ARTICLE Received 29 Jan 2013 | Accepted 23 Apr 2013 | Published 21 May 2013 DOI: 10.1038/ncomms2920 Thermally assisted quantum annealing of a 16-qubit problem N. G. Dickson1, M. W. Johnson1, M. H. Amin1,2, R. Harris1, F. Altomare1, A. J. Berkley1, P. Bunyk1, J. Cai1, E. M. Chapple1, P. Chavez1, F. Cioata1, T. Cirip1, P. deBuen1, M. Drew-Brook1, C. Enderud1, S. Gildert1, F. Hamze1, J. P. Hilton1, E. Hoskinson1, K. Karimi1, E. Ladizinsky1, N. Ladizinsky1, T. Lanting1, T. Mahon1, R. Neufeld1,T.Oh1, I. Perminov1, C. Petroff1, A. Przybysz1, C. Rich1, P. Spear1, A. Tcaciuc1, M. C. Thom1, E. Tolkacheva1, S. Uchaikin1, J. Wang1, A. B. Wilson1, Z. Merali3 &G.Rose1 Efforts to develop useful quantum computers have been blocked primarily by environmental noise. Quantum annealing is a scheme of quantum computation that is predicted to be more robust against noise, because despite the thermal environment mixing the system’s state in the energy basis, the system partially retains coherence in the computational basis, and hence is able to establish well-defined eigenstates. Here we examine the environment’s effect on quantum annealing using 16 qubits of a superconducting quantum processor. For a problem instance with an isolated small-gap anticrossing between the lowest two energy levels, we experimentally demonstrate that, even with annealing times eight orders of magnitude longer than the predicted single-qubit decoherence time, the probabilities of performing a successful computation are similar to those expected for a fully coherent system. Moreover, for the problem studied, we show that quantum annealing can take advantage of a thermal environment to achieve a speedup factor of up to 1,000 over a closed system. 1 D-Wave Systems Inc., 100-4401 Still Creek Drive, Burnaby, British Columbia, Canada V5C 6G9. 2 Department of Physics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6. 3 Foundational Questions Institute, PO Box 3022, New York, New York 10163, USA. Correspondence and requests for materials should be addressed to M.H.A. (email: [email protected]). NATURE COMMUNICATIONS | 4:1903 | DOI: 10.1038/ncomms2920 | www.nature.com/naturecommunications 1 & 2013 Macmillan Publishers Limited. All rights reserved. ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms2920 uantum computation1 is a computational paradigm that states, encountered at some point s ¼ s* known as an anticrossing harnesses quantum physics to solve problems. Theory (illustrated in Fig. 1). Using a two-state approximation, one can Qhas suggested that quantum computation could provide a write25,26,9: significant performance advantage over classical computation. À tf =ta 2 However, efforts to develop useful quantum computers have PGM 1 À e ; ta ¼ 2hn=pgmin; ð4Þ been blocked primarily due to environmental noise. Adiabatic where n ¼ |d(E1 À E0)/ds|s* is the relative slope of the energy levels quantum computation (AQC)2,3 is a scheme of quantum near s* and ta is the adiabatic timescale, which marks the computation that is theoretically predicted to be more robust 4–12 boundary between adiabatic (tf44ta) and non-adiabatic (tfoota) against noise . evolutions. Annealing such a closed system k times, with In AQC, a physical quantum system is initially prepared tf ¼ ttotal/k, yields the same Ptotal as annealing once with tf ¼ ttotal. in its known lowest-energy configuration, or ground state. The computation involves gradually deforming the system’s energy function, or Hamiltonian, in such a way that the system Open-system dynamics. In reality, all implementations of remains in the ground state throughout the evolution with high quantum algorithms are open systems, and thus subject to probability, and the ground state of the final Hamiltonian relaxation (transitions between energy eigenstates) and dephasing provides the solution to the problem to be solved. Quantum (randomization of relative phases between the eigenstates). In the annealing (QA)13–16 is a very similar, but more practical, scheme limit of weak coupling to the environment, dephasing in the of quantum computation that is different from AQC in two energy basis is irrelevant for QA, because only the ground-state aspects: the evolution is not required to be adiabatic, that is, probability matters and the relative phases between the energy the system may leave the ground state due to thermal or eigenstates do not carry any information during the computation. non-adiabatic transitions; and the final Hamiltonian is restricted Thermal excitation and relaxation, on the other hand, can reduce to be diagonal in the computation basis, with a ground state the instantaneous probability of the ground state by populating representing the solution to a hard optimization problem. Such the excited states. Nevertheless, in the limit of slow evolution, the optimization problems are integral to a wide range of applications ground state will always have the dominant probability, because from anthropology17 to zoology18. the excited states are occupied approximately with equilibrium QA has been experimentally demonstrated in non-program- (Boltzmann) probabilities. Unless there are an exponential 19,20 mable bulk solid state systems and also using 3-qubit nuclear number of states within the energy kBT (kB is Boltzmann’s magnetic resonance21. Recently, we have designed, fabricated and constant) from the ground state, the thermal reduction of the tested a programmable superconducting QA processor22,23. Here ground-state probability will not significantly affect the perfor- we report the first experimental exploration of the effect of mance. This picture changes in the strong coupling limit wherein thermal noise on QA. We explored this using a 16-qubit subset of it may no longer be possible to identify well-defined eigenstates of the processor used in Johnson et al.23 the system Hamiltonian independent of the environment27. It should be noted that, even if the environment is weakly coupled Results to the system and the equilibrium probability of the ground state Closed-system dynamics. Our processor implements the is not vanishingly small, the time to reach such a probability is a Hamiltonian concern for practical computation. For example, it is conceivable X that, due to thermal relaxation, one must wait orders of 1 x 1 HðsÞ¼ À DiðsÞsi þ EðsÞHP; ð1Þ magnitude longer than a typical closed-system evolution time 2 i 2 X X z z z HP ¼ hisi þ Jijsi sj ; ð2Þ i i o j A x;z where st/tf [0, 1], t is time, tf is the anneal time, si are Pauli |b 〉 δ 〉 matrices for the ith qubit, and Di(s) and E(s) are time-dependent E |a D 44 E energy scales. At t ¼ 0, i(0) E(0) 0, and the ground state is a Thermal Thermal superposition of all computation basis states, that is, all eigen- excitation relaxation z functions of s operators. At t ¼ t , E(1)44D (1)E0, and H(s)is Energy i f i gmin dominated by the Ising Hamiltonian HP, characterized by |a 〉 |b 〉 dimensionless local biases hi and pairwise couplings Jij. The Time ground state of H(1) thus represents the global minimum of HP. Finding this global minimum is known to be an NP-hard (non- Figure 1 | Illustration of thermal noise effects near an anticrossing. 24 deterministic polynomial time hard) optimization problem . An avoided crossing (anticrossing) of two lowest-energy eigenstates, An important feature of optimization problems is that, jia and jib , with a small minimum energy gap, gmin, separated from other although it may be challenging to find a global minimum, it is eigenstates by energy dE. Passing through the anticrossing very quickly easy to select the minimum from a set of candidate solutions, for swaps the probabilities of the ground and first excited states (blue and example, provided by a probabilistic algorithm such as QA. If one green dotted arrows), leaving the probabilities of jia and jib unchanged. is able to find a global minimum with probability PGM40 in one Thus, for a closed-system starting in jia , the final ground-state probability, trial taking time tf, the probability Ptotal of observing a global Pb, would be vanishingly small. However, with an environment at T40, minimum at least once in k-independent trials, and the total time thermal transitions can excite the system beforehand and relax it afterward required ttotal, is given by (red arrows), the net effect of which is to increase Pb (green arrow). Single-qubit tunnelling amplitudes are significantly larger before the P ¼ 1 Àð1 À P Þk; t ¼ kt ð3Þ total GM total f anticrossing (see Fig. 2b), making thermal excitations earlier in the annealing process much more likely than relaxation later. If T\dE/kB, For a noise-free (closed) system, PGM would depend on the higher excited states would also be occupied, reducing Pb; so a peak in B minimum energy gap, gmin, between the ground and first excited Pb is expected at T ¼ Tpeak dE/kB. 2 NATURE COMMUNICATIONS | 4:1903 | DOI: 10.1038/ncomms2920 | www.nature.com/naturecommunications & 2013 Macmillan Publishers Limited. All rights reserved. NATURE COMMUNICATIONS | DOI: 10.1038/ncomms2920 ARTICLE to reach an acceptable open-system probability. In that case, a despite the above argument, the computation cannot be hi = +1 considered robust against the environment. We therefore define h = 0 robustness against environmental noise as the ability of an i open quantum annealing system to yield the correct solution with hi = –1 acceptable probability within a time comparable to the closed- J = –1 system adiabatic timescale. ij 300 Experimental results. To experimentally investigate the effects of b noise on performance, we design an instance of HP that has an Δ i anticrossing with a small gmin between an eigenstate jiÆ , which is 200 (mK) a superposition of 256 equal energy (degenerate) local minima B k of HP, and an eigenstateji GM , which corresponds to the unique (nondegenerate) global minimum.