When I was in elementary school we learned that there are three phases of matter: solid, liquid, and gas. That’s it, no more. Which is a shame, because there are lots and lots of phases and their study is quite beautiful. But we did do some fun experiments with supersaturated liquids! (Though no one thought to call it a false vacuum…)
“Quantum computation” was definitely not on the list of phases. And you might be wondering what this would even mean. If you are, I’ve got just the paper for you: “Quantum computational renormalization in the Haldane phase”, a collaborative effort by Stephen D. Bartlett, Gavin K. Brennen, Akimasa Miyake, and yours truly.
Basically what we show is that something called “measurement-based quantum computation” (MQC) is robust in a certain part of the Haldane phase of a 1D chain of spin-1 particles, using some renormalization tricks to do this. Let me first give some background; experts might want to jump to the next paragraph. Quantum computation refers to using the rules of quantum mechanics to perform normal computations, like calculating your income tax or figuring out the fastest route from LA to Miami that doesn’t involve driving through, say, Arizona. (Assuming you don’t have an Arizona driver’s license.) Using the rules of quantum mechanics you can calculate the answers to some questions faster than if you use a normal computer, which uses the rules of classical mechanics. So we’re keen to build one. Due to the peculiarities of quantum mechanics, it’s possible to perform the computation by preparing a many-particle system in a certain way, and then measuring the particles one by one. If done properly, the measurement outcomes give you the output of the computation. This shows that all the quantum weirdness is due to the state of the many-particle system you start with, which is a very interesting conceptual point, since it doesn’t work this way for “classical” computers. But it’s potentially also an important practical point, since it might be easier to prepare this state and measure it appropriately than it is to do other styles of quantum computation. For instance, it might be possible to find a system which ends up in a state we could use for MQC when cooled below a threshold temperature. Sort of like how a ferromagnet cooled below its Curie temperature ends up in a magnetic state (though this particular example would be useless for MQC). That would make life easier. However, in general we’d need to control the parameters of the system very precisely in order to get just the right state out at the end. The hope of finding a phase, then, is to find a system for which it isn’t necessary to control these parameters so precisely; a system for which the computational ability of the system is a robust property of the phase. That’s in fact what we find to be the case for the Haldane phase of a 1-dimensional array of spin-1 particles.
Now, some more details for experts. Within the Haldane phase is the so-called AKLT (Affleck-Kennedy-Lieb-Tasaki) point, a Hamiltonian whose ground state can be exactly described by means of a matrix-product state. It’s a gapped system in the thermodynamic limit (which is one of the properties of the Haldane phase), so there’s hope that the ground state could be created just by sufficiently lowering the temperature of the chain. There’s also a nice MQC model where an AKLT chain can encode one logical qubit, such that measurements on the chain realize single-qubit operations. That’s not enough for a full quantum computation, so one can take many chains and couple them appropriately (which is cheating a little since the coupling is not a single-site measurement, but nevermind that for now).
But what happens away from the AKLT point? We investigated what happens to the quality of single-qubit and coupling operations when the actual state you’ve got comes from a different, deformed Hamiltonian and found that the quality (as measured by the fidelity) does in fact decrease. In other words, the computation becomes noisy. However, the deformed Hamiltonians also reside in the Haldane phase, and there is a real-space renormalization technique which takes such Hamiltonians to the AKLT point. This works because these ground states all have a common set of long-ranged degrees of freedom—renormalized spins that come from mapping a block of, say, L physical spins to a single new spin—and at this level the Hamiltonian looks a lot more like the AKLT Hamiltonian. So to perform a less-noisy quantum computation, we’d like to make our measurements on the renormalized spins, not the physical spins.
This violates the spirit of MQC, of course, since we’d now be measuring L physical spins simultaneously, which is considerably more difficult than single-site physical spin measurements. However, we were lucky to find a way around this! It is possible to implement a sequence of single-site measurements and use postselection to effectively measure the renormalized spin, a scheme we call buffering. Importantly, this procedure doesn’t depend on how the Hamiltonian is deformed, it’s the same all over the phase, except that you’ll want to choose a larger L the farther your system is from the AKLT point. Actually, we came up with buffering first, from a different analysis, and only later realized that the connection to renormalization explains why it works. Those details I’ll leave to the paper, but the upshot is that it’s not in principle necessary to have high-precision control of the Hamiltonian to create a ground state which is useful for quantum computation—the computational ability under a fixed set of measurements is a robust property of the phase.
Complementary Slackness 