A silicon-based surface code quantum computer

Individual impurity atoms in silicon can make superb individual qubits, but it remains an immense challenge to build a multi-qubit processor: There is a basic conflict between nanometre separation desired for qubit-qubit interactions, and the much larger scales that would enable control and addressing in a manufacturable and fault tolerant architecture. Here we resolve this conflict by establishing the feasibility of surface code quantum computing using solid state spins, or `data qubits', that are widely separated from one another. We employ a second set of `probe' spins which are mechanically separate from the data qubits and move in-and-out of their proximity. The spin dipole-dipole interactions give rise to phase shifts; measuring a probe's total phase reveals the collective parity of the data qubits along the probe's path. We introduce a protocol to balance the systematic errors due to the spins being imperfectly located during device fabrication. Detailed simulations show that the surface code's threshold then corresponds to misalignments that are substantial on the scale of the array, indicating that it is very robust. We conclude that this simple `orbital probe' architecture overcomes many of the difficulties facing solid state quantum computing, while minimising the complexity and offering qubit densities that are several orders of magnitude greater than other systems.

The problem of scalability remains one of the great challenges facing the development of quantum computers. The semiconductor revolution enabled the spectacularly successful scaling of classical information processing devices, and it is reasonable to hope that some of this vast expertise could be fruitfully brought to bear on quantum systems. An influential early paper exploring this possibility was that of Kane [1] in 1998, in which impurity atoms implanted in a pure silicon matrix would constitute the means of storing qubits. Operations between qubits would occur through direct contact interactions between such spins, which necessitated an inter-qubit spacing of at most nanometers (and therefore a precision considerably greater than this) together with exquisitely small and precisely aligned electrode gates to modulate the interaction. This architecture proved highly influential and stimulated a number of advances, including impurity positioning via STM techniques that have achieved nanometer precision [2,3]. However it remains extremely challenging as a path to practical quantum computing.
Since 1998 there has been dramatic progress in understanding the representation and processing of quantum information. Surface codes have emerged as an elegant and practical method for representing information in a quantum computer. The units of information, or logical qubits, can be encoded into a simple 2D array of physical qubits [4]. By measuring stabilizers, which essentially means finding the parity of nearby groups of physical qubits, errors can be detected as they arise. Moreover with a suitable choice of stabilizer measurements the encoded qubits can even be manipulated to perform logical operations. The act of measuring stabilizers over the array thus constitutes a kind of 'pulse' for the computer -it is a fundamental repeating cycle and all higher functions can be built upon it.
In view of the power and elegance of the surface code picture, one can now revisit the ideas of the Kane proposal and dramatically reimagine it as an engine designed 'from the bottom up' to efficiently perform stabilizer measurements. This is the task we undertake in the present paper. We find that one can abandon the need for direct gating between physical qubits, and with it the need for extreme precision in the location of impurities and the equally challenging demand for electrical gating of qubit-qubit interactions. This is replaced by a requirement for parity measurement of groups of four spins, which we argue can be performed by a simple repeating physical cycle. Crucially, we exploit long range dipole fields rather than contact interactions, and we are thus able to select the scale of the device according to our technological abilities. Presently we show that the tolerances in our scheme, i.e. the amounts by which dimensions can be allowed to vary, can be orders of magnitude greater than those demanded in the Kane proposal. A further advantage of our approach is that it requires active control of only the electron spins, rather than the nuclear spins (although the latter can be a useful additional resource, as we will note). These various advantages come with a new and unique challenge: the device consists of two mechanically separate parts, which are continually shifted slightly with respect to one another in a cyclic motion. Deferring a full discussion of practicality to later in the paper, here we simply note that the requirements in terms of the surface flatness and the precision of mechanical control are considerably less demanding than the tolerances achieved in modern hard disk drives.
We begin with a discussion of the physics of the parity measurement process, before moving on to analyse the robustness of the device against various kinds of imperfection. The essential elements of the scheme are shown in Fig. 1(a). Four spin-1 2 particles referred to as 'data qubits' are embedded in a static lattice. In practice these are likely to be electrons bound to isolated donor impurities in a semiconductor, which we describe in more detail later. Meanwhile another spin-1 2 particle is associated with a mechanically separate element which can move with respect to the static lattice. We assume this 'probe spin' is also electronic and associated with a impurity or defect. It will be necessary to prepare and measure the state of the probe; as we presently discuss, this might be achieved via spin-to-charge mapping or alternatively by optical means if the spin-bearing entity has conditional optical transitions, as for example an NV-centre in diamond. The length scales of the system could take a range of values -provided that one scales all the dimensions in proportion to one another, then the analysis we present here will still apply (making the system smaller increases the engineering challenge but would enhance the operating speed). With that in mind, without loss of generality we adopt the following scales in the remainder of this paper: The z-direction separation between a probe and a data qubit at closest approach is 40 nm. The separation between data qubits, on the other hand, is 400 nm so that the direct interaction between them is relatively negligible. Given this setup, our goal is to measure the parity of the four data qubits i.e. to make a measurement which reports 'even' and leaves the data qubits in the subspace {|0000 , |1100 , |0011 , |0110 , |1001 , |0101 , |1010 , |1111 }, or which reports 'odd' and leaves the four data qubits in the complementary subspace.
In the abstract language of quantum gates, building a parity measurement is straightforward. The following process is widely used in the quantum computing literature: we prepare an ancilla (the probe, in our case) in state |+ = (|0 + |1 )/ √ 2 and then perform phase gates G = diag{1, 1, 1, −1} between the ancilla and each data qubit in turn, before measuring the ancilla in the basis {|+ , |− }. If we see outcome |+ then the data qubits are in the 'even' space, while |− indicates 'odd'. (This is easy to see by reflecting that the ancilla state toggles |+ ↔ |− when it is phase-gated with data qubit in state |1 , but it is unchanged if that qubit is |0 ; thus the final state is |+ if and only if there have been an even number of such toggles.) Now in the present physical system, we can perform an operation that is essentially identical to the desired phase gate by exploiting the dipole-dipole interaction between the probe and the nearby data qubit. Spin-1/2 particles suffice for the protocol we describe and a generalisation to exploit higher spin systems should be possible. The Hamiltonian for two S=1/2 spins, each in a static B field in the Z direction and experiencing a dipole-dipole interaction with one another, is Here r is the vector between the two spins, andr is the unit vector in this direction. In the present analysis we assume that the Zeeman energy of the probe spin differs from the Zeeman energy of the data qubit by an amount ∆ = µ B B(g 1 − g 2 ) which is orders of magnitude greater than the dipolar interaction strength J/r 3 , a condition that prevents the spins from 'flip-flopping', as shown in [5]. In a reference frame that subsumes the continuous Zeeman evolution of the spins, their interaction is then simply of the form up to a global phase exp(−iθ/2) which we neglect. The condition that ∆ J/r 3 will certainly be met if the spin species for the data qubits differs from that of the probes, as for example if the data qubits are phosphorous atom in silicon, while the probes are NV centres in diamond. We have verified the form of S(θ) with exact numerical simulation for the parameters of our proposed system. Indeed even if the probe and data qubits are instances of the same physical system, then it should be possible to introduce a sufficient detuning by other means -for example by exploiting the hyperfine structure if there is a nuclear spin present.
Our goal now will be to acquire the maximum entangling value of θ = π/2 during the time that the two spins interact according to S(θ). For the present paper we consider two basic possibilities for the way in which the mobile probe spin moves past each static data qubit; these two cases are shown in Fig. 2(b). The first possibility is that the probe moves abruptly from site to site, remaining stationary in close proximity to each data qubit in turn. In this case, we simply have θ = αt i.e. the phase acquired increases linearly until the probe jumps away. The motion of the probe between sites is assumed to be on a timescale that is very short compared to the dwell time at each site; in practice this motion might be in-plane or it might involve lifting and dropping the probe.
An alternative which might be easier to realise is that the probe moves continuously with a circular motion (since this corresponds to in-phase simple harmonic motion of the probe stage in the x and the y directions). Because the data qubits are widely spaced, from the point of view of a data qubit the probe will come in from a great distance, pass close by and then retreat to a great distance. The interaction strength then varies with time; but by choosing the speed of the probe we can select any desired total phase acquisition during its transit. For our purposes U = S(π/2) is the ideal interaction: it is straightforward to show that the desired four-qubit parity measurement is achieved when the probe experiences an S(π/2) with each qubit in turn, followed by measurement (see Fig 2(d)). There is a residual phase shift to the data qubits but this is easily accounted for in the scheme e.g. by adjusting the next series of Hadamard rotations to absorb the phase.
The description above is in terms of ideal behaviour, but we should consider a wide variety of defects and errors in order to establish whether the device is realistic with present or near-future technology. This includes not  only the imperfections in state preparation, measurement and manipulation, but moreover also the systematic errors that result from the spins occupying positions that deviate from their ideal location. Given a full model of these errors, we can determine how severe the defects can be before the device ceases to operate as a fault tolerant quantum memory: this is the fault tolerance threshold. Presently we describe our numerical simulations which have determined this threshold.
In obtaining these results, we had to tackle a number of unusual features of this novel mechanical device. The most important point is that we must suppress the systematic errors that arise from fixed imperfections in the locations of the spins. Each data qubit is permanently displaced from its ideal location by a certain distance in some specific direction, and these details may be unknown to us -what is the effect of such imperfections on the idealised process of four-qubit parity measurement described above? Our analytic treatment (see Appendix II) reveals that the general result is to weight certain elements of the parity projection irregularly. Specifically, whereas the ideally even parity projector iŝ when the spins involved are misaligned then one finds that different terms in the projector acquire different weights, so that the projector has the form, whereŴ is a set of lower weighted projectors on odd states. Meanwhile, the odd parity projector,P odd be-comesP odd which is similarly formed of a sum of pairs; the pair (|0001 0001| + |1110 1110|) has a different weighting from the each of the other three permutations. Using these projectors to measure the stabilizers of the surface code presents the problem that the error is systematic: for a particular set of four spins, the constants A, B, C and D will be the same each time we measure an 'even' outcome. Each successive parity projection would enhance the asymmetry. In order to combat this effect, and effectively 'smooth out' the irregularities in the superoperator, we introduce a simple protocol that is analogous to the 'twirling' technique used in the literature on entanglement purification. Essentially we deliberately introduce some classical uncertainty into the process, as we now explain.
Suppose that one were to apply the imperfectP even projector to four data qubits, but immediately prior to the projection and immediately after it we flip two of the qubits. For example, we apply XX11 before and after, where X is the Pauli x operator and 1 is the identity. The net effect would still be to introduce (unwanted) weightings corresponding to A, B, C and D, however these weights would be associated with different terms than forP even alone; for example the A weight will be associated with |1100 and |0011 . Therefore consider the following generalisation: we randomly select a set of unitary single qubit flips to apply both before and after theP even projector, from a list of four choices such as That is, we choose to perform our parity projection as U iP even U i where i is chosen at random. We then note the parity outcome, 'odd' or 'even', and forget the i. The operators representing the net effect of this protocol,P smooth even andP smooth odd are specified in Appendix II. Essentially we replace the weightings A, B, C and D with a common weight that is their average, but at the cost of introducing Pauli errors as well as retaining the problem thatP smooth even has a finite probability of projecting onto the odd subspace, and similarlyP smooth odd has a risk of leaving the system in the even state. However these imperfections are tolerable -indeed they will occur in any case once we allow for the possibility of imperfect preparation, rotation, and measurement. Crucially the 'twirling' protocol allows us to describe the process in terms of a superoperator that we can classically simulate. It is formed from a probabilistically weighted sum of simple operators, each of which is eitherP even orP odd , together with some set of single qubit Pauli operations, i.e. S 1 S 2 S 3 S 4 where S i belongs to the set {1, X, Y, Z}. In our simulation we can keep track of the state of the many-qubit system by describing it as the initial state together with the accumulated Pauli errors.
In practice it may be preferable to achieve an equivalent effect to the U i twirling operators without actually applying operations to the data qubits. This is possible since flipping the probe spin before-and-after it passes over a given data qubit is equivalent to flipping that data qubit, i.e. it is only the question of whether there is a net flip between the probe and the data qubit which affects the acquired phase. Therefore we can replace the protocol above with one in which the probe is subjected to a series of flips as it circumnavigates its four data qubits, while those data qubits themselves are not subjected to any flips. Since we are free to choose the same i = 1 . . . 4 for all parity measurements occurring at a given time, these probe-flipping operations can be global over the device. In this approach the only operations that target the data qubits are the Hadamard rotations at the end of each complete parity measurement (these are required so that alternating rounds of parity measurement are in the X and Z basis). This is an appealing picture given that we wish to minimise noise on data qubits, and it is this variant of the protocol which we use in our numerical threshold-finding simulations, the results of which are shown in Fig. 3.
It is worth noting that in many real systems we may wish to use a spin echo technique to prevent the probe and data qubits from interacting with environmental spins. In this case we will want at least one flip applied to the spins (both the data and probe families) during Table I: Levels of infidelity assumed in our numerical simulations (see Fig. 3) compared with the best experimental results to date. Note we used two values for the measurement fidelity, depending on whether a |0 or |1 is observed. Jitter is random variation in dipolar coupling due to e.g. vibrations. The final row refers to the probability that a data qubit suffers an error during one cycle, either due to environmental decoherence or imperfect Hadamard gates. The experimentally reported numbers are abstracted from the detailed section in the main text. a parity measurement cycle; fortunately it is very natural to combine such echo flips with the flips required for twirling. Before concluding our description of this protocol we note that the idea of perturbing our system and then deliberately forgetting which perturbation we have applied, whilst perfectly possible, will not be the best possible strategy. We speculate that superior performance would result from cycling systematically though the U i choosing i = 1, 2, 3, 4, 1, 2.. over successive rounds; but for the present paper is suffices to show that even our simple random twirl leads to fault tolerance with a good threshold.

Results of numerical simulations
We combined the protocol described above, whose purpose is to smooth out systematic errors, with an error model that accounts for finite rates of random error in the preparation, control, and measurement of the spins involved. The error model we employed is the standard one in which, with some probability p, an ideal operation is followed by an error event: a randomly selected Pauli error, or simple inversion of the recorded outcome in the case of measurement. We specify the model more precisely in Appendix I, and in in Table I we tabulate the particular error probabilities used in our simulations.
Our threshold-finding simulation generates a virtual device complete with a specific set of misalignments in the spin locations. It tests this device to see whether it successfully protects a logical qubit for a given period, and then repeats this process over a large number of virtual devices generated with the same average severity of misalignments. Thus the simulation determines the probability that the logical qubit is indeed protected in these circumstances. By performing such an analysis for devices of different size (i.e. different numbers of data qubits) we determine whether this particular set of noise parameters is within the threshold for fault tolerance -if   Table I specifies the assumed error levels in preparation, control and measurement of the probe qubits. In (a) & (d) we have pure x-y plane displacement, so that the data qubits are ideally located in z. The data in (b) & (e) are for the pillbox distribution shown in the insert -data qubits are located with a circle of radius R in the x-y plane, and with a z displacement ± R 2 . In (c) & (f) the same ratio of 2 : 1 between lateral and vertical displacement is used, but with a normal distribution where R is now the standard deviation. Each data point in the figures corresponds to at least 50, 000 numerical experiments, there were over 10 million performed in total. so, then larger devices will have superior noise suppression. Repeating this entire analysis for different noise parameters allows us to determine the threshold precisely. The results are shown in Fig. 3, and are derived from over ten million individual numerical experiments.
The threshold results for the case of d = 40nm that we consider here are indicated on each plot. These show an extremely generous threshold in the deviation in the positioning of the implanted qubits, with displacements of up to 25nm being tolerable in the best case scenario Fig. 3(d). We note that the continuous motion mode leads to a higher tolerance than the abrupt motion mode, as the smooth trajectory means a lower sensitivity to positional deviations in the x-y plane.

Generalisation to quantum computation
Our simulations have found the threshold for a quantum memory. However surface code based quantum computing proceeds primarily via the same stabilizers which the memory uses (specific information processing operations are achieved, at the most basic level, by choosing not to measure certain subsets of stabilizers in a given time step). Therefore, essentially the same thresholds are relevant to the case of computation. There are a number of additional features that are required of the device in order to support computing. Some aspects, such as the state injection involved in magic state distillation, may be best performed by using operations on individual data qubits rather than the groups of four. This is possible within our constraint that we send only global pulses to our data qubits: control of individual probe spins implies the ability to control individual data qubits.
More generally one can ask about how multiple qubits should be encoded into a large array of data qubits, and what the impact of flaws such as missing data qubits would be on a computation. While a detailed analysis lies beyond the scope of the present paper, approaches such as lattice surgery can offer one simple solution that Figure 4: Example of how one could encode and process multiple logical qubits into a flawed array. Each white circle is a data qubit; each green patch is a subarray representing a single logical qubit. When we wish to perform a gate operation between two logical qubits then we begin making parity measurements at their mutual boundary, according to the lattice surgery approach protocol. If a region of the overall array contains multiple damaged or missing data qubits, we simply opt not to use it (red patches). Note that in a real device the patch structure would probably be several times larger in order to achieve high levels of error suppression.
is manifestly tolerant of a finite density of flaws. The approach is illustrated in Figure 4. Square patches of the overall array are assigned to hold specific logical qubits; stabilizers are not enforced (i.e. parity measurements are not made) along the boundaries except when we wish to perform an operation between adjacent logical qubits. Importantly, if a given patch is seriously flawed (because of multiple missing data qubits during device synthesis, or for other reasons) then we can simply opt not to use it -it becomes analogous to a 'dead pixel' in a screen or CCD. As long as such dead pixels are sufficiently sparse, then we will always be able to route information flow around them.

Practicality of the device
We now discuss the practicality of the proposed device, in light of the robustness to defects that our simulations have established.

Timescales and decoherence
We first direct our discussion to the operational timescales of the device. For a probe -data qubit separation of d = 40 nm as previously mentioned, a total interaction time of t int ≈ 1.2 ms is required to acquire the appropriate phase over all four qubits in a given stabilizer measurement. In the abrupt motion scenario (Fig.  2b),(i)) with negligible transfer times, this would allow operations in the 1 kHz regime. In the continuous circular motion picture, on the other hand, a significant time is required for the probe's transfer between data qubits, slowing down the device by roughly a factor of D/d = 10. It is worthwhile to consider a hybrid mode of operation with slow continuous motion in the vicinity of the data qubits and fast accelarated transfers, as an approach that could provide both fast operation and high positional error tolerance. We furthermore note, that due to the 1/d 3 dependence of the dipolar interaction, every reduction of d by a factor of two allows eight times faster operational frequencies. So, given that manufacturing tolerances continue to improve, this scheme will likely become readily scalable to faster speeds.
To prohibit decoherence of the data qubits from the dipole moment of its neighbours, we suggested a data qubit -data qubit separation D ≥ 10 × d = 400nm. The remaining dipolar interaction is a factor of 10 3 smaller than the probe qubit -data qubit interaction. Assuming that the dipolar interaction with the neighbours is the dominant source of decoherence, the fidelity for retaining the data qubit state unaltered after one stabilizer cycle can approach values > 99.98 %. In practice, D/d may take a range of values and optimising it will be a trade off between the smallest possible fabrication feature sizes, the achievable translation velocities and the decoherence time of the qubits.

Mechanics
The prototypical mechanical system that enables such motions with sub-nanometer positional accuracy is the tip of an atomic force microscope cantilever. In principle, an array of tips on a single cantilever could incorporate the probe qubits and a cyclic motion of the cantilever would allow the four qubit phase accumulation. Practical constraints such as height uniformity of the probe tips, however, impose severe challenges on the scalability of this approach up to larger qubit grid sizes.
A more viable mechanical system could be represented by x-y translation stages realised by micro electromechanical systems (MEMS). These devices are often manufactured from silicon-on-insulator wafers and could exploit the uniformity of the oxide layer to achieve a high homogeneity of the probe -data qubit separations d across the grid. Various designs for MEMS x-y translations stages have been put forward with travel ranges in the µm range or higher [9][10][11] and positional accuracies in the nm regime [12,13]. The translation times of the stages are primarily limited by the eigenfrequency of the stage and designs with frequencies > 10 kHz [14] permit translation times on the order of ∼ 100µs.

Material systems
We next direct our discussion to suitable solid state qubit systems for this orbital probe architecture. At the forefront of solid state spin qubits today are the NV centre in diamond and donor impurities in silicon, while new potential candidates, such as divacancy centers in silicon carbide, continue to emerge. The following paragraphs discuss both the qubit performance of these systems and the suitability of the host material in MEMS applications.
a. Silicon impurities Due to advanced fabrication processes and its excellent material properties, silicon is the predominant material for the realisation of highquality MEMS devices. Furthermore, silicon can be isotopically purified to a high degree, which reduces the concentration of 29 Si nuclear spins and creates an almost ideal, spin-free host system. Consequently, electron spins of donor impurities, such as phosphorous, show extraordinarily long coherence times of up to 2 s [8], thus enabling a very low data qubit memory error probability (sub 0.1%) over the timescale of a single parity measurement of 1.2 ms. Furthermore, initialisation, manipulation and read-out of the electron spin of single phosphorus impurities have been achieved, initially in devices made from natural silicon [15,16] and more recently, with higher performance in 28 Si-enriched epilayers [17]. The average measurement fidelity is reported as 97% with read-out timescales in the order of milliseconds. The single qubit control fidelity in these devices approaches 99.6%, which could be improved even further by the use of composite microwave pulses (99.93% reported in [7]). Furthermore, it was shown in [17] that the decoherence time of the qubit is not significantly affected by its proximity to the interface and can reach values up to 0.56 s with dynamical decoupling sequences. The footprint of the required electronic components to measure a single donor spin in silicon is typically on the order of 200 × 200 nm 2 and is thus small enough to achieve qubit grid separations of D = 400 nm.
In order to increase the read-out fidelity even further the quantum non-demolition character of nuclear spin measurements might be exploited [18]. By the end of a stabilizer cycle, the electron spin state could be transferred to the nuclear spin [19] from which it could then be deduced with measurement fidelities up to 99.99 % [17]. However, this advantage would have to be traded against the cost of longer read-out times (here 250 ms).
As shown by our threshold calculations, a key figure of this scheme is the implantation accuracy for the probe and data qubit grid. For silicon, in all three dimensions atomically-precise (±3.8Å) and reliable phospho-rous donor incorporation has been demonstrated using a hydrogen mask patterned by an STM tip [2,3]. This accuracy is more than an order of magnitude below our calculated thresholds of Fig. 3 and the challenge remaining is to maintain this precision over larger qubit grid sizes.
b. Diamond nitrogen-vacancy centers The electron spin qubit associated with the nitrogen-vacancy (NV) defect center of diamond features optically addressable spin states, which could be manipulated even at room temperature. By using resonant laser excitation and detection of luminescence photons, fast (∼ 40 µs [6])) and reliable (measurement fidelity of 96.3 % [20]) read-out of single NV center spins has been demonstrated. The nuclear spin of 14 N or of adjacent 13 C may again be exploited to enhance the measurement fidelity (99.6 % [21]).The coherence times in isotopically purified diamond samples (T 2 = 600ms at 77 K using strong dynamical decoupling [22]) are long enough to allow ms long stabilizer cycles While the qubit operations possible in NV centers are advanced, so far very few micro-electromechanical devices have been realised using diamond. Among them are resonator structures from single crystalline diamond-oninsulator wafers [23] and from nano-crystalline diamond [24]. In principle though, diamond possesses promising material properties for MEMS applications [25] and, given further research, could become an established material to built translatory stages.
Next, we consider the implantation accuracy: The highest accuracy NV centre creation method reported to date utilizes a hole in an AFM cantilever as a mask for focussed ion beam nitrogen implantation and achieves lateral accuracies of ∼ 25nm at implantation depths of 8 ± 3nm [26]. This precision is only slightly below threshold of this scheme and it is reasonable to hope that new implantation methods could meet the requirements of the scheme in the near future.
Another critical factor for all NV centre fabrication methods is the low yield of active NV centres per implanted nitrogen atom, which is typically well below 30 % [27]. Such low yield would result in a significant number of dead pixels within the grid and improvements of this figure are highly important to the scheme.
While there are still significant challenges remaining to an integrated diamond MEMs probe array, it is encouraging that the basic requirement of our proposed scheme, i.e. the control of the dipolar interaction of two electron spins by means of changing their separation mechanically, has already been achieved. Grinolds et al. were able to sense the position and the dipolar field of a single NV centre by scanning a second NV centre in a diamond pillar attached to an AFM cantilever across it -at a NV centre separation of 50 nm [28].
c. Silicon carbide vacancy defects In addition to NV centres, divacancy defects of certain silicon carbide (SiC) polytypes exhibit optically addressable spin states suitable for qubit operations [29]. Furthermore, SiC micro electromechanical devices [30] and the required fabrication techniques have evolved in recent years, which could open up the possibility of a material with both optical qubit read-out and scalable fabrication techniques. Some important aspects of qubit operation, however, such as longer decoherence times (360 µs reported in [31]), single shot qubit read-out and deterministic defect creation with high positional accuracy have yet to be demonstrated.

Conclusion
We have described a new scheme for implementing surface code quantum computing, based on an array of donor spins in silicon, which can be seen as a reworking of the Kane proposal to incorporate an inbuilt method for error correction. The required parity measurements can be achieved using continuous phase acquisition onto another 'probe' qubit, removing the challenging requirement for direct gating between physical qubits. Through simulations using error rates for state preparation, con-trol and measurement that are consistent with reported results in the literature, we find that this approach is extremely robust against deviations in the location of the qubits, with tolerances orders of magnitude greater than those seen in the origin Kane proposal. An additional benefit is that such a system is essentially scale independent, since the scheme is based on long range dipole interactions, so the dimensions of the device can be selected to match the available fabrication capabilities. 4. Measurement. We select a measurement error rate p m and then a particular outcome of the measurement, q ∈ {0, 1} corresponds to the intended projection P q applied to the state with probability (1 − p m ) and the opposite projection Pq applied with probability p m . This noisy projector can be written: In a refinement of this model, we can enter two different values of p m , one for the cause that j = 0 and one for j = 1. This reflects the reality of many experimental realisations of measurement where, e.g., |1 is associated with an active detection event and |0 is associated with that event not occurring (in optical measurement, the event is seeing a photon that is characteristic of |1 ). Because of the asymmetry of the process, once imperfections such as photon loss are allowed for then the fidelity of measurement becomes dependent on the state that is measured, |0 or |1 .

II. STABILIZERS AS SUPEROPERATOR
To characterise the entire process of the stabilizer measurement we carry out a full analysis of the measurement procedure including all sources of noise noted in the section above, and generate a superoperator from the result to completely describe the action of the stabilizing measurement procedure.
This probabilistic decomposition describes the operation as a series of Kraus operators, K i , applied to the initial state with probabilities p i , which depend on the chosen protocol, noise model and the error rates. The leading term i = 0 will have corresponding K 0 representing the reported parity projection, and large p 0 . For the protocols considered here, the other Kraus operations can be decomposed and expressed as a parity projection with additional erroneous operations applied.
Consider a known deterministic set of phase errors over a 4-qubit stabilizer-by-probe. The probe and data qubits mutually acquire phase through their dipole-dipole interaction. This interaction between probe and single data qubit leads to the following gate where θ = π/2 + δ(x, y, z) is a function of the position of the data qubit. This means that after the probe has passed over one of the four qubits the state of the system is where G = diag{1, i}, S k = diag{1, e iδ k }, S k = diag{e iδ k , 1}, Z = diag{1, −1} and the superscripts {a, b, c, d} label the data qubit on which the operator acts.
After the probe has passed four data qubits, each of which injects some erroneous phase δ i onto the probe qubit, the state of the system is proportional to (5) We want to measure the probe in the |± basis, the result of which will determine our estimate of the parity of the data qubits. Rewriting Equation 5, If measurement of the probe finds it in the |+ then we interpret this as an attempted even parity projection. We neglect for now the unconditional phases G d G c G b G a . The actual projection we have performed on the data qubits iŝ This is clearly not a true parity projection, as different even parity subspaces P Consider the following protocol to smooth out this systematic error in our parity measurement: we randomly select one of four patterns of X operators on the data qubits and apply it before and after theP even projector. We choose from the set U 1 = 1111, U 2 = 11XX, U 3 = 1X1X, U 4 = 1XX1 to smooth the weightings c i and s i of Equation 7.
The action of this protocol on the state ρ of the data qubits is thus, . (8) The operations U i have the effect of permuting the weightings of projecting into the different subspaces in P even . For example U 2P even U 2 has the same form asP even with the weightings redistributed according to the relabelling: 1 ↔ 2, 3 ↔ 4. Expanding out Equation 8 we find 16 'even' terms P odd . We add another level to our protocol, applying (1111) or (ZZZZ) with probability 1/2 to kill off the cross terms.
We then find that it is possible to re-express P smooth even (ρ) as the probabilistic sum of perfect odd and even parity projections, followed by Z errors on either one or two data qubits, P smooth even (ρ) = ω even P even ρP even + Γ 1ZZ1 P 1ZZ1 even ρP 1ZZ1 even + Γ 1Z1Z P 1Z1Z even ρP 1Z1Z even + Γ 11ZZ P 11ZZ where the Kraus operators are each applied with a certain probability. Writing Equation 9 in terms of P where C 1 = 1 4 c 2 1 + c 2 2 + c 2 3 + c 2 4 , C 2 = 1 2 (c 1 c 2 + c 3 c 4 ), Defining C i = cos( δi 2 ) and S i = sin( δi 2 ), the explicit forms of the resulting probabilities expressed as functions of the phase errors δ i are We have thus shown that random application of one of a set of four unitaries before and after an 'imperfect' parity projectionP even can be expressed as a superoperator on the data qubits. This has the form of the probabilistic application of 'perfect' parity projectors followed by pauli-Z errors on subsets of the data qubits. When the phase errors δ i are small the most probable operation is the desired perfect even parity projection P even with no errors. This information on stabilizer performance then enables classical simulation of a full planar code array, and its fault tolerance threshold can be assessed.
The above considers the superoperator for a noisy parity projection in our probabilistic protocol predicated on obtaining the 'even' result when measuring the probe. A similar result can be derived in the case that the probe is measured and found in the |− state.
The erroneous odd parity projection in this case is thuŝ Again randomly applying our four U i allows us to derive a superoperator P smooth odd (ρ) in terms of perfect parity projections and one-and two-qubit Z errors. This takes the form P smooth odd (ρ) = ω odd P odd ρP odd + λ 1ZZ1 P 1ZZ1 odd ρP 1ZZ1 odd + λ 1Z1Z P 1Z1Z odd ρP 1Z1Z odd + λ 11ZZ P 11ZZ odd ρP 11ZZ odd + ζ Z111 P Z111 even ρP Z111 even + ζ 1Z11 P 1Z11 even ρP 1Z11 even + ζ 11Z1 P 11Z1 even ρP 11Z1 even + ζ 111Z P 111Z even ρP 111Z even (13) where the probabilities are given by A similar analysis can be applied to the three-qubit stabilizers which define the boundaries of the planar code. The superoperators for these edge stabilizers are also expressed as perfect P even and P odd projections followed with some probability by one-and two-bit Z errors.