The QFT and the NISQ era

The Quantum Fourier Transform

Algorithm
Noise model
Correlations and Fidelity

QFT circuit

H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix}

QFT circuit

C(i,j)= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \theta \end{pmatrix}

H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix}

QFT circuit

\mathcal{U}_\mathrm{QFT} = \prod_{i=0}^n\big[[\prod_{j=i+1}^nC_\mathrm{rot}(i,j)]H_i\big]\\ \theta = e^\frac{i\pi}{2^{|i-j|}}

C(i,j)= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \theta \end{pmatrix}

H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix}

QFT circuit

\mathcal{U}_\mathrm{QFT} = \prod_{i=0}^n\big[[\prod_{j=i+1}^nC_\mathrm{rot}(i,j)]H_i\big]\\ \theta = e^\frac{i\pi}{2^{|i-j|}}

C(i,j)= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \theta \end{pmatrix}

H = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1\\ 1 & -1 \end{pmatrix}

The Quantum Fourier Transform

Algorithm
Noise model
Correlations and Fidelity

Basis of the study

Noise

\tilde{\mathcal{U}}_\mathrm{QFT} = \prod_{t=0}^T\tilde{U}_\delta(t),\qquad \tilde{U}_\delta(t) = e^{-i\delta V}U(t)

Fidelity

F(T)\;=\;1-\frac{\delta^{2}}{2}\sum_{t,t'=1}^{T} C(t,t')\;+\;O(\delta^{3})

C(t,t')=\frac{1}{N}\mathrm{Tr}\!\big[V(t',0)\,V(t,0)\big]

The Quantum Fourier Transform

Algorithm
Noise model
Correlations and Fidelity

Correlation accumulation

How to achieve this ?

Introducing a rotation
Reshuffling
Fidelity Scaling

The 'improved' QFT

1_{4\times 4}=R^\dagger(i,j) R(i,j) = R(i,j)R^\dagger(i,j)

[R(i,j),C_\mathrm{rot}(i,j)]=0

1_{4\times 4}=R^\dagger(i,j) R(i,j) = R(i,j)R^\dagger(i,j)

[R(i,j),C_\mathrm{rot}(i,j)]=0

[R(i,j),R(i,k)]=0

1_{4\times 4}=R^\dagger(i,j) R(i,j) = R(i,j)R^\dagger(i,j)

[R(i,j),C_\mathrm{rot}(i,j)]=0

[R(i,j),R(i,k)]=0

R = \begin{pmatrix} 0 & 0 & -1 & 0\\ 0&1&0&0\\ 1&0&0&0\\ 0&0&0&-1 \end{pmatrix}

Introducing a rotation
Reshuffling
Fidelity Scaling

The 'improved' QFT

\mathcal{U}_\mathrm{QFT} = \prod_{i=0}^n\big[[\prod_{j=i+1}^nC_\mathrm{rot}(i,j)]H_i\big]

\mathcal{U}_\mathrm{iQFT} = \prod_{i=0}^n\big[[\prod_{j=i+1}^nR(i,j)][\prod_{j=i+1}^nG(i,j)]H_i\big]\\ G(i,j) = R(i,j)^\dagger C_\mathrm{rot}(i,j)

\mathcal{U}_\mathrm{QFT} = \prod_{i=0}^n\big[[\prod_{j=i+1}^nC_\mathrm{rot}(i,j)]H_i\big]

The 'improved' QFT

Introducing a rotation
Reshuffling
Fidelity Scaling

Correlation accumulation

Overall fidelity

Dissipation

Superoperators
Analytical predictions
Numerical study

Vectorization

U(t) \rightarrow W(t) = U(t) \otimes \bar{U}(t),\\ \mathcal{W}_\mathrm{QFT} = \prod^T_\mathrm{t=0}W(t) = \mathcal{U}_\mathrm{QFT}\otimes\bar{\mathcal{U}}_\mathrm{QFT}

Vectorization

U(t) \rightarrow W(t) = U(t) \otimes \bar{U}(t),\\ \mathcal{W}_\mathrm{QFT} = \prod^T_\mathrm{t=0}W(t) = \mathcal{U}_\mathrm{QFT}\otimes\bar{\mathcal{U}}_\mathrm{QFT}

\tilde{W}_{\delta, \kappa}(t) = (1-\kappa)\tilde{U}_\delta(t) \otimes \tilde{\bar{U}}_\delta(t) + \kappa\sum_{i=1}^{r}K_i(t) \otimes \bar{K}_i(t)

Dissipative noise

\sum_{i=1}^{r}K^\dagger_i(t) K_i(t) = 1_{N}

Dissipative noise

\sum_{i=1}^{r}K^\dagger_i(t) K_i(t) = 1_{N}

\sum_{i=1}^{r>=2}K^\dagger_i(t)\otimes K_i(t)\xrightarrow{}\sum_{i\in\{1,2\}}K^\dagger_i\otimes K_i

Dissipation

Superoperators
Analytical predictions
Numerical study

Weingarten calculus

\overline{\mathcal{F}}\sim(1-T\kappa)\overline{F_\delta}+\frac{T\kappa}{N}=\overline{F_\delta}-T\left(\overline{F_\delta}-\frac{1}{N}\right)\kappa

Dissipation

Superoperators
Analytical predictions
Numerical study

Split models

Main results

Zone of interest
Scaling up
Overall fidelity increase

Main results

Zone of interest
Scaling up
Overall fidelity increase

QFT maximizes

iQFT maximizes

Main results

Zone of interest
Scaling up
Overall fidelity increase

\kappa^*=\frac{2 n (A n+B-C)-2 D}{n-1}\delta^2\ \overset{n\gg1}{\sim} 2An\delta^2

\begin{cases} &\epsilon\leq\delta ^2 \left(A n^3+B n^2\right)+\kappa \left(\dfrac{n^2+1}{2}\right)\\ &\epsilon\leq\delta ^2\left(C n^2+D n\right)+\kappa n^2 \end{cases}

Outlooks

Further noise models
Other shuffling gates / architectures