Some complex problems in physics can be recast as finding the ground state of an interacting quantum system. Not getting excited along the way can be the challenging part. See Letter p.194
Information processing based on quantum-mechanical interactions has the potential to yield a new and powerful set of computational tools and capabilities. In essence, replacing the classical bits of information in today's electronic devices with quantum bits (qubits) enables a computation to proceed on the basis of the rules of quantum mechanics. As one might surmise, however, if the quantum nature of this computational evolution were somehow corrupted along the way, the entire computation would be brought into question. On page 194 of this issue, Johnson et al.1 demonstrate that the evolution of a superconducting system of eight qubits is indeed consistent with the principles of quantum mechanics when a particular protocol called quantum annealing is implemented.
The solution for certain, challenging, optimization problems can be determined from the ground (minimal-energy) state of a system of interacting spins. Finding this ground state is itself a computationally demanding problem, and a form of adiabatic quantum computation called quantum annealing is one proposed means to do it2,3. Conceptually, an adiabatic process works as follows. The system starts in a known ground state with the interactions between neighbouring spins effectively turned off. The interactions are then slowly turned on such that the system evolves adiabatically — that is, without ever leaving its instantaneous ground state during this evolution2. If the system is never excited, always remaining in the instantaneous ground state, then it will surely end in the final, desired interacting ground state and we have the answer.
To illustrate how this process unfolds, consider a corrugated egg carton, with the ridges and furrows creating potential wells, and an egg placed in one of these wells. As the interactions are turned on, the egg carton slowly changes shape (never so fast that it jostles the egg into an adjacent well), and one of the wells becomes deeper than the others; this deeper well is the true ground state. A quantum egg will keep its cool and quantum mechanically tunnel through the ridges to reach this global minimum, whereas a classical thermal egg must resort to jumping excitedly over them. In the language of quantum annealing, the system continuously seeks the lowest-energy configuration by means of quantum tunnelling mediated by quantum fluctuations. This process should be distinguished from classical thermal annealing, in which thermal fluctuations lead the way.
In their study, Johnson et al.1 employ a one-dimensional Ising spin system, a chain of eight spins which can interact with one another in a nearest-neighbour pairwise manner. The spins are realized with superconducting flux qubits4, manufactured artificial spins whose values 'spin-up' and 'spin-down' correspond to clockwise and counter-clockwise circulating superconducting currents, which are respectively associated with the left and right wells of the qubit's potential energy diagram (Fig. 1). The authors went to great lengths to make the various qubit parameters tuneable, so that they could carefully calibrate the settings for each individual spin, as well as tune the spin–spin coupling strength and its sign (negative for ferromagnetic coupling and positive for anti-ferromagnetic coupling).
To demonstrate the distinction between quantum and thermal annealing, Johnson et al. leveraged ideas from the domain of macroscopic quantum tunnelling5,6. Consider first a single qubit and its double-well potential. The qubit starts in a symmetrical configuration with a low-potential tunnel barrier and an equal population in both wells — that is, with equal probability for spin-up or spin-down states to occur. As the tunnel barrier is continuously raised, the rates for quantum tunnelling (through the barrier) and thermal activation (over the barrier) continuously decrease until they effectively stop, freezing the population in place. The freezing for quantum tunnelling should depend primarily on the barrier, whereas the freezing for thermal activation should also scale with temperature (which provides the energy to 'jump over' the barrier). Which of the two processes causes the population to freeze in this system?
To answer this question, the authors raise the potential energy of the left well with respect to the right well at several times during the system's evolution to test whether, at these various times, the left-well population can find its way through the ever-increasing barrier to the lower-energy right well. They do this exercise not only as a function of time but also as a function of temperature (Fig. 1).
They find that the population freeze time (the time at which the barrier becomes so large that subsequently raising the left well has no further effect on the population distribution) saturates to a constant value as the temperature is reduced below 45 millikelvin, consistent with a quantum-tunnelling picture. Had a classical thermal activation picture been correct, one would expect the population to change continuously with temperature according to the value of the temperature and the height of the barrier. This conclusion presumes that the temperature itself is not stuck at 45 millikelvin. As a check, Johnson et al. show that at very early times, while the barrier is still low enough to allow thermal activation to occur, the freeze time does not saturate but instead scales with temperature down to 20 millikelvin. This serves as evidence that the population-freezing effect is not due to an inability to cool the device.
Johnson and colleagues1 then proceed to apply the quantum-annealing technique to the eight-qubit system. In this case, the first and last spins in the chain are forced to be spin-up and spin-down, respectively, while the remaining six spins are allowed to take on either value. With the spin–spin interactions turned on, the system is 'frustrated': although all spins would prefer to align with their neighbours (the authors use ferromagnetic coupling), this is not possible given that the first and last spins are fixed in opposite directions. As a result, a domain wall (an adjacent spin–up and spin–down) must appear somewhere, and its initial placement along the chain is equally likely. As before, the authors proceed to increase the barriers for each spin and, again, at various times and temperatures, raise the left wells for the inner six spins. Doing so makes one particular placement of the domain wall assume a lower energy than the others and, as before, the test is to see if the initial spin arrangements can find their way to this lowest-energy configuration. By finding the freeze time for the eight-qubit spin distribution as a function of temperature, the authors similarly determine that the observed spin-reconfiguration dynamics is most consistent with a quantum-tunnelling picture.
Given the number of dynamical parameters (changing the barrier height, raising the left wells, the time at which the wells are raised, and the temperature), the authors required simulations to draw their conclusions. Both the single-qubit and eight-qubit experiments were inconsistent with a classical model of thermal activation over the barrier to reach the ground state. Rather, the evolution was consistent with a theoretical quantum model that comprised four quantized energy levels in the presence of a thermal bath and that was simulated over a rather incredible 2 million computer-processing hours.
To be clear, this system was not used to perform any computational algorithm. And, whereas computing algorithms based on adiabatic quantum annealing have been demonstrated with small-scale model problems in nuclear magnetic resonance systems7, the practical benefits and scalability of quantum-annealing algorithms for large-scale problems remain unclear8,9. Furthermore, although Johnson and colleagues1 ruled out classical thermal activation over the barrier, they could not rule out thermal excitation within the potential wells. Such intra-well excitations certainly existed during the procedure employed by the authors (their purpose here was solely to distinguish quantum tunnelling from classical thermal activation over the barrier to reach the ground state), and these additional excitations would certainly complicate (if not prohibit outright) the implementation of an adiabatic quantum-annealing algorithm and reduce its effectiveness.
Nonetheless, the demonstration that an eight-qubit system evolves according to a quantum model in the presence of a thermal bath stands as a technical achievement, and its practical significance will ultimately be judged by the degree to which it enables the authors to address these and related questions. Whatever the outcome may be, there is still much to learn from a system that aims to remain firmly planted on the ground.
Johnson, M. W. et al. Nature 473, 194–198 (2011).
Farhi, E. et al. Science 292, 472–475 (2001).
Santoro, G. E., Martoňák, R., Tosatti, E. & Car, R. Science 295, 2427–2430 (2002).
Clarke, J. & Wilhelm, F. K. Nature 453, 1031–1042 (2008).
Voss, R. F. & Webb, R. A. Phys. Rev. Lett. 47, 265–268 (1981).
Devoret, M. H., Martinis, J. M. & Clarke, J. Phys. Rev. Lett. 55, 1908–1911 (1985).
Chen, H. et al. Phys. Rev. A 83, 032314 (2011).
Farhi, E., Goldstone, J. & Gutmann, S. Preprint at http://arxiv.org/abs/quant-ph/0201031 (2002).
Matsuda, Y., Nishimori, H. & Katzgraber, H. G. N. J. Phys. 11, 073021 (2009).
About this article
Chinese Journal of Physics (2019)