Abstract
Quantum annealing is a general strategy for solving difficult optimization problems with the aid of quantum adiabatic evolution. Both analytical and numerical evidence suggests that under idealized, closed system conditions, quantum annealing can outperform classical thermalizationbased algorithms such as simulated annealing. Current engineered quantum annealing devices have a decoherence timescale which is orders of magnitude shorter than the adiabatic evolution time. Do they effectively perform classical thermalization when coupled to a decohering thermal environment? Here we present an experimental signature which is consistent with quantum annealing, and at the same time inconsistent with classical thermalization. Our experiment uses groups of eight superconducting flux qubits with programmable spin–spin couplings, embedded on a commercially available chip with >100 functional qubits. This suggests that programmable quantum devices, scalable with current superconducting technology, implement quantum annealing with a surprising robustness against noise and imperfections.
Introduction
Many optimization problems can be naturally expressed as the NPhard problem of finding the ground state, or minimum energy configuration, of an Ising spin glass model^{1,2},
where the parameters h_{j} and J_{jk} are, respectively, local fields and couplings. The operators are Pauli matrices that assign values {±1} to spin values {↑,↓}. Two algorithmic approaches designed to address this family of problems are directly inspired by different physical processes: classical simulated annealing (SA) and quantum annealing (QA).
SA^{3} probabilistically explores the spin configuration space by taking into account the relative configuration energies and a timedependent (fictitious) temperature. The initial temperature is high relative to the system energy scale to induce thermal fluctuations that prevent the system from getting trapped in local minima. As the temperature is lowered, the simulation is driven towards optimal solutions, represented by the global minima of the energy function.
In QA^{4,5,6,7,8} the dynamics are driven by quantum, rather than thermal fluctuations. A system implementing QA^{9,10,11} is described, at the beginning of a computation, by a transverse magnetic field
The system is initialized, at low temperature, in the ground state of H_{trans}, an equal superposition of all 2^{N} computational basis states, the quantum analogue of the initial hightemperature classical state. The final Hamiltonian of the computation is the function to be minimized, H_{Ising}. During the computation, the Hamiltonian is evolved smoothly from H_{trans} to H_{Ising},
where the ‘annealing schedule’ satisfies A(0), B(T)>0 and A(T)=B(0)=0. If the change is sufficiently slow and there is no environment, the adiabatic theorem of quantum mechanics predicts that the system will remain in its ground state, and an optimal solution is obtained^{12,13}.
Realistically, one should include the effects of coupling to a thermal environment, that is, consider open system quantum adiabatic evolution^{14,15,16,17,18,19}. An implementation of open system QA has recently been reported in a programmable architecture of superconducting flux qubits^{20,21,22,23}, and applied to relatively simple protein folding and number theory problems^{24,25}. Although quantum tunnelling has already been demonstrated^{23}, the decoherence time in this architecture can be three orders of magnitude faster than the computational timescale^{23,26}, due in part to the constraints imposed by the scalable design. It is interesting to note that this decoherence is in the instantaneous energy eigenbasis (see below). In the circuit model of quantum computation this relatively short decoherence time would imply, without quantum error correction^{27,28}, that the system dynamics can be described by classical laws^{29}. In the context of open system QA, this might lead one to believe that the experimental results should be explained by classical thermalization, or that in essence QA has effectively degraded into SA.
Here, we address precisely this question: are the dynamics in open system QA dominated by classical thermalization with respect to the final Hamiltonian, as in SA, or by the energy spectrum of the timedependent quantum Hamiltonian? We answer this by studying an eightqubit Hamiltonian representing a simple optimization problem, and show that classical thermalization and QA make opposite predictions about the final measurement statistics.
Results
8Qubit degenerate Hamiltonian
Our Ising Hamiltonian, depicted in Fig. 1, has a 17fold degenerate ground state
sixteen of these states form a cluster of solutions connected by single spinflips of the ancillae spins (equation (4)), while the 17th ground state is isolated from this cluster in the sense that it can be reached only after at least four spinflips of the core spins (equation (5)). We analyse the spectrum of this Ising Hamiltonian in more detail next.
The spectrum can be analysed by first considering the spectrum of the Hamiltonian coupling a single ancilla spin to a core spin (such as spins 5 and 1 in Fig. 1). The spectrum, with the core (ancilla) spin written first (second) is
the minimum energy is −1 whether the core spin is up or down. Note that if the core spin is up, the minimum energy is −1 whether the ancilla is up or down; this will give rise to a 16fold degeneracy when we account for all spins below. We analyse the core spins’ energies by first taking into account only their couplings, that is, we analyse the ferromagnetic Hamiltonian of the four central core qubits (spins 1–4 in Fig. 1). Denoting by s the number of satisfied couplings (both spins linked by the coupling have the same sign), the energy is 4−2s, where sε{0, 2, 4}. The ground states of this Hamiltonian are the configurations ↑↑↑↑› and ↓↓↓↓›. As equation (6) shows that the minimum energy of a coreancilla pair is −1, when adding the lowenergy configurations of the couplings to the ancillae the minimum energy is −8. It also follows from equation (6) that the ground state configurations of the full 8qubit Hamiltonian are equations (4) and (5).
The allspins down case (5) results from the ↓↓› configuration in equation (6), while the 16fold degenerate case (4) results from the degeneracy of ↑↑› and↓↑›.
An important feature of the energy landscape of the 8qubit Hamiltonian is that it does not have any local minima. This can be easily proved by showing that a global minimum can always be reached from any state by performing a sequence of single spinflips and never raising the energy. To see this, consider an arbitrary state of the system. We can first flip all the ancillae spins to ↓› which, according to equation 6, can be done without raising the energy (independently of the state of the corresponding core spin). Then, we can flip the core spins to satisfy all the couplings between core spins, either making them all ↓› or all ↑›, whichever requires the fewest spinflips. Again, according to equation 6, this operation will not raise the energy of the coreancilla pair. Hence, the final state is either the isolated ground state ↓↓↓↓↓↓↓↓› or the state ↑↑↑↑↓↓↓↓› that belongs to the degenerate cluster of ground state configurations.
As we show below, classical thermalization predicts that the isolated solution will be found with higher probability than any of the cluster solutions, that is, it is enhanced. Furthermore, after an initial transient, faster thermalization corresponds to a higher probability of finding the isolated solution. Open system QA makes the exact opposite prediction: after an initial transient, the isolated solution is suppressed relative to the cluster, and faster quantum dynamics yields higher suppression (lower probability). Our experimental results are consistent with the open system QA prediction of the suppression effect, and inconsistent with classical thermalization. We next discuss these opposite effects, starting from the classical case.
Classical SA predictions
Let p_{i} denote the probability of state i in the cluster (4), and p_{s} the probability of the isolated state (5). The probabilities p_{i} are all similar in an ideal implementation, because states in the cluster are connected by single spinflips, so we consider the average cluster probability . This also helps remove experimental noise and systematic biases that affect the probability of each individual state in the cluster. Enhancement of the isolated state means that p_{s}≥p_{C}. Note that the general features of a thermalization process are determined by the spectrum of H_{Ising} and by the combinatorics of state interconversion. As seen above, each of the 17 degenerate ground states can be reached from any other state without ever raising the energy via a sequence of single spinflips, so that SA never gets trapped in local minima.
Consider standard classical thermalization given by single spinflip transitions whose rate depends only on energy differences (see Methods). The isolated ground state is connected to eight singly excited states via single spinflips, whereas all states in the cluster are connected via single spinflips to at most four singly excited states; the other four spinflips connect between other states in the cluster and hence conserve the energy. During classical annealing at constant low temperature, population feeds faster initially into the isolated state than into any state of the cluster, whence p_{s}≥p_{C}. As we report below, our SA numerics show that this is also the case for different cooling schedules throughout the thermalization evolution. Simultaneous double spinflips do not change this conclusion, and higher order simultaneous spinflips are less likely physically.
In the plots below we used the Metropolis update rule: if ΔE is the energy difference for the update, the transition probability is
We have also tested other update rules, such as Glauber’s^{30}. The main feature of interest (the isolated state is not suppressed) is unchanged, although the concrete probabilities do depend on the choice of annealing schedule, that is, the functional dependence of the temperature on the number of steps. The SA results for three different annealing schedules are shown in Fig. 2. The probability p_{s} of the isolated state is always above the average probability p_{C} for a state in the cluster.
It might be argued that thermalization at constant temperature corresponds most closely to the experimental situation, given that the experimental system remains at an almost constant 17 mK. The corresponding simulation can be seen in Fig. 2. On the other hand, the energy scale of the Ising model changes during the QA evolution (see the Fig. 4 insert), and the cooling schedule is determined not by the temperature alone but rather by the ratio between the energy scale and the temperature. We also show p_{s} and P_{C} for an exponential schedule with Metropolis updates and different numbers of steps in Fig. 3.
QA predictions
We next analyse the corresponding predictions of QA. A crucial difference with respect to SA is that now the relevant energy spectrum is given by a combination of the final Ising Hamiltonian and the transverse field. Consequently, as shown in Fig. 4, the degeneracy of the ground space is lifted for times t<T. The isolated state has support only on the highest eigenstate plotted during the second half of the evolution. Given that the system starts in the ground state, the isolated state is suppressed by the energy gap, until this gap vanishes at the end of the evolution. The isolated state remains suppressed nonetheless, as transitions to other lowenergy states require at least four spinsflips. The transverse field term, which drives simultaneous spinflip transitions, is small at large t. If the four spinsflips are not simultaneous, these transitions involve excited states with much higher energy and are suppressed. This predicted QA suppression of the isolated state is confirmed by our closed and open system quantum dynamical simulations (see Fig. 5).
We can understand the splitting of the degenerate ground subspace of the Ising Hamiltonian H_{Ising} by treating the transverse field as a perturbation of the Ising Hamiltonian H_{Ising} (thus, treating the QA evolution as that of a closed system evolving backward in time). According to standard degenerate perturbation theory, the perturbation P_{g} of the ground subspace is given by the spectrum of the projection of the perturbation on the ground subspace. Denoting by
the projector on the 17dimensional ground subspace, where ↕ means that the spin can be either up or down, we therefore wish to understand the spectrum of the operator
The isolated state is unconnected via single spinflips to any other state in the ground subspace, so we can write this operator as a direct sum of 0 acting on the isolated state and the projection on the ground subspace of the cluster
While σ^{x} acting on any of the four ancillae connects two cluster ground states, σ^{x} acting on any core spin of a cluster state is projected away. Therefore, the perturbation is given by the operator
where the sum is over the four ancillae spins.
Denoting the eigenbasis of σ^{x} by , with respective eigenvalues ±1, the transverse field splits the ground space of H_{Ising} lowering the energy of ↑↑↑↑++++›, and the four permutations of −› in the ancillae spins of ↑↑↑↑+++−›. None of these states overlaps with the isolated ground state, which is therefore not a ground state of the perturbed Hamiltonian. Furthermore, after the perturbation, only the sixth excited state overlaps with the isolated state. The isolated state becomes a ground state only at the very end of the evolution (with time going forward), when the perturbation has vanished.
Experimental results
A diagram of the experimentally achievable coupling configurations is shown in Fig. 6a. The experimental results are shown in Fig. 7. The key finding that is immediately apparent is that the isolated state is robustly suppressed, in agreement with the QA but not the SA prediction.
Is it possible that suppression has a physical explanation other than QA? The main physical argument along these lines is that a systematic or random bias due to experimental imperfections breaks the 17fold ground state degeneracy and energetically disfavours the isolated state, thus lowering p_{s} if the system thermalizes. We proceed to examine this and the robustness of the suppression effect.
First, note that spin numbers j=1,…,8 must be assigned to the flux qubits before each experimental run. One of the 4!4!2/8=144 possible such ‘embeddings’ allowed by the symmetries of the Hamiltonian and the hardware connectivity graph in each unit cell is shown in Fig. 6b. Second, note that spin inversion transformations commute with H_{trans}, and simply relabel the spectrum of both H_{Ising} and H(t): if a certain spin configuration has energy E, then the corresponding spin configuration with the jth spin flipped has the same energy E under .
Spin inversions also commute with the spinflip operations of classical thermalization. Therefore, all of our arguments for the suppression of the isolated state in QA and for its enhancement in classical thermalization are unchanged. Using spin inversions, we can check that the suppression effect is not due to a perturbation of the Hamiltonian such as a magnetic field bias. Indeed, by performing a spin inversion on all eight spins, we obtain a new Ising Hamiltonian where the isolated state is that with all spinsup. If a field bias suppressed the all spinsdown state, then it would enhance the all spinsup state. Figure 7 rules this out. We also tested cases with only antiferromagnetic couplings (only ancillae spins are inverted), and with random spin inversions (the set of inverted spins is chosen randomly from a uniform distribution). In all cases we found agreement with the QA prediction, but not with classical thermalization. The results for one such random inversion example are shown in Fig. 8.
Robust suppression holds even at the level of individual embeddings and spin inversions. We found that p_{s}3%, while p_{C}<~6% for each of the thousands of such cases we tested (the highest median p_{s} for the experiment in Fig. 7 is 0.004). Thus, suppression survives breaking of the ground state degeneracy, which certainly occurs because of the limited precision of ~10% (ref. 20) in our control of {h_{j}, J_{jk}}. The suppression effect is robust because it does not depend on the exact values of these parameters, but on the relatively large Hamming distance between the isolated state and the cluster.
Finally, we consider the effect of increasing the annealing time. Open quantum and classical systems converge towards thermal equilibrium. Therefore, if the cause of suppression is the QA spectrum, longer annealing times will result in p_{s} increasing, approaching its Gibbs distribution value. This increase would not be the case if p_{s} were governed by the spectrum of H_{Ising}. In Fig. 5 we compare a numerical simulation of open system QA, using an adiabatic Markovian master equation^{19}, with classical thermalization. The quantum prediction of increasing p_{s} is confirmed experimentally, as shown in Fig. 8.
Decoherence basis
It is interesting to highlight the difference between decoherence in the instantaneous energy eigenbasis versus the computational basis. We argue that the former is the relevant picture for our experiments. The difference is related, in turn, to whether the master equation is derived in the weak coupling limit (WCL) or singular coupling limit (SCL), the former being the limit where the system Hamiltonian dominates over the systembath coupling (see Methods).
The issue of the basis in which the system decoheres is an important one for QA. In our case the system is initialized to an excellent approximation in the pure ground state of the transverse field . In the computational basis, this corresponds to a density matrix whose elements are all 1/2^{N}. In the energy eigenbasis, however, only one element of the density matrix is nonzero, corresponding to the lowest energy eigenstate. During an adiabatic evolution, if the decoherence is in the energy eigenbasis, the offdiagonal elements of the density matrix in the energy basis are unimportant and their rapid decay is of no relevance to the final calculation^{31}. However, if the decoherence is in the computational basis, the filled density matrix very quickly decoheres to the maximally mixed (infinite temperature) classical state. Therefore, the SCL master equation predicts that the isolated state is not suppressed relative to the cluster of states, which is not in agreement with the experimental results.
Discussion
We thus arrive at our main conclusion: signatures of QA, as opposed to classical thermalization, persist for timescales that are much longer than the singlequbit decoherence time (from 5 μs to 20 ms versus tens of ns) in programmable devices available with presentday superconducting technology. Our experimental results are also consistent with numerical methods that compute quantum statistics, such as Path Integral Monte Carlo (Troyer and Røennow, personal communication). Our study focuses on demonstrating a nonclassical signature in experimental QA. Different methods are required to address the question of experimental computational speedups of open system QA relative to optimal classical algorithms.
There are no reported experimental measurements of the actual decoherence time for the system that we have considered. Hence, the best we can do at this point is to rely on theoretical estimates or experimental measurements on physical systems that are similar to ours. An estimate of T_{2}~150 ns based on measured features of the noise spectrum has been reported in the literature^{32}. There are two experimental reported results, one for a single phase qubit of T_{2}~21 ns (ref. 26), and another for a flux qubit of T_{2}~500 ns (ref. 33). The estimates thus range from a few tens to several hundred nanoseconds. The important point is that these values do not change our contention that these decoherence times do not destroy the quantum effects present for timescales that are much longer (>5 μs). After the completion of this work, a classical spin dynamics model appeared which also exhibits a suppression of the isolated state^{35}. This model can be interpreted as a meanfield model of coherent qubits. The primary purpose of our work has been to compare the predictions of classical simulated annealing to our experimental results and to the predictions of quantum open system dynamics (via the QA master equation), and hence we cannot directly rule out classical spindynamics models. However, it is an interesting problem for future study whether classical spin dynamics models can capture the thermalization trend measured experimentally and predicted by the QA master equation (Fig. 8).
Methods
Experimental system
Our experiments were performed using the DWave One Rainier chip at the USC Information Sciences Institute, comprising 16 unit cells of eight superconducting flux qubits each, with a total of 108 functional qubits. The couplings are programmable superconducting inductances. The qubits and unit cell, readout, and control have been described in detail elsewhere^{20,21,22,23}. The initial energy scale for the transverse field is 10 GHz (the A function in Fig. 4), the final energy scale for the Ising Hamiltonian (the B function) is 5.3 GHz, about 15 times the experimental temperature of 17 mK≈0.35 GHz. To gather our data, we ran each of the 144 embeddings 4,000 times, in batches of 1,000 readouts, resetting all the local fields and couplers after each batch.
Master equation
The SA master equation and the classical thermalization prediction p_{s}≥p_{C} can be derived from first principles from an adiabatic quantum master equation^{19}. Let H_{S}(t) and denote the system and systembath Hamiltonians. The Lindblad equation is
where , {a›} is the instantaneous eigenbasis of H_{S} for spin vector a, and . We are interested in the thermalization process in which the density operator is diagonal in the computational basis of spin vectors. The systembath coupling Hamiltonian then has the form , where σ^{±}=(σ^{x}±i σ^{y})/2. We denote by () the spin vector resulting from flipping the jth spin up (down). From here we arrive at the classical master equation for the populations p_{a} ≡ ρ_{aa}:
and the detailed balance condition . Equation 13 is the master equation that we used in our SA numerics. It can also be used to derive the classical thermalization prediction p_{s}≥p_{C}. To this end, it can be seen directly that the isolated state is connected to eight excited states with energy −4, giving the rate equation , and also that . Comparing, we conclude that given that initially ps≥pc, we always observe ps≥pc. A complete derivation is given in the Supplementary Methods.
Decay in the energy basis versus the computational basis
We demonstrate in detail the difference between decoherence in the energy versus the computational basis.
For the analysis in the energy basis it is more convenient to write the systembath Hamiltonian H_{SB} using Hermitian operators A_{α} and B_{α}. Recall that equation 12 was derived in the WCL. For simplicity, consider the case of nondegenerate energy eigenstates denoted by {a›}. Dropping the system Hamiltonian to simplify the notation, the terms that contribute to the population in the instantaneous energy eigenbasis are:
where is the transition matrix associated with the transition from the state a› to the state b›, and where we use the shorthand ρ_{ab}=‹aρ(t)b› and A_{ab,α}=‹aA_{α}b›. This result shows that the diagonal elements in the instantaneous energy eigenbasis relax towards the Gibbs distribution. This is crucial as it implies that most of the population will be found in the ground state.
The offdiagonal elements of the density matrix in the instantaneous energy eigenbasis are given by:
Therefore, we find that the decoherence is described by a system of coupled linear equations involving only the offdiagonal elements of the density matrix. As the Lindblad form of the master equation guarantees complete positivity, the instantaneous eigenvalues of the system of linear equations must be ≤0. The zero eigenvalues correspond to stationary states, which include the instantaneous Gibbs state and any decoherence free subspace. For our case of an independent dephasing model: and γ_{αβ}(ω)=δ_{αβ}γ(ω), the latter does not exist. Thus, we find that the offdiagonal elements of the density matrix in the energy eigenbasis decay to zero. As the gap is finite in our system and the singlequbit T_{2} is on the order of nanoseconds, this decay occurs on a timescale that is rapid compared with the annealing time.
We can contrast the WCL (energy basis) result to the case of the SCL. In this case, the dissipative part of the Lindblad master equation is given by:
If we assume that as before, then A_{α} is diagonal in the computational basis, which we denote by {x›}, and we choose to work in this basis rather than the instantaneous energy eigenbasis. In this case the dissipative part of the Lindblad master equation for ‹xdρ/dtx› vanishes, that is, the populations in the computational basis experience no net thermal excitations and do not equilibrate. For the offdiagonal elements, we obtain an expression similar to equation 15 if we restrict the sums to x′=x and y′=y:
where A_{xx,α}=‹xA_{α}x› and ρ_{xy}=‹xρ(t)y›. Thus, here too the density matrix becomes diagonal, but in stark contrast to the case of decoherence in the instantaneous energy eigenbasis, the ground state population is essentially equal to its initial value of 1/2^{N}. A more detailed derivation of the SCL master equation is presented in Supplementary Methods.
Additional information
How to cite this article: Boixo S. et al. Experimental signature of programmable quantum annealing. Nat. Commun. 4:2067 doi: 10.1038/ncomms3067 (2013).
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Acknowledgements
We thank M.H.S. Amin, T. Lanting, M.C. Thom, J. Preskill, T. Rønnow and M. Troyer for useful discussions. We particularly thank M.H.S. Amin for discussions that inspired our choice of the Ising Hamiltonian. This research was supported by the Lockheed Martin Corporation. S.B. and D.A.L. acknowledge support under ARO grant number W911NF1210523. D.A.L. was further supported by the National Science Foundation under grant number CHM1037992, and ARO MURI grant W911NF1110268.
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S.B. conceived and performed the experiment, and did most of the theoretical analysis of classical thermalization. T.A. performed the open system QA numerical analysis with assistance from N.C. F.S. analysed the possible effects of hardware imperfections on the experimental results. N.C. also performed closed quantum system calculations as well as other supplemental calculations. All authors participated in the interpretation of the experiments and theory development. T.A., D.L. and S.B. carried out the analysis of the decoherence basis. S.B. and D.L. wrote the manuscript. All authors discussed the results and commented on the manuscript.
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Supplementary Figures S1S2, Supplementary Methods and Supplementary References (PDF 221 kb)
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Boixo, S., Albash, T., Spedalieri, F. et al. Experimental signature of programmable quantum annealing. Nat Commun 4, 2067 (2013). https://doi.org/10.1038/ncomms3067
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