SCIENCE CHINA Information Sciences, Volume 62 , Issue 12 : 222501(2019) https://doi.org/10.1007/s11432-018-9847-0

## Implementing termination analysis on quantum programming

Shusen LIU 1,2,3,*, Kan HE 4,*,
• AcceptedMar 12, 2019
• PublishedNov 11, 2019
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### Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61672007, 11771011, 11647140). We were grateful to Ying DONG for his helpful dicussions.

### References

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• Figure 1

The flowchart of Qloop.

• Figure 2

Algorithms 1and 2are programmed onto Matlab 2017b. The experiment is conducted on the PC with Core i7 and 16 GB memory. Qloops 3and 4are executed repeated 1000 times for one setting and statistics over the total running time. The figure clearly illustrates Algorithm 2without the Jordan decomposition process has a significant acceleration on analysing of the termination. For both termination and almost-surely scenario, Algorithm 2speeds up almost 300 times than Algorithm 1.

•

Algorithm 1 The algorithm for checking termination with the Jordan decomposition

Require:$G$;

Function State, Instead = CheckTermination($G$);

${\left|~\Phi\right\rangle}~\gets$ MaxEntangledState where $d({\left|~\Phi\right\rangle})=d(G)$;

$[J,S]\gets~{\rm~Jordan}(G)$ s.t. $G=SJS^{-1}$, ${\left|~\phi\right\rangle}&apos;~\gets~S^{-1}{\left|~\Phi\right\rangle}$;

$[{\left|~u\right\rangle}~{\left|~v\right\rangle}~{\left|~w\right\rangle}]^{\rm~T}~\gets~{\left|~\phi\right\rangle}&apos;$ where $d({\left|~u\right\rangle})=\text{number~of~eignvalues~with~value~}~0$,$d({\left|~w\right\rangle})=\text{number~of~eignvalues~with~module~}~1$,$d({\left|~v\right\rangle})=d({\left|~\phi\right\rangle}&apos;)-d({\left|~u\right\rangle})-d({\left|~w\right\rangle})$;

if ${\left|~v\right\rangle}=0$ and ${\left|~w\right\rangle}=0$ then

$k~\gets~{\rm~CalSteps}(G)$;

State $\gets$ Finite termination;

Instead $\gets~\sum_{n=0}^{k-1}N_0G^n$;ELSIF${\left|~w\right\rangle}=0$

State $\gets$ Almost-surely termination;

Instead $\gets~\sum_{n=0}^{\infty}N_0G^n=N_0(I-G)^{-1}$;

else

State $\gets$ Non-termination or unknown;

Instead $\gets$ NULL;

end if

•

Algorithm 2 The algorithm for checking termination without the Jordan decomposition

Require:$G$;

Function State, Instead = CheckTermination($G$);

${\left|~\Phi\right\rangle}~\gets$ MaxEntangledState;

$[V,D]\gets$ eig($G^\dagger$);

$D_1~\gets~D~\text{~where~their~eigenvalues~with~module~}~1~$;

if $G^d~{\left|~\Phi\right\rangle}=0$ where $d$ is the dimension of $G$ then

$k\gets$ CalSteps($G$);

State $\gets$ Finite termination;

Instead $\gets~\sum_{n=0}^{k-1}N_0G^n$;ELSIF${\left|~\Phi\right\rangle}~\perp~D_1$

State $\gets$ Almost-surely termination;

Instead $\gets~\sum_{n=0}^{\infty}N_0G^n=N_0(I-G)^{-1}$;

else

State $\gets$ Non-termination or unknown;

Instead $\gets$ NULL;

end if

•

Algorithm 3 Qloop with Hadamard gate as a sub-program

Require:$\rho=\rho_1={\left|~1\right\rangle}{\left\langle~1\right|}$, $M=\{|0\rangle\langle~0|,|1\rangle\langle~1|\}$, ${\rm~HGate}=\frac{1}{\sqrt{2}}\big[\tiny{\begin{matrix} 1~&1\\ 1~&-1 \end{matrix}}\big]$;

Output:$\rho$.

while $M(\rho)$ do

HGate($\rho$);

end while

return $\rho$.

•

Algorithm 4 Qloop with bit flip gate as a sub-program

Require:$\rho=\rho_1={\left|~1\right\rangle}{\left\langle~1\right|}$, $M=\{|0\rangle\langle~0|,|1\rangle\langle~1|\}$, ${\rm~XGate}=\big[\tiny{\begin{matrix} 0&1\\ 1&0 \end{matrix}}\big]$;

Output:$\rho$.

while $M(\rho)$ do

XGate($\rho$);

end while

return $\rho$.

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