# The Limitations of Imperfect Quantum Cloning

# Is 4-sigma good enough?

This article is a brief follow up to my article titled, “Quantum Imperfect Cloning.” Under the “Future Work” section, I proposed ways to improve the fidelity of the results.

## Introduction

Although quantum mechanics forbids the precise cloning of unknown quantum states, it allows the imprecise cloning of these same states. The question before us is: how precise can we get?

## “Cheat” Method

I like to use OpenQASM for all things quantum. But, for a comparison, I took tomography measurements of the “unknown” quantum state, performed the calculations manually, and compared the resultant “cheat” state to the unknown state.

For this “cheat" comparison, I originally reported a SWAP Test result of 0.997. I have since been able to improve that result to 0.99768. However, the additional precision is merely a result of using every decimal place available for the calculations.

The limitation to this method, not withstanding quantum mechanics itself, is involuntary rounding. The tomography measurements in IBM Quantum Experience, for example, are rounded to the thousandth place. This imprecision alone negatively affects all subsequent calculations.

## OpenQASM-Restricted Method

The circuit diagram above shows the OpenQASM-restricted method, which I originally reported as achieving a SWAP Test result of about 0.7. The code is available as a GitHub repository. I have only been able to improve that to about 0.75, which is not much less unimpressive.

One method that I attempted was to use multiple copies of the unknown state. The idea was to use multiple syndrome bits to rotate fractions of pi, instead of just rotating by pi. However, even rotating by fifths of pi, which I tested, still allows for significant imprecision. Maybe this might work with a ridiculous number of identical unknown states, but, if we already have that many identical copies, why would we need imperfect clones?

## Conclusion

Since quantum computing is probabilistic, how perfect do cloned states need to be? If we measure an unknown state thousands of times and its 99.768%-identical copy thousands of times, would we notice a difference versus measuring the original unknown state thousands more times?

Furthermore, the imperfection is the result of rounding numbers. I wonder how much closer the comparison might be if additional precision could be made available.

Finally, I lament the inaccuracy of the OpenQASM-restricted circuit. As previously stated, maybe it’s still possible with a significant number of identical unknown states. However, and notwithstanding the impracticality of having so many identical copies, adding copies exponentially increases the number of conditional statements required; OpenQASM doesn’t have loops, so that’s an awful lot of copying, pasting, and manual editing. Consequently, this virtual project folder is closed.