Quantum annealing

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Quantum annealing (QA) is an optimization process for finding the

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 or essentially tunneling through peaks. If the rate of change of the transverse field is slow enough, the system stays close to the ground state of the instantaneous Hamiltonian (also see adiabatic quantum computation).^[6] If the rate of change of the transverse field is accelerated, the system may leave the ground state temporarily but produce a higher likelihood of concluding in the ground state of the final problem Hamiltonian, i.e., diabatic quantum computation.^[7][8] 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.^[9] Quantum annealing has also been proven to provide a fast Grover oracle for the square-root speedup in solving many NP-complete problems.^[10]

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^[11] and numerical^[12] evidence suggests that quantum annealing outperforms simulated annealing under certain conditions (see^[13] for a careful analysis, and,^[14] for a fully solvable model of quantum annealing to arbitrary target Hamiltonian and comparison of different computation approaches).

Quantum mechanics: analogy and 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 have 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.^[15] Since thermal transition probabilities (proportional to ${\displaystyle e^{-{\frac {\Delta }{k_{B}T}}}}$, with ${\displaystyle T}$ the temperature and ${\displaystyle k_{B}}$ the Boltzmann constant) depend only on the height ${\displaystyle \Delta }$ 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 in 1989 by Ray, Chakrabarti & Chakrabarti,^[1] the quantum tunneling probability through the same barrier (considered in isolation) depends not only on the height ${\displaystyle \Delta }$ of the barrier, but also on its width ${\displaystyle w}$ and is approximately given by ${\displaystyle e^{-{\frac {{\sqrt {\Delta }}w}{\Gamma }}}}$, where ${\displaystyle \Gamma }$ is the tunneling field.^[16] This additional handle through the width ${\displaystyle w}$, in presence of quantum tunneling, can be of major help: If the barriers are thin enough (i.e. ${\displaystyle w\ll {\sqrt {\Delta }}}$), quantum fluctuations can surely bring the system out of the shallow local minima. For an ${\displaystyle N}$-spin glass, the barrier height ${\displaystyle \Delta }$ becomes of order ${\displaystyle N}$. For constant value of ${\displaystyle w}$ one gets ${\displaystyle \tau }$ proportional to ${\displaystyle e^{\sqrt {N}}}$ for the annealing time (instead of ${\displaystyle \tau }$ proportional to ${\displaystyle e^{N}}$ for thermal annealing), while ${\displaystyle \tau }$ can even become ${\displaystyle N}$-independent for cases where ${\displaystyle w}$ decreases as ${\displaystyle 1/{\sqrt {N}}}$.^[17][18]

It is speculated that in a

• 1989 Idea was presented that quantum fluctuations could help explore rugged energy landscapes of the classical Ising spin glasses by escaping from local minima (having tall but thin barriers) using tunneling;[1]
• 1998 Formulation of quantum annealing and numerical test demonstrating its advantages in Ising glass systems;^[4]
• 1999 First experimental demonstration of quantum annealing in LiHoYF Ising glass magnets;^[21]
• 2011 Superconducting-circuit quantum annealing machine built and marketed by D-Wave Systems.^[22]

D-Wave 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 on its performance.^[22] The company claims this system uses a 128 qubit processor chipset.^[23] On May 25, 2011, D-Wave announced that Lockheed Martin Corporation entered into an agreement to purchase a D-Wave One system.^[24] On October 28, 2011 USC's Information Sciences Institute took delivery of Lockheed's D-Wave One.

In May 2013 it was announced that a consortium of

There are many open questions regarding quantum speedup. The ETH reference in the previous section is just for one class of benchmark problems. Potentially there may be other classes of problems where quantum speedup might occur. Researchers at Google, LANL, USC, Texas A&M, and D-Wave are working to find such problem classes.[34]

In December 2015, Google announced that the

"A cross-disciplinary introduction to quantum annealing-based algorithms" [37] presents an introduction to combinatorial optimization (NP-hard) problems, the general structure of quantum annealing-based algorithms and two examples of this kind of algorithms for solving instances of the max-SAT and Minimum Multicut problems, together with an overview of the quantum annealing systems manufactured by D-Wave Systems. Hybrid quantum-classic algorithms for large-scale discrete-continuous optimization problems were reported to illustrate the quantum advantage.^[38][39]

References