JJA and qubit calculation

I have now talking up a numerical project to calculate superconducting flux qubit. Much basic knowledge not known to me before is required. i wish to take this opportunity to tidy up a bit on the whole idea of the project, and hopefully through this short discussion a clearer and more concrete picture could emerged.

First of all, the proposed project, let's call it the flux qubit project (FQP), comprised of two components: the theoretical and the computational one. we have to know the theoretical concept first before attacking the problem at hand computationally.

The theoretical part is as follows. The problem at hand is that we have now a new type of flux qubit system, i call it 4 x 3 JJA. This is a proposal not seen anywhere in the literature because the proposer Hans Mooij has not really published it. The geometry of the 4x3 JJA is very much similar to the one mentioned in the Master thesis by Martijn (2009, Theoretical Group, Kavli Institute of Nanoscience, TU Delf). What we have is a quantum system (which is not microscopic but a macroscopic circuit of the micron size). This is a quantum system comprised of a large number of Josephson junctions (24 to be exact). To understand what this 4x3 JJA is good for, we have to understand how a single JJ works. And this in turns requires some basic knowledge in quantum mechanics, e.g. the idea of conjugate variables (e.g., {p, x}, {E,t}). In the case of a single JJ, the conjugate quantum variables are {q,phi}, where q is related to the excess number of cooper pair stored in the JJ [via q = n*(2e)], where phi is the phase difference across the JJ. In the practical implementation of a flux qubit, one normally does make some approximation to the quantum circuit to render the fluctuation in q irrelevant, so that only the variable phi is the dominant variable. q, however, is still important as its quantum fluctuation is necessary to induce quantum mixing. phi can be taken effectively as a quantised magnetic flux flowing through the JJ loop that can be detected. Its value is extremely tiny, phi \sim Phi_0, where Phi_0 = h/(2e) is the scale of quantum of magnetisation. phi (synonym to the quantum magnetic flux, or fluxon), due to its quantum character, is the quantum degree of freedom that plays the role of qubit in flux qubit. qubit is the quantity we want to manipulate finally (in a real quantum computer), hence the knowledge of how this qubit behaves is essential.

The behaviour of a flux qubit is governed by quantum mechanics. Any qubit must display a 'two-level' behavior. Under certain circumstance, the phase psi in JJ does behave like a two-level system (but this is not true in general). Hence, I really need to understand the quantum mechanical description of a two-level system, and how a flux in a JJA can be display such a two-level behavior quantum mechanically.

To describe the 4x3 JJA the quantum mechanical equations have to be written down (which has already done by Mooij). The most important quantity is the Hamiltonian. In general, once the Hamiltonian of a quantum system is written down, the behavior of the system is completely determined. What one does is then to first find out the ground state of the system. To this end, one solves the time independent Shrodinger equation to obtain the energy spectrum of the system. Basically, what is gained from this calculation is the energy of the system as a function of the external environment variables in time independent case. The degree of freedoms here are the phases (9 of them in total) in the JJA. In a single JJ, there is only one such DOE. In a JJA, there are many, and the solution to the quantum problem becomes complicated. Referring to the statement mentioned above (that 'the system must display two level behavior), one ought to make sure that, under certain restricted conditions (e.g the external magnetic bias, the so-call frustration), the energy spectrum obtained displays a profile of two weakly coupled ground states for the dof concerned (see e.g. the energy profile in page 10, Figure 2.2 of Floor Paauw's PhD thesis, 2009, Kavli Institute of Nanoscience, TU Delft). Such ground state is in turn conveniently described by the Hamiltonian Eq. (2.4) of Paauw's thesis in page 12. Essentially this Hamiltonian (which is a reduced version of the more general Hamiltonian for the whole JJA) describes the quantum mechanics of a quantum spin that has two nearly degenerate lowest ground states which are weakly coupled via the coupling terms \Delta. This \Delta term is directly related to the fluctuation of the charge q alluded to earlier. Whereas the term \epsilon controls the energy barrier that separate the two ground states of the spin. \epsilon in practice is controllable in experiment via the external magnetic frustration. Due to the presence of the coupling term \Delta, the two approximately degenerate ground states of the flux 'mix' giving rise to superposition of states. The coupling term \Delta is the 'source' of quantum behavior in the spin, otherwise, if \delta → 0 the system is reduced to a classical Ising model. The description of mixing mechanism in quantum mechanics is very much similar in spirit to the coupled quantum oscillators or Zeeman effect in atomic physics mentioned in many standard text book.

This spin model of the qubit is taken from the quantum Ising Model. If I understand how quantum mechanics describes the Ising model, I can just take over the well-known results and the interpretation of the quantum Ising Model to apply it here. In a generic Ising model, the spin only takes on two values, labelled |uparrow> and |downarrow>. Hence a qubit that has only two quantum states can be 'mapped' to a spin (which also has only two states). The mathematical description of a spin can equally well be applied to a qubit. In other words, the behavior of a qubit can be imagined to be well represented by that of a quantum spin. In basic quantum mechanical text book one often read about illustrative examples on quantum mechanical calculation done on Ising Model and also on coupled harmonic oscillators. That's why I say I need basic knowledge in quantum mechanics to understand the behavior of the flux qubit.

OK. the task comes in steps. The first step is to establish the two-level behavior of the qubit so that it can be described by the spin Hamiltonian of Eq. (2.4) in Pauuw's thesis. This initial step is by no way a trivial one although it's also not that difficult to establish. Mooij has made this step for us. There are papers published by others who reported solely on the discovery of 'yet another two-level system' suitable as a potential qubit. In these paper they merely reported that they have found another JJA system which in certain limit reduced to the two-level spin Hamiltonian of Eq. (2.4), leaving the details to the others.

The time independent calculation of the GS energy is only the first step. I also need to work out the excited state energy spectrum. The GS of the JJA can be calculated using a version of quantum Monte Carlo (QMC) called Diffusion Monte Carlo (DMC), and it is not a difficulty task. Martijn has already done it in his Master thesis. The excited state is slightly difficult. It will be tackled using excited state Monte Carlo Monte Carlo (ESMC).

Once the time independent ground state energy of the JJA is solved, I will proceed further to trace out the time evolution of the qubit system starting from its stationary (i.e. time-independent) solution. Technically, if the Hamiltonian H and the solution of the quantum state at t=0 is known, then the evolution of its states can be obtained via

|state(t)> = exp(iH*t)|state(t=0)>.

The theories, methodology and other technical details of this I am still learning, but in principle it's nothing more that a routine calculation for those who know how to use quantum mechanics at their finger tips (no me though). Any basic quantum mechanics textbook will talk about this. I hope that I will be able to write to more about this part on how to trace the time evolution of qubit soon (i.e when I understand them better).

There are still one more thing i have not clarify. As mentioned in the beginning of this 'article', this project also contains a 'computational aspect'. Pragmatically, this is the real task that I have to work on. Understanding the theoretical framework and methodologies in QM calculation are only the first step. The real dirty job is to solve the quantum equations. In our case concerned, we have to use computer to solve it by writing Fortran codes. This is the kind of dirty job I am working on now. Numerical and computational strategies to solve QM system are a big field by itself. For this matter, the book by Thijssen is essentially useful. For small system, often there are standard numerical techniques/methods available to solve our problem. For example, one often heard about some jargons in computational quantum mechanics e.g. Ritz variational method, Hartree-Fock, Density functional theory, Lanczos method, Quantum Monte Carlo, generalisd eigen value problem, optimisation etc. Two years ago i am still very ignorant of all these jargon. But now i think i am now more or less (at least in principle) understand what these things are and how to implement them in my computer. These are all 'standard technical trick' that one uses in routine basis to solve quantum mechanical problems. However, i suppose that the ability to know how to use computer to solve a generalised quantum mechanical Hamiltonian is not a very common knowledge even for experienced researchers.

The main issue one often encounter is that for a quantum system having too large a number of degree of freedoms, or a complicated interactions (as described by the Hamiltonian), computer power simply becomes not enough since the Fock space in which the solution lies increases exponentially with every additional degree of freedom. The larger the Fock space is the search for the solution simply needs more step to iterate, hence longer time. One often have to use alternative or clever computational tricks to circumvent this bottle neck. One common 'master tool' is to resort to QMC method when everything else does not work.

Well, the above is a simplified (and maybe erroneous) description of how i understand the project. I had no rigorous training in quantum mechanics in the past. However i really find it intellectually satisfying to learn, understand and actually calculate with quantum mechanics. QM is such an important and essential subject for anyone who want to deal with the fundamental aspects of our materialistic universe. I consider it a very very important tool that every serious theoretical physicist must master. It is unfortunate that USM treats QM as an optional subject and never want to stress on this. QM is so powerful that one cannot do without when dealing with physics at their fundamental level. For example, for anyone who wishes to research in material's physics at the nanoscale (nano science), without QM one can only scratch the surface of the core problem. So is semiconductor science, magnetism or optoelectronics. Our current materialistic development all owes their success to our ability to use QM to investigate the behavior of matter at their most fundamental level. I recall that a student once mentioned enthusiastically that he wants to learn QM. Now i think i have also developed similar kind of excitement. I gradually start to realise how ESSENTIAL QM is for fundamental physics research. With it one can do many many things in physics research.

As a last remark, one feels very empowered when envisaging the scenario where one has mastered the technical details of quantum theoretical technology, and further, is able to use the computational techniques learn to solve a generic quantum Hamiltonian. Give me a Hamiltonian which is otherwise difficult to probe analytically, and i will tell u how the system behaves using the computational methods.

August 09