Gates for the Kane Quantum Computer in the Presence Of

Home , Kane quantum computer

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Charles . 3 × × 10 10 − 1, 5 − ∗ eas opt h dlte fZ ,Sa,and Swap, X, Z, of fidelities the compute also We . ia, /- 4 n s-hn Goan Hsi-Sheng and rhtcue[]i ie ee h aeacietr con- architecture Kane The here. given is [3] architecture aeadcnrle ae nsbeto VB Finally, V. B. IV Section swap subsection in the in drawn and are gates A, conclusions Z IV in- controlled subsection These and in gate IV. gate qubit Section CNOT Two the in C. shown clude III subsection subsec- are in in gates rotations rotations non-adiabatic X Z and Sec- B A, III III including tion subsection gates, system. qubit in one the evolution for for free results use the equation presents we III master which tion the decoherence of present model and simple the describe computation. quantum comparable tolerant or fault than for less required is that analyzed that de- to gates find spin the We in of architecture. error Kane values the the typical in for expected equation phasing master sim- We the computing. ulate quantum tolerant fault for threshold adiabatic corresponding sim- the faster, than scheme. much fidelity is higher which it and [34], construct pler sequence to pulse single able a are construct. in we to schemes qubit gates here non-adiabatic CNOT two analyzed Using adiabatic gate arbitrary three swap an require to the would create example, times For to non-adiabatic three directly. possible using gate to is gate, up it qubit applied adiabatic two schemes the arbitrary be an whereas a to create off addition, required run In is be easily CNOT could cycle. gates clock these adiabatic digital to of contrast timing In the applied. voltages gates, turn- the simply gates off but or These shapes, on pulse ing [34]. complicated on gate rely adiabatic not do the potentially and than gates faster, fidelity The simpler, higher are gate. paper CNOT this adiabatic in the analyzed for [35] al et Fowler The pulse non-adiabatic 33]. use [32, here [34]. has presented schemes system Refs. gates the qubit in of two analytically modeling on investigated stochastic fluctuations and been Voltage gate ‘A’ for qubits. 31] the individual [30, (NMR) of Resonance rotations Magnetic Nuclear in used re nrdcint h aeqatmcomputing quantum Kane the to introduction brief A hsppri raie sflos nScinI we II Section In follows. as organized is paper This error the to analyzed gates the of each compare We of that to analogue direct a is here given analysis The lrn unu computation. quantum olerant uhWls yny S 02 Australia 2052, NSW Sydney, Wales, outh eso htteenmrclresults numerical these that show We I H ATREQUATION MASTER THE II. dqatmgtsfrtesolid-state the for gates quantum ed h eoeec,w n htthe that find we decoherence, the f 2, † hysics, ing ‡ 2 sists of P donor atoms embedded in Si. The orientation Description Term Value Unperturbed Hyperfine 3 of the nuclear spin of each P donor represents one qubit. A 0.1211 × 10− meV When placed in a magnetic field applied in the z di- Interaction Hyperfine Interaction −3 rection, Zeeman splitting occurs. This is given by the Az 0.0606 × 10 meV During Z Rotation Hamiltonian Hyperfine Interaction −3 Ax 0.0606 × 10 meV during X Rotation H = g µ BZ + µ BZ , (1) B n n n B e Constant Magnetic Field − B 2.000T Strength where Z is the Pauli Z matrix, and the subscripts e (n) Rotating Magnetic Field Bac 0.0025T indicate electronic (nuclear) spin. The magnetic field, Strength B, may be controlled externally. The application of a Hyperfine Interaction −3 AU 0.1197 × 10 meV resonant rotating magnetic field adds the following terms during Interaction to the spin Hamiltonian: Exchange Interaction JU 0.0423 meV during Interaction Hac = gnµnBac [Xn cos(ωact)+ Yn sin(ωact)] − +µBBac [Xe cos(ωact)+ Ye sin(ωact)] . (2) TABLE I: Typical parameters used for numerical calculations.

The electronic spins couple to their corresponding nuclear spins via the hyperfine interaction, for P : Si at T =1.4K in isotopically enriched 28Si. Chiba and Harai [16] have also measured the electronic decoher- HA = Aσe σn, (3) · ence times of P : Si, finding a rate of T2e = 100µs. For 29 where A is the strength of the interaction. The design the nucleus, recent results for the nuclear spin of a Si of the Kane quantum computer calls for control of the nucleus show a maximum value of T2n = 25s [36]. strength of the hyperfine interaction externally by ap- Recently Tyryshkin et al [18] obtained an experimen- plying appropriate voltages to ‘A’ gates. The electronic tal measurement of T2e = 14.2ms at T = 8.1K and spins couple to adjacent electrons via the exchange inter- T2e = 62ms at T = 6.9K for a donor concentration of action 0.87 1015cm−3 in isotopically pure Si. At millikelvin temperatures× the decoherence time is likely to be even

HJ = Jσe1 σe2 , (4) longer. Additionally these measurements were carried · out in a bulk doped sample, and in our case we will where e1 and e2 are two adjacent electrons, and J is the be considering a specifically engineered sample. Inter- strength of the exchange interaction which may be con- actions such as exchange and dipole-dipole interactions trolled externally through the application of voltages to contribute to the coupling between electrons. In some ex- the ‘J’ gates. periments, such as in Ref. [18], these potentially benefi- Altogether the spin Hamiltonian of a two donor system cial coupling have been treated as sources of decoherence, is given by but in the operation of a quantum computer these interactions can be either be decoupled or used to generate 2 entanglement, useful for quantum computation. Hence, Hs = HBi + HAi + HJ + Haci . (5) it is expected that T2e may even be longer than those re- Xi=1 ported in [18]. Nevertheless, we use the value of 60ms as a The times and fidelities of the gates naturally depend conservative estimate for electronic dephasing time. We on exactly which parameters are used to calculate them. expect the nuclear dephasing times to be several orders For many of the gates in this paper the typical parameters of magnitude bigger than electronic dephasing times. We shown in Table I were used. These parameters are similar choose the following parameters to be typical of P : Si the to the parameters used for the pure state calculations in systems we are considering: [34].

A simple model of decoherence was used for these cal- T2e = 60ms, (6) culations. There are many diﬀerent decoherence mech- T2n = 1s. (7) anisms, but our model only considers pure dephasing (without engergy relaxation). Whereas dephasing is certainly not the only source of decoherence, it likely to The typical errors presented in the tables contained in be the dominant eﬀect on a time scale shorter than the the next three sections are evaluated at these typical de- energy relaxation (dissipation) time, T1. For example, phasing times.

Feher and Gere [12] measured T1n > 10 hours for nu- The simple decoherence model we consider corresponds clear spin at a temperature of T =1.25K, B =3.2T and to the master equation

T1e 30 hours under similar conditions. In contrast, ≈ experimentally measured times for T2 have been much i ρ˙ = [Hs,ρ] [ρ], (8) shorter. Gordon and Bowers [11] measured T2e = 520µs −~ − L 3 where the dephasing terms are given by In the rotating frame, the master equation has the solution 2 −4ΓN t [ρ] = Γe [Zei , [Zei ,ρ]] + Γn [Zni , [Zni ,ρ]] (9) ρ00(0) ρ01(0)e L ρ(t)= −4ΓN t . (17) Xi=1 ρ10(0)e ρ11(0)

Characteristic dephasing rates, Γ2e and Γ2n are related This has the effect of exponentially decaying the off di- to the dephasing rates by the equations: agonal terms of the density matrix, but leaves the diago- 1 nal components unchanged. For a single isolated nuclear T2e = , (10) spin, the simple model has no effect on eigenstates of 4Γe 1 Zn (i.e., there is no relaxation process for these states). T2n = . (11) In contrast, it has a dramatic influence on superposition 4Γn states whose off diagonal terms decay exponentially (i.e., We define fidelity (and therefore error) in terms of the dephasing). Two such states are actual state after applying an operation, ρ, and the in- 1 tended state after that operation, ρ′. Due to systematic + = ( 0 + 1 ) , errors and decoherence these states will not necessarily | i √2 | i | i ′ 1 be the same. When comparing against a pure state ρ , = ( 0 1 ) . the fidelity F of an operation is defined as |−i √2 | i − | i F (ρ,ρ′) = Tr (ρρ′) ; (12) We can easily calculate the expectation value of the Pauli X matrix, X , for the + state Error is defined in terms of fidelity h i | i X = Tr (Xρ) , (18) E(ρ,ρ′)=1 F (ρ,ρ′). (13) h i − = exp( 4ΓN t). (19) − Typically we would like to know the greatest error pos- For a single nuclear spin coupled to an electronic spin sible for any input state. This is a computationally dif- via the hyperfine interaction the Hamiltonian is given by ficult problem. In the results which follow, the approach taken is to calculate the fidelity for each of the computational basis states, and each of the input states which Hs = HB + HA. (20) would ideally generate a Bell state. This has two main benefits. The first is that a high fidelity indicates that We assume that electron is initially polarized by the large the gate is successfully creating or preserving entangle- magnetic field, B. The evolution of this Hamiltonian ment. The second is that Bell states are superposition was calculated for their typical values [given in Eq. (6), states, which are susceptible to dephasing. We also calcu- Eq. (7) and Table I]. The fidelity after different times lated the effect of each gate on the four Bell input states is shown in Fig. 1. This figure shows the Bloch sphere for the CNOT gate. For typical parameters, Bell input radius, given by states give similar fidelities to those shown in this paper. 2 2 2 Throughout this paper we will use the states 0 and r = X + Y + Z . (21) | i ph i h i h i 1 to represent the nuclear spin up and spin down states A pure state has a radius of one, and a radius of less than |respectively.i We will use the and to represent | ↑i | ↓i one indicates a mixed state. An initial state of + electronic spin up and spin down states respectively. was used. The radius decays at a rate governed by| ↓ thei

nuclear decoherence time, which in this case is T2n = 1s. III. ONE QUBIT GATES As is expected, this decay is the same as the well known solution to the Bloch equations. A. Free Evolution B. Z Rotations The spin of an isolated nucleus undergoing Larmor precession in the presence of a magnetic field, B [37]. In this Z rotations on the Kane architecture may be performed case it is easy to solve the master equation with dephas- by varying the Larmor precession frequency of a single ing exactly. Considering only the nuclear spin, we have qubit [34]. Graphs showing the error in the Z gate at different dephasing rates are shown in Figs. 2 and 3. Figure 2 shows the error at different rates of decoher- Hs = gnµnBZn. (14) ence for both electrons, Γe, and nuclei, Γn, for a single The decoherence terms has only the single term qubit in the 0 state. The calculated error does not significantly depend| ↓ i on the electronic or nuclear dephas- [ρ] = Γn[Zn, [Zn,ρ]], (15) ing rate. The pure 0 state is not affected by decoher- L |↓ i = 2ΓN (ρ ZnρZn). (16) ence terms in the master equation. Therefore the only − 4

0 10 0.8

−2 10 0.6

Error −4 10 Radius 0.4

−6 10 0.2

0 0 10 10 Γ −1 0 (s ) Γ (s−1) 0 0.5 1 1.5 2 2.5 E N Time (s) FIG. 2: Error in the Z gate for | ↓ 0i initial state at diﬀering FIG. 1: Eﬀect of dephasing on the free evolution of a single rates of dephasing. embedded P atom in the Kane architecture.

State Systematic Error Typical Error 6 6 6 −5.5 |0i 3.8 × 10− 3.8 × 10− 10 10 6 6 |+i 1.9 × 10− 1.9 × 10− 10−4.5 10−1 −6 −6 Maximum 3.8 × 10 3.8 × 10 4 −2 10 10 10−3.5 )

TABLE II: Summary of Z gate error. −1 2 10

(s −4 −3 E 10 10 Γ −2.5 −1.5 0 10 10 effect of dephasing occurs when the hyperfine interaction 10 10−5 rotates this state. The effect of dephasing on this state 10−0.5 −2 is negligible. 10 The error in the Z operation for an initial state of 0 , −6 |↓ i −4 is primarily due to systematic error of 3.8 10 . This 10 −4 −2 0 2 4 6 × 10 10 10 10 10 10 error is due to the hyperfine interaction coupling between Γ (s−1) electrons and nuclei. This allows a small probability of N finding the electrons in an excited state. At typical rates of decoherence for the short duration of a Z rotation (ap- FIG. 3: Contour plot showing the error in the Z Gate for | ↓ +i initial state evolving at differing rates of dephasing. proximately 21ns), systematic error is the dominant ef- Contour lines connect, and are labelled by, points of equal fect. For the 0 state, the typical error is 3.8 10−6. |↓ i × error. In the previous section we noted that superposition states, + and , are affected by dephasing terms more than|↓ eigenstatesi |↓−i of Z. This is illustrated in Fig. 3. Electronic dephasing times have little effect on the C. X Rotations overall fidelity. As the nuclear dephasing rate increases, the fidelity decreases, with a maximum error of approx- X rotations are performed using a resonant magnetic imately 0.5. This indicates all quantum coherence was field, Bac. The error in the X rotation was found at dif- lost and we are in classical mixture of the states 0 | ↓ i ferent rates of dephasing. This is shown for the two basis and 1 . For typical rates of dephasing [given in Eq. states in Figs. 4 and 5. Evident on both of these graphs (6),| and ↓ i Eq. (7)], the error is found to be 1.9 10−6 × is a valley which levels out at a minimum error. This er- which is largely due to systematic error. ror is primarily due to systematic error in the gate. For The maximum error, for typical rates of dephasing, the 0 initial state the systematic error is found to of any of the states tested for the Z gate is 3.8 10−6. be 2|.3 ↓ i10−6, and for the 1 initial state the sys- This error is largely due to systematic effects rather× than tematic× error is found to be| 4. ↓9 i 10−6. As the nuclear × dephasing. This error suggests it is theoretically possible dephasing rates, Γ2n , increases, the fidelity of the opera- to do a Z rotation with an error of less than the 1 10−4 tion decreases. The fidelity of the operation drops to 0.0 limit suggested for fault tolerant quantum computing.× indicating that, in the limit of large dephasing rates, the The results for the Z gate are summarized in Table II. rotating magnetic field does not have the desired effect 5

State Systematic Error Typical Error 10−3 −6 −6 |0i 2.3 × 10 3.8 × 10 −4.5 −2 −1 6 10 10 10 |1i 4.9 × 10−6 6.4 × 10−6 10 Maximum 4.9 × 10−6 6.4 × 10−6 4 10 10−0.5 TABLE III: Summary of X gate error. ) −3.5 −1 2 10 10 10 −3.5 10 (s −2.5 E 10 −4.5 Γ 6 10 0 10 10 −2 10 −1.5 10−5.5 10 4 −2 10 10 10−1.5 10−4 )

−1 −5 −4 2 −4 10 10 10 10 −4 −2 0 2 4 6

(s 10 10 10 10 10 10 E Γ (s−1) Γ N 0 10 10−1 10−2.5 FIG. 5: Contour plot showing the error in the X Gate for 10−0.5 −2 10 | ↓ 1i initial state at differing rates of dephasing. Contour lines connect, and are labelled by, points of equal error. 10−5 −4 10 −4 −2 0 2 4 6 10 10 10 10 10 10 Γ (s−1) N which may be created using the steps specified in Ref. FIG. 4: Contour plot showing the error in the X gate for | ↓ 0i [34]. In the following discussion of two qubit gates, if initial state evolving at differing rates of dephasing. Contour not explicitly stated, all initial electron spin states are lines connect, and are labelled by, points of equal error. assumed to be . | ↓↓i The error in the CNOT in the presence of dephasing was found by numerically solving the master equation [Eqs. (8) and (9)] for the appropriate pulse sequence of a resonant magnetic field. [34]. For different rates of dephasing, different fidelities In Figs. 4 and 5, we see that the fidelity of the X ro- are obtained. Fidelities were calculated for each of the tation depends weakly on the electronic dephasing rate, four computational basis states, and for the evolution

Γ2e . The solutions to the equations under these condi- leading to the four Bell states. Fidelities for the four tions show that the principal cause of error is the electron computational basis states are shown in Fig. 6. becoming excited to a higher energy level. Each of the states shows a minimum error when the Under typical conditions, the 0 initial state has rates of electronic and nuclear dephasing are slow. The an error of 3.8 10−6. The 1| ↓statei has an error of remaining error is due to systematic error in the gate. 6.4 10−6. In this× operation dephasing| ↓ i has a much more Some sources of systematic error for the CNOT oper- important× role than in Z rotations. One reason for this ation include off-resonant effects, excitation into higher is that an X rotation takes longer, approximately 6.4µs. electronic energy levels, and imperfections in the pulse The error in the X gate is summarized in Table III. sequences (such as the break-down of the second or- The maximum error from the two basis states tested for der approximations used to derive appropriate times for the X gate was 6.4 10−6. The error induced in this pulse sequence [34]). Systematic error for each of the operation is less than× a threshold of 1 10−4 required for states 00 , 01 , 10 and 11 are 4.0 10−5,2.6 10−5, | −i5 | i | i −| 5 i × × fault tolerant quantum computing. × 1.9 10 , and 2.9 10 respectively. For evolution starting× in an initial Bell× state, we find the systematc error is 3.5 10−5, 3.4 10−5, 1.9 10−5 and 2.6 10−5. Sytematic× error for states× resulting× in a Bell state× are IV. TWO QUBIT GATES shown in Table IV. In each of the four states, as the dephasing rate in- A. The CNOT Gate creases, the fidelity decreases. In the limit of large dephasing rates, the computational basis states tend to stay The CNOT gate is specified by in their original states. This is particularly evident from the graphs of 00 and 01 which have higher fidelity 1000 (lower error) at| highi dephasing| i rates. In contrast, the 0100 states 10 and 11 have lower fidelities (highter error) Γ X =   , (22) 1 0001 at high| ratesi of| dephasing.i For example, the 00 state  0010  stays in the 00 state after the CNOT operation.| Ati such   | i 6

10 10−1

6 6 10 10 10−0.5

4 4 −1 10 10 10 10−2 10−2 ) ) −1 −1 2 −3.5 2 −3.5 10 10 10 10 10−110−1.5 (s (s

E E −3 Γ −0.5 Γ 10 0 10 0 10 10 10−4 10−3 10−1.5 10−4 10−2.5 10−4.5 −2 −2 10 10 10−1.5 10−1.5 10−1 10−2.5 −0.5 −4 −4 10 10 10 −4 −2 0 2 4 6 −4 −2 0 2 4 6 10 10 10 10 10 10 10 10 10 10 10 10 Γ (s−1) Γ (s−1) N N

(a) |00i initial state (b) |01i initial state

6 6 10 10 10−0.5 10−0.5 10−1 4 4 −1.5 10 10 10 −2.5 ) 10−2.5 ) 10 −1 2 −1 2 10 −2 10 10−3.5 −4 10

(s 10 (s −3 E 10 E Γ Γ −3 −1 0 0 10 10 10 10−1.5 10 10−4.5 10−4 10−3.5 10−2 −2 −2 10 10−4.5 10

−4 −4 10 10 −4 −2 0 2 4 6 −4 −2 0 2 4 6 10 10 10 10 10 10 10 10 10 10 10 10 Γ (s−1) Γ (s−1) N N

FIG. 6: Contour plots showing error in the CNOT operation for diﬀerent rates of electronic and nuclear dephasing. Contour lines connect, and are labelled by, points of equal error. high rates of dephasing, not even the single qubit rota- of one of the four Bell states is found to be 6.0 10−5, tions described in Section III C apply, which form part 6.0 10−5, 4.4 10−5, and 5.1 10−5. Typical× errors of the CNOT gate operation. For these states quantum for× states resulting× in a Bell state× are shown in Table coherence has been lost. When we consider, for example, IV. This implies that under our very simple decoherence the state ψ =1/√2( 00 + 01 ) we ﬁnd that at a high model, the maximum error in the CNOT gate in any basis | i | i | i 6 −1 −5 dephasing rate of Γn = Γe = 54 10 s that the error state is 8.3 10 . This is only marginally under the of the gate is 0.5. In this case,× quantum coherence has threshold of× 1 10−4 required for fault tolerant quantum been lost between the two states, and the qubit evolves computation. × to a completely mixed state. Errors for each of the computational basis states are Electronic dephasing rates play a much bigger role in shown in Table IV, and the maximum error of any of the two qubit gates than in single qubit gates. four computational basis states or states with evolution

For the typical dephasing times, T2e and T2n [given in leading to a Bell state is plotted in Fig. 7. Eq. (6), Eq. (7)], we ﬁnd the error for the states 00 , These results are directly analogous to calculations 01 , 10 and 11 are 8.3 10−5,6.8 10−5,6.2 10| −5i, made for the adiabatic CNOT gate [35]. We have used |andi 7|.2 i 10−5| respectively.i × The error× for an initial× state the same noise model as was used in their numerical sim- × 7

State Systematic Error Typical Error The advantage of nonadiabatic gates over adiabatic −5 −5 |00i 4.0 × 10 8.3 × 10 gates is that the pulse schemes required are much simpler, −5 −5 |01i 2.6 × 10 6.8 × 10 faster, and as at the conditions considered in Ref. [35], |10i 1.9 × 10−5 6.2 × 10−5 −5 −5 the two schemes have approximately the same fidelity. |11i 2.9 × 10 7.2 × 10 Considering the electronic decoherence times measured |00i + |11i 2.9 × 10−5 7.0 × 10−5 5 5 in Ref. [18] it is likely that electronic dephasing rates are |00i − |11i 3.2 × 10− 7.3 × 10− 5 5 not as large as considered in Ref. [35]. At lower rates |01i + |10i 3.1 × 10− 7.2 × 10− −5 −5 of dephasing, we approach the systematic error, which |01i − |10i 2.3 × 10 6.4 × 10 Maximum 4.0 × 10−5 8.3 × 10−5 may be smaller for nonadiabatic gates than adiabatic gates [34]. Another distinct advantage of the nonadia- TABLE IV: Summary of CNOT gate fidelities. batic gates is that any two qubit gate may be made. This allows us to construct two qubit gates directly (such as the swap gate), which are faster and higher fidelity than expressing them as combinations of CNOT gates and sin- 6 10 gle qubit rotations. If the Kane computer was being run 10−0.5 from a digital clock cycle, non-adiabatic two qubit gates 10−1 4 10 could be controlled at discrete times, and do not require 10−2 −2.5 the continuous and sophisticated pulse shapes required

) 10 −1 2 −3.5 for adiabatic gates operating at this speed and ﬁdelity. 10 10 10−1.5 (s 10−4 E Γ 0 10 B. The Swap Gate and Controlled Z Gate 10−3

−2 10 Similar calculations to those calculated for the CNOT gate were carried out for the swap gate and the controlled

−4 Z gate. The swap gate is specified by the matrix 10 −4 −2 0 2 4 6 10 10 10 10 10 10 Γ (s−1) 1000 N 0010 USwap =   . (23) FIG. 7: Contour plot of the maximum error in basis and 0100   Bell output states of the CNOT gate shown as a function of  0001  electronic and nuclear dephasing rates. Contour lines connect, The controlled Z gate is specified by the matrix and are labelled by, points of equal error. 100 0 010 0 Γ Z =   . (24) 1 001 0 ulations. In calculations for the adiabatic CNOT gate at  0 0 0 1  decoherence times, for electronic and nuclear dephasing  −  rates of T2e = 200µs and T2n = 10s respectively, the The circuit which may be used to create the swap gate maximum error of any of the four basis states for the may be found in Ref. [34]. adiabatic CNOT gate was found to be ‘just over 10−3’ The master equation was solved numerically for each [35]. In comparison, for the same conditions, we find that pulse sequence. The error in the swap gate was calculated the maximum error in the non-adiabatic gate is 7 10−3. for each basis state, and each state whose output state is a Under these conditions both non-adiabatic and adiabatic× Bell state. Note that for these two gates the states which gates give similar fidelities. give Bell states as output are themselves Bell states. A Where does this error come from at the dephasing separate simulation was completed for each combination rate specified above, for the nonadiabatic CNOT gate? of nuclear and electronic dephasing times. The maximum The error in an X rotation under these conditions is error of any of the basis states has been plotted in Fig. 3 10−6 and the error in the entangling operation of 8 for the swap gate, and Fig. 9 for the control Z gate. × π −3 the gate Um , is 4 10 and therefore it is clear Similar features that were evident for the CNOT gate 4 × that the two qubit entangling operation is the major are visible in these figures. The corresponding errors are source of error. When electronic decoherence times are shown in Table V for the swap gate, and Table VI for the short, any electron mediated operations will be affected control Z gate. by this decoherence. By increasing the exchange interaction strength, the time required for the electron mediated operation may be reduced. For example, at a strength of V. CONCLUSION J = 0.0529meV the error decreases to 1 10−3 and the maximum error in the CNOT gate is also× 1 10−3 for In conclusion, we have investigated the effect of de- any of the computational basis states. × phasing on the Kane quantum computing architecture. 8

State Systematic Error Typical Error |00i 3.9 × 10−5 9.0 × 10−5 6 |01i 1.4 × 10−5 7.9 × 10−5 10 −5 −5 |10i 1.6 × 10 8.0 × 10 −1 5 5 10 − − 4 |11i 3.8 × 10 8.9 × 10 10 −5 −4 10−2 |00i + |11i 5.3 × 10 1.4 × 10 −2.5 −5 −4 ) 10

|00i − |11i 7.4 × 10 1.6 × 10 −1 2 −3.5 |01i + |10i 1.7 × 10−5 1.0 × 10−4 10 10 (s −4

−5 −4 E 10

|01i − |10i 1.5 × 10 1.0 × 10 Γ −5 −4 0 . . −0.5 Maximum 7 4 × 10 1 6 × 10 10 10 10−3

TABLE V: Summary of swap gate error. −2 10

10−1.5 −4 10 −4 −2 0 2 4 6 6 10 10 10 10 10 10 10 Γ (s−1) N 10−0.5 −1 4 10 10 FIG. 9: Contour plot of the maximum error in basis states 1 ) and Bell output states of the Γ Z gate shown as a function of 10−2.5 −1 2 −3 electronic and nuclear dephasing rates. Contour lines connect, 10 10

(s −3.5 and are labelled by, points of equal error.

E 10 Γ 0 10 10−4 −2 10 Gate Typical Error Systematic Error Time −2 −6 −6 10 Z 3.8 × 10 3.8 × 10 0.02µs 6 6 10−1.5 X 6.4 × 10− 4.9 × 10− 6.4µs −5 −5 −4 CNOT 8.3 × 10 4.0 × 10 16.0µs 10 4 5 −4 −2 0 2 4 6 − − 10 10 10 10 10 10 Swap 1.6 × 10 7.4 × 10 19.2µs Γ (s−1) −5 −5 N Γ1Z 9.4 × 10 3.8 × 10 16.1µs

FIG. 8: Contour plot of the maximum error in basis states TABLE VII: Times and errors for each of the gates investi- and Bell output states of the swap gate shown as a function of gated. electronic and nuclear dephasing rates. Contour lines connect, and are labelled by, points of equal error. a decoherence process as in Ref. [18], it is likely that the typical decoherence times for the Kane architecture may be further reduced, and therefore unambiguously We used a simple model of decoherence and investigated under the threshold required for fault tolerant quantum how this model affected proposed gate schemes on the computation. Kane quantum computer. For typical decoherence rates Construction and operation of the Kane quantum com- [given in Eq. (6), Eq. (7)], these results are summarized puter is extremely challenging. In the actual physical in Table VII. system there will undoubtedly be noise and decoherence Each of the errors, for typical rates of dephasing, found processes not considered in our simple physical model. here are close to the error threshold required for fault This substantial effort would never be able to achieve tolerant quantum computation. If the temperature is its ultimate goal of a working quantum computer if there lowered, and coupling between qubits is not considered were fundamental reasons why such a computer could not operate. In this paper we investigated the one such effect on proposed gates for the Kane quantum computer. Our State Systematic Error Typical Error simulations indicate that errors due to dephasing, the 5 5 |00i 3.0 × 10− 8.1 × 10− dominant form of decoherence in the Kane architecture, −5 −5 |01i 1.5 × 10 6.6 × 10 do not rule out fault tolerant quantum computation. |10i 1.1 × 10−5 6.2 × 10−5 |11i 3.8 × 10−5 8.9 × 10−4 |00i + |11i 3.6 × 10−5 9.4 × 10−5 |00i − |11i 3.3 × 10−5 9.2 × 10−5 Acknowledgments |01i + |10i 1.7 × 10−5 7.5 × 10−5 5 5 |01i − |10i 1.1 × 10− 7.0 × 10− The authors would like to thank G. J. Milburn for 5 5 Maximum 3.8 × 10− 9.4 × 10− support. This work was supported by the Australian Research Council, the Australian government and by the TABLE VI: Summary of controlled Z gate error. US National Security Agency (NSA), Advanced Research 9 and Development Activity (ARDA) and the Army Re- 01-1-0653. H.-S.G. would like to acknowledge financial search Office (ARO) under contract number DAAD19- support from Hewlett-Packard.

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0 10 Γ (s−1) N