Abstract
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 welldefined 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 smallgap anticrossing between the lowest two energy levels, we experimentally demonstrate that, even with annealing times eight orders of magnitude longer than the predicted singlequbit 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.
Introduction
Quantum computation^{1} is a computational paradigm that harnesses quantum physics to solve problems. Theory has suggested that quantum computation could provide a significant performance advantage over classical computation. However, efforts to develop useful quantum computers have been blocked primarily due to environmental noise. Adiabatic quantum computation (AQC)^{2,3} is a scheme of quantum computation that is theoretically predicted to be more robust against noise^{4,5,6,7,8,9,10,11,12}.
In AQC, a physical quantum system is initially prepared in its known lowestenergy configuration, or ground state. The computation involves gradually deforming the system’s energy function, or Hamiltonian, in such a way that the system remains in the ground state throughout the evolution with high probability, and the ground state of the final Hamiltonian provides the solution to the problem to be solved. Quantum annealing (QA)^{13,14,15,16} is a very similar, but more practical, scheme of quantum computation that is different from AQC in two aspects: the evolution is not required to be adiabatic, that is, the system may leave the ground state due to thermal or nonadiabatic transitions; and the final Hamiltonian is restricted to be diagonal in the computation basis, with a ground state representing the solution to a hard optimization problem. Such optimization problems are integral to a wide range of applications from anthropology^{17} to zoology^{18}.
QA has been experimentally demonstrated in nonprogrammable bulk solid state systems^{19,20} and also using 3qubit nuclear magnetic resonance^{21}. Recently, we have designed, fabricated and tested a programmable superconducting QA processor^{22,23}. Here we report the first experimental exploration of the effect of thermal noise on QA. We explored this using a 16qubit subset of the processor used in Johnson et al.^{23}
Results
Closedsystem dynamics
Our processor implements the Hamiltonian
where s≡t/t_{f}∈[0, 1], t is time, t_{f} is the anneal time, are Pauli matrices for the ith qubit, and Δ_{i}(s) and Ε(s) are timedependent energy scales. At t=0, Δ_{i}(0)>>Ε(0)≈0, and the ground state is a superposition of all computation basis states, that is, all eigenfunctions of operators. At t=t_{f}, Ε(1)>>Δ_{i}(1)≈0, and H(s) is dominated by the Ising Hamiltonian , characterized by dimensionless local biases h_{i} and pairwise couplings J_{ij}. The ground state of H(1) thus represents the global minimum of . Finding this global minimum is known to be an NPhard (nondeterministic polynomial time hard) optimization problem^{24}.
An important feature of optimization problems is that, although it may be challenging to find a global minimum, it is easy to select the minimum from a set of candidate solutions, for example, provided by a probabilistic algorithm such as QA. If one is able to find a global minimum with probability P_{GM}>0 in one trial taking time t_{f}, the probability P_{total} of observing a global minimum at least once in kindependent trials, and the total time required t_{total}, is given by
For a noisefree (closed) system, P_{GM} would depend on the minimum energy gap, g_{min}, between the ground and first excited states, encountered at some point s=s* known as an anticrossing (illustrated in Fig. 1). Using a twostate approximation, one can write^{25,26,9}:
where ν=d(E_{1}−E_{0})/ds_{s*} is the relative slope of the energy levels near s* and t_{a} is the adiabatic timescale, which marks the boundary between adiabatic (t_{f}>>t_{a}) and nonadiabatic (t_{f}<<t_{a}) evolutions. Annealing such a closed system k times, with t_{f}=t_{total}/k, yields the same P_{total} as annealing once with t_{f}=t_{total}.
Opensystem dynamics
In reality, all implementations of quantum algorithms are open systems, and thus subject to relaxation (transitions between energy eigenstates) and dephasing (randomization of relative phases between the eigenstates). In the limit of weak coupling to the environment, dephasing in the energy basis is irrelevant for QA, because only the groundstate probability matters and the relative phases between the energy eigenstates do not carry any information during the computation. Thermal excitation and relaxation, on the other hand, can reduce the instantaneous probability of the ground state by populating the excited states. Nevertheless, in the limit of slow evolution, the ground state will always have the dominant probability, because the excited states are occupied approximately with equilibrium (Boltzmann) probabilities. Unless there are an exponential number of states within the energy k_{B}T (k_{B} is Boltzmann’s constant) from the ground state, the thermal reduction of the groundstate probability will not significantly affect the performance. This picture changes in the strong coupling limit wherein it may no longer be possible to identify welldefined eigenstates of the system Hamiltonian independent of the environment^{27}. It should be noted that, even if the environment is weakly coupled to the system and the equilibrium probability of the ground state is not vanishingly small, the time to reach such a probability is a concern for practical computation. For example, it is conceivable that, due to thermal relaxation, one must wait orders of magnitude longer than a typical closedsystem evolution time to reach an acceptable opensystem probability. In that case, despite the above argument, the computation cannot be considered robust against the environment. We therefore define robustness against environmental noise as the ability of an open quantum annealing system to yield the correct solution with acceptable probability within a time comparable to the closedsystem adiabatic timescale.
Experimental results
To experimentally investigate the effects of noise on performance, we design an instance of that has an anticrossing with a small g_{min} between an eigenstate , which is a superposition of 256 equal energy (degenerate) local minima of , and an eigenstate , which corresponds to the unique (nondegenerate) global minimum. By studying a smallgap problem, we are addressing what are expected to be the most difficult problems for closedsystem AQC (although for an open system, problems with exponentially many lowenergy excited states may represent the hardest problems). Moreover, we are interested in exploring evolution during which the minimum gap is passed nonadiabatically, and investigating dependence on annealing time and temperature. To experimentally violate adiabaticity, we require g_{min}/k_{B}<<1 mK. For 16qubit problems, such small gaps are quite uncommon, and it can be challenging to engineer an instance with sufficiently small g_{min}. Our designed instance is illustrated in Fig. 2a and further described in the Methods section and Supplementary Note 1. The same type of anticrossing has been argued^{28,29} to render QA ineffective because of the extremely small g_{min}, though methods have been proposed to eliminate such anticrossings^{30,31,32}.
Of key importance are the energy scales Δ_{i}(s) and Ε(s) in equation (1), which can be calculated from independently calibrated device parameters (see, for example, Harris et al.^{33} and Johansson et al.^{34}). Results for the 16 qubits used in this study are plotted in Fig. 2b. Using these quantities, we calculate the eigenspectrum of H(s). Features relevant to this work are found in a narrow region around s≈0.64, where the anticrossing is expected, as shown in Fig. 2c. The minimum gap, g_{min}/k_{B}=0.011 mK, is more than three orders of magnitude smaller than T (≳20 mK). The global minimum (green) of is the dominant component of the ground state after the anticrossing, but dominates the first excited state before it. The opposite is true for (blue). Our first objective is to experimentally verify that there is an anticrossing at the predicted position.
In the limit of infinitely slow evolution, the instantaneous probability of occupying each eigenstate is approximately given by the Boltzmann distribution. Therefore, the probability P_{GM}(s) of occupying should be small before the anticrossing but large after the anticrossing. The opposite should hold for the probability P_{Σ}(s) of occupying . The two probabilities should coincide at the anticrossing, where P_{GM}(s*)=P_{Σ}(s*)≈0.5. Measuring the instantaneous probabilities P_{GM}(s) and P_{Σ}(s) would therefore provide information about the approximate position of the anticrossing.
We measure the instantaneous probabilities P_{GM}(s) and P_{Σ}(s) by annealing the system slowly, with t_{f}=100 ms, but interrupting it at s=s_{d} by rapidly moving to s=1 within 20 μs. This rapid evolution takes approximate snapshots of P_{GM}(s=s_{d}) and P_{Σ}(s=s_{d}). P_{Σ} is determined by summing over the probabilities of observing all 256 local minima of at the end of the evolution. Such a measurement gives a representation of the equilibrium distribution up to some s, beyond which thermal relaxation timescales become exceedingly long (see Supplementary Fig. S5 and Supplementary Note 3 for more details).
Figure 2d shows measurements of P_{GM}(s) and P_{Σ}(s). As expected, P_{GM}≈1−P_{Σ}≈0 before the anticrossing. The total probability of states other than the global minimum and the 256 local minima is <0.1%. The two data sets cross near the theoretically predicted s*, where P_{GM}=P_{Σ}≈0.5. For s>s*, P_{GM}≈1−P_{Σ} becomes large. As the tunnelling amplitudes Δ_{i}(s) are reduced towards the end of annealing, the relaxation between the energy levels becomes slower and slower, and finally the probabilities freeze due to extremely slow relaxation^{23}. The groundstate probability at the freezeout point, therefore, determines the final success probability. As a result, the probabilities saturate for s≳0.66 because of the diminishing relaxation rates between and . (The saturation point and value depend on t_{f}; see Supplementary Fig. S5 and Supplementary Note 3.)
To examine the impact of noise on QA, we consider the effects of varying T and t_{f}. In the case where t_{f}<<t_{a}, passing through the smallgap anticrossing quickly will approximately swap the probabilities of the two crossing states, as depicted in Fig. 1. For T=0, this would give vanishingly small P_{GM}. Because dominates the excited state just before the anticrossing, increasing T is expected to increase P_{GM}. As T approaches δE/k_{B}, the energy separation between and , and higher eigenstates, those latter states will be populated, thus reducing P_{GM}. Therefore, a peak in P_{GM} as a function of T is expected at T_{peak}≲δE/k_{B}.
Figure 3 shows experimental measurements of the final success probability P_{GM} as a function of T for different t_{f}. All curves, except for those with t_{f} ≥200 ms, show an initial increase with T up to a maximum. For T≳40 mK, the benefits of thermal noise diminish as the system becomes thermally excited to the eigenstates that are at energy δE/k_{B}∼50 mK above the lowest two states, depicted in Fig. 2c.
It should be emphasized that the enhancement with temperature observed in this experiment is a result of having a smallgap anticrossing separated from all other states by a large energy gap. In more general cases, when several excited states have comparable energy gaps with respect to the ground state with no smallgap anticrossings, increasing T would decrease the probability of success, as observed in Fig. 3 for T>40 mK.
The degradation of P_{GM} for high temperatures is instructive, but there is something important to be learned from the lowtemperature limit as well. An open system with T=0 has been theoretically predicted to behave similarly to a closed system for such an anticrossing^{10,35,36,34}. In this experiment, it was infeasible to reduce T below ∼20 mK. Instead, to obtain a crude estimate of P_{GM} at T=0, we extrapolate the curves in Fig. 3 (blue squares). In Fig. 4a, the data shown in Fig. 3 are plotted as a function of t_{f} for different T. We fit (blue dashed line) the formula for closedsystem probability, equation (4), to the extrapolated points (blue squares) using t_{a} as the fitting parameter, giving t_{a}=57.2 ms. The fact that the extrapolated points (T=0) could be fit with equation (4) supports the prediction above. All T>0 curves fit very poorly to this equation.
Discussion
Despite the influence of thermal noise, it can be seen in Fig. 4a that 0.45≲P_{GM}≲0.8 at t_{f}=t_{a} for all T>0 studied. These probabilities are comparable to P_{GM}=0.63 expected for the closed system. This is because the timescale to reach equal thermal occupation of two anticrossing states is determined by a relaxation time proportional to g_{min}^{−2} (refs 9, 10), similar to t_{a} in equation (4). It is important to note that for a single, unbiased qubit near s*, we estimate a decoherence time that is millions of times shorter than t_{a} (see Supplementary Note 2). The fact that P_{GM} similar to that of a closed system can be reached in time t_{a}, despite the significantly shorter decoherence time, supports theoretical predictions that QA can be performed in the presence of small environmental noise^{4,5,6,7,8,9,10,11,12}.
We can also extract the value of g_{min} for this instance based on the value of t_{a} found from the fitting above. In equation (4), ν is only weakly dependent on Hamiltonian parameters, whereas g_{min} is exponentially sensitive^{37}. We therefore use the value of ν/k_{B}≈5.3 K, obtained from the computed spectrum in Fig. 2c, to calculate g_{min} based on the above t_{a}. The result, g_{min}/k_{B}=0.021 mK, is about twice as large as that predicted in Fig. 2c. This is within the expected uncertainty, considering the exponential sensitivity of g_{min} (see Supplementary Note 1).
Given the values of P_{GM} in Fig. 4a, one can also compare the relative performance of repeated annealing of the QA processor to that of an ideal closed system. Figure 4b shows the total time t_{total} required to achieve P_{total}=0.99 by repeated annealings, as a function of t_{f}, calculated using equation (3). This calculation ignores the overhead for processor preparation and readout, which do not reflect underlying physics. As noted earlier, for the closed system (blue squares and dashed line), t_{total} is independent of t_{f}. For T>0, t_{total} decreases with decreasing t_{f}, and can be almost three orders of magnitude smaller than that expected for the closed system with t_{f}=0.01 ms. Clearly, in the case of a smallgap anticrossing, such as the one studied here, annealing an open system fast multiple times can have a significant performance advantage over annealing slowly once or annealing a closed system, as predicted^{11}. Such an efficiency enhancement due to coupling to the environment has been predicted to have an important role in nature, for example, in photosynthetic quantum energy transfer^{38}.
In summary, we have found experimental evidence that for a 16qubit instance with an avoided crossing having an extremely small gap in energy, QA can be robust against thermal noise, well beyond the decoherence time, in line with theoretical predictions. Using a QA processor comprising superconducting flux qubits, we show that the presence of a small amount of thermal noise does not hinder QA in the studied example, and can, in fact, significantly enhance its performance. The thermal enhancement of the performance is restricted to instances with a smallgap anticrossing well separated from other excited states. For general problems with many energy levels thermally occupied during the evolution, increasing T is expected to decrease the final success probability because energy levels within k_{B}T of the ground state may become thermally occupied during the annealing. However, unless the number of these nearby levels increases exponentially with the number of problem variables, the corresponding reduction in groundstate probability can be compensated by repeating the annealing a number of times. These results suggest that QA offers a practical and promising way to perform quantum optimization.
Methods
The processor
The devices studied in this experiment are from a QA processor made of 128 superconducting flux qubits, all controlled by onchip Single Flux Quantum superconducting digital circuitry as well as filtered analog signal input lines. The qubits are briefly discussed in this section and in detail in Harris et al.^{33} The onchip control circuitry is described in detail in Johnson et al.^{39} The qubits are numbered 0–127 starting from the topleft corner of the chip, and the 16qubit subset used in this experiment is depicted in Supplementary Fig. S2. Supplementary Fig. S2b represents the qubits as line segments, intersecting at couplers, reflecting the qubit’s physical layout as long, thin radio frequency Superconducting Quantum Interference Device (rfSQUID) loops. Some of these qubits have been used in previously published experiments. For example, qubit 48 was used in a singlequbit experiment^{40}, qubits 48–55 were used in twoqubit experiments^{41}, as well as the 8qubit experiments discussed in Harris et al.^{22} and Johnson et al.^{23} The annealing procedure we use here is the same as that described in Harris et al.^{22}
The rfSQUID flux qubit
The qubits used in this experiment are superconducting compound Josephson junction rfSQUID flux qubits described in detail in Harris et al.^{33} A simplified version of the qubit is illustrated in Supplementary Fig. S3. It has two superconducting loops and therefore two flux degrees of freedom Φ_{1} and Φ_{2}, subject to external flux biases Φ_{1x} and Φ_{2x}, respectively. At Φ_{1x}≈Φ_{0}/2, the rfSQUID has two bistable states with persistent current flowing clockwise or counterclockwise through the large loop. These two states form the qubit’s logical ‘0’ and ‘1’ states. The value of the persistent current (I_{p}) and the tunnelling amplitude (Δ_{i}) between the two bistable states are controlled by Φ_{2x}. The energy bias between the two states is controlled by Φ_{1x}. Supplementary Fig. S4 plots the measured values of I_{p} and Δ_{i} as a function of s for all 16 qubits. Measurement details are provided in Harris et al.^{33} The qubits are calibrated to have approximately the same persistent currents, but Δ_{i} is not uniform among the qubits, as is clear from the figure. The overall energy scale E(s) in equation (1) is related to the persistent current through , where M_{0} is a characteristic mutual inductance determined by the qubit–qubit couplers. Annealing is accomplished by linearly changing Φ_{2x} from −0.59Φ_{0} at t=0 to −0.65Φ_{0} at t=t_{f}. To have a uniform Hamiltonian throughout the evolution, other timedependent biases are also applied as detailed in Harris et al.^{22}
The problem instance
The instance, shown in Fig. 2a, is designed to have a small energy gap anticrossing, using a phenomenon similar to that presented in Dickson^{32}. The eight central qubits in the figure, if unbiased, would have as ground states the alldown and allup ferromagnetically ordered states. Additional qubits are attached in such a way as to cause 255 more states, close to the allup state and far from the alldown state, to become degenerate with these states. Then, biases on the central qubits are adjusted to raise the energy of the 256 nearby states, relative to the alldown state, . Repulsion of the 256 degenerate states due to tunnelling between them, away from the end of the evolution, causes the uniform superposition of them, , to be lower in energy than for large enough Δ_{i}(s). Because there is a small amount of tunnelling between and , they anticross with a small gap, while being separated from higher energy states by about Δ_{i}(s*), as shown in Fig. 2c. Additional details can be found in Supplementary Fig. S1 and Supplementary Note 1.
Additional information
How to cite this article: Dickson, N. G. et al. Thermally assisted quantum annealing of a 16qubit problem. Nat. Commun. 4:1903 doi: 10.1038/ncomms2920 (2013).
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Acknowledgements
We thank I. Affleck, D. V. Averin, H. G. Katzgraber, D. A. Lidar, R. Raussendorf and A. Yu. Smirnov for useful discussions.
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N.G.D. and M.H.A. developed the idea for the experiment. N.G.D. and M.W.J. conducted the experiment. M.W.J., R.H., F.A., A.J.B., E.H., T.L., R.N., A.P., C.R., E.M.C., P.C., S.U. and A.B.W. conducted supporting device parameter measurements and developed measurement algorithms. P.B. and E.T. implemented the chip mask design. F.C. and P.S. developed the drive electronics. E.L., N.L., T.O. and J.W. fabricated the devices. A.T. and M.C.T. designed portions of the measurement apparatus. J.P.H. selected the devices for testing. C.E., C.R., P.dB. and C.P. installed the device and supported operation of the measurement apparatus. M.D.B., I.P., J.C., T.C. and P.dB. developed the software interface for the measurement apparatus. N.G.D., K.K., F.H. and M.H.A. analysed the problem instance for suitability. M.H.A., N.G.D., M.W.J., R.H., Z.M., E.M.C., A.B.W., T.M., S.G. and G.R. wrote and edited the manuscript.
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Dickson, N., Johnson, M., Amin, M. et al. Thermally assisted quantum annealing of a 16qubit problem. Nat Commun 4, 1903 (2013). https://doi.org/10.1038/ncomms2920
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DOI: https://doi.org/10.1038/ncomms2920
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