# Noise on the ancilla for syndrome measurement

To detect noise through stabiliser measurement, in the circuit formalism, there is usually an ancilla which stores a parity check result. However, what happens if also the ancilla suffers from the same noise. Does this nullify all the theory? Isn't the assumption of a perfect ancilla too impractical?

• this is the difference between error correction (in particular as described upon first studying it) and fault-tolerant error correction, which adjusts for all the ancillas etc being noisy as well. Commented Jan 24, 2022 at 9:54

## 2 Answers

The ancilla being noisy has two effects: making the measurement output noisier, and creating "hook errors" which look like multiple data qubit errors. An $$n$$-qubit stabilizer can have hook errors equivalent to up to $$\lfloor n/2 \rfloor$$ data errors (which occurs when the error on the ancilla occurs exactly halfway through the stabilizer measurement operations), with the specific subset of data qubits depending on the order of operations. Hook errors can be mitigated by sticking to small stabilizers, or by using techniques like flag qubits to distinguish the hook errors from other errors.

Does this nullify all the theory?

What? Of course not. Of course people have thought about the ancilla being noisy!

Whenever a paper says it is doing simulations with "circuit noise", they are considering noise on the ancilla interspersed with the operations the stabilizer measurement has been decomposed into. There are tons of examples of this... an extremely recent one is "A circuit-level protocol and analysis for twist-based lattice surgery".

Isn't the assumption of a perfect ancilla too impractical?

Yes, it would be a ridiculous thing to assume would hold in practice.

Often research will ignore noise on the ancilla as a starting point, to make a problem easier to get a hold on. But that's different from assuming it doesn't have to be dealt with or that it doesn't exist in reality.

Syndromes are analyzed in space and time.

An error on ancilla (also equivalent in effect to measurement error) seems as a sign flip of the syndrome in time, in only 1 ancilla.

While error on data qubit will affect the 2 ancillas next to him:

In the general case, data qubits will show even number of errors, while ancilla error, will create an odd number of errors - just a flip over time.

You can see here X/Y/Z error effect on the syndrome, and M is an effect of the error on ancilla/ measurement error