kpmg-qnu-labs-report-on-quantum-cryptography-2021

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The process of correcting the errors in quantum channel and arriving at identical keys between ALICE and BOB is called as post processing. By this time, the communication between ALICE and BOB using quantum channel is over and any communication during this step is carried over classical channel (usually Ethernet).

Post processing addresses the following errors:

- BOB detects only very few of the bits that ALICE sends (often in the range of 1 – 5 %). This is due to the low efficiency of the detectors.
- Some of the bits that BOB detects are wrong because of temperature effects, false detection & unknown errors.
- The clocks of ALICE and BOB are not synchronized. So the bits BOB detects has an offset with respect to the clock that ALICE sends. This lack of synchronization could be due to

- The difference in the path lengths of quantum channel and clock pulse fibre
- The time difference introduced during processing the clicks by detector & time to digital convertor (TDC).

There are three major steps in post processing.

**Offset correction:**Calculating the offset between ALICE and BOB’s clock and correcting for it.**Sifting:**Choosing the bits from ALICE corresponding to the bits that BOB has detected.**Error correction:**Correcting the errors in the bits that both ALICE and BOB have detected.

For each bit that BOB detects, we get two pieces of information - the value of the bit (0 or 1), the time at which the bit was detected. If the clock of ALICE and BOB are synchronous, then all we need to do is look up the bit sent by ALICE at that time and then see whether the bit value is same or not.

Unfortunately, the clock of ALICE and BOB are not synchronous to start with. Even If we assume they are synchronous, the time taken by the pulse to travel from ALICE to BOB introduces an offset between ALICE and BOB. So the time reported by ALICE for sending a bit and the time reported by BOB for receiving the same bit will be different. So before comparing the bits of ALICE and BOB, we need to calculate this offset and then correct BOB’s timestamps by reducing this offset. But how do we calculate this offset?

The key to calculating the offset is the assumption that this offset remains constant for all bits in one cycle (and all cycles). What it means is, that if the offset for the first bit detected by BOB is calculated at T1 nano-seconds, the offset for the 100th (or 1000th) bit will also be T1 nano-seconds. Once we make this assumption, then the offset calculation becomes easy as illustrated in the below diagram.

**Question:** What bit of ALICE corresponds to BOB’s first bit (0 that is marked with arrows) in the above diagram?

**Answer:** It is that bit, which produces the least amount of error when we compare all the bits of BOB with ALICE. So we move the first bit of BOB across ALICE’s bits and compute how many bits of BOB turn out to be wrong. The position in which the number of errors is minimum is the actual offset.

As we discussed earlier, BOB detects much fewer bits than what ALICE sent. Sifting is the process of discarding ALICE bits that BOB did not detect. After this process, both ALICE and BOB will have the same number of bits.

BOB will not only detect few of the bits that ALICE sent, but will also detect some of the bits incorrectly. Error correction is the process of correcting the bits that were detected incorrectly by BOB. A protocol called cascade protocol is usually adopted to correct the error bits.

In cascade protocol, BOB splits its bits into smaller sets and then compares the parity of the set with ALICE. When the parities don’t match, BOB splits that set into two and compares the bits of each subset with ALICE. This method of splitting and drilling down is continued till BOB finds one bit whose parity is not matched. BOB will continue this process for all sets till it detects all the errors.

After error correction, ALICE and BOB will have same number of matching bits. This, after a suitable amount of compression, will be the final Quantum KEY.

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