Quantum Annealing by Hidetoshi Nishimori
Executive Summary
Quantum annealing is a quantummechanical metaheuristic (generic and approximate method) to solve combinatorial optimization and sampling problems. It was proposed by Kadowaki and Nishimori in 1998 in the form now widely used.
Many problems of practical importance can be formulated as combinatorial optimization, and the development of efficient methods to solve combinatorial optimization problems has enormous practical significance.
Also of recent focus of research activities are to use quantum annealing for Boltzmann machine learning and to simulate quantum magnets by quantum annealing devices.
A Canadian company DWave Systems has developed a hardware to realize quantum annealing. Google, NASA, and LockheedMartin, and Los Alamos National Laboratory are among their customers. Also, many corporations and organizations, including Oak Ridge National Laboratory, Volkswagen, Recruit Communications, and Denso are using the machine over the cloud.
Researchers are discussing if the DWave machine can solve optimization problems more quickly/efficiently than ordinary computers do. The answer depends on the problem and criteria of speed. For more details, see here for a list of recent papers on this and related topics of quantum annealing.
The speed of gate model (circuit model) quantum computers far exceeds that of usual (classical) computers for some problems. Gatemodel quantum computers are universal because they can in principle process any computational tasks. Notice that the speed does not exceed that of classical computers except for a special set of problems. An important example is the simulation of small molecules. In this sense, gatemodel quantum computers are better regarded as specialpurpose machines in practice.
Several projects are running worldwide to develop key technologies for highperformance quantum annealers, including IARPA QEO and NEDO (in Japanese).
Introduction
Quantum annealing is a generic approximate method to search for the minimum of a cost function (multivariable function to be minimized) through a control of quantum fluctuations. Quantum annealing is used mainly for combinatorial optimization problems with discrete variables. Many practically important problems can be formulated as combinatorial optimization, including machine learning for clustering, distribution of components in factories, and route optimization in traffic. Finding efficient methods to solve such optimization problems is of enormous social significance, which is the key reason why quantum annealing attracts much attention. Also of current research interest are the sampling problem for machine learning and simulation of quantum magnets.
To perform quantum annealing, one writes the cost function in terms of the Ising model of magnetism. The Hamiltonian (energy function) of the Ising model should be chosen such that its lowestenergy state (ground state) represents the solution to the combinatorial optimization problem. A term representing quantummechanical fluctuations is then added to the Hamiltonian to induce quantum transitions between states.
One first chooses a very large value for the coefficient for the quantum term, large relative to the coefficient of the cost function. This leads to a uniform probability of the existence of states over all states as in the left panel (blue line) of the figure. This is a trivial state and can be realized easily. Then one gradually decreases the magnitude of the coefficient of the quantum term. The state of the system follows the timedependent Schrödinger equation, the natural time evolution of a physical system. The coefficient of the quantum term finally reaches zero, and only the classical term of the Hamiltonian, the Ising model whose ground state is to be identified, remains at the end of the process. It is expected that the probability is by far largest for the solution (the rightmost panel of the figure) to the combinatorial optimization problem at the end of the annealing process.
Figure: The process starts from the state on the leftmost figure with a uniform probability distribution over all classical states. The system follows the Schrödinger equation to reach the final state on the right, in which the probability focuses on the solution to the combinatorial optimization problem. The abscissa represents the classical state, and the ordinate is for the value of the cost function or the classical Ising Hamiltonian to be minimized (black) and the quantummechanical probability of states (blue).
Formulation
The Hamiltonian of the Ising model with a transverse field (the transversefield Ising model) is written as follows,
where σ is the Pauli matrix. The first term on the right side is the (classical) Ising model. The second term, the transversefield term, causes quantum fluctuations between classical states. The coefficient Γ(t) starts with a very large value and decreases toward zero at the end of the annealing process, following the natural time evolution of the Schrödinger equation.
Historical note
This prescription of quantum annealing as presently widely used was first proposed in 1998 [A1], where the method was shown to be very effective for several combinatorial optimization problems. An experiment for a magnetic system soon followed [A2] as well as largescale numerical studies of complex systems [A3]. Notice that Kadowaki had already done in his Thesis [A4] something similar to [A3]. Also, a continuousspace formulation, different from the Isingmodel formulation for combinatorial optimization, had appeared in [A5]. A paper published in 2001 [A6] focused attetion to computational complexity using the following Hamiltonian, under which the system evolution terminates after a finite time T ,It was proposed in [A6] to use the term "quantum adiabatic evolution" because the system is supposed to follow the instantaneou ground state adiabatically from the initial state to the final state.
[A1] T. Kadowaki and H. Nishimori, Phys. Rev. E58 (1998) 5355.
[A2] J. Brooke, D. Bitko, T.F. Rosenbaum and G. Aeppli, Science 284 (1999) 779.
[A3] G. E. Santoro, R. Martonak, E. Tosatti, and R. Car, Science 295 (2002) 2427.
[A4] T. Kadowaki, Thesis, Tokyo Institute of Technology (1998); arXiv:quantph/0205020
[A5] A. B. Finnila, M. A. Gomez, C. Sebenik. C. Stenson, and J. D. Doll, Chem. Phys. Lett. 219 (1994) 343.
[A6] E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Lundgren, and D. Preda, Science 292 (2001) 472. See also arXiv:quantph/0007071.
Quantum annealing is sometimes understood as a "noisy version" of quantum adiabatic computation in the sense that the former allows for environmental noise as well as diabatic transitions in contrast to the latter. A careful scrutiny of the abovelisted papers reveals that these two ideas are essentially the same, at least in their initial formulations. Nevertheless, they differ in several aspects including the strict imposition of adiabaticity in quantum adiabatic computation. Quantum annealing, in contrast, allows for a diabatic time evolution under the expectation that the system reaches the target state with a nonvanishing probability even without the adiabatic condition. Notice that noise is not intrinsic in quantum annealing, although it is not excluded.
Also, references [A2,A3,A6] all cite [A1], and a paper by DWave researchers [A7] states "DWave processors solve Ising problems by quantum annealing in the form proposed by Kadowaki and Nishimori".[A7] J. King et al, arXiv:1701.04579.
Recent studies show that quantum annealing is also useful to sample the BoltzmannGibbs distribution for machine learning. See [A8] for review. Also worth attention are simulations of quantum magnets by quantum annealers [A9].
[A8] J. Biamonte et al, Nature 549, 195 (2017); V. Dunjko and H. J. Briegel, arXiv:1709.02779
[A9] A. King et al, Nature 560, 456 (2018); R. Harris et al, Science 361, 162 (2018).
Strengths and Weaknesses
Let us compare quantum annealing with a few related methods.
Simulated annealing and quantum Monte Carlo
Simulated annealing is a generic method to solve optimization problems
using classical probability, or thermal effects in analogy with statistical
mechanics. It has been shown that quantum annealing, in theory, allows
the system to approach the correct solution more quickly than simulated
annealing if we control the coefficient of the transverse field Γ(t) as a function of time t in the same manner as the temperature T(t) in simulated time [B1]. Recent extensive benchmark results indicate that
quantum annealing has scaling advantages to reach the correct solutions
than simulated annealing in many cases of spin glass type problems [B2].
It is also shown analyutically for a simple problem that quantum annealing
is exponentially faster than simulated annealing [B3]
Quantum annealing in its naive formulation can often be simulated on a
conventional computer using quantum Monte Carlo simulations. One of the
advantages of the dedicated quantumannealing device in its present form
is to perform the annealing process much more quickly, in terms of the
absolute time, than by simulations on the convetional computer. This is
an advantage of a constant numerical factor, not necessarily in the scaling
advantage which matters when the problem size increases. Nevertheless,
when this constant numerical factor is large, it means practical usefulness.
See [B2] for a recent benchmark.
[B1] S. Morita and H. Nishimori, .J. Math. Phys. 49, 125210 (2008).
[B2] T. Albash and D. A. Lidar, Phys. Rev. X8, 031016 (2018).
[B3] Y. Susa et al, arXiv:1808.01582
Gate (circuit) model
Many of the studies of quantum computation concern the gate model, in which one applies quantum gates one by one to the state of a system toward the desired solution of the problem. A large number of research papers have been published, but experiments have been plagued by decoherence, i.e., destruction of quantum states through interactions with the environment. It will take some more time before a device of practical usefulness is built. Also, practical applications of the gatemodel quantum computer are relatively limited: e.g., integer factoring, simulations of quantum systems, and solving a class of linear algebra problems. Quantum algorithms for these problems have been proven to show an exponential speedup. For other problems, gatemodel quantum computers are not known to be fast. Also of interest is quantum approximate optimization algorithm (QAOA) for combinatorial optimization [B4], though it is not known at present if this algorithm runs fast. Watch this excellent lecture by Peter Shor.
[B4] E. Farhi and J. Goldstone, arXiv:1411.4028
One of the realistic applications of gate model quantum computers is quantum simulation of smallsize molecules [B5]. Also, methods to efficiently solve linear equations by gate model quantum computers may be applied to machine learning and related tasks [B6] though some care must be taken [B7].
[B5] B. Bauer et al, Phys. Rev. X6, 031045 (2016); P.J.J. O'Malley et al, Phys. Rev. X6, 031007 (2016); Proc. Nat. Acad. Sci. USA., 114, 7555 (2017)
[B6] J. Biamonte et al, Nature 549, 195 (2017)
[B7] S. Aaronson, Nature Physics, 11, 291 (2015).
A quantum annealing machine can be used for combinatorial optimization, which has a wide range of practical applications. Also, it is known that an extended form of quantum annealing is theoretically equivalent to the circuit model [B8, B9]. It would also be remarkable that the same extention leads to an exponential speedup of quantum annealing in comparison with the conventional simple protocol, at least for a special problem [B10].
[B8] D. Aharonov et al, SIAM J. Comp. 37 (2007) 166.
[B9] J. D. Biamonte and P. J. Love, Phys. Rev. A78, 012352 (2008).
[B10] H. Nishimori and K. Takada, Front. ICT 4, 2 (2017) and references therein.
Quantum annealing is less susceptible to decoherence than the circuit model is since the former follows the inherentlystable ground state or low excited states.
Gate (circuit) model 
Quantum annealing 


Strengths 
Several algorithms of practical importance are proven to be exponentially faster than classical methods. 
Many problems of practical importance, such as machine learning, can be represented as combinatorial optimization. Resilient against noise. 
Weaknesses 
Qubits are very susceptible to decoherence, i.e. easily destroyed by noise. Error correction needs a very heavy overhead. Faster than conventional machines only for a few problems as long as practical relevance is concerned. 
Problems are yet to be found that are proved to be solved exponentially more efficiently than by classical methods and are of practical significance.Errorcorrection scheme yet to be established. 
Current Status of Implementation 
About 50 qubits （superconducting circuits） 
About 2,000 qubits（superconducting circuits） 
Prospects 
Needs extremely many qubits, tens of millions or more, if one implements error corrections for problems of pracitical significance. Will take decades to realize such. If an error correction is set aside for the moment, a machine with hundreds of qubits may be realized in several to ten years. 
Applications by hybrid operation with conventional computers are reported. 
Relation with classical (conventional) computers
The goal of quantum annealing is to solve combinatorial optimization and
related problems. Quantum annealing machines will not replace conventional
computers for other problems. This fact is shared by the gate model. One
should not misunderstand that conventional computers will be replaced by
quantum computers for any problems.
As long as optimization problems are concerned, there exists some evidence
that, for some problems, quantum annealing outperforms simulated annealing
carried out on classical computers [B2, B3]. It is nevertheless unclear
whether or not there exist problems of significant practical impact where
quantum annealing achieves a significant speedup over conventional methods.
Notice that there indeed exist a problem in which an exponential speedup by quantum annealing (not by adiabatic
quantum computation) is proved although this problem is of limited practical
usefulness.
Related notes
Quantum annealing needs an exponentially long time to find the exact solution to most difficult problems among all combinatorial optimization problems.
Indeed, it is explicitly shown from the analysis of energy spectra that quantum annealing spends an exponentially long time to solve several specific problems.
It has nevertheless been well known from a number of numerical and analytical studies that quantum annealing improves the performance, if not exponentially, of solving many combinatorial optimization problems in comparison with classical simulated annealing.
It is not expected that quantum annealing would drastically reduce the time to reach solutions for many problems for which only exponentially slow classical algorithms are known. It is also incorrect, on the other hand, that there would exist no such cases. Indeed this paper shows that an exponential speedup is reached for the binarytree problem. A small downside of this result is that the problem is of limited practical interest.
It is not necessary for quantum annealing to follow the instantaneous ground state. Evaluation of computational complexity based on the energygap analysis would clarifly only part of the capabilities of quantum annealing. Indeed, the abovementioned solution to the binarytree problem does not follow the instantaneous ground state. In other words, adiabatic quantum computation (AQC) is a subset of quantum annealing (QA) but the converse is not true. Also, quantum annealing may be understood to allow for interactions with the environment including temperature effects.
Theoretical equivalence is proved between the circuit (gate) model and quantum annealing with extended interactions [B8, B9]. It is nevertheless difficult in practice to translate the language of one to that of the other. The two approaches are better considered to be different in practice.
Several papers show that the DWave machine operates under the principle of quantum mechanics, at least as long as a part of the system is involved [B11, B12].
[B11] T. Lanting et al, Phys. Rev. X 4, 021041 (2014)
[B12] S. Boixo et al, Nature Commun. 4, 2067 (2013), Nature Commun. 7, 10327 (2016)
It may seem counterintuitive that data from the DWave machine shows quantumness when the coherence time is of the order of 10 nano seconds whereas the computation time is of the order of 10 micro seconds or longer. Nevertheless, a theoretical argument suggests that, when decoherence applies in the energy eigenbasis, noise is not as detrimental as naively expected [B13].
[B13] T. Albash and D. Lidar, Phys, Rev. A 91, 062320 (2015)
Further Readings
Listed below are review articles. I refer the reader to papers cited therein for original papers. Contributions from our group at Tokyo Institute of Technology are shown in my list of papers.
Watch also those presentations at AQC2018, the Adiabatic Quantum Computing Conference, in June 2018, where a number of important recent developments have been reported.
T. Albash and D. A. Lidar, Rev. Mod. Phys. 90, 015002 (2018)
A. Das and B. K. Chakrabarti, Rev. Mod. Phys. 80, 1061 (2008)
G.E. Santoro and E. Tosatti, J. Phys A 39, R393 (2006)
S. Suzuki, J. Inoue, and B. K. Chakrabarti, Springer, 2012
A. Ghosh and S. Mukherjee, arXiv:1310.1339
Watch here a series of lectures on quantum computing at a conference at IPAM, UCLA, in August, 2018.
Lecture by Richard Harris of DWave Systems