Cat state preparation failure

Now it remains to estimate the probability of failure for the cat state and the logical state _|+_iL. Under our simplified procedure for choosing the “winning” syndrome, we first reject any syndrome that does not have a consistent interpretation— that is, an even number of nontrivial syndrome bits. Recall that the syndromes ideally detect the boundaries of error chains, so syndromes with an odd number of bits are inconsistent with this interpretation. Out of the remaining consistent syndromes, we choose the syndrome that occurs most often to be the “winning” syndrome.

If the winning syndrome leads us to exactly identify the correct Pauli frame, then we have succeeded. But we could instead identify some other syndrome which is slightly di↵erent than the correct Pauli frame. If the number of locations l where the inferred Pauli frame di↵ers from the correct Pauli frame is smaller than the number of non-diagonal faults that occurred during the preparation, we can reinterpret this as being the same as a perfect syndrome decoding followed by non-diagonal faults which add the errors. These kinds of failures are already accounted for in

Section 3.6.1; we call them “valid” syndromes. We are therefore interested in the probability that the winning syndrome is “catastrophically” di↵erent, such that the previous condition fails.

Suppose that the winning syndrome occursttimes due to faults, and that there areuadditional rounds that each contain at least one fault. Some of these additional rounds might be rejected, and some may not, but we can assume that they are all rejected because, by definition, they do not produce the winning syndrome. Suppose that non-diagonal faults occur insof ther0 rounds of syndrome measurement; these faults can alter the syndrome. There arer0 t urounds without any faults, and the number of distinct syndromes detected in these rounds is at mosts+ 1.

Now we can use the pigeonhole principle to obtain a lower bound on t, expressed in terms of r0_{,u, ands. There are at mosts+ 1 “valid” syndromes that can occur in syndrome measurement} rounds that have no non-diagonal faults. Combining these with the winning syndrome, there are at most s+ 2 possible syndromes that can occur in the r0 _{u accepted rounds. Of these s+ 2} syndromes, the winning syndrome must occur at least as many times as any other syndrome; hence,

t

⇠_{r0 u} s+ 2

⇡

, (3.25)

wheredxedenotes the smallest integer greater than or equal tox.

To bound Perr(P_|cati), we sum overs anduin each cat state preparation step, estimating the

number of possible fault histories using the upper bound Equation (3.25) ont. In the first of the t winning rounds, a particular winning syndrome is found, which di↵ers in at least two bits from the actual syndrome in the beginning of that round. Then this same syndrome is found again in each of the remaining winning rounds. The sum over all possible winning syndromes, weighted by their probabilities, is bounded above by the probability that at least two measured syndrome bits are faulty in the first of thetwinning rounds. Each ZZ measurement is performed using one |+_ipreparation, two CZ gates, and oneX measurement; therefore, the probability of error in the measurement of a single syndrome bit is bounded above by 4"+ 2"0, and the probability that at least two syndrome bits are in error is bounded above by p₂ (4"+ 2"0)2_.

In each of thesrounds that containXerrors, we must sum over all the possibleX-error patterns that can occur in that round. The sum over allX-error patterns, weighted by the probabilities, is bounded above by the probability that at least one nondiagonal fault occurs in that round. Since the round contains 2p CZ gates, this probability is in turn bounded above by 2p"0_.

Once the X-error pattern has been chosen in each round that contains X errors, we know the actual syndrome at the beginning of each round. And once the winning syndrome is chosen in the first winning round, we know which syndrome bits must have errors in each of the remainingt 1 winning rounds. If the cat state preparation fails, then, by definition, at least two syndrome bits have errors in each of these rounds; hence, each winning round after the first has a probability weight bounded above by (4"+ 2"0)2_.

Taking into account that the rounds with non-diagonal faults can be chosen in at most r_s0 ways, and enumerating the ways to choose whicht rounds produce the winning syndrome and which u additional rounds have faults, we obtain

Perr(P_|cati)nr r0 X s=0 r0 X u=0 ✓_r0 s ◆✓ _r0 u+t ◆✓_u+t u ◆✓_p 2 ◆ (4"+ 2"0)2t(4p"+ 2p"0)u(2p"0)s, (3.26) and, similarly, Perr(P|+i)m r+ X s=0 r+ X u=0 ✓_r + s ◆✓ _r + u+t ◆✓_u+t u ◆✓_n 2 ◆ (4"+ 2"0)2t(4n"+ 2n"0)u(2n"0)s, (3.27) where t denotes dr0 _u

s+2e. The prefactor nr in Equation (3.26) arises because we use n length-p

cat states in each measurement, and each measurement is repeated r times. The prefactor m in Equation (3.27) arises because the encoded state_|+_iL is a product ofmlength-ncat states.

We should add another contribution to the failure probability, because if half of the qubits (or more) have X errors, we might decode the cat-state syndrome incorrectly. A syndrome of the repetition code points to two possible X-error patterns, one low weight and one high weight. We always assume the low-weight interpretation is correct, so if the high-weight interpretation is actually correct, then a Pauli-frame errorX⌦_{pis applied to the cat state (or worse if the cat state} qubits are used multiple times). The additional contribution, then, is bounded by the probability that each ofdp/2equbits in the cat state are each hit by X errors at least once in at least one of the cat state preparations. This upper bound is

Phigh weightnr

✓ _p dp/2_e

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