Abstract
The realization of a scalable quantum information processor has emerged over the past decade as one of the central challenges at the interface of fundamental science and engineering. Here we propose and analyse an architecture for a scalable, solidstate quantum information processor capable of operating at room temperature. Our approach is based on recent experimental advances involving nitrogenvacancy colour centres in diamond. In particular, we demonstrate that the multiple challenges associated with operation at ambient temperature, individual addressing at the nanoscale, strong qubit coupling, robustness against disorder and low decoherence rates can be simultaneously achieved under realistic, experimentally relevant conditions. The architecture uses a novel approach to quantum information transfer and includes a hierarchy of control at successive length scales. Moreover, it alleviates the stringent constraints currently limiting the realization of scalable quantum processors and will provide fundamental insights into the physics of nonequilibrium manybody quantum systems.
Introduction
The majority of realistic approaches to quantum information processing impose stringent requirements on the qubit environment, ranging from ultrahigh vacuum to ultralow temperature^{1,2,3}. Such requirements, designed to isolate the qubit from external noise, often represent major experimental hurdles and may eventually limit the potential technological impact of a quantum information processor. For these reasons, developing a realistic framework for a feasible solidstate quantum processor capable of operating at room temperature is of both fundamental and practical importance. Nitrogenvacancy (NV) colour centres in diamond stand out among other promising qubit implementations^{4,5,6,7} in that their electronic spins can be individually polarized, manipulated and optically detected under roomtemperature conditions. Each NV centre constitutes an individual twoqubit quantum register as it also contains a localized nuclear spin. The nuclear spin, which has an extremely long coherence time, can serve as a memory qubit, storing quantum information, while the electronic spin can be used to initialize, readout, and mediate coupling between nuclear spins of adjacent registers. Magnetic dipole interactions allow for coherent coupling between NV centres spatially separated by tens of nanometers. Although, in principle, a perfect array of NV centres would enable scalable quantum information processing, in practice, the finite creation efficiency of such centres, along with the requirements for parallelism, necessitate the coupling of registers separated by significantly larger distances.
Recent advances involving the quantum manipulation of NV defects have allowed researchers to achieve subdiffraction limited resolution, singleshot readout, and dipolecouplingmediated entanglement between neighbouring NV electronic spins^{8,9,10,11,12,13,14,15,16,17,18}. Despite such substantial developments, it remains unclear whether these individual pieces, each of which invariably require a unique set of experimental conditions, can be seamlessly unified into a scalable roomtemperature architecture^{19}. Thus, the development of an architectural blueprint that combines the associated experimental facets, while demonstrating that such a combination can enable highfidelity quantum operations, is of crucial importance.
In what follows, we describe and analyse a feasible architecture for a roomtemperature, diamondbased quantum information processor. Our approach makes use of an array of single NV centres, created through ion implantation and subsequent annealing^{11,20}. To overcome the challenge of coupling remote NV registers, we develop a novel method that enables coherent longrange interactions between NV centres, mediated by an optically unaddressable 'dark' spin chain data bus (DSCB)^{21}. For concreteness, within our architecture, we will consider the specific implementation of such a DSCB by utilizing implanted nitrogen impurities (P1 centres) with spin 1/2, as shown in Fig. 1a^{8,22}. We analyse realistic imperfections and decoherence mechanisms, concluding that the implementation of this architecture is feasible with current experimental technology. Moreover, we demonstrate the possibility of highfidelity remote coupling gates, whose error rates fall below the threshold for quantum error correction in a twodimensional (2D) surface code^{23}.
Results
The NV qubit register
Single NV registers contain a spin triplet electronic ground state (S=1) and can be optically pumped and initialized to the spin state, which has no magnetic dipole coupling with other NV registers or impurities. After optical initialization, the electronic spin of each register remains in the state, unless coherently transferred to the state by a resonant microwave (MW) pulse, as shown in Fig. 1a^{10,11,12,13}. The NV nuclear spin associated with nitrogen atoms (I=1/2 for ^{15}N) possesses an extremely long coherence time (^{13}C nuclear spins could also in principle be utilized) and will serve as the memory qubit in our system^{24,25}; manipulation of the nuclear spin is accomplished with radio frequency (RF) pulses^{26}. The Hamiltonian governing the electronic and nuclear spins of the NV register is
with zerofield splitting Δ_{0}=2.87GHz, electronic spin gyromagnetic ratio μ_{e}=−2.8 MHz/Gauss, nuclear spin gyromagnetic ratio μ_{n}=−0.43 kHz/Gauss, and hyperfine coupling A=3.0 MHz^{10}. The application of a magnetic field along the NVaxis () ensures full addressability of the twoqubit system, resulting in the energy levels shown in Fig. 1a. A universal set of twoqubit quantum operations can easily be achieved with only MW and RF controls, as shown in Fig. 1b and detailed in Methods^{26}.
Furthermore, it is possible to selectively readout the state of the NV register; for example, to readout the nuclear qubit of a register, we apply a C_{n}NOT_{e} gate to couple the electronic and nuclear spins, thereby allowing for readout of the electronic spin based on fluorescence detection. Our approach to scalability will ultimately involve a hierarchical design principle that ensures a spatial separation between NV registers, which is of order the optical wavelength; while this will, in principle, enable individualized readout, additional use of a red LaguerreGaussian donut beam can further enhance readout fidelities^{14}. Indeed, the readout of individual registers may be complicated by the strong fluorescence background from neighbouring NV centres. To suppress this background fluorescence, a red donut beam can be used, with its minimum located at the particular NV centre being readout^{14}. Whereas the fluorescence signal from the NV register located at the minimum persists, the remaining illuminated registers will be dominated by the stimulated emission induced by the red donut beam. In addition to suppressing the background noise, the red donut beam can also suppress the nuclear decoherence of the remaining NV registers, by reducing the amount of time these registers spend in the electronicexcited state (Supplementary Note 1)^{27}. Moreover, this approach may be particularly applicable in the case where NV registers are separated by subopticalwavelength distances. After each round of fluorescence detection, the electronic spin is polarized to the state, while the I_{z} component of the nuclear spin, a quantum nondemolition observable, remains unchanged^{28}. Therefore, it is possible to repeat this readout procedure multiple times to improve the readout fidelity^{15,16}. A strong magnetic field Tesla along the NV axis should be used to decouple the electronic and nuclear spins to achieve highfidelity singleshot readout of NV registers^{16}. In addition to subwavelength readout, optical donut beams also introduce the possibility of selectively manipulating individual NV registers with subwavelength resolution. In this case, we envision the use of a green LaguerreGaussian donut beam; whereas unilluminated NV centres may respond to a resonant MW pulse, illuminated registers undergo a strong optical cycling transition that suppresses their response to MW pulses due to the quantum Zeno effect^{29,30}.
Approach to scalable architecture
One of the key requirements for faulttolerant quantum computation is the ability to perform parallel gate operations. In our approach, this is achieved by considering a hierarchy of controllability. The lowest level of the hierarchy consists of an individual optically addressable plaquette with horizontal and vertical spatial dimensions ∼100–500 nm, containing a single computational NV register, as shown in Fig. 2a. The plaquette dimensions are chosen such that register control and readout can be achieved using conventional farfield or subwavelength optical techniques^{10,14,24,30,31}. The second level, termed a superplaquette (∼10 μm ×10 μm), consists of a lattice of plaquettes whose computational registers are coupled through DSCBs. At the highest level of the hierarchy, we consider an array of superplaquettes, where individual superplaquettes are controlled by confined MW fields^{32}. In particular, microsolenoids can confine fields to within superplaquettes, allowing for parallel operations at the superplaquette level. For example, as shown in Fig. 2, independent MW pulses can allow for simultaneous operations on the electronic spins of all computational NV registers within all superplaquettes. To control registers at the superplaquette boundaries, we define a dual superplaquette lattice (Fig. 2a). Localized MW fields, within such a dual lattice, can provide a smooth transition between the boundaries of neighbouring superplaquettes.
Taking advantage of the separation of length scales inherent to optical control and MW confinement provides a mechanism to achieve parallelism; indeed, the hierarchical control of plaquettes, superplaquettes and superplaquette arrays allows for simultaneous single and twoqubit gate operations, which are fundamental to faulttolerant computation. One of the key differences between the currently proposed architecture and previous proposals^{6,33}, is that the design here does not rely on optically resolved transitions, which are only accessible at cryogenic temperatures.
The required 2D array of NV centres can be created via a twostep implantation process. We envision first implanting single nitrogen atoms along particular rows within each plaquette, as shown in Fig. 2b; subsequent annealing occurs until the creation of an NV centre, after which, a second nitrogen implantation step generates the spin chain data bus. The selective manipulation of individual registers within our 2D array is enabled by the application of a spatially dependent external magnetic field ; this 1D magnetic field gradient is sufficiently strong (α∼10^{5} T m^{−1}) to allow for spectroscopic MW addressing of individual NV registers, each of which occupies a unique row in the superplaquette, as shown in Fig. 2b^{32,34,35}.
Dark spin chain data bus
To coherently couple two spatially separated NV centres, we consider two distinct approaches. First, we consider an approach, which is appropriate for spinstate transfer along the direction of the magnetic field gradient, in which individual addressing of spins is possible. This allows for an adiabatic sequential swap between neighbouring qubits and, consequently, between the ends of the chain. Alternatively, in the situation where individual addressing of spins is not possible (that is, in the direction transverse to the field gradient), we show that global control pulses achieve effective Hamiltonian evolution, which enables quantum state transfer through the spin chain. In both cases, we show that perfect state transfer and remote coupling gates are possible even when the intermediate spin chain is completely unpolarized (infinite spin temperature).
We begin by analysing the adiabatic sequential swap in a spin1/2 chain. This approach is suitable to couple registers in plaquettes that are vertically adjacent, relying on the individual addressability of qubits and utilizing the magnetic dipole coupling between spinchain elements. As shown in the Supplementary Methods, under the secular approximation, the magnetic dipole coupling between a pair of neighbouring spins can be reduced to Ising form
where κ is the relevant component of the dipole tensor, ω_{0} captures the electronic Zeeman energy, and δ_{i} characterizes both the hyperfine term (nuclear spin dependent) and the magnetic field gradient. From the Ising Hamiltonian, an XX interaction between qubits can be distilled by driving with , leading to (under the rotating wave approximation, in the rotating frame, with , and in a rotated basis with (x,y,z)→(z, −y, x))
The spinflip process in H_{int} is highly suppressed in the limit of , whereas the same process is dominant in the case of . Hence, by slowly ramping the Rabi frequencies Ω_{1} and Ω_{2} through one another, adiabatic swap of the quantum states of the two impurities can be achieved through rapid adiabatic passage, as shown in Fig. 3a. Generalizing to arbitrary length spin chains yields , whereby the sequential adiabatic swap of quantum states along the spin chain can be achieved by successively tuning individual Rabi frequencies across one another. During the adiabatic swap of a single pair of spins, higher order interactions, such as those resulting from nexttonearest neighbours, will be suppressed because of the differences in Rabi frequencies. By including the magnetic dipole coupling between the electronic spin of the NV register and the spin chain quantum channel, we arrive at an effective mixed spin chain with the DSCB connecting the two electronic spins of the vertically separated NV registers.
Crucially, such an adiabatic sequential swap is robust against variations in the coupling strength κ, which can be induced by the imprecise implantation of impurities that form the spin1/2 chain; in particular, even for the case of varying κ_{i,i+1}, perfect adiabatic swap occurs so long as the rate at which Ω_{i} and Ω_{i+1} are ramped through one another is sufficiently small. Within the proposed architecture, the impurities forming the horizontal spin chain will not induce operational errors during the vertical adiabatic sequential swap as the design principle allows for selective spin echoing (Fig. 5).
Next, we consider a second method, termed freefermion state transfer (FFST) developed in ref. 36, to coherently couple NV registers in the horizontal direction. In contrast to the adiabatic sequential swap, the method utilizes only global control over dark impurities and effective Hamiltonian evolution. The relaxation of the requirement of individual control over elements of the dark spin chain renders this second method, applicable for coherent coupling between NV registers in horizontally adjacent plaquettes, transverse to the direction of the field gradient. In particular, the protocol achieves coherent coupling through an unpolarized, infinitetemperature spin chain, employing purely Hamiltonian evolution under
as shown in Fig. 3b. This Hamiltonian, obtained in a similar fashion equation (3), results in coherent interactions between NV centres, which is best understood through an analogy with eigenmode tunneling in a manybody system. Specifically, the spin chain described by H_{FFST} can be viewed as a system of noninteracting fermions. As described in ref. 36, by tuning the NV centres into resonance with a single fermionic eigenmode, an effective threestate system can be realized. Mediated by this fermionic eigenmode, the electronic states of two remote NV centres can be coherently swapped. Coupled with arbitrary twoqubit gates between the nuclear and electronic spin (Fig. 1b; Methods), an electronic swap gate enables universal computation between spatially remote nuclear spin memories, as shown in Fig. 3c. Crucially, such a swap gate is insensitive to the polarization of the intermediate dark spins and highfidelity quantum state transfer can be achieved, provided that the fermionic mode is delocalized and that the coupling, g, of the NV qubit to the spin chain is controllable. As detailed in the Methods, by utilizing the threelevel NV groundstate structure (Fig. 3b), it is possible to fully control the NVchain coupling. This tunability also ensures that FFST is fundamentally robust to experimentally relevant couplingstrength disorder, which could be induced by implantation imprecision. Indeed, by separately tuning the NVchain coupling on either side of the DSCB, it is possible to compensate for both disorderinduced asymmetry in the fermionic eigenmode as well as altered eigenenergies (Supplementary Methods)^{36,37,38}.
Implementation, operational errors and gate fidelities
The specific implementation of the DSCB can be achieved with implanted nitrogen impurity ions. Dipole coupling between neighbouring nitrogen electronic spins forms the DSCB, whereas dipole coupling between the NV and nitrogen electronic spins forms the qubit–DSCB interaction; nonsecular terms of this magnetic dipole coupling are highly suppressed owing to the spatially dependent external magnetic field B_{z}(y), resulting in the effective interaction found in equation (2). In addition, the nitrogen impurities possess a strong hyperfine coupling, the principal axis of which can take on four possible orientations due to tetrahedral symmetry^{39,40,41}. Dynamic Jahn–Teller (JT) reorientation of the nitrogen impurity's hyperfine principal axis results in two particular considerations, namely the addressing of additional JT frequencies yielding a denser superplaquette frequency spectrum and the JTgoverned spinlattice relaxation time (Supplementary Methods). As is characterized by an Arrhenius rate equation^{40} at ambient temperatures, a combination of a static electric field and slight cooling by ∼50 K allows for a substantial extension of the relaxation time to ∼1 s; hence, in the following consideration of operational errors, we will assume that we are limited by , the spinlattice relaxation time of the NV centre.
We now consider various imperfections, which may introduce operational errors. In particular, we consider the errors associated with the sequential swapmediated coupling between vertically adjacent registers and the FFST between horizontally adjacent registers. We begin by discussing the analytic error estimate associated with each method, after which, we summarize the results of full numerical simulations (Supplementary Methods).
First, we consider the accumulated infidelity associated with the adiabatic sequential swap,
The first term, , represents offresonant crosstalk induced by MW manipulations with Rabi frequency Ω_{i}. Here Δ_{g} characterizes the gradientinduced splitting achieved within the superplaquette frequency spectrum (Supplementary Methods). The second term, p_{adia}, corresponds to the nonadiabatic correction resulting from an optimized adiabatic ramp profile^{42,43,44}. The third term, , is directly obtained from equation (3) and corresponds to additional offresonant errors. The fourth error term, corresponds to the depolarization error induced by the finite NV T_{1} time, while the final error term, corresponds to the infidelity induced by dephasing. As each error term is considered within the context of a single adiabatic swap, the total error contains an additional factor of N, representing the chain length, which is plaquette size dependent.
We can similarly consider the accumulated infidelity associated with FFST,
In direct analogy to , the first term in corresponds to the excitation of an NV register by offresonant MW fields. The second term, p_{f}, corresponds to the undesired coupling with offresonant fermionic modes. Since the coupling strength is characterized by ^{36}, while the splitting of the eigenenergy spectrum ∼κ/N, such an offresonant error induces an infidelity . The third error term, p_{g}, results from the protocol designed to control, g, the NVchain coupling (see Methods for details). Directly analogous to , the fourth and fifth terms correspond to errors induced by the operational time, t_{FFST}, which causes both depolarization and dephasing.
Finally, we perform numerical simulations, taking into account the nitrogen JT frequencies, to characterize the infidelity of both the adiabatic sequential swap and FFST within the NV architecture, as shown in Fig. 4. The results of these calculations are in excellent agreement with the above theoretical predictions. In particular, these simulations reveal that, for sufficiently long , operational infidelities in both DSCB methods can be kept below 10^{−2}.
These simulations clearly show that the T_{1} time of the NV electronic spin is of critical importance in obtaining highfidelity quantum operations. While at room temperature, T_{1} appears to vary depending on the particular sample and on the specific properties of the local NV environment, such as strain, values on the order of 10 ms are generally obtained^{24,39}. However, the spinlattice relaxation mechanism governing T_{1} is most likely related to an Orbach process^{45,46}, which is strongly temperature dependent. In such a case, modest cooling of the sample by ∼50 K, is likely to extend T1 by more than an order of magnitude, thereby making highfidelity gates possible.
Given that such numerical estimations suggest the possibility of achieving highfidelity twoqubit operations between remote NV registers, the proposed architecture seems well suited to the implementation of topological quantum error correction. In particular, we imagine implementing a 2D surface code, which requires only nearestneighbour twoqubit gates and singlequbit preparation and measurement^{23,47,48}. Recent progress in optimizing this surface code has yielded an error threshold of ɛ∼1.4% (ref. 49), which is above the estimated infidelity corresponding to both the adiabatic sequential swap and FFST; thus, in principle, implementation of such a 2D surface code can allow for successful topological quantum error correction, and hence, fault tolerant quantum computation^{50}.
Discussion
The above considerations indicate the feasibility of experimentally realizing a solidstate quantum computer capable of operating under ambient conditions at or near room temperature. We emphasize that a majority of the elements required for the realization of individual qubits in our architecture have already been demonstrated. In our approach, these techniques are supplemented by both a new mechanism for remote register coupling between NV centres as well as a hierarchical design principle, which facilitates scalability. The remote coupling mechanisms discussed can naturally be implemented via nitrogen ion implantation in ultrapure diamond crystals and are robust to realistic imperfections and disorder^{36}. Moreover, single errors during quantum state transfer are localized to individual transport channels and do not have a propagating effect on the remaining computation.
While we hope that the proposed architecture evinces the feasibility of room temperature quantum information processing, the implementation and integration of the various proposed elements still require significant advances in areas ranging from engineering to materials science. Crucially, recent results have demonstrated substantial progress towards overcoming challenges such as the optimization of planar microcoil arrays^{51,52} and efficient beam steering in micromirror systems^{53}. Furthermore, by eliminating requirements for cryogenic temperatures, our blueprint aims to make the realization of a scalable quantum computer significantly more practical.
The present work opens a number of new directions that can subsequently be explored. In particular, although we have considered the direct errors associated with DSCBmediated coupling, it is instructive to note that the fidelity of such quantum gates can often be significantly improved, using techniques from optimal control theory^{54,55}. For example, such methods of optimal control, while negating the detrimental effects of decoherence, can also simultaneously allow for the implementation of highfidelity gates, despite both frequency and coupling disorder as induced by ion implantation errors. Indeed, the ability to precisely guide the quantum evolution via optimal control, even when the system complexity is exacerbated by environmental coupling, provides an alternative solution to improve single and twoqubit gate fidelities^{56}. In addition, it is well known that the local strain field surrounding each NV centre can significantly alter the register's properties; hence, through a detailed understanding of electric field induced strain, it may be possible to improve the coherence properties of the qubit. Moreover, the long coherence times of individual P1 centres, each of which harbours an associated nuclear spin, suggest the possibility of utilizing these dark spins as computational resources in and of themselves^{57}. Beyond these specific applications, a number of scientific avenues can be explored, including, for example, understanding and controlling the nonequilibrium dynamics of disordered spin systems.
Methods
Controlling qubitchain coupling in the NV architecture
To achieve an effective Hamiltonian of the form given by equation (4), it is essential to control the coupling strength between the NV register and the neighbouring impurity. Here we utilize the three levels of the NV electronic spin^{58} to effectively control g, as shown in Fig. 3b, whereby the Hamiltonian (under MW driving) can be written as
where Ω represents the Rabi frequency on the NV register, Δ represents the associated detuning, and Ω_{N} represents the Rabi frequency on the nitrogen impurity. In this case, as the NV twophoton detuning is zero, it is convenient to define bright and dark states, and ; further, in the resulting twolevel picture, the associated dressed states are and , in the limit . Hence, rewriting the Hamiltonian in this limit yields
where correspond to the two eigenstates of the nitrogen impurity. The coupling term can be further reexpressed as
Thus, by working within the NV subspace {D〉,−〉}, it is possible to completely control the coupling between the NV register and Nitrogen impurity, , by tuning the Rabi frequency and detuning. It is possible to work in the required twostate subspace by ensuring that and hence, that the +〉 state remains unpopulated, with corresponding offresonant error κ^{2}/Δ^{2}.
Furthermore, we evince a possible scheme to coherently map the quantum information that is stored in the nuclear memory into the desired electronic subspace. For example, consider mapping 0〉(α↑〉+β↓〉) to (α−〉+βD〉)↑〉, where the first (tensor) factor corresponds to the electronic state and the second corresponds to the nuclear state of a single NV. The proposed mapping can be achieved in a twostep process. First, by simultaneously performing a π−pulse on the transitions 0〉 ↓〉→−1〉↓〉 and 0〉↓〉→1〉↓〉 with oppositely signed Rabi frequencies, one can map 0〉↓〉 to D〉↓〉. Next, one utilizes an RF pulse to flip the nuclear spin, which yields D〉↓〉→D〉↑〉. Finally, turning Ω on in an adiabatic fashion ensures that the state preparation populates only D〉 and −〉, thereby mapping the quantum information into the desired electronic subspace.
Arbitrary twoqubit gates within the NV register
Whereas the DSCB enables longrange quantum logic between spatially separated NV electronic spins, universal twoqubit gates between the nuclear spin quantum memories require additional local logic between nuclear and electronic spins, as outlined in Fig. 1b. Here we specify, in detail, the implementation of such local logic gates. A strong MW pulse (with ) can perform an arbitrary electronic spin rotation independent of the nuclear spin state. This can be mapped to a nuclear spin rotation by implementing a swap gate between the nuclear and electronic spins. Such a swap gate can be decomposed into three controlledNOT gates: C_{e}NOT_{n} C_{n}NOT_{e}C_{e}NOT_{n}. A C_{e}NOT_{n} gate can be accomplished by utilizing an RF πpulse (with ), which flips the nuclear spin conditioned on the electronic spin being in 1〉_{e}. Finally, a hyperfinedriven controlledphase gate enables the remaining C_{n}NOT_{e} gate (up to single Hadamards on the electronic spin). This demonstrates that a universal set of local twoqubit gates between NV electronic and nuclear spins can easily be achieved with only MW and RF controls^{26}. Such arbitrary local logic is crucial to enable the preparation of nuclear spin quantum information and forms an important portion of DSCBmediated remote coupling (Fig. 3c).
Specific implementation of architecture
In this section, we offer a specific implementation of the architectural design principle and discuss the various ingredients required to achieve DSCBmediated coherent coupling between spatially separated NV registers. In particular, we consider the refocusing of nonnearest neighbour interactions along the horizontal spin chain.
The effective Hamiltonian evinced in equation (4) has nearest neighbour form; although nextnearestneighbour interactions will represent a correction, we show that such interactions can be refocused within the current architectural design, and further, that, in principle, interactions beyond nextnearestneighbour can also be refocused. In particular, the horizontal spin chain (N total spins) is arranged in a staggered sawtooth fashion, as shown in Fig. 5. Within such an architecture, nearest neighbour coupling terms correspond to all pairs of adjacent spins, each separated by ∼20 nm, with corresponding interaction Hamiltonian
where the sum runs over all nearest neighbour pairs in a given dark spin chain. Thus, nexttonearest neighbour terms for each spin correspond to the subsequent strongest interaction
where the prime denotes the next link in the sawtooth chain as shown in Fig. 5. In addition to the impurity spins, FFST incorporates the electronic spin of the NV register into a mixed spin chain. It is important to note that the spinflip Hamiltonians H_{N} and H_{NN} are derived from the secular approximated Ising coupling by the application of driving fields as per equation (3). As each row (1,2,3,4) is separately addressable by virtue of the magnetic field gradient (applying four frequencies per row to ensure that all JT and nuclear spin states are addressed), it is possible to apply a spinecho procedure to refocus the nextnearestneighbour terms. In particular, by flipping the spins in rows 1 and 2 (Fig. 5) after time T_{d}/2, where T_{d} is a small fraction of the desired evolution duration, the nextnearestneighbour interactions are refocused since each term contains spins from only row 1 or 2. However, half of the nearest neighbour interactions are also refocused, leaving effective evolution under the Hamiltonian Analogously, by flipping the spins in rows 2 and 3, effective evolution under the Hamiltonian is achieved, again with H_{NN} refocused. Combining the evolution according to H_{eff1} and H_{eff2} yields the desired nearestneighbour Hamiltonian with nexttonearest neighbour interactions refocused. However, as H_{eff1} and H_{eff2} do not commute, it will be necessary to employ piecewise evolution according to the Trotter–Suzuki formalism^{59}. Further refocusing of higher order nonnearest neighbour interactions can also be achieved by extending the number of rows corresponding to the sawtooth design; such an extension allows for the isolation of each specific pair of nearest neighbour interactions, thereby achieving the desired nearestneighbour evolution through a Trotter sequence.
Additional information
How to cite this article: Yao, N. Y. et al. Scalable architecture for a room temperature solidstate quantum information processor. Nat. Commun. 3:800 doi: 10.1038/ncomms1788 (2012).
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Acknowledgements
We gratefully acknowledge conversations with G. Goldstein, J. Maze, E. Togan, Y. Chu, J. Otterbach, Z.X. Gong, L.M. Duan, C. Laumann, C. Mathy, A. Zhai, J. Preskill, N. Schuch and Y.T.Siu. This work was supported by the NSF, DOE (FG0297ER25308), CUA, DARPA QUEST, AFOSR MURI, NIST, the DFG within SFB631 and the Nano Initiative Munich (NIM), the Lee A. DuBridge Fellowship and the Sherman Fairchild Foundation.
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Affiliations
Physics Department, Harvard University, Cambridge, Massachusetts 02138, USA.
 N.Y. Yao
 , A.V. Gorshkov
 , P.C. Maurer
 & M.D. Lukin
Institute for Quantum Information, California Institute of Technology, Pasadena, California 91125, USA.
 L. Jiang
 & A.V. Gorshkov
MaxPlanckInstitut für Quantenoptik, HansKopfermannStrasse 1, Garching, D85748, Germany.
 G. Giedke
 & J.I. Cirac
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Contributions
N.Y. Y., L.J. and A.V.G. contributed equally to this work. All authors contributed extensively to all aspects of this work.
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The authors declare no competing financial interests.
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Correspondence to N.Y. Yao.
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