_{1}

^{*}

This report describes a fundamental model of a quantum circuit for finding complex eigenvalues of Hamiltonian matrices on the quantum computers through the use of an iteration algorithm for estimation of the phase. In addition to this, we demonstrate the use of the model for simulating the resonant states for quantum systems.

In the circuit model of quantum computations the change of the quantum states is implemented through quantum operators. A given algorithm, or computation, can be run on a quantum computer by using a sequence of quantum operators. These operators can be represented by unitary matrices in computational basis. These matrices represent a transformation of the linear space, the rotation is an example of such transformation. As a consequence of the transformation most of the vectors in the space are converted into new vectors, which are with a different direction. But for each transformation, there are some special vectors, which, after being transformed, continue to point in the same direction, although their length can be changed. These vectors are called eigenvectors, and the factors which depend on the change of their lengths are called eigenvalues. And now let’s look what this means in the mathematical language.

In order to transform a given vector, it must be multiplied by the transformation matrix. So, if we have a matrix H and vector ψ, the transformation vector will be Hψ. If the transformation vector has the same direction as the original vector, this is an eigenvector. Mathematically, this means that the eigenvector must be divisible to the original vector so that they can be written as Eψ, where E is a number. Thus the eigenvectors must satisfy the condition: Hψ = Eψ.

The quantum state is something that encapsulates the entire necessary information for the physical system. Moreover, the quantum system has an infinite number of possible states, but they all can be expressed as linear combinations from a certain number of base states. For example, if the states are marked with kets (|∙∙∙⟩), then they can be used numbered kets (

Regardless of the type of the states, the coefficient a_{i} can be stored in a vector,

If the vector is an eigenvector of any matrix, then the state, which corresponds to the vector, is an eigenstate of the operator, which corresponds to the matrix.

Although this unitary image is suitable for many tasks, it prevents the application of a quantum computation at new tasks, where the computations can only be described through Hamiltonian matrices. In addition, each attempt for approximation of those Hamiltonian parts will affect negatively on the accuracy of the results. The model of the circuit, created for the Harrow et al.’s algorithm [

It is known that the quantum algorithms are more efficient than their classic matches [

The simulation of the non-Hermitian matrices in the quantum theory requires the simulation of the Hamiltonian evolutionary matrices. However, the standard quantum computations are made by the use of unitary operators of the evolution. It has been proven that the quantum circuit can be built from Hamiltonian operators, representing quantum measurement. In addition, it was demonstrated that a specific type of single-qubit Hamiltonian operators, the controlled-NOT operators and all single-qubit unitary operators constituted a formalized group of operators for Hamiltonian quantum circuits without the need of additional qubits [

In this report, we present a systematic approach for estimation of the complex eigenvalues of the general matrix, using а standard iteration algorithm with a programmable model of the circuit [

In the following sections the computational model will be discussed and described in [

We will develop a similar fast method for the finding of all eigenvalues of a complex Hamiltonian matrix M. The difference between the proposed in this publication algorithm and the known analogous ones [

Let

where

i.e. F is the set of all complex 2n × 2n Hamiltonian matrices.

We define the set

which in fact is a set of all Hamiltonian 2n × 2n matrices, raised to the power of 2.

The matrix, which appears to be a square of a Hamiltonian matrix, satisfies the conditions for J-symmetry, i.e. it may have one of the following two types:

1)

2)

As seen, in the square of the Hamiltonian matrix the angle cells B and C are asymmetric, and the diagonal ones transposed relative to each other.

If _{i}―the eigenvalues), then

These observations determine the base of the next SR algorithm for computation of the eigenvalues L_{1}, L_{2}, ∙∙∙, L_{n} of

1) We define

2) We compute the unitary matrix Q, so that

where H is a right Hessenberg matrix

3) We use a QR algorithm for computing

4) For i from 1 to n we compute

Point 2 of the plan of the SR algorithm is the only action in this process, which requires immediate clarification.

We will consider one step of the algorithm by which we will show how the unitary symplect

We will describe the original matrix, as follows:

(The zero diagonals in B and C are due to the asymmetricity). It turns out that the block C of this matrix can be zeroed by the use of a sequence of Householder and Givens symplectic similarity transformations.

The first action is to zero C_{31} and C_{41}, using the Householder matrix

The working algorithm H in short is used for determining the vector

We compute

We construct

We construct the Householder matrix

where

The exchange of the locations of y and z can be used for determining the matrix H_{k}, so that v_{i} = 0, for

It may be noticed that when the similarity transformation is executed with the first matrix, only the rows and columns 2, 3 and 4 of A, B and C affect the results:

The zeroing of positions (13) and (14) of C follows from the asymmetricity of this matrix.

The next action is to zero

The algorithm in brief consists of the following:

1) We compute

2) If

3) We construct the matrix

where

Here the argument

After performing the transformation can be noted that only the second row and column of A, B and C are affected by the result:

The next action of the SR algorithm is to zero a_{31} and a_{41} from the matrix. This can be done through the use of the working algorithm H, but with a correction at the construction of the matrix

The result of the transformation is the following:

It may be noted that only rows and columns 2, 3 and 4 of A, B and C are transformed from this transformation.

With this is finished the zeroing in the first column of N (at k = 1).

After

In conclusion of this method may be mentioned that it is used for development of the simplex method for solving algebraic and differential equations of Riccati.

For a conditional n-qubit system, represented by the N-dimensional matrix U the input data, accepted to be |α›, to the matrix U and the output data

Since

Recently we have described a model of a formalized programmable circuit, which can be used for simulating the action of the matrix U in Equation (1) only by determining the angles of the matrix elements. The aim is to establish circuit equivalents for the rows of matrix in N × N block quantum operations; and then to combine the blocks through the use of 2^{n}^{+1} different states of the (n + 1) ancilla control qubits (

Here, each submatrix V_{i} have the ithe row from the given matrix as its first row in the following form

In order to use the matrix V instead of the matrix U in Equation (1), we modify the input data

where

The other states in the output data represented with “•”, are subject to ignoring. Therefore, the likelihood of success of the postselection of the zero state in the main qubits, which is a success of the simulations, is given by к^{2}. For example, if N = 2, Equation (3) becomes as follows:

The separation of the main and ancilla qubits in the circuit is entirely artificial, since their roles may be exchanged in the circuit. In order to facilitate the process of verification, the circuit is divided into: Input Modification, Formation and Combination.

In the process for estimation of the phase for facilitation, we exchange the computing qubits at the end with the auxiliary qubits at the beginning and we use the following output data, instead of those of Equation (3):

where the first N states are selected to be the important states.

By measuring all ancilla qubits in a computer way and the postselection for the result^{2} = 1/2^{2n}.

The polar form of the complex eigenvalue λ_{j}, belonging to the matrix U, can be written as:

where _{j}. The circuit, shown in

the number of iterations m justify the precision. Xk is the binary value, derived from the kth iteration of the IPEA through the use of the

For the Hamiltonian matrices, however, the algorithm for estimation of the phase by itself is not enough. To compute the eigenvalue both parameters φ_{j} and r_{j} must be found. Through the use of the IPEA with the main circuit, described in the preceding section, the parameters φ_{j} and r_{j} can be found, and hence the eigenvalues of the Hamiltonian matrices can also be found by the algorithm for estimation of the phase without additional cost. Since the fixed circuit size the usage of the IPEA to the model of the circuit also has a fixed model. Therefore, we only need to set the values of the angles in each iteration of the IPEA.

In order to use the basic model of the circuit shown in

the phase qubit is zero the controlled Hadamard operator on the ancilla qubit is used a scaling operator which operates. This operator has the following structure:

In

where κ = 1/N. Therefore, where the ancilla is zero in the output data the phase of the selected N states can be calculated

In the evolution of the system for each loop of the algorithm for estimation of the phase, the effect of the eigenvalue can be observed in the evolution:

where

When the

where

and the states that do not include_{k} in^{i}^{2π(0∙xk)}:

When the last Hadamard operator is applied to the phase qubit, the last state is as follows:

or in shorter form:

Here, because of the size of the system κ = 1/2. For the general case κ is directly related to the size of the ancilla qubit and is 1/n, if there are (n + 1) qubits in the ancilla. Thus, the upper equation can be presented in a general form, as follows:

Since x_{k} is a bit value, it can be 0 or 1. When x_{k} = 0, since e^{i}^{2π0} = 1, the above equation is reduced to:

In the case when x_{k} = 1, since e^{i}^{π} = −1, Equation (17) becomes:

Since_{k} = 0, the probability to be found the phase qubit in _{k} = 1, then the probability to be found the phase qubit in _{k}.

In each iteration of the algorithm for estimation of the phase in this way is determined a different bit of the phase. For the κth iteration of the algorithm for estimation of the phase for determining the κth digit of the phase, we see

As we have shown above, the phase qubit output probability, when the ancilla is zero, is determined by

Since κ and P are known (for matrix 2 × 2 the value of κ is 1/2 because both Hadamard operators in the circuit),

By taking the average estimates from different iterations of the estimation of the phase, the estimate of _{j} in Equation (6) we can compute the eigenvalue of the Hamiltonian matrix.

In order to generate the matrix elements through the use of rotation operators around the y- and z-axis, in the algorithm for estimation of the phase the absolute values of the elements must be less than one. One of the ways to achieve this is to separate all the elements by the maximum element absolute value. This ensures that on the basis of the relationship of the norm and the eigenvalue, the largest absolute eigenvalue is less than the absolute sum of the row or the column of the matrix, which can be maximum N (i.e. since the absolute maximum value is 1). It is not recommended the eigenvalue to be less than one. If the eigenvalues, however, are larger than 1, this approach may require scaling within the very algorithmiteration, since in the powers there is a possibility the elements to become again greater than one. Another approach to be scaled the elements is to be used the first norm or the infinite norm of the matrix, which are easy for computing. The scaling of the matrix through the norm makes the eigenvalue less than 1, and thus in the matrix power the elements go towards zero. Hereinafter the scaling is done only at the beginning. Therefore in the κth iteration in the algorithm for estimation of the phase we have

The accuracy for determining the value of r_{j} requires to be determined the states (1 + r_{j}) and (1 − r_{j}) in the probabilities for finding the phase qubit in

The complexity of the algorithm for estimation of the phase with one iteration is dominated mainly by the complexity of the implementation of the given operator. It is also necessary to be made a tomography of the state of the phase qubit, in order to be determined the absolute eigenvalue, which is different on the basis of the basic quantum machine used in the algorithm for estimation of the phase.

It is known that the complexity of implementation of the operator of the quantum computer requires O(N^{2}) number of operations with one and two qubits. In addition, more effective circuits are possible for the sparse matrices. The general matrix, described in ^{2} rotation operators, constantly controlled by the first 2n qubits. The decomposition of this network requires 2^{2n} number of CNOTs and 2^{2n} single rotation operations, which are explained in Annex B. Hence, the circuit in total requires 2^{2n} CNOT, 2^{2n} single rotation, 2n Hadamard and n SWAP operators. And so the overall complexity is O(N^{2}). Please also note that for the zero elements there is no need to be used rotation operators. Thus for the sparse matrices the operations number can be reduced.

Because the complexity of one iteration of the algorithm for estimation of the phase is dominated by the complexity of the implementation of a given operator, each algorithm iteration requires O(N^{2}) operations. Then, for m number of iterations we obtain O(mN^{2}) as the complexity.

When the powers of the matrix cannot be effectively computed, the complexity for computing the power of matrix U at classical computers must also be taken into account. The power of the matrix may be found by using the method of successive squaring, which requires only one dot matrix multiplication. Then the total complexity becomes a combination of these two:

where m is the number of the iterations, M is the complexity of the matrix multiplication. Note that, even if it is necessary the powers of the matrix to be computed, at least this is still at least as effective, as the classical algorithms, which require to a great extent O(N3) time for finding an eigenvalue. Also note that the complexity of the algorithm in Equation (22) does not include the complexity of measurement and determination of the value of r_{j}. It depends on the measurement protocol, the quantum state and the basic quantum machine. Additionally, the total number of iterations m can be large in cases in which, for

In comparison with the work done by Wang et al., the likelihood of success decreases exponentially in both algorithms described in this report and in their report. In our case, however, by scaling the matrix elements in each iteration of the algorithm, the likelihood can be increased. Moreover, here we can find the values of the angles for the rotation operators very easily for implementation of the given operator. However, in their method, the subsystems H_{A} and H_{B} and their interaction H_{AB} must be found in order to be implemented the given H: H = H_{A} + H_{B} + H_{AB}. In terms of the quantum complexity, because their method is based on Hamiltonian successive projective measurements on H_{A}, which cannot be implemented deterministically, the complexity exponentially increases. Put in other words, in their case the finding of the given operator powers is easier, but when the size of the subsystem H_{A} is large, the complexity is dominated by the complexity of the measurements.

Certain tasks can be extremely difficult or even impossible to be solved within the standard formalism of the quantum mechanics, where the obvious properties of the dynamic nature are real and are associated with the eigenvalues of the Hermitian operators. In most implementations of the standard quantum Hermitian formalism the use of a Hermitian operator is presented as an equivalent to a non-Hermitian operator. The phenomena resonance, in which the particles are temporarily caught by the potential, can be examined in the scope of the non-Hermitian quantum mechanics. In the non-Hermitian quantum mechanics the resonant states’ lifetime is proportionally dependent of the behavior of the imaginary part of the eigenvalue. Let’s suppose that we have a non-unitary operator U = e^{−iHt/ħ} for a non-Hermitian Hamiltonian H with eigenstates of the energy _{j}, i.e._{j} is an eigenvalue of H; if t and ħ are set to 1, then _{j} of the matrix U can be evaluated. This also allows us to define the respective eigenvalue E_{j} of the Hamiltonian H from

We have presented a basic scheme for the execution of an algorithm for phase estimation for matrices of Hamiltonian (Appendix). Because the circuit model for matrices of Hamiltonian represents a basic programmable circuit, the circuit for the phase estimation algorithm is also formalized: it may be used for each matrices type for computing its eigenvalues. We showed that the complex eigenvalues of the non-Hermitian quantum systems may be found by means of this method. As an example we have shown how the resonant states of a model of the Hamiltonian are examined.

NikolayRaychev, (2016) Formalized Quantum Model for Solving the Eigenfunctions. Journal of Quantum Information Science,06,16-30. doi: 10.4236/jqis.2016.61003

By way of example, we look at the next radial Hamiltonian:

wherever J and a are free potential parameters (J is the potential depth in the potentialgraph. a is associated with the potential barrierwidth.) The potential element

which has its own eigenvector:

and the basic state eigenenergy (0.6249 − 0.4139i). Because the upper matrix of Hamiltonian is non-Hermitian, the time evolution operator, found from U = e^{−iHt/ħ}, is non-unitary, and is calculated as:

with the eigenvalue 1.2268 + 0.8849i. We used the above-described phase estimation method, to simulate this non-unitary operator and to determine the complex eigenvalues of H. In the simulation, because the matrix elements absolute values of U are greater than 1, we scale the elements by means of the first matrix norm. This causes the elements of the matrix of U to proceed towards zero in the limit. We use 11 iterations in the phase estimation algorithm. After the 11th iteration the probability difference between seeing the phase qubitas 0 or 1 is almost equal.

Therefore, from the simulation results, we calculate the eigenvalue of U as 1.22255 + 0.88355i, and thereby also the eigenvalue of H as 0.6259 − 0.4089i, which gives an error in the series of 10^{−}^{3}. This iscoming from the error of rounding and the limited available powers of U (when the power proceed towards zero, we can no longer distinguish