Adiabatic quantum computation [3] has been shown to be equivalent to the usual circuit model [4]. This is only known to hold for ideal systems without noise. While there are effective techniques for fault-tolerance in the circuit model [5], it remains unknown whether adiabatic quantum computation can be made fault-tolerant.

In spite of this, there has been some enthusiasm for implementing restricted forms of adiabatic quantum computation in very noisy hardware based on the hope that it would be naturally robust. Even in the original proposal for quantum adiabatic computation [3] it was suggested it might be a useful technique for solving optimization problems. Recent papers about D-Wave hardware have studied a particular sort of optimization problem, namely finding the ground state of a set of Ising spins. These spins are taken to live on a graph. The problem instance is determined by a choice of a graph and either ferromagnetic or antiferromagnetic interactions between each pair of bits connected by an edge of the graph. Finding this ground state is NP-hard if the graph is arbitrary [6] and efficiently approximable when the graph is planar [7]. The connectivity of the D-Wave machine is somewhere in between and it is not known whether the associated problem is hard.

The D-Wave machine is made of superconducting “flux” qubits [8] (first described in Mooij et al. [9]). Because of the high decoherence rates associated with these flux qubits, it has been unclear whether the machine is fundamentally quantum or merely performing a calculation equivalent to that of a classical stochastic computer. Boixo et al. [1, 2] attempt to distinguish between these possibilities by proposing tests for quantumness that the D-Wave machine passes but a purely classical computer should fail. This letter presents a classical model that passes the tests, exhibiting all the supposedly quantum behaviors.

Is it surprising that results of the D-Wave experiments differ greatly from classical simulated annealing? And should this be considered evidence that the machine is quantum or more powerful than classical computation in some way? We argue here that the answer to these questions is “no.”

What is called “quantum annealing” is often compared to classical simulated annealing [11], or to physical annealing itself. Though both quantum annealing and simulated annealing are used to find lowest-energy configurations, one is not the quantum generalization of the other. More properly, quantum annealing should be considered quantum adiabatic ground state dragging.

Simulated annealing proceeds by choosing a random starting state and/or a high temperature, then evolving the system according to a set of rules (usually respecting detailed balance) while reducing the simulated temperature. This process tends to reduce the energy during its evolution but can, of course, become stuck in local energy minima, rather than finding the global ground state.

Quantum annealing, on the other hand, involves no randomness or temperature, at least in the ideal. Rather, it is a particular type of adiabatic evolution. Two Hamiltonians are considered: The one for which one desires to know the ground state H_f, and one which is simple enough that cooling to its ground state is easy, H_i. At the start of the process, the system is initialized to the ground state of H_i by turning on H_i and turning off H_f and waiting for thermal equilibration. Then H_i is gradually turned off while H_f is gradually turned on. If this is done slowly enough, the system will at all times remain in the ground state of the overall Hamiltonian². Then, at the end of this process the system will be in the ground state of H_f³. It remains to measure the ground state. For the problems considered in Boixo et al. [1, 2], the ground state is typically known to be diagonal in the z basis. Thus, a measurement need not introduce any randomness.

Since classical simulated annealing is intrinsically random and “quantum annealing” is not, the differences reported in Boixo et al. [1, 2] are not surprising. For the eight-bit suppression of finding the isolated state, two things could have happened: Either the ideal machine would find the isolated state always, or never. It happens that, due to the structure of the state, the ideal outcome is “never,” which is certainly a suppression. The bimodal distribution found for the 108-bit computations is also just what one would expect: In the perfect case of no noise either the calculation gets the correct answer or it does not. The outcome is deterministic so there should be exactly two peaks, at probability of success zero and one. Classical annealing, which begins from a random state on each run, is not expected to succeed with probability one, even for cases where the system is not frustrated.

The bimodality of the D-Wave results, in contrast to the unimodality of simulated annealing, can be seen as evidence not of the machine's quantumness, but merely of its greater reproducibility among runs using the same coupling constants, due to its lack of any explicit randomization. The simulated annealing algorithm, by contrast uses different random numbers each time, so naturally exhibits more variablity in behavior when run repeatedly on the same set of coupling constants, leading to a unimodal historgram. Indeed if the same random numbers were used each time for simulated annealing, the histogram would be perfectly bimodal. To remove the confounding influence of explicit randomization, we need to consider more carefully what would be a proper classical analog of quantum annealing.

If, as we have argued, classical simulated annealing is not the correct classical analog of quantum annealing, what is? The natural answer is to classically transform a potential landscape slowly enough that the system remains at all times in the lowest energy state. In the next section we give a model classical system and, by running it as an adiabatic lowest-energy configuration finder, demonstrate that it exhibits the same computational behavior interpreted as a quantum signature in Boixo et al. [1, 2].

The flux qubits in the D-Wave machine decohere in a time considerably shorter than the time adiabatic evolution experiment runs. The decoherence times are stated to be on the order of tens of nanoseconds while the adiabatic runtime is 5–20 ms [2]. For this reason, one would expect that no quantum coherences should exist. We therefore model the qubits as n classical spins (“compass needles”), each with an angle θ_i and coupled to each other with coupling J_ij = 0,± 1. Each spin may also be acted on by external magnetic fields h_i in the z-direction as well as an overall field in the x-direction B_X. The potential energy function is then given by

and

Compare these to Equations (1) and (2) in Boixo et al. [1].

The adiabatic computation is performed (or simulated) by running the dynamics while gradually changing these potentials from V_trans to V_ising over a time T according to

with A(0) = B(T) = 1 and A(T) = B(0) = 0. The equations of motion are simply:

It is straightforward to integrate this system of ordinary differential equations.

We have argued that quantum annealing and simulated annealing are very different procedures. The deterministic nature of quantum annealing leads to rather different behaviors than the random processes of simulated annealing. However, other deterministic procedures can also lead to behavior very similar to that observed in the D-Wave device. Our classical model reproduces all the claimed signatures of quantum annealing. We recommend using the term “ground-state adiabatic dragging” or simply “adiabatic computation” for such nonrandom processes.

Note that in Johnson et al. [12], under highly diabatic conditions, evidence was shown that the D-Wave device exhibits quantum tunneling. We emphasize that this does not constitute evidence for an essential use of quantum effects during adiabatic dragging and indeed an effectively classical model may well capture the physics and computational behavior seen in Boixo et al. [1, 2]. Tunneling is a possible additional path for a system to anneal into a lower energy state without needing enough energy to cross a potential barrier. If the decoherence is such that each spin is projected into a local energy eigenstate, then in the adiabatic limit, where each spin is at all times in its local ground state, tunneling locally can be of no assistance. The tunneling in Johnson et al. [12] is indeed only shown locally.

Furthermore, there is nothing preventing the implementation of our simulated compass model in hardware. It would be possible to build an analog classical machine that could simulate it very quickly, but it would be simpler to use a digital programmable array with one processing core per simulated spin. Since each spin requires knowledge of at most six of its neighbors along the connectivity graph, the algorithm can be easily parallelized. A 108 core computer specialized to running our algorithm could easily run hundreds or thousands of times faster than simulating it on a desktop computer, and could be built for modest cost using off-the-shelf components. Similarly, any classical physics can be efficiently simulated on a classical machine.

This is not to suggest that simulating classical physics directly on a classical computer is a good way to solve optimization problems. Classical simulated annealing [11] and other heuristic techniques have been extremely successful, and can also be very fast on special-purpose hardware. It is shown in Boixo et al. [2] that simulated annealing is competitive with the D-Wave One machine even on desktop CPUs (this has expanded on in great detail including new experiments on the D-Wave Two processor [13]). Stronger evidence of both quantumness and a resulting algorithmic speedup is needed before quantum adiabatic computers will have proved their worth.

In the year since the original posting of this work on the quantum physics archive (arxiv.org), much has happened. The original preprint on which we were commented have been published [14, 15] and [15] in particular was updated to incorporate to some of our criticisms, also discussed in Boixo et al. [16] The main point of the response was that our model captured the bimodality, but not more subtle features of the data. An improved version of our model addressing these details appears as Shin et al. [17] This has also been responded to and re-responded to Vinci et al. [18] and Shin et al. [19] The current status of the question is that for a broad range of parameter space there is a simple classical model that captures the D-wave machine's input-output behavior. There is also a corner of parameter space where this model fails to reproduce certain aspects of the input-output behavior that a quantum master equation does capture. This quantum master equation fails to capture other aspects of the machine's behavior which the classical model does reproduce. As is to be expected for a complex system, there is no known model, either classical or quantum, that captures all aspects of the machine. We are gratified that our modest model has stimulated a more careful study of D-Wave's machine.