The Logical Phase Gate

We first note that since the logical_cnot and Hadamard gates are transversal, the logical_cphase is also transversal; it can be implemented by doing a logical Hadamard on the target, followed by a logical_cnot, followed by another logical Hadamard on the target.

A destructive measurement of the logical X (respectively, Z) operator can be performed by transversally measuring the operator X (respectively, Z) on each qubit in the code block. (With the logical cnot and cphasetransversal, we can also measure non-destructively the logicalX or the logicalZ operator by using as control an ancilla prepared in the logical|+istate.) If we were also able to easily measure the logical Y,YL =iXLZL, we would be able to implement the logical phase gate by using, e.g., the circuit in figure 5.12.

Although YL =⊗n2

|ψi •

>= Y |0i Z S|ψi

Figure 5.12. Circuit simulation of the phase gate,S, by using a measurement of Y. Contrary to the standard notation, here the correction operatorZ is applied when the measurement outcome is +1 and no correction is applied when the outcome is−1.

|ψi • • eiπ/4_S|ψi |+ii • |+ii

Figure 5.13. Circuit simulation of the phase gateS using the ancillary state|+ii.

give a measurement 1-Ga which satisfies property 2(b). This is because the Bacon-Shor code has no stabilizer operators that can be written as tensor products ofY operatorsalone. Therefore, the problem is that we cannot perform error correction on the transversal measurement outcomes and, so, the eigenvalue we would deduce for the logicalY could be erroneous even if a single one of the transversal measurements failed.

We could instead measure the logicalY operator nondestructively using cat states similar to our discussion in §5.2.1.1. Implementing this measurement would require controlled-Y gates, which at the next level of the recursive simulation would have to be implemented in an encoded form. But then, the problem is that the controlled-Y gate is complex and all the transversal operations we have discussed so far (_cnot, _cphaseand Hadamard) are real. Therefore, we do not have a direct transversal method for implementing the logical controlled-Y gate.

Fortunately, there is a method for simulating the logical S gate by using only logical _cnot and _cphase gates provided we can prepare a certain logical ancillary state: Consider the states |±ii ∝ |0i ±i|1iwhich are the±1 eigenstates ofY. Given the state|+ii, we can simulateS (up to an irrelevant phase) with the circuit in figure 5.13. Hence, the problem of constructing a 1-Ga for the logical phase gate reduces to the problem of preparing thelogicalancillary state|+ii.

Simulating S and S∗ in Superposition

Based on the circuit in figure 5.13, we will now first describe a simple procedure by which the simulation of the logical S gate is still possible even if the logical ancillary state |+ii is replaced by another logical ancillary state which is easier to prepare [93]. This procedure makes use the following two observations: (i) theSgate is the only complex gate in our universal gate set, and (ii) if the state|−iiis used instead in the the circuit in figure 5.13, then the operationS† =S∗ will be simulated.

5.13. We can expand|0i ∝ |+ii+|−iiand, for each of the two terms, we can consider the two paths of the subsequent computation which are executed in superposition. In one path S is simulated and in the other S∗. Thus, if every time we want to simulate S in our circuit we use the same

ancillary |0i state and because S is the only complex gate in our gate set, the final state of the computation will be a linear superposition of one term where the desired computation unitary, U, has been implemented and a second term whereU∗has been implemented instead. In other words, if the initial computation state is|ψinitiali, then the final computation state will be

|ψfinali= |+ii ⊗U|ψinitiali+|−ii ⊗U

∗_|ψinitiali

√

2 . (5.20)

In the end of the computation some operator Awill be measured which we can take to be real (and so, due to hermiticity,AT _{=A). We now want to see that the expectation value forAwill be} the same as if the desiredU had been simulated all along. Indeed, we compute

hAi=hψinitial|U

†_{AU+ (U}∗₎†_AU∗

2 |ψinitiali=hψinitial|U

†_{AU|ψinitiali, (5.21)}

since,∀|ψi,hψ|(U∗₎†_AU∗_|ψi=hψ|U†_AT_U|ψi=hψ|U†_AU|ψi.

We can use this procedure at the logical level as well. The only penalty we pay for simulating the logical S gate in this way is that we need to swap around the ancillary logical |0i block we use for the simulation if we are constrained to use only local interactions; but this will only give us a linear penalty in the size of the computation. Of course, we should emphasize that this trick works because theS gate is not used in implementing error correction. As a consequence, we only need to implement logicalS gates at thehighestlevel of the recursive fault-tolerant simulation and, moreover, it is not necessary that we execute different logicalS gates in parallel anywhere in our computation. Therefore, provided the physical noise strength is below the accuracy threshold for CSS operations, we may obtain any desired accuracy in the simulation of the logicalS gates at the highest level of our recursive simulation by choosing this highest level appropriately.

Noisy|+iiDistillation

In cases when S is not the only complex gate in our gate set, there is also a straightforward procedure for fault-tolerantly preparing the required ancillary “quantum software:” We can begin by preparing many noisy logical|+iistates and use them to progressively distill less and less noisy copies. A possible distillation circuit is shown in figure 5.14.

The measurement outcome in this circuit is ideally +1, and an error on one of the two |+ii states can be detected. We note that becauseY stabilizes the state|+ii, we need only worry about Z errors—we can write X =iZY which is, up to the irrelevant phase i, equivalent to Z when it acts on|+ii. Postselecting on the +1 measurement outcome, the fidelity of the output|+iistate is

|+ii • |+ii |+ii • •

>= X

Figure 5.14. Circuit for the nondeterministic distillation of the state|+ii. Postselecting on the +1 measurement outcome, the output|+iistate has quadratically improved fidelity.

increased quadratically relative to the fidelity of the input state since it takes errors inboth input |+iistates for an error in the output state to go undetected.

To be concrete, assume noise is local and stochastic and letpancdenote the probability that some fault has occurred during the preparation of one of the initial copies of the distillation protocol. Then by “twirling,” i.e., by applying at random with probability 1/2 aY operator (and by doing nothing with probability 1/2), we may describe the initial copies as having a Z error with probability at mostpanc. If the CSS operations of the protocol are executed ideally, then the probability of a Z error on the output copy conditioned on getting the +1 measurement outcome and accepting the state is p(1)anc≤ p2anc p2 anc+ (1−panc)2 ≤2p2anc, (5.22)

since both input copies must have aZ error in order for an erroneous output state to be accepted.4 If we repeat the distillation protocol l ≥ 2 times and since the input copies for each round are prepared independently at the previous round, we find p(l)anc ≤ 1₂(2panc)

2l

which indicates a distillation threshold,pdist,_thr |+ii= 1/2.