Surface code

[WTG+13], and fast coherent donor/dot SWAP operations. Finally, in Sec. 6.8 we deal with the most important decoherence processes involved, and quantify their consequences for the fidelity of the SWAP transfer.

6.2

Surface code

In this section we outline succinctly the main properties of a surface code architecture that could be implemented within the quantum computer which is the subject of this chapter. We focus on the key aspects of its physical realizations and the advantages it could pro- vide in terms of improved fault tolerance in the computation. More extended descriptions can be found, for example, in Refs. [DKLP02, FMMC12]. Remarkably, a small size in- stance of a surface code, based on fewer than nine qubits, has been realized very recently [KBF+15].

The underlying principle of surface codes, which have been first studied in the some- what separate context of topological order [Kit03], consists of separating the notions of the physical qubit, i.e. a two level system addressable in a controlled manner, from the logical qubit, which is the true repository of the quantum information to be stored and processed. Namely, if the logical qubit is cleverly encoded within some combination of states of physical qubits, the logical manipulations can be intertwined with efficient error correction algorithms involving a large number of physical qubits, thus enhancing the threshold decoherence that can be tolerated by the full computation. In fact, surface codes have been evaluated theoretically to tolerate per-operation error rates of the order of ∼1%, larger than any other error correction algorithm developed so far. Broadly speak- ing, the other major advantage of this kind of algorithm is the automatic scalability of the scheme.

More specifically, Ref. [FMMC12] proposes a planar array of physical qubits, as shown in Fig. 6.1, where solid circles represent so-called measurement qubits, and open circles are the data qubits. A further distinction has to be made within the measurement qubits, in that half are intended to perform σz-kind of local measurements, with the other half

doingσx, following the procedure schematized in the lower panels of Fig. 6.1 and that

we will describe in some detail later on. While single qubit initialization and driving are required, the only two-qubit manipulation needed is a CNOT gate between nearest neigh- bours: how to realize the latter within the donor/dot quantum computer discussed in this chapter will in fact be investigated in Sec. 6.7. The surface code is based on the cre- ation of entangled collective states that result from sequences of such CNOT operations, which are performed between neighbouring physical qubits: such states host the actual logical qubits. CNOT operations are intermingled with non-demolition measurements of

CHAPTER6: DONOR-DOT SCHEME

Figure 6.1: Figure taken from Ref. [FMMC12], showing a schematic planar surface code. The architecture relies on alternating data qubits, drawn as open circles, and measurement qubits corresponding to solid circles. The logical circuits in the lower panels describe the sequence of ordered operations that should be performed in order to stabilize the array. The basic ingredients are addressable CNOT gates between a data qubit and its four neigh- bouring measurement qubits, which can be devised either to measure theZˆaZˆbZˆcZˆdor the

ˆ

XaXˆbXˆcXˆdstabilizers, the latter with two extra Hadamard gatesH. Final non-demolition

measurementsI of the data qubits complete the recipe, which should be realized in paral- lel across the whole array.

6.2. SURFACE CODE

the logical states, that allow one to track eventual errors occurring during the processing. Crucially, it is possible to gain information about the logical qubits without destroying (i.e., projecting) their quantum states because the latter are hosted by a collection (that is, more than one) of physical qubits.

Let us examine how errors can be dealt with: we consider two basic kinds of error, that can be represented by the action ofXˆ bit-flip and Zˆ phase-flip operators acting locally, and stochastically, on the physical qubits. If there is a way to detect where such errors have taken place, at each stage of the surface code, then extra single-qubit gates may be applied externally, in a controlled manner, to compensate their outcomes. In fact, for example an

ˆ

X-error could be corrected simply by an extra, intentionalXˆ operation. Moreover, not all the errors occurring at the physical qubits should be taken into account, but only those that affect the measurements involved in the construction and preservation of the collec- tive logical states: this feature provides, in fact, a much more efficient way of dealing with errors than the naive application of extra gates. Ref. [FMMC12] suggests that this way can rely entirely on classical computing, as it encompasses only the ability to take into account the detections of errors when extracting the final information from the measure- ments of the logical qubits, rather than to correct them within the computation (something that would require quantum computing abilities).

Let us specify more carefully the working principles of a surface code: the placement of the two planar arrays of measurement (MQ) and data qubits (DQ) is structured so that each DQ, as shown in the upper panel of Fig. 6.1, is coupled to two MQ forX-like measurements, and two MQ forZ-like measurements, while each MQ adjoins four DQ. Collective operators acting simultaneously on four physical qubits, that are called stabiliz- ers, are defined:ZˆaZˆbZˆcZˆdandXˆaXˆbXˆcXˆd, where the labels identify MQ as in the lower

panel of Fig. 6.1. Hence theZ(X)-like measurements can now be better described as the projection of the state of a DQ onto an eigenstate of a collectiveZ(X) stabilizer, where the projection is accomplished by subsequent CNOT operations between the DQ and its four neighbouring MQ, plus a final projective measurement (with two extra Hadamard gates for theX stabilizer). Thus the benefit produced by the MQ is now easily pointed out in that they allow the formation and preservation of a quiescent state, i.e. a simultane- ous eigenstate of both kinds of stabilizers. If the sequence of such operations is carefully chosen (see Appendix B of Ref. [FMMC12]), the state of all the data qubits in the array is eventually an eigenstate of all the plaquette stabilizers defined throughout the array. A quiescent state is stable because it is easily built up after one run of the code just described, irrespective of the initial states of the local physical qubits, and is then pre- served, assuming that no decoherence takes place, in all subsequent runs: this remarkable property is a consequence of the commutation of the two kinds of stabilizers introduced before, that is in turn made possible since those operators ‘live’ on multiple sites, so that

CHAPTER6: DONOR-DOT SCHEME

even though[ ˆXa,Zˆa]6= 0, those stabilizers that have qubits in common still commute. A

quiescent state, more importantly, is useful because it allows the detection of errors. It can be shown, in fact, that the occurrence of a phase-flip error on a data qubit at some stage of the surface code is either cancelled by a subsequent run, that projects the ‘wrong’ state back to the quiescent one, or signalled by the simultaneous sign change of the measure- ment results of the twoX-measurement qubits adjacent to that particular data qubit. This latter property is essentially explained by the anticommutation relation{Xa, Za}= 0, and

the crucial feature that each data qubit is flanked both byX- andZ-measurement qubits. Thus, even classical software can record the location of that error on a data qubit, which is then easily erased by ‘manually’ flipping the sign of the measurement outcomes of its two adjacentX-measurement qubits. Of course, mutatis mutandis, the same procedure can be applied for bit-flip errors. The panorama of errors that may affect the qubits is surely larger, including the possibility of errors in the measurements themselves, or correlated errors between adjacent plaquettes, but their treatment is beyond the scope of this brief review.

So far we have discussed how unwanted deviations of an initial quiescent state can be either disabled or diagnosed by the surface code, but of course a full quantum computer needs to deal with (robust) manipulation of an initialized quantum state. As it turns out, the presence of boundaries to the array, or ‘local cuts’ realized e.g. by shutting off one or more measurement qubits, gives the chance to enlarge the stabilized logical degrees of freedom, i.e. to define logical operators whose results are still protected by the continuous run of surface code cycles. We present briefly the second way of creating extra logical qubits, as it is more easily implemented in the donor/dot architecture presently investi- gated.

Fig 6.2 displays an hole in the array, that follows after shutting off a Z-measurement qubit, which can be done simply by not performing the corresponding CNOTs with its neighbouring data qubits. This way, the stabilizer relative to the hole is not measured any longer, and the resulting extra degrees of freedom can be encoded in a Hilbert space avail- able for the action of logical operators. The latter can be defined as the tensor product of chains of local measurement operators that either link the hole with a boundary, or enclose the hole in a loop.

As regarding a quantitative evaluation of the effectiveness of a surface code, an insight- ful figure of merit is provided by the dependence ofPL, the number of errors occurring

on the logical qubits per each surface code cycle, versus p, the rate of errors affecting thephysicalqubits at each step, which is obviously correlated to the coherence timesT2.

Ref. [FMMC12] points out that such dependence is strongly determined by the so-called distance d of the array, i.e. the minimum number of physical qubit operators compos- ing a logical operator. Simulations therein show that the approximate scaling relationship

6.2. SURFACE CODE

Figure 6.2: Figure taken from Ref. [FMMC12] that shows how a hole in the array can be exploited to define a logical qubit. The hole is a consequence of the local disabling of a Z-measurement qubit, that is thus prevented from participating in the sequence of CNOTs with its neighbouring data qubits. The extra degrees of freedom coming from the smaller number of projections performed within the array allows the definition of two logical operators,XL =X1X2X3andZL =Z3Z4Z5Z6, that can act on the quiescent state within

a surface-code-protected Hilbert space. The chain ofX operators links the cut with theX boundary of the array, while theZ chain forms a loop around it. More complicated ways to create extra logical qubits can be devised, that do not require involving the boundary of the array.

CHAPTER6: DONOR-DOT SCHEME

PL ≈ 0.03(p/pth)d describes the situation for a wide range of the parameters in play, at

least for the basic kinds of error considered in this brief discussion. pth is a threshold

error rate for the physical qubits, about 0.6%, so that ifp < pth andd is large enough,

PL is easily seen to drop to exponentially small values. Anyway, in terms of physical

implementations, the total number of physical qubits needed to achieve a target low PL

increases quickly ifp is less than but close to pth, thus a trade-off between gate fideli-

ties on the physical qubits and dimensions of the array should guide future experimental development.