Implementation
In order to understand the potential applications of the D-Wave computer, it is
important to understand the unique properties inherent in quantum computing (QC). The
unit of information in QC is the quantum bit or qubit. While classical bits store
information as either 1 in the on state or 0 in the off state, qubits can be both on and off
simultaneously due to the superposition of states in a quantum system. The superposition
of states persists until the qubits are observed and the wave function collapses at which
point they are seen to be in one state or the other. This means that information requiring
2n classical bits can theoretically be stored in just n qubits and operations on all
combinations of n classical bits (2n operations) can be done in just one operation on n
qubits. Because of this property, quantum computers have a significant potential
advantage over classical computers when finding an optimal solution from many possible
options.
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The theory of adiabatic quantum computation (AQC) is a model that focuses on
using superposition and other QC properties to arrive at the lowest energy of a complex
energy function. In AQC, a quantum system is initialized in its lowest possible energy
state called the ground state. Adiabatically, meaning without heat transfer, the energy
function known as the Hamiltonian is evolved from the simple Hamiltonian that
described the initial system to a more complex Hamiltonian in such a way that the system
remains in its lowest possible energy configuration [3]. By ‘fixing’ the complex
Hamiltonian in a particular way such that it models a problem of interest, one can
interpret the final ground state of the AQC system as the problem’s solution. The
quantum processor at the core of the D-Wave, using superconducting flux qubits and
spin-spin qubit couplings, attempts to find the lowest energy states of a system by
utilizing a process intrinsically related to the AQC model called Quantum Annealing
(QA) [4 p. 3]. Because purely adiabatic processes are not practical in real world
situations, QA (illustrated in Figure 2 below) allows a degree of non-adiabatic energy
(heat) transfer and does not require the system to remain in the ground state for the entire
process [3].
Figure 2: (a) Comparison of Classical
annealing in which potential energy
barriers can only be crossed by thermal
excitation to higher energy states and
Quantum annealing which allows
tunneling through potential barriers. (b)
Weight of Disordering Hamiltonian (Γ)
and weight of Encoded Hamiltonian
(Λ) as a function of time throughout
the annealing process [6].
Nature 473
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In addition to the encoded Hamiltonian that describes the problem, QA also introduces a
disordering Hamiltonian that increases uncertainty in the system (as described by the
Heisenberg Uncertainty Principle). This allows the system to pass through high-energy
states to low energy states without ever occupying the high-energy state – a process
known as quantum tunneling (Figure 2.a) [4 p. 3]. By decreasing the weight of the
disordering Hamiltonian (Γ) from 1 to 0 and increasing the weight of the encoded
Hamiltonian (Λ) from 0 to 1 (Figure 2.b), the system will settle to a state that is a local
minimum of the encoded Hamiltonian and a good solution to the encoded problem,
though not necessarily the optimal solution if the annealing time is small [4 p. 3], [6].
Discrete Combinatorial Optimization
Some of the most challenging problems in applied mathematics and computer
science are problems that involve finding the optimal object from a set of objects each
consisting of a combination of discrete components [7]. Of specific interest is a subset of
optimization problems called Non-Deterministic Polynomial-time (NP)-Hard problems.
The classic NP-Hard problem is known as the traveling salesman problem (TSP) and is
described as such: A salesman needs to visit some number of cities and must visit each
city once and only once. What path should he take so that the total distance he travels is
minimized [8]? It is clear that using brute-force to find the solution to this problem
involves calculating and comparing the distances of every possible route and that as the
number of cities increases, the number of possible routes grows exponentially.
Any path optimization between nodes in an interconnected network, from the
quickest path to send a network packet over the Internet to DNA sequencing, can be
modeled as a variation of the TSP. In general, if we consider a graph G = (V, E) of
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vertices (V) / nodes (these would be the cities in the original example) connected by
edges (E), each with an associated weight corresponding to some arbitrary relationship
between the nodes it connects which models the desired problem (this would be distance
in the original example), the path that minimizes the sum of its constituent edge weights
is the optimal solution [9 p. 12].
Ising Model Formulations and the Quantum Hamiltonian
For even a reasonable number of nodes, the computing power required to solve
NP-Hard optimization problems grows so fast it becomes impossible for classical
computers to come even remotely close to finding optimal solutions [8]. For AQC based
quantum computers, however, these sorts of problems are much simpler. The key is a
statistical model originally developed to describe ferromagnetic systems known as the
Ising Model. Using this model, theoreticians have been able to formulate many discrete
combinatorial optimization problems as the Ising Hamiltonian of a distribution of
disordered interacting magnets called spin glasses [9 p. 3]. Spin glasses are perfect
classical analogues to qubits making it possible to interpret the Ising Hamiltonian as a
Quantum Hamiltonian whose ground state is the optimal solution to the embedded NPHard optimization problem [9 p. 3]. By encoding this Hamiltonian into the QA processor,
low energy solutions to NP-Hard problems may be arrived at. As the annealing time
becomes large, the probability that the result obtained by QA is the ideal solution that
would be obtained by AQC approaches 1. In reality however, it is desirable for the
annealing time to be as small as possible and thus the system is much more likely to settle
at an energy state that is a local minima rather than the ground state located at the
absolute minima. Because of this single iterations of QA tend to produce heuristics rather
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than the optimal solution itself [4 p. 2]. Additionally, the limited number of qubits in the
D-Wave’s processor (512 qubtis in the D-Wave Two) makes it impossible to directly
embed quantum Hamiltonians for problems with more than 512 nodes. Because of these
limitations, the D-Wave must rely heavily on the assistance of classical algorithms able to
divide large problems with many nodes into multiple smaller problems that can be
directly embedded in a quantum Hamiltonian.
How Revolutionary Are the D-Wave Systems?
For more than a year after the D-Wave One was released, the big debate was over
whether or not the computer was quantum at all. Detractors claimed that the there was no
evidence that the annealing was quantum rather than thermal (see Figure 2.a) and that
while D-Wave had created an interesting computer made of superconducting magnets,
they had not created a quantum computer. The first significant evidence in support of DWave’s claim that the computer was quantum came from a USC research group at ISI.
The USC research group was able to show that the “experimental signature” created from
the aggregate of multiple annealing trials with varying parameters was consistent with the
signature produced by simulated quantum annealing, while being markedly different
from the signature produced by both simulated and experimental thermal annealing [10].
Despite the evidence that at least some degree of the computer operates on quantum
principles, the extent is still unclear. With the D-Wave Two now in the capable hands of
NASA, Google, and Universities Space Research Association, it is likely that more
information on the true nature of the D-Wave system will emerge. The ease with which
some of the world’s hardest problems can be put into forms easily solved by quantum
annealing has huge implications and will likely drive an explosion of interest in these
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systems over the next few years. Problems with nodes small enough to for the
Hamiltonian to be embedded directly can already be solved with high probability in just a
few iterations of QA. Additionally, some of the world’s brightest researchers and
computer scientists are now working on developing algorithms to optimize large
problems for D-Wave embedding. This, in conjunction with continued exponential
scaling of quantum processors, may very soon allow quantum computers to begin making
significant contributions towards finding the solutions of problems that cannot be solved
today. The ability to optimize huge amounts of data quickly and effectively would
profoundly change what we are technologically capable of -- things like simulation of the
brain, the earth’s ecosystem, and even the entire universe would all enter the realm of
possibility. Futurists and hopefuls predict that humanity is nearing a technological
singularity after which humanity would be so fundamentally changed that, like a black
hole, prediction beyond is impossible, and although such sweeping proclamations can
seem a bit fanciful, with such a potentially powerful technology just at its very dawn it’s
hard not to wonder.
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References:
[1] D-Wave Systems Homepage. D-Wave Systems Inc. [Online]. Available:
http://www.dwavesys.com
[2] N. Jones. (2013, May 16). Google and NASA Snap Up Quantum Computer. Nature.
[Online]. Available: http://www.nature.com/news/google-and-nasa-snap-up-quantumcomputer-1.12999
[3] N.G. Dickson, M.W. Johnson, M.H. Amin, et al. (2013, May 21). Thermally Assisted
Quantum Annealing of a 16-qubit Problem. Nature Communications. Volume 4. SP
1903. Nature Publishing Group [Online]. Available:
http://dx.doi.org/10.1038/ncomms2920
[4] G. Rose, W.G. Macready. (2007, August 10). An Introduction to Quantum Annealing.
D-Wave Systems Inc. [Online]. Available:
http://dwave.files.wordpress.com/2007/08/20070810_dwave_quantum_annealing.pdf
[5] R. Harris, M.Amin, N. Dickson, et al. (2010, September). Experimental Investigation
of an Eight Qubit Unit Cell in a Superconducting Optimization Processor. D-Wave
Systems Inc. [Online]. Available:
http://dwave.files.wordpress.com/2010/10/20100909_ dwave_unit_cell_overview_ii_richard_harris_1.pdf
[6] M. W. Johnson, M. H. S. Amin, S. Gildert, et al. (2011, May 12). Quantum Annealing
With Manufactured Spins. Nature 473, 194–198. DOI: 10.1038/nature10012. Nature
Publishing Group [Online]. Available: http://dx.doi.org/10.1038/nature10012
[7] J. Lee; A First Course in Combinatorial Optimization. Cambridge University Press.
2004. [Print]. ISBN 0-521-01012-8.
[8] Schrijver. (2013, February 3). A Course in Combinatorial Optimization. Pg. 97.
[Online]. Available: http://homepages.cwi.nl/~lex/files/dict.pdf
[9] A. Lucas. (2013, February 26). Ising Formulations of many NP Problems. Harvard
University. [Online]. Available: http://arxiv.org/abs/1302.5843
[10] B. Sergio, T. Albash, F. Spedalieri. (2012, December 7). Experimental Signature of
Programmable Quantum Annealing. Nature Communications. Volume 4. SP 3067.
DOI: 10.1038/ncomms3067. Nature Publishing Group [Online]. Available:
http://arxiv.org/pdf/1212.1739v1.pdf