Some thoughts on Landau-Zener Quantum Computation

Adiabatic quantum computation was originally inspired by, and got its name from, the adiabatic theorem. This theorem states that a physical system initialized in its lowest energy state will, even in the presence of significant external poking and prodding, remain in its lowest energy state as long as this poking and prodding is done slowly enough. What “slowly enough” means can be written as a function of the minimum gap between the system’s ground state and its first excited state. This concept can be used for computation by encoding the solution to a hard computational problem in the ground state of some physical system (the hardware used to embody the AQC computational model).

The original idea is quite brilliant in several respects. One of these is that there is no need to change any of the parameters of the QC quickly–all of the control lines are DC. This is a significant advantage in practice, as high frequency control is exceptionally difficult to scale to thousands/millions of physical qubits, where QC needs to be to compete with conventional computers. Another is that this model has innate robustness against decoherence. Sometimes the reason for this robustness is explained by pointing out that in AQC the state the system is in is protected from other states by an energy gap. I think though that this is more a symptom of where the real advantage is coming from, which is that dephasing occurs in the energy eigenbasis, not the readout basis. Yet another is that to my knowledge the only really scalable design ever proposed in the literature for any model of quantum computation is based on AQC.

As more and more thought was given to how AQC works in practice several issues arose. The one I’d like to focus on here has to do with the original motivation for the computational model–namely the adiabatic theorem. While the original idea was showing that there was a theoretical mechanism for solving hard problems by remaining in the ground state throughout an entire computation, what we really care about is a little more subtle than this. A key point is that while people often think of AQC as a complete solver, in practice running an AQC obviously gives no guarantee of global optimality. Because of this AQC is explicitly heuristic regardless of how it is operated. What we really care about for characterizing the performance of AQC is how the output probability distribution changes as a function of the way the Hamiltonian is morphed from initial to final configurations.

This is a very hard problem, but there is a way to begin picking at the threads by thinking about the content of the adiabatic theorem in a different way. Somewhere during the computation the energy gap is smallest. This minimum gap sets the runtime for solving the problem and is therefore the “danger point” for the system leaving the ground state. For a large class of problems, the evolution through this minimum gap point is equivalent to a Landau-Zener crossing, where the two lowest eigenstates are the levels in an effective two-level system. The LZ problem has a long history and analytical results for the probability of leaving the ground state–even in the presence of strongly coupled environments–are available. Therefore the probability of leaving the ground state for AQC can be analyzed by treating the physics of the anticrossing in the LZ framework, even for real AQCs in the presence of noise.

How does thinking of a transition through the minimum gap point as a LZ transition help with understanding AQC? The main advantage to looking at things this way is the large body of literature on the effects of environments on the transition probability. The LZ model (which is very closely related to AQC) has been around since 1932, and could be considered the first model of quantum computation. A fascinating fact, which is probably obvious with heaping spoonfuls of hindsight, is that the probability of remaining in the ground state after an LZ transition doesn’t depend on the environment at all for broad classes of QC-environment couplings. This is another facet of the wonderful inherent noise-resistant properties of the AQC model, and is a great example of how many-body physics can help reveal deep truths about quantum information theory.

While we continue to call the model AQC, I think it’s fair to say that maybe because of our solid-state physics bias we’re thinking of it more and more as LZQC.

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