Limited fidelity

quorum¹ is the following:

|ψ₁i=|↑↑i,

|ψ2i=|↑↓i,

|ψ3i= SWAP|↑↓i=|↓↑i,

|ψ4i=e^iπ2σ2z√

SWAPe^iπ4σ1x|Si=|↑x↑xi,

|ψ5i=√

SWAPe^−iπ4σ1x|Si=|↓x↑xi,

|ψ6i=√

SWAPe^iπ4σ1x|Si=|↑^x↓^xi,

|ψ7i=e^iπ2σ2z√

SWAPe^iπ4σ1y|Si=|↑^y↑^yi,

|ψ8i=√

SWAPe^−iπ4σ1y|Si=|↓^y↑^yi,

|ψ9i=√

SWAPe^iπ4σ1y|Si=|↑^y↓^yi,

|ψ10i=e^−iπ4σ2ze^iπ4σ1x|Si,

|ψ₁₁i=e^iπ4σ2ze^−iπ4σ1x|Si,

|ψ12i=e^iπ4σ2ze^iπ4σ1x|Si,

|ψ₁₃i=e^−iπ4σ2ze^iπ4σ1y|Si,

|ψ14i=e^iπ4σ2ze^−iπ4σ1y|Si,

|ψ₁₅i=e^iπ4σ2ze^iπ4σ1y|Si,

(6.30)

The necessary quantum circuits are drawn in figure 6.3. This sequences of elementary quantum gates needed here are relatively short, at most one measurement needs three elementary gates. Note that no more than oneπ/2 rotation of one of the spin qubits has to be performed per measurement. In reference [120], where both, exchange interaction and ESR have been demonstrated in the same DQD, the ESR rotations were slower than the exchange based gates. Under the assumption that the potential error by ESR gates is higher than the error by exchange operations, it is important to limit the ESR gates included in the scheme to a minimum. Furthermore, the ESR pulses have to be applied to one of the spins only. This is of practical relevance as Brunner et al. [120] found that in their sample the ESR pulses performed better in one dot than in the other. The potential errors are considered in the next section.

6.4 Limited fidelity

In this section, we want to determine the influence of non-perfect operations on the reli-ability of the experimental outcome. To do so, the perfect projectionsPj from equation (6.25) are replaced by by imperfect operatorsP_j⁰, which are not necessarily projection op-erators anymore, i.e., (P_j⁰)²6=P_j⁰ is possible. The imperfect projections can still be used for reconstruction of the density matrix applying self-consistent tomography [102, 271].

1Note that reference [44] contains a typo regarding the construction of|ψ14i, where the gates are are applied to the state|ψ13ibut have to applied to the state|Si.

|ψ2i

| ↑i

| ↓i

SWAP

| ↑i

| ↓i

|ψ3i (a)

|Si

√ SWAP

|ψji j= 4, . . . ,9 RAjθ₁^j

ESR

RZθ^j₂ (b)

|Si |ψji

j= 10, . . . ,15

R_Zθ₂^j R_Ajθ₁^j

ESR (c)

Figure 6.3: Quantum circuits for realizing the quorum in equation (6.30). (a) A SWAP gate translates the state |↑↓i to |↓↑i. No single-qubit rotations are neces-sary. (b) The circuits for the states |ψ4i, . . . ,|ψ9i. The axes of the ESR rotation are A4 = A5 = A6 = X and A7 = A8 = A9 = Y while the an-gles of the rotations are θ⁽⁴⁾₁ = θ₁⁽⁶⁾ = θ⁽⁷⁾₁ = θ₁⁽⁹⁾ = −θ₁⁽⁵⁾ = −θ⁽⁸⁾₁ = π/2 and θ₂⁽⁴⁾ = θ⁽⁷⁾₂ = π, θ⁽⁵⁾₂ = θ₂⁽⁶⁾ = θ⁽⁸⁾₂ = θ₂⁽⁹⁾ = 0. (c) The circuits for the states |ψ₁₀i, . . . ,|ψ₁₅i. The rotation axes for the first qubit are A10 = A11 = A12 = X and A13 = A14 = A15 = Y. The rotation an-gles are θ⁽¹⁰⁾₁ = θ⁽¹²⁾₁ = θ⁽¹³⁾₁ = θ⁽¹⁵⁾₁ = −θ₁⁽¹¹⁾ = −θ⁽¹⁴⁾₁ = π/2 and θ⁽¹¹⁾₂ = θ⁽¹²⁾₂ = θ₂⁽¹⁴⁾ = θ₂⁽¹⁵⁾ = −θ⁽¹⁰⁾₂ = −θ₂⁽¹³⁾ = π = π/2. For panels (b) and (c), the initial state of the circuit is the spin singlet |Si. Note that

|ψ₁i, . . . ,|ψ₉iare product states while|ψ₁₀i, . . . ,|ψ₁₅iare maximally entan-gled as single-qubit gates are applied to the maximally entanentan-gled state |Si and single-qubit gates do not change the entanglement. The ESR rotations, which allow for rotations about thex- and they-axes, are limited to oneπ/2 rotation applied to the first spin. In the measurement scheme, the circuits have to be performed in reverse order because the state at the left side of the circuit is the one, which allows for a projective measurement. Panel (a) first published in N. Rohling and G. Burkard, Phys. Rev. B 88, 085402 (2013), http://dx.doi.org/10.1103/PhysRevB.88.085402 c2013 American Physical Society.

6.4 Limited fidelity For this, the states operatorsP_j⁰ are considered to be unknown and have to be estimated from experiment as well. An experimenter might prepare the system in the state P_i⁰, which is achieved just by reversing the measurement procedure number i in time with an initial state |S(0,2)i and then perform the measurement for operatorP_j⁰. From this, the matrix elements

Mij = P_i⁰|P_j⁰

M (i, j= 1, . . . ,15) (6.31) can be obtained. Regarding the matrix M, maximum likelihood algorithms [102, 271]

can be used to estimate P_j⁰. Note that Mis invariant under a unitary operation. This means that the estimation of the operators is ambiguous. However, this is actually no serious problem as also other expectation values and, thus, the experimental results are invariant under this unitary operation. In other words, the basis of the Hilbert space which is fixed in the estimation can be used throughout all experiments done with the system. Further note that this self-consistent tomography needs less measurements than quantum process tomography [271].

While self-consistent tomography is a general concept, one might also use knowledge about the physical processes which are applied within the measurement scheme. The quantum gates and, thus, the measurements might depend on a set ofnnoisy parameters α₁, . . . , α_n. Experimentally, the distributions of these parameters, p(α₁, . . . , α_n), have to be determined. In practice, those parameters might describe electric and magnetic fields, see reference [102]. In reference [23], an ESR experiment is fitted with suitable parameters. The measurement operators P_j⁰(α₁, . . . , α_n) depend on α₁, . . . , α_n. Actu-ally, for reconstruction only the mean operator, which is denoted again by P_j⁰ without dependence, has to be known,

P_j⁰ = Z

dα₁. . . dα_np(α₁, . . . , α_n)P_j⁰(α₁, . . . , α_n). (6.32) If the parameters have independent Gaussian distributions, only the mean values hα_ii and hα²_ii for i = 1, . . . , n have to be determined. Determining the distributions of the parameters incorrectly leads to systematic errors in the later performed state tomogra-phy. If the distribution is well-known but broad, the state tomography is more difficult than in the ideal case. To see this, we consider the example where the operators P_j⁰ are given by

P_j⁰ = 1−f_j²

3 1+4f_j²−1

3 Pj (6.33)

where f_j = Tr[(p P_jP_j⁰p

P_j)^1/2] is the fidelity of the, in general, mixed quantum state P_j⁰ with respect to the pure stateP_j =|ψ_ji hψ_j|. Forf_j = 1, the ideal case ofP_j⁰ =P_j is realized. Forfj = 1/2, equation (6.33) yields the completely mixed stateP_j⁰ =1/4, which is useless for state tomography. Let us consider, as an example within the reconstruction, the component ρ₃, see equation (6.27). For the imperfect measurements, we gain under the assumption f1 =f2 =:f

ρ₃ = 3{[Tr(P₁⁰ρ) + Tr(P₂⁰ρ)]−1}

2(4f²−1) . (6.34)

The reconstruction using the measurement resultsm⁰₁ andm⁰₂ for the operators Tr(P₁⁰ρ) and Tr(P₂⁰ρ), leads to ˜ρ3 = (3{[m⁰₁+m⁰₂]−1})/(2(4f²−1)). Furthermore, it is assumed thatm⁰₁ andm⁰₂are statistically independent with the same value for the statistical error denoted byδ⁰. Then the statistical error δ of ˜ρ₃ is related toδ⁰ via

δ⁰ = (4f²−1)δ 3√

2 . (6.35)

A Chernoff bound [272, 273] provides an upper limit for the probability Pout that the inequality|m⁰_1/2−Tr(P_1/2⁰ ρ)|> δ⁰ is fulfilled for a given number of runsN_run[274, 275],

P_out ≤2 exp

−2δ⁰² Nrun

. (6.36)

If the reconstruction should provide a precision in a way that|ρ˜₃−ρ₃|> δ does occur with a probability smaller thanP_limit, this means that, for measuringm⁰₁ and m⁰₂, the number of experimental runs has to be

N_run= ln(2/P_limit)

2δ⁰² = 9 ln(2/P_limit)

δ²(4f²−1)² (6.37)

each. To summarize, repeating the experiment often compensates for a low fidelity f > 1/2. We also see that for the extreme case f = 1/2 such a compensation is not provided, because using the completely mixed state 1/4 as the measurement operator does not provide any information about the quantum state.

6.5 Conclusions

In this chapter, we have considered a state tomography scheme for two spin qubits hosted in one DQD. The measurements are spin-to-charge conversions followed by a measurement of the charge state. This leads to a situation where the measurement operators are projections on quantum states in contrast to a setup where each spin, or any other qubit, can be measured individually, see for example [101] for spin qubits and [85] for photons. In this context, a quorum consists of 15 such measurements. It has been shown in this chapter that no quorum exist where the traceless parts of the projection operators are a basis in the space of traceless 4×4 matrices, which would be ideal with respect to a small statistical error. A set of mutually unbiased bases provides states for constructing a quorum. It is of significant practical relevance that the quantum circuits suggested for the realization of the included measurements are relatively short.

Each measurement contains no more than one π/2 rotation which has to be performed by applying an ESR pulse to one of the spin qubits. In general, all operations are erroneous and the error has to be determined experimentally as well. A statistical error within the measurement operations enlarges the statistical error of the measurement outcomes, which is anyway not zero because quantum mechanics does in general not provide precise predictions for a measurement but measurement probabilities. While

6.5 Conclusions this can be compensated by a larger number of experimental runs, errors in determining the distributions of the operations or of the parameters describing them, would lead to systematic errors in the state tomography scheme.

It remains an open question what is the optimal scheme with respect to minimal errors in the reconstruction. The answer should significantly depend on the errors of the different operations, which have to be performed during the measurement process.

Another question for further investigations is whether the reconstruction can be improved by performing more than 15 different measurements.

7 Outlook

The results presented in chapters 4, 5, and 6 are solutions to selected problems within the field of quantum computing with quantum dots: the quest for universal quantum gates in quantum registers with spin and valley qubits, optimizing echo sequences for an exchange-only qubit, and the search for a quantum state tomography scheme tailored for two spin qubits in a DQD. For all these results, generalizations and extension are conceivable.

In chapter 4, the exchange interaction acts as a four-qubit interaction if each spin and each valley state of a two-electron DQD is considered to be a qubit. Nevertheless, in combination with single-qubit operation, a two-qubit gates can be constructed by this interaction. But a general description for the four-qubit interaction is not provided.

Especially it is unclear whether the exchange interaction can be used to directly entangle one single-spin and one single-valley qubit. The coupling in the registers considered in sections 4.2 and 4.3 was achieved in a subspace where a qubit is defined by the spin or valley states of two electrons. In contrast, two-qubit operations are characterized by no more than three real parameters given by the Makhlin invariants, see section 2.2.

More general insights in four- and other multi-qubit couplings can be very useful for constructing quantum gates from these interactions.

Due to its relevance for silicon quantum dots, which are currently under intense inves-tigation because of their long spin lifetimes, the valley DOF in these systems will almost certainly be subject to further research in these systems. The model which was consid-ered in this thesis to describe the spin and valley quantum computing might be extended for example by including the effects of spin-orbit interaction, a model for the dependence of the valley states on the interface roughness and disorder in silicon quantum dots, see e.g. [214, 245], or by considering additional terms coupling the spin and the valley DOFs of the same electron directly, e.g., by the valley dependent g-factor found in [186]. The latter term can even result in spin-valley two-qubit gates in a single quantum dot.

The echo sequences of chapter 5 were dedicated to the noise originating from a nuclear spin bath. In isotopically purified silicon quantum dots, this effect is suppressed by eliminating the nuclear spins in advance [276]. For charge noise, which is then most likely the dominating source of decoherence, echo sequences are also tools to reduce decoherence, but is not efficient for fast fluctuating terms [276]. Echo sequences can also be used for noise spectroscopy along the lines of [172].

The state tomography scheme in chapter 6 is proposed for two single-spin qubits in a DQD where the measurement of the spin states is realized by spin-to-charge conversion within this DQD in contrast to individual qubit readout, which allows for state tomog-raphy by analyzing the correlation data as implemented in [101]. The optimal choice of a tomography scheme highly depends on the performance of the available quantum

gates and measurements. This means the scheme should be optimized for the device in which it is supposed to be implemented. Possible extensions of the scheme presented in this thesis are state tomography for spin and valley states stored in the same DQD and quantum process tomography.

Basic concepts of quantum computing with quantum dots are well-established nowa-days. One might have the impression that the main remaining challenges are mainly of experimental and engineering character regarding the realization of high fidelity quan-tum gates and the construction of scalable quanquan-tum dot arrays. Nevertheless, on the path to this achievements, theoretical physics is of considerable relevance with respect to understanding of the phenomena limiting the coherence and the fidelity and finding suitable solutions to overcome these difficulties.

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