^{*}

Edited by: Jacob Biamonte, Institute for Scientific Interchange Foundation, Italy

Reviewed by: Alexandre M. Zagoskin, Loughborough University, UK; Scott Aaronson, Massachusetts Institute of Technology, USA

*Correspondence: John A. Smolin, IBM Research, 1101 Kitchawan Road, Yorktown, NY 10598, USA e-mail:

This article was submitted to Interdisciplinary Physics, a section of the journal Frontiers in Physics.

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A pair of recent articles [

Adiabatic quantum computation [

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 [

The D-Wave machine is made of superconducting “flux” qubits [

In Boixo et al. [

The probability of finding the isolated (all down) state _{s}_{C}_{s}_{s}_{C}_{s}_{C}

In Boixo et al. [^{1}

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 [

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 _{f}_{i}_{i}_{i}_{f}_{i}_{f}^{2}_{f}^{3}

Since classical simulated annealing is intrinsically random and “quantum annealing” is not, the differences reported in Boixo et al. [

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. [

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 [_{i} and coupled to each other with coupling _{ij}_{i}_{X}

and

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

The adiabatic computation is performed (or simulated) by running the dynamics while gradually changing these potentials from _{trans} to _{ising} over a time

with

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

The simulated adiabatic dragging time

For this case we programmed 108 spins with the same connectivity and same random ± couplings as in Boixo et al. [

We show in Figure

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. [

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,

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 [

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 [

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The authors thank Charles H. Bennett, Jay Gambetta, Mark Ritter, and Matthias Steffen for helpful comments on our manuscript.

^{1}They also look at what happens with fewer than 108 spins, so the total number of experiments they performed is actually much larger.

^{2}If the ground state is degenerate at some point during the process then this may not be true.

^{3}Of course, “slowly enough,” depends on the gapbetween the ground state and other nearby energy eigenstates. If the system is frustrated, there are many such states and the evolution must proceed exponentially slowly, just as frustration hinders classical simulated annealing. Though it has often been suggested that quantum annealing is a panacea, whether it can outperform classical simulated annealing in such cases is unknown.