No. The mass of a proton is about $1.67times10^{-27}$ kg. Therefore the total maximum energy released by its annihilation with an antiproton is $2mc^2= 2times1.67times10^{-27}times9times10^{16} = 3times10^{-10}$ Joule.

This is not much. Even if all this energy would be deposited inside the victims brain, it is a very small amount. But it would not. As this paper (focusing on the possible applications of antimatter in space propulsion) elaborates, most of the energy is released in the form of fast moving, penetrating pions (which can fly 10 cm even in solid tungsten, and presumably much more in tissue), and some in the form of neutrinos, which are almost non-interacting and useless.

But let us look aside, and estimate the effect it could have with all energy discharged in the victims brain:

Although the released energy are not pure gamma rays, the basic damage mechanism is the same for all high-energy, ionising particles: They kick out electrons that form atoms, severing molecular bonds. Therefore it is useful to calculate the dose. (energy deposited per unit mass) As the mass of human brain is around 1.5 kg, we get $2times10^{-10}$ Grays. For comparison, a single session of radiotherapy can deposit a dose of 1-2 Grays.

The electrons in the atom I have totally neglected, since they have rest mass about 1830 times smaller, and are so harmless, that in PET diagnosis, people can be injected with anti-electron (positron) releasing radioactive materials.

So I am quite sure that this would not kill or incapacitate a person, and would mean little contribution even to his/her long term cancer risk. If the government wants killer implants, go with explosives or electrical gadgets.

No. Positron emission tomography is regularly used to scan brains.

PET detects gamma rays created when positrons, emitted by an injected radio tracer undergoing positron emission decay, annihilate with electrons in the patients tissue.

For example, a brain scan using 18F-FDG has an effective radiation dose of 14 mSv [1], which is on the order of the natural background radiation you are exposed to over a year in Denver, Colorado.

[1] http://hps.org/documents/Medical_Exposures_Fact_Sheet.pdf

This mechanism is a poor one to control populations. To be sure of killing the target, you need a pretty big bang because most of the result of matter/anti-matter annihilation is ionising radiation rather then brain damaging explosion. This means you have to put nearby people and property at significant risk which is counter productive. For this type of big brother technological control of population, it might be better to consider a simpler embedded device which is critical for everyday living in your society (making its presence acceptable to the population) and which has secondary control and punitive roles such as triggering pain, immobilizing, and if necessary, terminating the host. Now rebels who wish to bypass its controls must find alternate ways to survive in their society, offering you lots of additional story options.

No...but how much would it take?

This puts a Chinese firecracker at about 30 Joules and Wikipedia puts a gunshot's kinetic energy at 1.8×10³

So we'll assume about 100 Joules as a necessary amount of energy to kill a person when released directly into the brain. Orders of magnitude here are the important factor.

b.Lorenz's answer has a single proton annihilation at 3×10⁻¹⁰ Joules.

Dividing the first by the second tells us that we need approximately 3×10¹² hydrogen atoms worth of antimatter (on the order of ten billion times more than CERN has collected in one place at the same time). A mole is 6.022x10²³ atoms (and weighs about 1 gram), so we need about 5×10^-11 grams worth of hydrogen (or really, any anti-element: the neutrons required do increase the mass, but we're talking about so little that even a hundred times as much is still on the order of a single nanogram).

Your containment device would probably take up more mass as well as requiring external power (you need to contain it in an electromagnetic field). This is, of course, assuming that you can keep it trapped for very long at all.

You may as well just use gunpoweder.

No, because we have real-world examples. Astronauts are outside the shield of our atmosphere and occasionally get hit with high energy cosmic rays. These carry a bigger punch, yet they don't kill.

Almost certainly not. As in, "you'll have more chance of winning the Lottery than you will of killing them by this method".

Antimatter annihilation of a single atom - we'll be good here and say one with a hefty nucleus like, say, iron - releases

$$left(2 mathrm{atoms}right) times left(frac{55.8452 mathrm{g}}{1 mathrm{mol}}right) times left(frac{1 mathrm{kg}}{1000 mathrm{g}}right) times left(frac{1 mathrm{mol}}{6.022 times 10^{23} mathrm{atoms}}right) times c^2 approx 1.67 times 10^{-8} mathrm{J}$$

which is 16.7 nanojoules, or over 100 GeV, of energy. (The "2 atoms" factor is because you need a second atom's worth in equivalent - not necessarily in the form of a literal single atom - of ordinary matter to complete the annihilation.) The release of this will likely not be all at once, but rather will basically consist of the heavy anti-iron atom, upon teleportation to the brain center, annihilating with some lighter atom which will cause it to explode catastrophically into a shower of lighter particles and anti-particles as well as VERY hard (100 MeV+) gamma rays for the anti-nucleon annihilations, and these anti-particles will also collide with and cause similar explosions of the atoms they encounter elsewhere, producing even more showers of tertiary, quaternary, etc. ionizing particles. Essentially it's a demolition derby on an atomic scale with billions of bits of high-energy matter flying around and knock apart everything in their wake - DNA, proteins, and more. Keep in mind that a chemical bond has energy only on the order of 1 eV, so this is enough to break on the order of 100 billion chemical bonds.

Now that sounds rather extreme. But there's two things to keep in mind here: Even a single cell, if we for simplicity [and wrongly] treat it as a sphere of water 10 µm in diameter, contains about 17 trillion molecules and thus 34 trillion chemical bonds. Effectively there's only enough energy to break about 0.3% of them. Granted, that could be considerably destructive to that single cell, and thus you might expect we could at least kill one neuron with this (you cannot turn a neuron to cancer, because they cannot divide, though if you get something like a glial cell, then it's possible in theory, and this is a real and actually common type of brain tumor, called a glioma). However, that assumes all the particles are absorbed in the neuron, and that will almost surely not be the case, because that would mean total absorption within 5 micrometers assuming it appears dead center, and these forms of radiation are far more penetrating. The result is maybe you might break a few thousand or million of bonds all over the entire brain - something with maybe over $10^{24}$ atoms in it. That will be virtually unnoticeable.

Which is what our second point is. The 100 GeV of energy released here corresponds to about a thousand typical 1 MeV particles of the type that naturally exist in background radiation, not taking into account the possibly increased penetration of some of the highest-energy products which will make it even less damaging(*). As a dose to the brain tissue itself, it corresponds to (assuming it like gamma, which will actually not, again, be right, but we just want the order of magnitude, and using 1.5 kg for the mass of a brain) around 10 nanosieverts (nSv) of dose. The average background exposure in the United States is 3.1 millisieverts (mSv) per year (cite: https://www.nrc.gov/reading-rm/doc-collections/fact-sheets/bio-effects-radiation.html) or about 99 nSv/Ms. Thus your brain is dosed with about this much about every 0.1 Ms, or 100 ks, or a bit over a day (86.4 ks). In effect, you get all of an extra day's and change worth of normal background dosage for this stint. Very unlikely to kill, and impossible to kill "instantly". In fact such ultra-low doses may even have a protective, and not harmful, effect due to possible radiation hormesis (not sure what the evidence on this is as of now).

Nonetheless, there is a potentially useful lateral angle to this that might be worth considering, and that's that if people generally have a fear of things like "antimatter" that they've seen in movies and don't necessarily understand very well except that they make things go "boom", such a thing could be a useful psychological control tactic on at least some of the population. If you want to make the threat credible, I'd suggest instead having some kind of device in the brain that creates a small artificial aneurysm. A burst aneurysm can kill very fast, and if the device can also self-destruct so as not to leave residue, could look like a "natural" event to an unsophisticated autopsier. Such a thing might work by, for example, being placed near a suitable blood vessel and then, upon triggering, would start a release of some kind of chemicals that partially break down the vessel wall, weakening it and thus allowing for a swelling or hernia of blood (the aneurysm) to form, that then bursts and causes massive brain damage. All the better since you can control the placement in the brain to target the areas most likely to cause death or at least major disability.

(*) You might think highly-penetrating radiation is "worse" than lower-penetrating, e.g. gamma is "worse" than alpha, but this is only with regard to the fact that an external source of alpha is "better" in that it can only burn the skin, but gamma, due to being penetrating, can "burn" all tissues through the full thickness of the body uniformly, leading to radiation poisoning, essentially a "systemic radiation burn". But that's only for an external source, with the skin blocking. In fact, if the source is ingested, alpha particles are much worse, because they have much more ionizing punch per particle. Effectively you're now comparing them both on a fair playing field as full-body irradiators, and the gammas are considerably less damaging due to the fact that greater penetration means less chance of interaction. This is part of why that polonium-210, and not, say, cobalt-60 [a strong, and relatively "pure", gamma emitter, and much easier (and cheaper!) to get ahold of], was used to assassinate the late Russian defector Alexander Litvinenko a few hundred megaseconds ago. The needed lethal dose was much less due in part to this fact.