A Quantum Annealer, such as a D-Wave machine is a physical representation of the Ising model and as such has a 'problem' Hamiltonian of the form $$H_P = \sum_{J=1}^nh_j\sigma_j^z + \sum_{i, j}J_{ij}\sigma_i^z\sigma_j^z.$$

Essentially, the problem to be solved is mapped to the above Hamiltonian. The system starts with the Hamiltonian $H_I = \sum_{J=1}^nh'_j\sigma_j^x$ and the annealing parameter, $s$ is used to map the initial Hamiltonian $H_I$ to the problem Hamiltonian $H_P$ using $H\left(s\right) = \left(1-s\right)H_I + sH_P$.

As this is an anneal, the process is done slowly enough to stay near the ground state of the system while the Hamiltonian is varied to that of the problem, using tunnelling to stay near the ground state as described in Nat's answer.

Now, why can't this be used to describe a gate model QC? The above is a Quadratic unconstrained binary optimization (QUBO) problem, which is NP-hard... Indeed, here's an article mapping a number of NP problems to the Ising model. Any problem in NP can be mapped to any NP-hard problem in polynomial time and integer factorisation is indeed an NP problem.

Well, the temperature is non-zero, so it's not going to be in the ground state throughout the anneal and as a result, the solution is still only an approximate one. Or, in different terms, the probability of failure is greater than a half (it's nowhere near having a decent probability of success compared with what a universal QC considers 'decent' - judging from graphs I've seen, the probability of success for the current machine is around $0.2\%$ and this will only get worse with increasing size), and the anneal algorithm is not bounded error. At all. As such, there's no way of knowing whether or not you've got the correct solution with something such as integer factorisation.

What it (in principle) does is get very close to the exact result, very quickly, but this doesn't help for anything where the exact result is required as going from 'nearly correct' to 'correct' is still an extremely difficult (i.e. presumably still NP in general, when the original problem is in NP) problem in this case, as the parameters that are/give a 'nearly correct' solution aren't necessarily going to be distributed anywhere near the parameters that are/give the correct solution.

Edit for clarification: what this means is that a quantum annealer (QA) still takes exponential time (albeit potentially a faster exponential time) to solve NP problems such as integer factorisation, where a universal QC gives an exponential speed up and can solve the same problem in poly time. This is what implies a QA cannot simulate a universal QC in poly time (otherwise it could solve problems in poly time that it can't). As pointed out in the comments, this is *not* the same as saying that a QA cannot give the same speedup in other problems, such as database search.

1

If I understand correctly, you are basically saying that a quantum annealer cannot describe a quantum circuit because the problem of finding the minimum of an arbitrary Hamiltonian is NP-hard. I don't understand this implication. Simulating quantum circuits is also in general hard to simuate classically (see e.g. 1610.01808)

– glS – 2018-03-15T12:41:17.4571

Also, some problems solvable via algorithms expressed as quantum circuits are known to be also solvable via quantum annealing. A notable example is database search (see e.g. section II of 1006.1696). This means that in some sense one

– glS – 2018-03-15T12:51:22.477canin some circumstances map a q circuit into an q annealing problem. Doesn't this also invalidate your third paragraph (specifically, the claim thatthis [can't] be used to describe a gate model QC)1@glS no, not at all - it still takes exponential time to find the min (as per the paper in your second comment) of an NP-hard problem, so while there are problems in P (e.g. database search) where the speedup may be able to match that of universal QC, solving an NP problem still takes exponential time to be within bounded error, where a universal QC may be able to solve the same problem in poly time, e.g. integer factorisation. As QA can't do this, a QA cannot simulate a universal QC in poly time – Mithrandir24601 – 2018-03-15T13:28:34.610

Ok, but that is not what you are saying in the answer (or at least, not explicitly). From the answer it looks like you are saying that QA can never be used to solve a problem solved via gate model QC. This is very different than saying that QA cannot efficiently solve an NP-hard problem (which

couldsometimes be solved by a quantum circuit... though I don't think this has been proven, as we don't know whether Factoring is really NP-hard, and most other problems in which a quantum advantage has been shown are not decision problems, to my knowledge). – glS – 2018-03-15T13:31:59.747I've made an edit that hopefully clarifies things. It's not known whether P=NP or not, sure, but it's still a specific example of QC being exponentially faster, according to current knowledge – Mithrandir24601 – 2018-03-15T14:05:15.833

And I've just realised/remembered that it's not

quiteexponentially faster for factorising, although to paraphrase Nathan Wiebe, it's exponential in spirit – Mithrandir24601 – 2018-03-15T16:39:57.120