Contribuciones originales

1. There is a Clifford circuit which takes an input qubit in arbitrary state |φi, along with an ancilla qubit in the magic state(|0i+eiπ/4|1i)/√2, and outputs the stateT |φi[see Fig.5.9(a)].

2. For all sufficiently large min and fin, there is a Clifford circuit which takes

an input of min copies of the (fidelity fin) magic state (|0i+e

iπ/4_|1i)/√₂ , and outputsmoutcopies of the (fidelityfout) magic state(|0i+e

iπ/4_|1i)/√₂ , such thatfout > fin, but wheremout < min. Moreover, if the Clifford circuit

is implemented without error, then there is a threshold fidelity fT such that

fout can be made arbitrarily small for fixedmout by increasing min provided

f_in < f_T[see Fig.5.9(b)].

It is therefore possible to achieve universal fault-tolerant quantum computing using a fault-tolerant Clifford computer, provided one has access to a large number of encoded magic states of sufficiently high fidelity.

|Ti |Ti |Ti |T_i |Ti | i a T| i (SX)a Z

Clifford

(a)

(b)

Figure 5.9: (a) A Clifford circuit that applies the T gate to its input qubit. (b) Magic state distillation is performed in consecutive rounds, with the output of one round forming the input to the next. In each round, a larger number of lower fidelity encoded magic states is transformed by a Clifford circuit into a smaller number of higher fidelity magic states.

An example of a Clifford circuit which can be used to implement state distillation is described below and in figure 5.10. In the description, we assume the circuit is perfect, and any noise is that associated with the noisy magic states.

1. Apply the encoding circuit for the [[15,1,3]] code on fifteen encoded qubits on one half of a Bell pair. The state of the system is(|0i |¯0i+|1i |¯1i)/√2. 2. Apply the transverse T gate (using 15 copies of the |Ti state, each with

an associated error rate p). If p = 0, the state of the system is (|0i |¯0i+

eiπ/4_{|1i |¯1i)/}√_{2 = ((|0i −ie}iπ/4_{|1i)|+¯i+ (|0i+ie}iπ/4_|1i)|+¯i)/2 .

3. Measure all the qubits comprising the [[15,1,3]] code in the X basis. The outcomes allow us to infer the measurement of theX-type stabilizers, and the

X-type logical operator, which we denotes=−1,1.

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Figure 5.10: The circuit used for|Tidistillation can be understood by first encoding one half of a Bell pair in the fifteen qubit code, using fifteen (noisy) |Ti states to apply the transverseT gate in the fifteen qubit code, followed by measurement of the logicalX operator giving outcomes. This forces the state of the other half of the Bell pair into something which differs from|Tiby ans-dependent Clifford gate. By also measuring theX-stabilizers of the fifteen qubit code, and discarding the output of the process when the stabilizers are not satisfied, one can improve the fidelity of the output state. Note that since the encoding circuit is composed of Clifford gates, and theT gate preserves the Clifford group, we could pull theT gates through the encoding circuit to give a new circuit with equivalent action but where the ancilla states for the fifteen qubit code are |Ti states rather than stabilizer states, and the encoding clifford circuit is replaced by a modified clifford circuit.

Ifp= 0, then the stabilizers must be satisfied, and the state of the only unmeasured qubit will be (|0i+iseiπ/4|1i)√2, and note that|Ti ∝ Ss(|0i+iseiπ/4|1i)√2. Forp > 0, since the fifteen qubit code has distance three, all weight one and two

Z- type errors will result in some unsatisfied stabilizers. No X-type errors will affect the state since they commute with theXmeasurement. If we discard the state unless all theX-type stabilizers are satisfied, there can be contributions only from the logicalZ operators of weight three and above. NeglectingO(p4)terms, the35 distinct weight-three logicalZ operators result in an error rate in the post-selected state of35p3. Note that the post-selection results in a reduction of yield: on average, approximately a fraction 15p of the states will be discarded. This circuit has an asymptotic rater = 1/15, since (forp→0) fifteen input magic states are replaced by one magic state of higher fidelity in each round.

In an implementation one must produce an encoded resource state with a sufficient fidelity. In a topological stabilizer code such as the toric code or the color code, one can first prepare an unencoded magic state in a singe qubit, encode this in a small code, and then grow the code distance while preserving the encoded state. The result will be a (noisy) encoded magic state.

Note that there will be additional errors introduced by the Clifford circuit which is used to distill the magic states. The additional errors are reduced by encoding the information in a larger code, which increases the overhead.