2.1 QFT Algorithm

The Fourier transform is used in a wide range of classical problems from signal processing to complexity theory. The QFT is the quantum generalization of the Discrete Fourier Transform (DFT) where the inputs are the amplitudes on a computational basis of a wave function. It is part of many quantum algorithms, most notably Shor’s factoring algorithm ^[5] which makes use of the QFT for period finding.

The classical DFT maps an input vector (x0,...,xN-1) to the output vector (y0,...,yN-1) according to Eq. (1):

where

Meanwhile the QFT acts on an input pure n-qubit quantum state x

where N=2n, and {j} is the standard computational basis, and maps it to the output state y according to

where ωNjk is the same as the classical case, Eq. (2).

2.2 Circuit implementation

Experimental quantum computers nowadays generally use a small set of universal quantum gates in order to generate quantum circuits. The quantum IBM computer (IBM Q) is no exception and only works with certain quantum gates, which made translating the matrix in Eq. (4) to quantum gates an essential task.

As an example, for a two qubit state, the quantum gate representation of the algorithm requires a few basic gates: the Hadamard transform and a controlled rotation of angle π/2 around the z axis, Rz, whose matrix representations are shown in Eq. (5) and (6), respectively.

where Rz is

the QFT circuit UQFT=(I⊗H).cRz.(H⊗I) is shown in Fig. 1.

Figure 1 Two qubit QFT circuit for input state 00. Illustration given by ^[9]. 

3.1 Error model using the density matrix and the isotropic index

In order to model the systematic error, the first step is to reconstruct the resulting density matrix ρexp, measuring the experimental projections from the data obtained from IBM Q, using Quantum State Tomography (QST) ^[4]. To recover the density matrix state, measurements must be made in the three basis (axes) of space, X,Y and Z whose matrices are defined:

For example, for any 2 qubit state its matrix ρ is given by

where σ0=I, σ1=X, σ2=Y, σ3=Z and the values of tr(ρσi⊗σj) are obtained, measuring the state several times in order to approach the expected value by the statistical average.

3.2 Isotropic index

The isotropic index is useful to separate the global loss of information, called weight ω, from the misalignment A with respect to a reference pure state. While nothing can be done about the loss of information, the misalignment (coming from systematic errors), could be corrected, hence the importance of separating these two type of errors.

Considering the pure reference state ψ, ρψ = ψψ and the decomposition of a state ρ after a noisy process,

the double index is defined as:

where λ is the smallest eigenvalue of ρ,

where Fid is the fidelity between states ^[4], and ρψ⊥=I-ρψ/2n-1 is the isotropic mixed state orthogonal to ψ.

The misalignment takes values in the interval [-1,1], where 1 means the resulting state is completely aligned with the reference state ρψ and -1 means the resulting state is completely misaligned i.e. aligned with ρψ⊥. Moreover, ω takes values in the interval [0,1]: when ρ is a pure state ω=0, and when the loss of information is total, ω=1.

Consider now the state ψ as the reference state of the resulting theoretical state from the application of QFT to some basis state, and ρexp the obtained experimental matrix state from the same input state. Decomposing the ρexp

it is always possible to obtain ω and ρ^.

In the particular case where ρ^ is a pure state, i.e., ρ^=ϕϕ, one could partially correct the error with a unitary matrix that takes ϕ to the state ψ.

Unfortunately, generally the state ρ^ is not a pure state, so it is necessary to find the pure state ϕ, closest to ρ^, and with maximal alignment with it, i.e., ϕ should satisfy Eq. (14)

where A is the alignment index. These states can be graphically seen in Fig. 2.

Figure 2 Representations of the states in Bloch’s Sphere. 

In order to find ϕ that satisfied the requirement, optimization methods were used. Once ϕ was found, a rotation matrix U that takes ϕ to theoretical ψ, is easy to write out, for any basis initial state. It is a corresponding matrix U _i and ϕi for each input canonical basis state i. Hence, the complete model is defined as a control unitary rotation U_c , and n ancillary qubits, that simultaneously correct the n basis states, and so any input state. An example of U_c for two qubits Hilbert space, is given in Sec. 4 (see Fig. 3). Although this method is more exact, it has a high computational cost since the resulting state needs to be measured in all 4ⁿ local projections, where n is the number of qubits. In Sec. 3.3 we will analyze another method that manages to estimate the necessary correction by measuring only 3n projections.

Figure 3 Unitary correction U _c for any input state. 

3.3 Error model using only 3n projection’s

Since determining the density matrix ρexp using the QST method is computationally expensive, another method is proposed to avoid this step. The idea is to find the pure state ϕ, closest to the ρexp measuring only those projections relevant for the correct functioning of the algorithm. This is, those that maximally differentiate each of the states resulting from the application of the QFT, on each of the 2 ⁿ initial basis states.

Fortunately, in the case of the QFT algorithm, the theoretical output state for each input basis state is separable on all qubits. Although the mixed state ρexp is generally not, we assumed the hypothesis that the error is expected to affect in such a way that the resulting state stays close to the theoretical one, and we shall discuss if this assumption proved right experimentally. We assume that a good estimate in this case is to find the reduced density matrix of ρexp in each qubit i, ρexpi, i.e., measuring the projections X,Y,Z, of this qubit, in total 3n measurements.

Once we have obtained the reduced density matrices of each qubit i, the eigenvector with the highest eigenvalue is extracted ϕi without optimization. In the case of a one qubit matrix state, the decomposition in (13), ρ^ is a pure state and is always the most aligned.

Similarly to the previous method, the unitary rotation matrices are determined for each input basis state U=U1⊗U2...Un, that takes the state ψ to ϕ. Finally the total U _c matrix is defined similarly to previous method as in Fig. 3.

As an example, the resulting unitary matrix for two qubits on IBM Q computer ibm_santiago, are shown in the next section and the results are compared to those of the first method.

4.1 Experimental results with the isotropic index

Performing the QST on this data we obtained the experimental reconstructed density matrix for ibmq_santiago, which was deemed too large to fit this article.

The theoretical output state corresponding to input 01 is ψ01=-⊗+y, where +y=0+i1. For this state, the pure state ϕ01 with the maximum alignment with ρexp is found:

Finally the unitary rotation matrix U ₀₁ is calculated which maps the state ϕ01 to the output theoretical state ψ01. Similarly, the pure states ϕij obtained for the other input basis states are presented in the Table III.

This procedure was performed for the four resulting states, obtaining the four unitary transformations. Using two ancillary qubits, a unitary transformation U _c is defined that controls which element of the unitary set should be applied, so that the model works for any input state. This model is shown in Fig. 3.

In order to model the decoherence, a total Depolarizing Chanel is applied on the theoretical state ψ, and the resulting state

is compared with the transformation of the experimental state under U:ρ'=UρexpU†. Although the decoherence error generated by the computer does not have to fit with a chosen Depolarizing Channel, this simple error model does not change the alignment of the states and can estimate the effect of decoherence.

Finally in Table IV and as shown in Fig. 4, the results are compared. This comparison is done by calculating the fidelity in each step F _1, F_2, F₃ defined as:

where ψ is the expected theoretical state and ρ'=UρexpU†.

Figure 4 Fidelities Ec. (??) for the method using the isotropic index. F₁ in white bar, F₂ in gray bar, and F₃ in black bar.  

4.2 Experimental results Using only 3n projections

In contrast with the previous method, now we look for an estimate of the state ϕ, closer to the state ρexp without the explicit representation of the density matrix, using only 3n projective measurements. Those projections are chosen so that the unitary matrix brings the state ρexp closer to the particular basis state, and as far as possible from the other three basis states. The idea is to look for the pure state that is most aligned with each of the reduced one qubit matrices. With this purpose we measured the projections X,Y,Z in each of the qubits, and found the one qubit reduced matrices. To get the pure state with most alignment, no optimization is needed, because for one qubit matrices the principal eigenvector ϕi (the eigenvector with maximal eigenvalue) is the state with most alignment, getting ϕ=ϕ1⊗ϕ2.

Finally the desired separable unitary rotation for each basis state U=U1⊗U2 is found, and similarly to the previous method a Depolarizing Chanel is applied on the theoretical state ψ, and the resulting state ρDch=ϵ(ψ) is compared with the transformation of the experimental state under U: ρ'=UρexpU†. The fidelity between ρDch and ρ' is shown for the four input states in Table V and shown in Fig. 5.

Figure 5 Fidelities Eq. (??) for the method using 3n projections. F ₁ in gray bar, F ₂ in white bar, and F ₃ in black bar.