Abstract
Closedsystem quantum annealing is expected to sometimes fail spectacularly in solving simple problems for which the gap becomes exponentially small in the problem size. Much less is known about whether this gap scaling also impedes opensystem quantum annealing. Here, we study the performance of a quantum annealing processor in solving such a problem: a ferromagnetic chain with sectors of alternating coupling strength that is classically trivial but exhibits an exponentially decreasing gap in the sector size. The gap is several orders of magnitude smaller than the device temperature. Contrary to the closedsystem expectation, the success probability rises for sufficiently large sector sizes. The success probability is strongly correlated with the number of thermally accessible excited states at the critical point. We demonstrate that this behavior is consistent with a quantum opensystem description that is unrelated to thermal relaxation, and is instead dominated by the system’s properties at the critical point.
Introduction
Quantum annealing (QA)^{1,2,3,4,5,6,7,8}, also known as the quantum adiabatic algorithm^{9,10} or adiabatic quantum optimization^{11,12} is a heuristic quantum algorithm for solving combinatorial optimization problems. Starting from the ground state of the initial Hamiltonian, typically a transverse field, the algorithm relies on continuously deforming the Hamiltonian such that the system reaches the final ground state—typically of a longitudinal Ising model—thus solving the optimization problem. In the closedsystem setting, the adiabatic theorem of quantum mechanics^{13} provides a guarantee that QA will find the final ground state if the runtime is sufficiently large relative to the inverse of the quantum ground state energy gap^{14,15}. However, this does not guarantee that QA will generally perform better than classical optimization algorithms. In fact, it is wellknown that QA, implemented as a transverse field Ising model, can result in dramatic slowdowns relative to classical algorithms even for very simple optimization problems^{12,16,17,18,19}. Generally, this is attributed to the appearance of exponentially small gaps in such problems^{20}.
A case in point is the ferromagnetic Ising spin chain with alternating coupling strength and open boundary conditions studied by Reichardt^{12}. The “alternating sectors chain” (ASC) of length N spins is divided into equally sized sectors of size n of “heavy” couplings W_{1} and “light” couplings W_{2}, with W_{1} > W_{2} > 0. Since all the couplings are ferromagnetic, the problem is trivial to solve by inspection: the two degenerate ground states are the fully aligned states, with all spins pointing either up or down. However, this simple problem poses a challenge for closedsystem QA since the transverse field Ising model exhibits an exponentially small gap in the sector size n^{12}, thus forcing the runtime to be exponentially long in order to guarantee a constant success probability. A related result is that QA performs exponentially worse than its imaginarytime counterpart for disordered transverse field Ising chains with open boundary conditions^{21}, where QA exhibits an infiniterandomness critical point^{22}.
As a corollary, we may naively expect that for a fixed runtime, the success probability will decrease exponentially and monotonically with the sector size. While such a conclusion does not follow logically from the adiabatic theorem, it is supported by the wellstudied Landau–Zener twolevel problem^{23,24,25}. How relevant are such dire closedsystem expectations for realworld devices?
By varying the sector size of the ASC problem on a physical quantum annealer, we find a drastic departure from the above expectations. Instead of a monotonically decreasing success probability (at constant runtime), we observe that the success probability starts to grow above a critical sector size n^{*}, which depends mildly on the chain parameters (W_{1}, W_{2}). We explain this behavior in terms of a simple opensystem model whose salient feature is the number of thermally accessible states from the instantaneous ground state at the quantum critical point. The scaling of this “thermal density of states” is nonmonotonic with the sector size and peaks at n^{*}, thus strongly correlating with the success probability of the quantum annealer. Our model then explains the success probability behavior as arising predominantly from the number of thermally accessible excitations from the ground state, and we support this model by adiabatic master equation simulations. Our result does not imply that opensystem effects can lend an advantage to QA, and hence it is different from proposed mechanisms for how opensystem effects can assist QA. For example, thermal relaxation is known to provide one form of assistance to QA^{26,27,28,29}, but our model does not use thermal relaxation to increase the success probability above n^{*}. We note that Ref.^{28} introduced the idea that significant mixing due to opensystem effects (beyond relaxation) at an anticrossing between the first excited and ground states could provide an advantage, and its theoretical predictions were supported by the experiments in Ref.^{30}. In Ref.^{28}, an analysis of adiabatic Grover search was performed (a model which cannot be experimentally implemented in a transverse field Ising model), along with numerical simulations of random field Ising models. In contrast, here we treat an analytically solvable model that is also experimentally implementable using current quantum annealing hardware. We also compare our empirical results to the predictions of the classical spinvector Monte Carlo (SVMC) model^{31}, and find that it does not adequately explain them. Our study lends credence to the notion that the performance of realworld QA devices can differ substantially from the scaling of the quantum gap.
Result
The alternating sectors chain model
We consider the transverse Ising model with a timedependent Hamiltonian of the form:
where t_{f} is the total annealing time, s = t/t_{f} ∈ [0, 1], and A(s) and B(s) are the annealing schedules, monotonically decreasing and increasing, respectively, satisfying B(0) = 0 and A(1) = 0. The alternating sectors chain Hamiltonian is
where for a given sector size n the couplings are given by
Thus the b + 1 oddnumbered sectors are “heavy” (J_{i} = W_{1}), and the b evennumbered sectors are “light” (J_{i} = W_{2}) for a total of \(2b + 1 = \frac{{N  1}}{n}\) sectors. This is illustrated in Fig. 1.
We briefly summarize the intuitive argument of Ref.^{12} for the failure of QA to efficiently solve the ASC problem. Consider the \(N \gg 1\) and \(n \gg 1\) limit, where any given light or heavy sector resembles a uniform transverse field Ising chain. Each such transverse field Ising chain encounters a quantum phase transition separating the disordered phase and the ordered phase when the strength of the transverse field and the chain coupling are equal, i.e., when A(s) = B(s)J_{i}^{32}. Therefore, the heavy sectors order independently before the light sectors during the anneal. Since the transverse field generates only local spin flips, QA is likely to get stuck in a local minimum with domain walls (antiparallel spins resulting in unsatisfied couplings) in the disordered (light) sectors, if t_{f} is less than exponential in n. We note that this mechanism, in which large local regions order before the whole is wellknown in disordered, geometrically local optimization problems, giving rise to a Griffiths phase^{22}.
This argument explains the behavior of a closedsystem quantum annealer operating in the adiabatic limit. To check its experimental relevance, we next present the results of tests performed with a physical quantum annealer operating at nonzero temperature.
Empirical results
As an instantiation of a physical quantum annealer we used a DWave 2X (DW2X) processor. We consider ASCs with sector size n ∈ [2,20]. Since the number of sectors b = (N − 1)/n must be an integer, the chain length varies slightly with n. The minimum gap for these chains is below the processor temperature. Additional details about the processor and of our implementation of these chains are given in Section Methods.
Figure 2a–c shows the empirical success probability results for a fixed annealing time t_{f} = 5 μs. Longer annealing times do not change the qualitative behavior of the results, but do lead to changes in the success probability (we provide these results in Supplementary Note 7). A longer annealing time can result in more thermal excitations near the minimum gap, but it may also allow more time for ground state repopulation after the minimum gap. The latter can be characterized in terms of a recombination of fermionic excitations by a quantumdiffusion mediated process^{33}. Unfortunately, we cannot distinguish between these two effects, as we only have access to their combined effect in the finaltime success probability.
In stark contrast to the theoretical closedsystem expectation, the success probability does not decrease monotonically with sector size, but exhibits a minimum, after which it grows back to close to its initial value. The decline as well as the initial rise are exponential in n. Longer chains result in a lower P_{G} and a more pronounced minimum, but the position of the minimum depends only weakly on the chain parameter values (W_{1}, W_{2}) (the value of n^{*} shifts to the right as (W_{1}, W_{2}) are increased) but not on N.
What might explain this behavior? Clearly, a purely gapbased approach cannot suffice, since the gap shrinks exponentially in n for the ASC problem^{12} [see also the inset of Fig. 2a]. However, for all chain parameters we have studied, the temperature is greater than the quantum minimum gap. In this setting not only the gap matters, but also the number of accessible energy levels that fall within the energy scale set by the temperature. In an opensystem description of quantum annealing^{26,34,35,36,37,38}, both the Boltzmann factor exp(−βΔ) (β denotes the inverse temperature and Δ is the minimum gap) and the density of states determine the excitation and relaxation rates out of and back to the ground state. As we demonstrate next, the features of the DW2X success probability results, specifically the exponential fall and rise with n, and the position of the minimum, can be explained in terms of the number of singlefermion states that lie within the temperature energy scale at the critical point.
Fermionization
We can determine the spectrum of the quantum Hamiltonian [Eq. (1)] by transforming the system into a system of free fermions with fermionic raising and lowering operators \(\eta _k^\dagger\) and η_{k}^{32,39}. The result is^{12}:
where E_{g}(s) is the instantaneous ground state energy and {λ_{k}(s)} are the singlefermion state energies, i.e., the eigenvalues of the linear system
where the matrices A and B are tridiagonal and are given in Supplementary Note 1 along with full details of the derivation. The vacuum of the fermionic system \(\vert{0}\rangle\) is defined by \(\eta_k0\rangle=0\forall{k}\) and is the ground state of the system. Higher energy states correspond to single and manyparticle fermionic excitations of the vacuum. At the end of the anneal, fermionic excitations corresponds to domain walls in the classical Ising chain (see Supplementary Note 2).
The Ising problem is \({\Bbb Z}_2\)symmetric, so the ground state and the first excited state of the quantum Hamiltonian merge toward the end of evolution to form a doubly degenerate ground state. Since any population in the instantaneous first excited state will merge back with the ground state at the end of the evolution, the relevant minimum gap of the problem is the gap between the ground state and the second excited state: Δ(s) = λ_{2}(s), which occurs at the point s^{*} = argmin_{s∈[0,1]}Δ(s). In the thermodynamic limit, this point coincides with the quantum critical point, where the geometric mean of the Ising fields balances the transverse field, \(A\left( {s^ \ast } \right) = \sqrt {W_1W_2} B\left( {s^ \ast } \right)\)^{40,41}. Henceforth, we write Δ ≡ Δ(s^{*}) for the minimum gap.
Spectral analysis
Let k^{*} be the number of singlefermion states with energy smaller than the thermal gap at the critical point, i.e.,
As can be seen by comparing Fig. 2d–f to Fig. 2a–c, we find that the behavior of k^{*} correlates strongly with the ground state success probability for all ASC cases we tested, when we set T = 12 mK = 1.57 GHz, the operating temperature of the DW2X processor (we use k_{B} = ℏ = 1 units throughout). Specifically, k^{*} peaks exactly where the success probability is minimized, which strongly suggests that k^{*} is the relevant quantity explaining the empirically observed quantum annealing success probability. Longer chains result in a larger value of k^{*} and a more pronounced maximum. Of all the ASC sets we tried, we only found a partial exception to this rule for the case (1, 0.5, and 200), where k^{*} peaks at n^{*} = 5 [Fig. 2e], but the empirical success probability for n = 5 and n = 6 is roughly the same [Fig. 2b]. We show later that this exception can be resolved when the details of the energy spectrum are taken into account via numerical simulations.
In contrast, the total number of energy eigenstates (including multifermion states) that lie within the thermal gap [E_{g}(s^{*}), E_{g}(s^{*}) + T], while rising and falling exponentially in n like the empirical success probability in Fig. 2a, does not peak in agreement with the peak position of the latter [see the inset of Fig. 2d].
Why and how does the behavior of k^{*} explain the value of n^{*}? Heuristically, we expect the success probability to behave as
where d is the “thermal density of states” at the critical point s^{*}. Note that the role of the gap here is different from the closedsystem case, since we are assuming that thermal transitions dominate over diabatic ones, so that the gap is compared to the temperature rather than the annealing time. Contrast this with the closedsystem case, where the Landau–Zener formula for closed twolevel systems and Hamiltonians analytic in the time parameter (subject to a variety of additional technical conditions) states that: \(P_{\mathrm{{G}}}\sim 1  e^{  \eta {\mathrm{\Delta }}^2t_{\mathrm{f}}}\), where η is a constant with units of time that depends on the parameters that quantify the behavior at the avoided crossing (appearing in, e.g., the proof of Theorem 2.1 in Ref.^{25}). Since then P_{G} = O(ηΔ^{2}t_{f}), we expect the success probability to decrease exponentially at constant runtime t_{f} if the gap shrinks exponentially in the system size.
Our key assumption is that the thermal transitions between states differing by more than one fermion are negligible. That is, thermal excitation (relaxation) only happens via creation (annihilation) of one fermion at a time (see Supplementary Note 3 for a detailed argument). Additionally, the Boltzmann factor suppresses excitations that require energy exchange greater than \(\lambda _{k^ \ast }\). Starting from the ground state, all singlefermion states with energy \(\le E_{\mathrm{{g}}} + \lambda _{k^ \ast }\) are populated first, followed by all twofermion states with total energy \(\le E_{\mathrm{{g}}} + \lambda _{k^ \ast } + \lambda _{k^ \ast  1}\), etc. In all, \(\mathop {\sum}\nolimits_{k = 1}^{k^ {\ast} } \left( {\begin{array}{*{20}{c}} {k^ {\ast} } \\ k \end{array}} \right) = 2^{k^ {\ast} }  1\) excited states are thermally populated in this manner. Thus \(d \sim 2^{k^ \ast }\) states are thermally accessible from the ground state.
For a sufficiently small gap we have \(1  e^{  \beta {\mathrm{\Delta }}}\sim \beta {\mathrm{\Delta }}\), so that \(P_{\mathrm{{G}}}\sim \beta {\mathrm{\Delta }}/d\). As can be seen from Fig. 2d–f, k^{*} rises and falls steeply for n < n^{*} and n > n^{*}, respectively. For the ASCs under consideration, d varies much faster with n than the gap Δ (see Fig. 3). Thus \(P_{\mathrm{{G}}}\sim 2^{  k^ {\ast} }\). This argument explains both the observed minimum of P_{G} at n^{*} and the exponential drop and rise of P_{G} with n, in terms of the thermal density of states. In Supplementary Note 4 we give a more detailed argument based on transition rates obtained from the adiabatic master equation, which we discuss next.
Master equation model
We now consider a simplified model of the opensystem dynamics in order to make numerical predictions. We take the evolution of the populations p = {p_{a}} in the instantaneous energy eigenbasis of the system to be described by a Pauli master equation^{42}. The form of the Pauli master equation is identical to that of the adiabatic Markovian quantum master equation^{35}, derived for a system of qubits weakly coupled to independent identical bosonic baths. The master equation with an Ohmic bosonic bath has been successfully applied to qualitatively (and sometimes quantitatively) reproduce empirical DWave data^{43,44,45,46}. However, it does not account for 1/f noise^{47}, which may invalidate the weak coupling approximation when the energy gap is smaller than the temperature^{48}.
After taking diagonal matrix elements and restricting just to the dissipative (nonHermitian) part one obtains the Pauli master equation^{42} describing the evolution of the population p = {p_{a}} in the instantaneous energy eigenbasis of the system:^{38}
Here all quantities are timedependent and the matrix elements are
where we have assumed an independent thermal bath for each qubit α and where the indices a and b run over the instantaneous energy eigenstates of the system Hamiltonian [Eq. 4] in the fermionic representation [i.e., \(H(s)a(s)\rangle = E_a(s)a(s)\rangle\)] and ω_{ab} = E_{a} − E_{b} is the corresponding instantaneous Bohr frequency. Since the basis we have written this equation in is timedependent, there are additional terms associated with the changing basis^{35}, but we ignore these terms here since we are assuming that the system is dominated by the dissipative dynamics associated with its interaction with its thermal environment.
The rates γ(ω) satisfy the quantum detailed balance condition^{49,50}, γ(−ω) = e^{−βω}γ(ω), where ω ≥ 0. In our model each qubit is coupled to an independent puredephasing bath with an Ohmic power spectrum:
with UV cutoff ω_{c} = 8π GHz and the dimensionless coupling constant ηg^{2} = 1.2 × 10^{−4}. The choice for the UV cutoff satisfies the assumptions made in the derivation of the master equation in the Lindblad form^{35}. Note that we do not adjust any of the master equation parameter values, which are taken from Ref.^{43}. Details about the numerical solution procedure are given in Section Methods, and in Supplementary Note 5 we also confirm that the validity conditions for the derivation of the master equation are satisfied for a relevant range of n values given the parameters of our empirical tests.
Numerically solving the master equation while accounting for all thermally populated \(2^{k^ \ast }\) states is computationally prohibitive, but we can partly verify our interpretation by restricting the evolution of the system described in Eq. (8) to the vacuum and singlefermion states. This is justified in Supplementary Note 3, where we show that transitions between states differing by more than a single fermion are negligible. In other words, the dominant thermal transitions occur from the vacuum to the singlefermion states, from the singlefermion states to the twofermion states, etc. The restriction to the vacuum and singlefermion states further simplifies the master Eq. (8) to:
where \(\left\{ {p_b} \right\}_{b = 1}^N\) are the singleparticle fermion energy populations and {λ_{b}} their energies found by solving Eq. (5), and M_{ab} [Eq. (9)] \(M_b = \mathop {\sum}\nolimits_{\alpha = 1}^N \left \langle 0 \vert\sigma _\alpha ^z\vert b\rangle \right^2\). For a better approximation that accounts for more states, we can also perform a twofermion calculation where we keep the vacuum, the first k^{*} onefermion states and the next k^{*}(k^{*} − 1)/2 twofermion states. For twofermion simulations the master equation becomes Eqs. (11) and (12) along with
where all summations run from 1 to k^{*}, and p_{ij} denotes the population in the twoparticle fermion energy state \(\eta _i^\dagger \eta _j^\dagger \vert0\rangle\).
We can now numerically solve this system of equations. As seen in Fig. 4a, where we plot the final populations in the different singleparticle fermion states at t = t_{f} for onefermion simulations, only the first k^{*} singlefermion levels are appreciably populated. This agrees with our aforementioned assumption that states with energy greater than \(\lambda _{k^ \ast }\) are not thermally populated. In Fig. 4b we plot the population in the instantaneous ground state as a function of time for twofermion simulations. The system starts in the gapped phase where the ground state population is at its chosen initial value of 1. The ground state rapidly loses population via thermal excitation as the system approaches the critical point, after which the population essentially freezes, with repopulation via relaxation from the excited states essentially absent (see inset). Thus, it is not relaxation that explains the increase in ground state population seen in Fig. 2a–c for n > n^{*}. Instead, we find that the ground state population drops most deeply for n = n^{*}. This, in turn, is explained by the behavior of k^{*} seen in Fig. 2d–f, as discussed earlier.
We show in Fig. 5 the predicted final ground state population under the one and twofermion restriction. This minimal model already reproduces the correct location of the minimum in P_{G}. It also reproduces the nonmonotonic behavior of the success probability. It does not correctly reproduce the exponential fall and rise. However, including the twofermion states gives the right trend: it leads to a faster decrease and increase in the population without changing the position of the minimum, suggesting that a simulation with the full \(2^{k^ \ast }\) states would recover the empirically observed exponential dependence of the ground state population seen in Fig. 2a–c.
Discussion
A commonly cited failure mode of closedsystem quantum annealing is the exponential closing of the quantum gap with increasing problem size. It is expected, on the basis of the Landau–Zener formula and the quantum adiabatic theorem, that to keep the success probability of the algorithm constant the runtime should increase exponentially. As a consequence, one expects the success probability to degrade at constant runtime if the gap decreases with increasing problem size. Our goal in this work was to test this failure mode in an opensystem setting where the temperature energy scale is always larger than the minimum gap. We did so by studying the example of a ferromagnetic Ising chain with alternating couplingstrength sectors, whose gap is exponentially small in the sector size, on a quantum annealing device. Our tests showed that while the success probability initially drops exponentially with the sector size, it recovers for larger sector sizes. We found that this deviation from the expected closedsystem behavior is qualitatively and semiquantitatively explained by the system’s spectrum around the quantum critical point. Specifically, the scaling of the quantum gap alone does not account for the behavior of the system, and the scaling of the number of energy eigenstates accessible via thermal excitations at the critical point (the thermal density of states) explains the empirically observed ground state population.
Does there exist a classical explanation for our empirical results? We checked and found that the SVMC model^{31} is capable of matching the empirical DW2X results provided we finetune its parameters for each specific chain parameter set {W_{1}, W_{2}, N}. However, it does not provide as satisfactory a physical explanation of the empirical results as the fermionic or master equation models, which require no such finetuning; see Supplementary Note 6 for details.
Our work demonstrates that care must be exercised when inferring the behavior of opensystem quantum annealing from a closedsystem analysis of the scaling of the gap. It has already been pointed out that quantum relaxation can play a beneficial role^{26,27,28,29,30}. However, we have shown that relaxation plays no role in the recovery of the ground state population in our case. Instead, our work highlights the importance of a different mechanism: the scaling of the number of thermally accessible excited states. Thus, to fully assess the prospects of opensystem quantum annealing, this mechanism must be understood along with the scaling of the gap and the rate of thermal relaxation. Of course, ultimately we only expect opensystem quantum annealing to be scalable via the introduction of error correction methods^{51,57,57,58,59}.
Methods
Alternating sector chains
We generated a set of ASCs with chains lengths centered at \(N\sim \left\{ {55,135,175,200} \right\}\) and with sector sizes n ranging from 2 to 20. Since the chain length and sector size must obey the relation (N − 1)/n = 2b + 1 with integer b, there is some variability in N. Table 1 gives the (N, n) pair combinations we used for chain set with mean length 175.
Quantum annealing processor used in this work
The DWave 2X processor (DW2X) is an 1152qubit quantum annealing device made by DWave Systems, Inc., using superconducting flux qubits^{52}. The particular processor used in this study is located at the University of Southern California’s Information Sciences Institute, with 1098 functional qubits and an operating temperature of 12 mK. The total annealing time t_{f} can be set in the range [5, 2000] μs. The timedependent Hamiltonian the processor is designed to implement is given by
with dimensionless time s = t/t_{f}. Figure 6 describes the annealing schedules A(s) and B(s). The coupling strengths J_{ij} between qubits i and j can be set in the range [−1, 1] and the local fields h_{i} can be set in the range [−2, 2].
We used t_{f} = 5 μs. For each ASC instance we implemented 10 different embeddings, with 10 gauge transforms each^{53}. In total, 10^{5} runs and readouts were taken per instance. The reported success probability is defined as the fraction of readouts corresponding to a correct ground state. For additional details on the DW2X processor we used see, e.g., Ref.^{54}.
Numerical procedure for solving the master equation
We solve the coupled differential Eqs. (11)–(14) using a fourth order Runge–Kutta method given by Dormand and Prince^{55} with nonnegativity constraints^{56}. We compute the transition matrix elements via Supplementary Eq. (35) and the bath correlation term via Eq. (10).
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgments
We thank Ben W. Reichardt for useful discussions. The computing resources used for this work were provided by the USC Center for High Performance Computing and Communications. A.M. was supported by the USC Provost Ph.D. Fellowship. The research is based upon work (partially) supported by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA), via the U.S. Army Research Office contract W911NF17C0050. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon.
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A.M., T.A., and D.A.L. designed the experiments, which A.M. performed. A.M., T.A. and D.A.L. contributed to the analysis and the writing of the manuscript. A.M.computed the thermal transition probabilities and performed the numerical simulations.
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Mishra, A., Albash, T. & Lidar, D.A. Finite temperature quantum annealing solving exponentially small gap problem with nonmonotonic success probability. Nat Commun 9, 2917 (2018). https://doi.org/10.1038/s41467018052399
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DOI: https://doi.org/10.1038/s41467018052399
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