The other day Jingbo Wang from our Physics department asked me about circulant Hamiltonians.
I had previously come across circulant orbitals from the work of Bob Parr.
Circulant orbitals are noncanonical HartreeFock or density functional theory (DFT) molecular orbitals which have been transformed unitarily in order to have a density as close as possible to the electron density divided by the number of electrons. The name arises from the fact that circulant orbitals transform the effective Hamiltonian into a Hermitian circulant matrix – that is a matrix where the rows (or columns) are shifted by a unit cyclic permutation.
I was told by Jingbo that such circulant Hamiltonians are really useful for performing quantum computations.
What this is about
The purpose of this blog is to trivially generalize the work of Bob Parr to show that any group of quantum states can be unitarily mixed to ensure that the Hamiltonian has a circulant structure in the new basis. The structure depends only on the actual energies of the states involved.
There are two use cases for the final result:

Given a general circulant matrix, what are the eigenavlues of a physical system needed to construct it?

Given a real system which gives certain eigenvalues, what circuant matrix fesults from it, and can it be used to do any quantum computation?
In order actually make a real device (at least!) two serious problems need to be addressed:

How do we design a molecule or system with appropriate excited state energies?

How do we introduce a coherent excited state into the sodesigned system at an initial time, make sure errors and noise are corrected, and probe it at a later time.
I was assured that the error correction part of the second problem is soluble.
The circulant basis
We begin by introducing the circulant basis. It is a discrete Fourier transform of a set of system eigenstates.
So: Let \( \hat{H} \) be some physical Hamiltonian with eigenstates \( E_n\) and with corresponding orthonormal eigenstates \( \Psi_n \), \( n=1,\ldots,N \). Then we have as usual
The circulant basis is defined by the transformation
In other words, \(\Phi_I\) is a discrete Fourier transform of a finite set of eigenstates. \( \omega \) is an \(N\)th root of unity, \( \omega^N = 1 \).
Theorem
The circulant transformation is unitary
Proof
Since the definition of a unitary transformation is that is preserves orthonormality, we only need to show that the new basis \( \Phi_I \) is also orthonormal, i.e. that \( \braket{\Phi_I}{\Phi_J} = \delta\sub{IJ} \).
Now there are two cases. When \( I=J \) the summand is always equal to one and the right hand side is equal to 1. When \( I\neq J \) the summation is a geometric series with initial term 1 and ratio \( \omega \), so the sum is given by \( (1\omega^n)/(1\omega) \), which is zero since \( \omega \) is an \(N\)th root of unity.
The Hamiltonian in the circulant basis
Theorem
Proof
Just substitute in the definitions for the circulant states.
The Hamiltonian in the circulant basis is a circulant matrix
A circulant matrix has rows (or columns) which are 1cyclic permutations of the previous row (or column) e.g.
Or in other words \( H\sub{IJ} = H\sub{I+1,J+1} \) where the indices are taken modulo \( N \) the size of the matrix. The circulant matrix is actually only defined by one row or column.
Theorem
The Hamiltonian in the circulant basis is a circulant matrix
Proof
Obvious by direct inspection. Also we define:
The latter is the defining vector for the circulant matrix.
Designing a Hamiltonian with a certain circulant structure
By “designing” the matrix we mean choosing the energy eigenvalues for the Hamiltonian in order to achieve a desired circulant structure. We do not mean actually finding a real physical system with these eigenvalues. That is a much more difficult task alluded to in the introduction.
Obtaining the eigenvalues associated with a given circulant matrix is actually fairly trivial. All that is requited is to take the inverse discrete Fourier transform of the defining vector of the circulant matrix.
Theorem
Proof
Substitute the definition of the defining vector.
Where to now?
I don’t really know, but here were some ideas from the discussion after the talk I gave in physics:

Do some accurate quantum chemical calculations on real systems, find their energies \( E_m \) and from them find the circulant matrix \( H_K \). Can anything be done with this matrix? This approach seems a bit ad hoc.

Take some examples of circulant matrices known to be useful for quantum computing. Find the energy eigenvalues \( E_m \) corresponding to this system. Try to construct a real system, perhaps a real molecule, perhaps a quantum dot or other one dimensional system with the required energy levels.

An earlier post shows how the eigenvalues of a one dimensional system can be systematically computed, in general. This approach could be used to rationally design such a “quantum line” system with the desired eigenvalues.

Alternatively, use approach 1. to develop some familiarity with the type of systems giving rise to certain energy levels.

Jingbo noted that, using localized basis functions, some Hamiltonians naturally have a circulant structure e.g. benzene. One is reminded of the Huckel type theories. Initially I agreed. However, I now think that the Hamiltonians with these connectivities are one electron effective Hamiltonians – they are not actual real Hamiltonians which model the dynamics of real multielectron wavefunctions – except possibly for the motion of one electron excited from the highest occupied molecular orbital (HOMO). It is not clear that the excited states of such effective systems will have the same circulant structure. However, the idea deserves further consideration because it could give insight into how the connectivity of atoms in a molecules affects its circulant structure. This could be probed by more accurate quantum chemical calculations.
