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In mathematics and applications, quantum annealing (QA)^[1]^[2] is a general method for finding the global minimum of a given objective function over a given set of candidate solutions (candidate states), by a process using quantum fluctuations. It is used mainly for problems where the search space is discrete (combinatorial optimization problems) with many local minima; such as finding the ground state of a spin glass.

Quantum annealing starts from a quantum-mechanical superposition of all possible states (candidate states) with equal weights. Then the system evolves following the time-dependent Schrödinger equation, a natural quantum-mechanical evolution of physical systems. The amplitudes of all candidate states keep changing, realizing a quantum parallelism, according to the time-dependent strength of the transverse field, which causes quantum tunneling between states. If the rate of change of the transverse-field is slow enough, the system stays close to the ground state of the instantaneous Hamiltonian, i.e., adiabatic quantum computation ^[3]. The transverse field is finally switched off, and the system is expected to have reached the ground state of the classical Ising model that corresponds to the solution to the original optimization problem. An experimental demonstration of the success of quantum annealing for random magnets was reported immediately after the initial theoretical proposal. ^[4]

Comparison to simulated annealing

Quantum annealing can be compared to simulated annealing, whose "temperature" parameter plays a similar role to QA's tunneling field strength. In simulated annealing, the temperature determines the probability of moving to a state of higher "energy" from a single current state. In quantum annealing, the strength of transverse field determines the quantum-mechanical probability to change the amplitudes of all states in parallel. Analytical ^[5] and numerical ^[6] evidence suggests that quantum annealing outperforms simulated annealing under certain conditions.

Quantum mechanics: Analogy & advantage

The tunneling field is basically a kinetic energy term that does not commute with the classical potential energy part of the original glass. The whole process can be simulated in a computer using quantum Monte Carlo (or other stochastic technique), and thus obtain a heuristic algorithm for finding the ground state of the classical glass.

In the case of annealing a purely mathematical objective function, one may consider the variables in the problem to be classical degrees of freedom, and the cost functions to be the potential energy function (classical Hamiltonian). Then a suitable term consisting of non-commuting variable(s) (i.e. variables that has non-zero commutator with the variables of the original mathematical problem) has to be introduced artificially in the Hamiltonian to play the role of the tunneling field (kinetic part). Then one may carry out the simulation with the quantum Hamiltonian thus constructed (the original function + non-commuting part) just as described above. Here, there is a choice in selecting the non-commuting term and the efficiency of annealing may depend on that.

It has been demonstrated experimentally as well as theoretically, that quantum annealing can indeed outperform thermal annealing (simulated annealing) in certain cases, especially where the potential energy (cost) landscape consists of very high but thin barriers surrounding shallow local minima. Since thermal transition probabilities (~ $\exp {(-\Delta /k_{B}T)}$ ; $T$ => Temperature, $k_{B}$ => Boltzmann constant) depend only on the height $\Delta$ of the barriers, for very high barriers, it is extremely difficult for thermal fluctuations to get the system out from such local minima. However, as argued earlier ^[7], the quantum tunneling probability through the same barrier depends not only the height $\Delta$ of the barrier, but also on its width $w$ and is approximately given by $\exp {(-\Delta ^{1/2}w/\Gamma )}$ ; $\Gamma$ => Tunneling field.^[8] If the barriers are thin enough $(w\ll \Delta ^{1/2})$ , quantum fluctuations can surely bring the system out of the shallow local minima. This $O(N^{1/2})$ advantage in quantum search (compared to the classical effort growing linearly with $\Delta$ or $N$ , the problem size) is well established.^[9]

It is speculated that in a quantum computer, such simulations would be much more efficient and exact than that done in a classical computer, because it can perform the tunneling directly, rather than needing to add it by hand. Moreover, it may be able to do this without the tight error controls needed to harness the quantum entanglement used in more traditional quantum algorithms.

Implementations

In 2011, D-Wave Systems announced the first commercial quantum annealer on the market by the name D-Wave One and published a paper in Nature ^[10] on its performance. The company claims this system uses a 128 qubit processor chipset.^[11] On May 25, 2011 D-Wave announced that Lockheed Martin Corporation entered into an agreement to purchase a D-Wave One system.^[12] On October 28, 2011 USC's Information Sciences Institute took delivery of Lockheed's D-Wave One, where it has become the first operational commercial "quantum computer".^[13]

In May 2013 it was announced that a consortium of Google, NASA AMES and the non-profit Universities Space Research Association purchased an adiabatic quantum computer from D-Wave Systems with 512 qubits.^[14]^[15]

D-Wave's architecture differs from traditional quantum computers in that it has noisy, high error-rate qubits, since it is designed specifically for quantum annealing.

References

^ T. Kadowaki and H. Nishimori, "Quantum annealing in the transverse Ising model" Phys. Rev. E 58, 5355 (1998)
^ A. B. Finilla, M. A. Gomez, C. Sebenik and D. J. Doll, "Quantum annealing: A new method for minimizing multidimensional functions" Chem. Phys. Lett. 219, 343 (1994)
^ E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Ludgren and D. Preda, "A Quantum adiabatic evolution algorithm applied to random instances of an NP-Complete problem" Science 292, 472 (2001)
^ J. Brooke, D. Bitko, T. F. Rosenbaum and G. Aeppli, "Quantum annealing of a disordered magnet", Science 284 779 (1999)
^ S. Morita and H. Nishimori, "Mathematical foundation of quantum annealing", J.Math. Phys. 49, 125210 (2008)
^ G. E. Santoro and E. Tosatti, "Optimization using quantum mechanics: quantum annealing through adiabatic evolution" J. Phys. A 39, R393 (2006)
^ P. Ray, B. K. Chakrabarti and A. Chakrabarti, "Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations", Phys. Rev. B 39 11828 (1989)
^ A. Das, B.K. Chakrabarti, and R.B. Stinchcombe, "Quantum annealing in a kinetically constrained system", Phys. Rev. E 72 art. 026701 (2005)
^ J. Roland and N.J. Cerf, "Quantum search by local adiabatic evolution",Phys. Rev. A 65, 042308 (2002)
^ M. W. Johnson et al., "Quantum annealing with manufactured spins", Nature 473 194 (2011)
^ "Learning to program the D-Wave One". Retrieved 11 May 2011.
^ "D-Wave Systems sells its first Quantum Computing System to Lockheed Martin Corporation". 2011-05-25. Retrieved 2011-05-30.
^ "USC To Establish First Operational Quantum Computing System at an Academic Institution". 2011-10-28. Retrieved 2011-10-30.
^ N. Jones, Google and NASA snap up quantum computer, Nature (2013), doi: 10.1038/nature.2013.12999, http://www.nature.com/news/google-and-nasa-snap-up-quantum-computer-1.12999.
^ V. N. Smelyanskiy, E. G. Rieffel, S. I. Knysh, C. P. Williams, M. W. Johnson, M. C. Thom, W. G. Macready, K. L. Pudenz, "A Near-Term Quantum Computing Approach for Hard Computational Problems in Space Exploration", arXiv:1204.2821

Review Articles and Books

G. E. Santoro and E. Tosatti, "Optimization using quantum mechanics: quantum annealing through adiabatic evolution" J. Phys. A 39, R393 (2006).
A. Das and B. K. Chakrabarti, "Colloquium: Quantum annealing and analog quantum computation" Rev. Mod. Phys. 80, 1061 (2008).
S. Suzuki, J.-i. Inoue & B. K. Chakrabarti,"Quantum Ising Phases & Transitions in Transverse Ising Models", Springer, Heidelberg (2013), Chapter 8 on Quantum Annealing.
V. Bapst, L. Foini, F. Krzakala, G. Semerjian and F. Zamponi, "The quantum adiabatic algorithm applied to random optimization problems: The quantum spin glass perspective", Physics Reports 523 127 (2013).
Arnab Das and Bikas K Chakrabarti (Eds.), "Quantum Annealing and Related Optimization Methods", Lecture Note in Physics, Vol. 679, Springer, Heidelberg (2005).
Anjan K. Chandra, Arnab Das and Bikas K Chakrabarti (Eds.),"Quantum Quenching, Annealing and Computation", Lecture Note in Physics, Vol. 802, Springer, Heidelberg (2010).
A. Ghosh and S. Mukherjee, "Quantum Annealing and Computation: A Brief Documentary Note", arXiv:1310.1339.

Category:Stochastic optimization Category:Optimization algorithms and methods Category:Quantum algorithms

[1] T. Kadowaki and H. Nishimori, "Quantum annealing in the transverse Ising model" Phys. Rev. E 58, 5355 (1998)

[2] A. B. Finilla, M. A. Gomez, C. Sebenik and D. J. Doll, "Quantum annealing: A new method for minimizing multidimensional functions" Chem. Phys. Lett. 219, 343 (1994)

[3] E. Farhi, J. Goldstone, S. Gutmann, J. Lapan, A. Ludgren and D. Preda, "A Quantum adiabatic evolution algorithm applied to random instances of an NP-Complete problem" Science 292, 472 (2001)

[4] J. Brooke, D. Bitko, T. F. Rosenbaum and G. Aeppli, "Quantum annealing of a disordered magnet", Science 284 779 (1999)

[5] S. Morita and H. Nishimori, "Mathematical foundation of quantum annealing", J.Math. Phys. 49, 125210 (2008)

[6] G. E. Santoro and E. Tosatti, "Optimization using quantum mechanics: quantum annealing through adiabatic evolution" J. Phys. A 39, R393 (2006)

[7] P. Ray, B. K. Chakrabarti and A. Chakrabarti, "Sherrington-Kirkpatrick model in a transverse field: Absence of replica symmetry breaking due to quantum fluctuations", Phys. Rev. B 39 11828 (1989)

[8] A. Das, B.K. Chakrabarti, and R.B. Stinchcombe, "Quantum annealing in a kinetically constrained system", Phys. Rev. E 72 art. 026701 (2005)

[9] J. Roland and N.J. Cerf, "Quantum search by local adiabatic evolution",Phys. Rev. A 65, 042308 (2002)

[10] M. W. Johnson et al., "Quantum annealing with manufactured spins", Nature 473 194 (2011)

[11] "Learning to program the D-Wave One". Retrieved 11 May 2011.

[12] "D-Wave Systems sells its first Quantum Computing System to Lockheed Martin Corporation". 2011-05-25. Retrieved 2011-05-30.

[13] "USC To Establish First Operational Quantum Computing System at an Academic Institution". 2011-10-28. Retrieved 2011-10-30.

[14] N. Jones, Google and NASA snap up quantum computer, Nature (2013), doi: 10.1038/nature.2013.12999, http://www.nature.com/news/google-and-nasa-snap-up-quantum-computer-1.12999.

[15] V. N. Smelyanskiy, E. G. Rieffel, S. I. Knysh, C. P. Williams, M. W. Johnson, M. C. Thom, W. G. Macready, K. L. Pudenz, "A Near-Term Quantum Computing Approach for Hard Computational Problems in Space Exploration", arXiv:1204.2821

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