The AbramsLloyd quantum algorithm computes an eigenvalue and the corresponding eigenstate of a unitary matrix from an approximate eigenvector
Calculation of eigenvalues and eigenvectors of the Hamiltonian operator is one of the most frequent problems of physics. Most of the classical algorithms require an exponential amount of time for this. However, not all eigenvalues are of physical interest, but only few lower lying ones. Abrams and Lloyd’s quantum algorithm [
AbramsLloyd algorithm starts with an approximation to the eigenvector. Normally, it is the result of a classical calculation. The algorithm needs two quantum registers. The first, an index register, is prepared in a uniform superposition of all the computational basis states. The second, a target register, is prepared with approximate eigenvector. After the application of AbramsLloyd algorithm, the eigenvalue is calculated from the measurement on the index register. The corresponding eigenstate is in the target register but is not measured.
There are three disadvantages with the approximation. First, if the component of an eigenstate in the approximate eigenvector is small, then many trials and measurements will be needed to get the eigenvalue. Second, there is no advice on how to select the approximate eigenvector containing important eigenstates. Third, if an eigenvalue measured by AbramsLloyd algorithm is the same as the one obtained earlier, then without seeing the eigenstates, it is not possible to say whether the eigenvalue is a repeat of the previous or is a degenerate eigenvalue with a different eigenstate.
A method based on coarse and fine grids [
The AbramsLloyd algorithm transforms the approximate eigenvector to an actual eigenvector. Can we use this eigen state to obtain a better approximate eigenvector for another iteration of AbramsLloyd algorithm? The eigenvectors of a normal matrix are orthonormal. Thus, the first eigenvector, obtained by AbramsLloyd algorithm gives a handle to the orthonormal eigenspace. By sequentially orthogonalizing the eigen vectors, we get the other eigenvalues in sequence. If we nonorthogonalize the eigenvector in all the dimensions, we get a state that is not orthogonal to any eigenstate. In other words, we get a state containing all the eigenstates. This can now be used as the initial state for the Bang et al. algorithm [
AbramsLloyd algorithm needs an index register of
The state
When the condition (
Both the AbramsLloyd and the Bang et al. algorithms require that the initial approximate eigenvector has a nontrivial component of the interesting eigenvectors. Even if we roughly know the important eigenvector, we have to prepare
A uniform superposition of all computational basis states as the initial
If the objective is to get another eigenvalue, then the next
If the aim is to obtain a
The eigenvector in the AbramsLloyd method is in the target register of
It is tempting to think that the computational basis states can be used sequentially as the approximate eigenvector. The drawback is easily seen. If
The state obtained after the application of AbramsLloyd algorithm is a basis state in the orthonormal eigen basis. Rotation of each computational basis of this eigenstate by
It is interesting to analyse whether there exists any relationship between the orthogonalising operator and the unitary operator whose eigenstates are to be orthogonalized. It is obvious that they should not commute as commuting operators have simultaneous eigenvectors. The operators that do not commute too may not orthogonalize. Can we find a set of operators that will indeed orthogonalize?
Let
Given a unitary operator
Consider
Let
As
Equating the
This happens only when each side is equal to zero. Since
The anticommuting property rigorously specifies the orthogonalizer. Let us measure the effectiveness in orthogonalization of some wellknown operators that may not anticommute. Let us investigate how close an operator
A single qubit orthogonalization operator for states with real amplitudes is
Some single qubit orthogonalisation operators suitable for various types of eigenvectors.





0.97493 
0  −0.43388  −0.30680 
0.97493 
−0.43388  0  −0.30680 

−0.3909 
−0.18825  0.1433 

−0.35869 
0.24441  0.42632 
We wish to exactly orthogonalize, but, because of the complex amplitudes or nonproduct form of the obtained eigenvector, we may end up with a quasi orthogonal approximate eigenvector. The AbramsLloyd algorithm will then, without certainty, result in new eigenvalues and eigenvectors for these approximations.
We have tested our method with a
Comparison of eigenvalues obtained numerically and through AbramsLloyd quantum algorithm sequentially. The approximate eigenvector is obtained by orthogonalization of eigenstate of 3 qubits. The single qubit orthogonalization operator is
Numerical  AbramsLloyd  

eigenvalue of 

eigenvalue of  





−2 

22 

−1.9635 
−1 

27 

−0.9818 
1 

5 

0.9818 
0  1  0  1  0 
0  1  0  1  0 
0  1  0  1  0 
The method to generate a state containing all the eigenvectors is tested on a unitary matrix
The eigenvectors









The eigenvalues









For this unitary matrix, we allocate an index register of 3 qubits. For the
The simulated quantum state is seen as
The eigenvalues obtained using AbramsLloyd algorithm are not the exact eigenvalues (Table
The AbramsLloyd quantum algorithm computes an eigenstate of a unitary operator whatever be the initial approximate eigenvector. This eigenvector is a basis vector in the orthonormal eigen space. Orthogonalization of this eigenvector, followed by AbramsLloyd algorithm, can result in the other eigenvalues without repetitions. Simple sequential orthogonalization of a qubit of the eigenvector at a time gives the eigenvalues sequentially, but better methods of choosing the qubits are needed so that important eigenvalues are obtained. The nonorthogonalization of the first eigenvector obtained using AbramsLloyd algorithm results in a state with all the eigenvectors and the corresponding eigenvalues. This state is then a suitable candidate for amplitude amplification algorithms like the Bang et al. algorithm.
The author acknowledges the financial and management support provided by the Department of Science and Technology (DST), Goverment. of India, through the Women Scientist SchemeA (SR/WOSA/PS19/2009) to carry out this work. The author is thankful to K. Vinod Kumar, CDAC, Mumbai, for the help in the computation. The author is grateful to the referee, whose valuable comments led her to characterizing the orthogonalization operator.