Phonon-induced decoherence of a charge quadrupole qubit

The past two decades have witnessed remarkable advances in the development of qubits in semiconductor-based systems [1–17]. Qubits constructed using electrons confined in lateral quantum dots (QDs) have been realized experimentally in different forms, including single-spin qubits [4–6, 8, 18, 19], singlet-triplet qubits [20, 21], QD hybrid qubits [22–24], exchange-only qubits [25–27], and charge qubits with different numbers of electrons [23, 28–30]. One of the main sources of qubit decoherence in each of these systems is charge noise, which can arise from high-temperature electrical noise in the voltage controls or the motion of defects and free charges in the heterostructure. The recently proposed charge quadrupole qubit, formed of one electron in a triple QD, is robust against uniform electric field fluctuations due to the high symmetry of its basis states [31, 32]. The qubit states are symmetric with respect to the central QD, thus forming a decoherence-free subspace, which is well-known for spin qubits [33–35], but novel for charge qubits. However, as for all qubits relying on a symmetric operating point, phonons can break this symmetry and cause decoherence. Indeed, phonons have been shown to be a significant source of decoherence for many types of QD qubits [13, 36–46].

Here we study theoretically the decoherence of a charge quadrupole qubit arising from its coupling to phonons. In addition to decoherence between qubit states, we account for the effects of a leakage state. The leakage state is not coupled to the qubit subspace in the ideal case, but phonons break the symmetry of the qubit and consequently induce a coupling. The effects of phonons are greatest when the energy separation between the states is large, because the phonon density of states increases with energy and the electron–phonon matrix elements depend on momentum. To characterize the effects of phonon-induced decoherence and to identify favorable operating parameters for the qubit, we study Larmor and Ramsey pulse sequences, which can be used to implement arbitrary single-qubit gate operations. An important parameter for qubit control is the tunnel coupling, which strongly affects the qubit energy. Since phonon-induced decoherence increases as the tunnel coupling increases, and the decoherence arising from charge noise [31, 32] decreases as the tunnel coupling increases, both sources of decoherence must be considered to identify the optimal working regime for the qubit. We demonstrate that there is a working regime where qubit infidelity from both these processes is less than 0.01%.

The paper is organized as follows. In section 2 we present the Hamiltonian of a charge quadrupole qubit interacting with a phonon bath. In section 3 we consider the time evolution of the density matrix describing the qubit subspace and a leakage state. We discuss phonon-induced decoherence rates in section 4. Then we compare phonon-induced infidelity to the infidelity due to charge noise in section 5. A discussion of how to reduce other possible sources of decoherence in such devices is presented in section 6. Details of the calculations and additional information are provided in the appendices.

2.1. Electron in a triple QD

The charge quadrupole qubit is formed in a triple QD that hosts one electron [31], as illustrated in figures 1(a) and 1(b). The qubit is operated in a symmetric fashion, such that the energies of the first and third QDs are the same and the middle QD has the same tunnel coupling to the two outer QDs. However, phonons can perturb the potential of the triple QD, as shown in figure 1(c), giving rise to decoherence.

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To model this system, we consider electron wave functions in the x–y plane of a quantum well, which are constructed from the ground states of harmonic oscillator potentials that approximate the QD confinement [47]. For the left, right, and central QDs these are given by

$\begin{eqnarray}&&{\psi }_{L,R}=\displaystyle \frac{1}{\sqrt{\pi }l}\exp \left[-\displaystyle \frac{{(x\pm a)}^{2}+{y}^{2}}{2{l}^{2}}\right],\end{eqnarray} \tag{ 1 }$

$\begin{eqnarray}&&{\psi }_{C}=\displaystyle \frac{1}{\sqrt{\pi }l}\exp \left[-\displaystyle \frac{{x}^{2}+{y}^{2}}{2{l}^{2}}\right],\end{eqnarray} \tag{ 2 }$

respectively. Here, a is the interdot distance and l characterizes the radius of the dots. Both a and l are assumed to be the same for all three QDs. As the dots are situated rather close to one another, there are overlaps between these states, $S=\langle {\psi }_{L}| {\psi }_{C}\rangle =\langle {\psi }_{R}| {\psi }_{C}\rangle$. To construct orthonormal wave functions of an electron in a triple QD we must take these overlaps into account. Note that we include the overlaps $\langle {\psi }_{L}| {\psi }_{C}\rangle$ and $\langle {\psi }_{R}| {\psi }_{C}\rangle$ but not $\langle {\psi }_{L}| {\psi }_{R}\rangle$, because the separation between the left and right dots is much larger than between an outer dot and the central dot. Generalizing a well known procedure for double dots [47, 48] to the three-dot problem, we construct the orthonormal wave functions for an electron to be in the left, right or central QDs of the triple QD (see appendix B for details)

$\begin{eqnarray}&&{{\rm{\Phi }}}_{L,R}=({\psi }_{L,R}-2S{\psi }_{C})\varphi ,\end{eqnarray} \tag{ 3 }$

$\begin{eqnarray}&&{{\rm{\Phi }}}_{C}=\displaystyle \frac{{\psi }_{C}-g{\psi }_{L}-g{\psi }_{R}}{\sqrt{1-4{gS}+2{g}^{2}}}\varphi ,\end{eqnarray} \tag{ 4 }$

where $g=S/(4{S}^{2}-1)$, and φ(z) represents the component of the wave function in the heterostructure growth direction, as described in detail in appendix A. We have checked that the results do not depend on the detailed form of φ(z). This insensitivity comes from the fact that the phonons responsible for transitions have wavelengths that are much greater than the thickness of φ(z). In this work, we therefore take φ(z) to have a simple Gaussian envelope with a full width at half maximum of 4.7 nm.

As described in [31], the Hamiltonian of an electron in a triple QD in the $\{| {{\rm{\Phi }}}_{L}\rangle ,| {{\rm{\Phi }}}_{C}\rangle ,| {{\rm{\Phi }}}_{R}\rangle \}$ basis is

$\begin{eqnarray}{H}_{0}=\left(\begin{array}{ccc}{U}_{L} & {t}_{A} & \ 0\\ {t}_{A} & {U}_{C} & {t}_{B}\\ 0\ \ & {t}_{B} & {U}_{R}\end{array}\right)=\left(\begin{array}{ccc}{\epsilon }_{d} & {t}_{A} & \ 0\\ {t}_{A} & {\epsilon }_{q} & \ {t}_{B}\\ 0\ \ & {t}_{B} & -{\epsilon }_{d}\end{array}\right)+\displaystyle \frac{{U}_{L}+{U}_{R}}{2},\end{eqnarray} \tag{ 5 }$

where t_A, t_B are tunnel couplings between the dots, and U_L, U_C, and U_R are the potentials of the left, central, and right QDs, which are defined by the gate voltages. The dipolar and quadrupolar detuning parameters are defined as ${\epsilon }_{d}=({U}_{L}-{U}_{R})/2$ and ${\epsilon }_{q}={U}_{C}-({U}_{L}+{U}_{R})/2$, respectively. When ${t}_{A}={t}_{B}=t$ and _d = 0, H₀ describes a charge quadrupole qubit [31]. Gate operations can be performed by varying the quadrupolar detuning _q in time, as illustrated in figure 1(d).

In addition to the dipolar and quadrupolar detunings, several other experimental parameters can be tuned, including the dot separation a, the wave function extent that characterizes the radius of the dots, l, the tunnel coupling t, and the temperature T. To estimate the value of t, we model t as the tunnel coupling between the two sides of a double well potential ${{\hslash }}^{2}{({x}^{2}-{a}^{2}/4)}^{2}/(2{m}_{\mathrm{eff}}{l}^{4}{a}^{2})$, where ${m}_{\mathrm{eff}}=1.73\times {10}^{-31}\,\mathrm{kg}$ is the transverse effective mass of an electron in Si. With details given in appendix C, we obtain

$\begin{eqnarray}&&t=-\displaystyle \frac{3{{\hslash }}^{2}({a}^{2}+4{l}^{2})}{64{m}_{\mathrm{eff}}{l}^{4}\sinh [{a}^{2}/(4{l}^{2})]}.\end{eqnarray} \tag{ 6 }$

Note that for this double well potential form, the detuning between the wells is zero, and we have ignored any dependence of t on the detunings. However it allows us to estimate how strongly the change in a or l affects tunnel coupling. From equation (6), we see that $| t|$ grows rapidly with l. Changing the tunnel coupling is found to have a strong effect on decoherence rates, as we will show below.

The eigenstates of the Hamiltonian H₀ when _d = 0 form a convenient basis for the quadrupole qubit, given by

$\begin{eqnarray}&&| 0\rangle =\displaystyle \frac{-4t| {{\rm{\Phi }}}_{C}\rangle +(| {{\rm{\Phi }}}_{L}\rangle +| {{\rm{\Phi }}}_{R}\rangle )({\epsilon }_{q}+{\rm{\Omega }})}{2\sqrt{\tfrac{1}{1+\tfrac{{\epsilon }_{q}}{{\rm{\Omega }}}}}({\epsilon }_{q}+{\rm{\Omega }})},\end{eqnarray} \tag{ 7 }$

$\begin{eqnarray}&&| 1\rangle =\displaystyle \frac{2t(| {{\rm{\Phi }}}_{L}\rangle +| {{\rm{\Phi }}}_{R}\rangle )+| {{\rm{\Phi }}}_{C}\rangle ({\epsilon }_{q}+{\rm{\Omega }})}{4t\sqrt{\tfrac{1}{1-\tfrac{{\epsilon }_{q}}{{\rm{\Omega }}}}}},\end{eqnarray} \tag{ 8 }$

$\begin{eqnarray}&&| L\rangle =\displaystyle \frac{1}{\sqrt{2}}(| {{\rm{\Phi }}}_{R}\rangle -| {{\rm{\Phi }}}_{L}\rangle ),\end{eqnarray} \tag{ 9 }$

Here, $| 0\rangle$ and $| 1\rangle$ represent the qubit subspace and ${\rm{\Omega }}=\sqrt{8{t}^{2}+{\epsilon }_{q}^{2}}$ is the qubit energy splitting. The state $| L\rangle$ represents a leakage state that decouples from the qubit subspace in the absence of environmental noise. The wave functions are illustrated schematically in figure 1(b). In the following, we use the $\{| 0\rangle ,| 1\rangle ,| L\rangle \}$ basis to study the time evolution of the qubit coupled to phonons.

2.2. Interaction with the phonon bath

In this subsection we show how we include phonons in our model. The Hamiltonian of an electron in a triple QD coupled to the bath of phonons is given by $H={H}_{0}+{H}_{{\rm{el}}-{\rm{ph}}}+{H}_{{\rm{ph}}}$, where H_ph is the phonon bath Hamiltonian defined as ${H}_{{\rm{ph}}}={\sum }_{{\boldsymbol{q}},s}\hslash {\omega }_{{\boldsymbol{q}}s}\left({b}_{{\boldsymbol{q}},s}^{\dagger }{b}_{{\boldsymbol{q}},s}+\tfrac{1}{2}\right)$ with ${b}_{{\boldsymbol{q}},s}$ and ${b}_{{\boldsymbol{q}},s}^{\dagger }$ are annihilation and creation operators for phonons of mode $s\in \{l,{t}_{1},{t}_{2}\}$ (corresponding to one longitudinal and two transverse acoustic modes) and wave vector ${\boldsymbol{q}}$, with the phonon frequency ${\omega }_{{\boldsymbol{q}}s}$. The electron–phonon interaction is described by ${H}_{{\rm{el}}-{\rm{ph}}}$.

In this work, we will study triple QDs constructed in Si-based heterostructures, which can be, for example, Si/SiO₂ or Si/SiGe. The electron–phonon interaction Hamiltonian in bulk Si is given by [49–51]

$\begin{eqnarray}&&{H}_{{\rm{el}}-{\rm{ph}}}={{\rm{\Xi }}}_{d}{\rm{Tr}}{\boldsymbol{\varepsilon }}+{{\rm{\Xi }}}_{u}{\varepsilon }_{{zz}},\end{eqnarray} \tag{ 10 }$

where ${\boldsymbol{\varepsilon }}$ is a strain tensor defined as [45, 51]

$\begin{eqnarray}&&{\varepsilon }_{{ij}}=\displaystyle \frac{1}{2}\left(\displaystyle \frac{\partial {u}_{i}}{\partial {r}_{j}}+\displaystyle \frac{\partial {u}_{j}}{\partial {r}_{i}}\right),\end{eqnarray} \tag{ 11 }$

$\begin{eqnarray}&&{\boldsymbol{u}}=\displaystyle \sum _{{\boldsymbol{q}},s}\sqrt{\displaystyle \frac{\hslash }{2{\rho }_{{\rm{Si}}}{{Vqv}}_{s}}}{{\boldsymbol{e}}}_{{\boldsymbol{q}}s}({b}_{{\boldsymbol{q}},s}{\mp }_{s}{b}_{-{\boldsymbol{q}},s}^{\dagger }){{\rm{e}}}^{{\rm{i}}{\boldsymbol{q}}\cdot {\boldsymbol{r}}},\end{eqnarray} \tag{ 12 }$

where Ξ_d and Ξ_u are deformation potential constants, ${\boldsymbol{u}}$ is a displacement operator, ${\boldsymbol{r}}$ denotes the position of the electron, $s\in \{l,{t}_{1},{t}_{2}\}$, v_s is the speed of sound for each of these modes, ρ_Si is the mass density of bulk Si, and V is the volume we consider. The polarization vectors are chosen as follows: ${{\boldsymbol{e}}}_{{\boldsymbol{q}},l}={\boldsymbol{q}}/q$, ${{\boldsymbol{e}}}_{-{\boldsymbol{q}},{t}_{1}}=-{{\boldsymbol{e}}}_{{\boldsymbol{q}},{t}_{1}}$, and ${{\boldsymbol{e}}}_{-{\boldsymbol{q}},{t}_{2}}={{\boldsymbol{e}}}_{{\boldsymbol{q}},{t}_{2}}$, so that in equation (12) one has ∓_l = −, ${\mp }_{{t}_{1}}=-$, and ${\mp }_{{t}_{2}}=+$. Explicit expressions for the polarization vectors are given in appendix D. We assume the following parameters for Si: deformation potential constants 5 and 8.77 eV [50], speed of sound for longitudinal phonons ${v}_{l}=9\times {10}^{3}$ m s⁻¹ and transverse phonons ${v}_{t1}={v}_{t2}=5.4\times {10}^{3}$ m s⁻¹ [52, 53], the bulk mass density 2.33 g cm⁻³.

Evaluating H in the regime used for the charge quadrupole qubit in the $\{| {{\rm{\Phi }}}_{L}\rangle ,| {{\rm{\Phi }}}_{C}\rangle ,| {{\rm{\Phi }}}_{R}\rangle \}$ basis yields

$\begin{eqnarray}H=\left(\begin{array}{ccc}0 & t & 0\\ t & {\epsilon }_{q} & t\\ 0 & t & 0\end{array}\right)+\left(\begin{array}{ccc}{P}_{{LL}} & {P}_{{LC}} & \ \ 0\\ {P}_{{LC}}^{\dagger } & {P}_{{CC}} & {P}_{{CR}}\\ 0\ \ \ & {P}_{{CR}}^{\dagger } & {P}_{{RR}}\end{array}\right)+{H}_{{\rm{ph}}},\end{eqnarray} \tag{ 13 }$

where P_LL, P_CC, P_RR, P_LC, P_CR are electron–phonon interaction matrix elements, defined as ${P}_{\alpha ,\beta }=\langle {{\rm{\Phi }}}_{\alpha }| {H}_{{\rm{el}}-{\rm{ph}}}| {{\rm{\Phi }}}_{\beta }\rangle$. Note that we have neglected the matrix elements P_LR here because the corresponding wave functions have a negligible overlap.

In the following, we treat ${H}_{{\rm{el}}-{\rm{ph}}}$ as a perturbation and study how the electron–phonon interaction affects the dynamics of the three-level system $\{| 0\rangle ,| 1\rangle ,| L\rangle \}$.

To investigate the time evolution and decoherence of the qubit, we employ the Bloch–Redfield formalism [39, 54, 55]. First, we apply to the full Hamiltonian H the transformation U_d that diagonalizes H₀, yielding $\tilde{H}={U}_{d}^{-1}{{HU}}_{d}$. Similarly, we define ${\tilde{H}}_{{\rm{el}}-{\rm{ph}}}={U}_{d}^{-1}{H}_{{\rm{el}}-{\rm{ph}}}{U}_{d}$. We then derive the equation for the time evolution of the density matrix of our three-level system in the $\{| 0\rangle ,| 1\rangle ,| L\rangle \}$ basis following the procedure described in [54] (for the full derivation see appendix E): (1) write down the von Neumann equation, (2) trace over phonon degrees of freedom, (3) apply the Born–Markov approximation. The Markov approximation is justified here by the fact that the phonons transit the triple QD in a period $\lesssim {\tau }_{\mathrm{corr}}=2(l+a)/{v}_{t1}$ [39], which is by several orders of magnitude shorter than the decoherence times we present below. The gate times we consider in section 5 are ∼2–11 times larger than τ_corr. The resulting equations for the elements of the density matrix ρ are

$\begin{eqnarray}&&{\dot{\rho }}_{{nm}}=-{\rm{i}}{\omega }_{{nm}}{\rho }_{{nm}}+\displaystyle \sum _{k,j}[({{\rm{\Gamma }}}_{{jmnk}}^{+}+{{\rm{\Gamma }}}_{{jmnk}}^{-}){\rho }_{{kj}}-{{\rm{\Gamma }}}_{{kjjm}}^{-}{\rho }_{{nk}}-{{\rm{\Gamma }}}_{{nkkj}}^{+}{\rho }_{{jm}}],\end{eqnarray} \tag{ 14 }$

where the indices n, m, j, k refer to the eigenstates of H₀, $\{| 0\rangle ,| 1\rangle ,| L\rangle \}$, and the frequency ω_nm is the splitting between states n and m. The decoherence rates due to the electron–phonon interaction, ${{\rm{\Gamma }}}_{{jmnk}}^{+}$ and ${{\rm{\Gamma }}}_{{jmnk}}^{-}$, are

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{jmnk}}^{+}=\displaystyle \frac{1}{{{\hslash }}^{2}}{\int }_{0}^{\infty }{\rm{d}}\tau \langle {\bar{H}}_{{\rm{el}}-{\rm{ph}}}^{{jm}}(\tau ){\bar{H}}_{{\rm{el}}-{\rm{ph}}}^{{nk}}(0)\rangle {{\rm{e}}}^{-{\rm{i}}{\omega }_{{nk}}\tau },\end{eqnarray} \tag{ 15 }$

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{jmnk}}^{-}=\displaystyle \frac{1}{{{\hslash }}^{2}}{\int }_{0}^{\infty }{\rm{d}}\tau \langle {\bar{H}}_{{\rm{el}}-{\rm{ph}}}^{{jm}}(-\tau ){\bar{H}}_{{\rm{el}}-{\rm{ph}}}^{{nk}}(0)\rangle {{\rm{e}}}^{-{\rm{i}}{\omega }_{{jm}}\tau },\end{eqnarray} \tag{ 16 }$

where τ is time. Here, the angular brackets denote an average over phonon degrees of freedom, $\langle \hat{O}\rangle ={{\rm{Tr}}}_{{\rm{ph}}}\{\hat{O}{\rho }_{{\rm{ph}}}\}$, where ρ_ph is the thermal equilibrium state of the phonon bath. The notation ${\bar{H}}_{{\rm{el}}-{\rm{ph}}}(\tau )$ indicates that ${\tilde{H}}_{{el}-{ph}}$ is expressed in the interaction representation, defined as ${{\rm{e}}}^{{{\rm{i}}{H}}_{{\rm{ph}}}\tau }{\tilde{H}}_{{\rm{el}}-{\rm{ph}}}{{\rm{e}}}^{-{{\rm{i}}{H}}_{{\rm{ph}}}\tau }$. From equations (15) and (16) we see that Bloch–Redfield formalism takes into account all transitions between the states.

In the following, we use equation (14) to calculate qubit evolution during Larmor and Ramsey pulse sequences, implemented for qubit manipulations [1, 56] and shown in figure 1(d). The Larmor pulse sequence is performed in the following way: the quadrupolar detuning is suddenly pulsed from a large negative value to _q = 0, where the qubit freely precesses for time τ, and then suddenly pulsed back to its initial value. It is used to perform rotations about the $\hat{x}$ axis of the Bloch sphere. For the Ramsey pulse sequence, we first pulse from large negative detuning to _q = 0, allowing it to evolve long enough, ${\tau }_{\tfrac{\pi }{2}}=\pi /(2{\omega }_{10})$, to perform an X(π/2) rotation, then the system is pulsed to a required positive detuning, where it evolves freely for time τ, and then is pulsed back to _q = 0 for another X(π/2) rotation and back to the base detuning. The Ramsey pulse sequence allows us to perform rotations about an axis in the $\hat{x}\mbox{--}\hat{z}$ plane, whose orientation depends on the quadrupolar detuning value during the free evolution step. The combination of Larmor and Ramsey sequences can be used to perform any rotation on the Bloch sphere, i.e. an arbitrary single-qubit gate operation.

To gain insight into the physics of phonon-induced decoherence processes, we consider the evolution of the system during the Larmor and Ramsey pulse sequences in this section. In both cases, we take the initial state to be $| 0\rangle$ (i.e. ${\rho }_{\mathrm{initial}}=| 0\rangle \langle 0|$), defined at _q = −80 μeV. Thus, the Bloch sphere we consider below is defined in the basis of $| 1\rangle$ and $| 0\rangle$ at this detuning. At the end of the pulse sequences the quadrupolar detuning is pulsed back to −80 μeV, where it is measured via projecting the resulting state, ρ_final, onto its initial state, $\mathrm{Tr}\{{\rho }_{\mathrm{final}}{\rho }_{\mathrm{initial}}\}$. We assume that pulsing is very quick and the qubit experiences phonons only during the free precession time τ. Since the decoherence during Larmor rotations is addressed below separately, we neglect interactions with phonons during the X(π/2) portions of the Ramsey pulse sequence. The value of the quadrupolar detuning for the free-evolution part of the Ramsey pulse sequence we consider below, ${\epsilon }_{q}^{\mathrm{opt}}$, corresponds to a rotation about the $(\hat{z}+\hat{x})/\sqrt{2}$ axis. There are two reasons for choosing this value of ${\epsilon }_{q}^{{\rm{opt}}}$ . First, a rotation about the $\hat{z}$ axis would require pulsing ${\epsilon }_{q}\to \infty$, which is not experimentally feasible; a more realistic approach is to perform a composite sequence that yields an effective Z rotation [57]. Second, as we show below, larger values of ${\epsilon }_{q}^{{\rm{opt}}}$ yield faster decoherence rates, so we choose the minimum value of ${\epsilon }_{q}^{{\rm{opt}}}$ that is consistent with [57]. Since we ignore decoherence during the X(π/2) portions of the Ramsey calculations, equation (14) is therefore only solved for the ${\epsilon }_{q}={\epsilon }_{q}^{\mathrm{opt}}$ portion of the sequence. We define decoherence rates Γ_Lar and Γ_Ram for the Larmor and Ramsey pulse sequences, respectively, by fitting the decay of the overlap of the resulting measured state with the target state as a function of evolution time to the exponential form e^−Γτ.

Figure 2 shows the dependence of ${{\rm{\Gamma }}}_{\mathrm{Lar}}$ and Γ_Ram on l and $| t|$. Here, the growth of the rates is mainly due to the growth of $| t| ;$ the change in l apart from its effect on $| t|$ is insignificant. By considering the different contributions to equation (14), we find that the dominant decoherence processes for both pulse sequences are the transitions between the states $| 1\rangle$ and $| 0\rangle$, and between the states $| 1\rangle$ and $| L\rangle$, as indicated in the inset of figure 3. The main contribution comes from the parts of the corresponding matrix elements of ${\tilde{H}}_{{\rm{el}}-{\rm{ph}}}$ containing the terms ${P}_{{RR}}-{P}_{{LL}}$ for $| 1\rangle \leftrightarrow | L\rangle$ and ${P}_{{LL}}+{P}_{{RR}}-2{P}_{{CC}}$ for $| 1\rangle \leftrightarrow | 0\rangle$. These combinations arise naturally from the transformation ${\tilde{H}}_{{\rm{el}}-{\rm{ph}}}={U}_{d}^{-1}{H}_{{\rm{el}}-{\rm{ph}}}{U}_{d}$ and the full expressions for the most important matrix elements are presented in appendix D. Consequently, for the setting _q = 0, which is used for Larmor oscillations, the decoherence rate is proportional to the functions ${J}_{1L}(t)=| t{| }^{3}| \exp \left[\tfrac{-{\rm{i}}{a}\sqrt{2}| t| }{{\hslash }{v}_{t1}}\right]-\exp \left[\tfrac{{\rm{i}}{a}\sqrt{2}| t| }{{\hslash }{v}_{t1}}\right]{| }^{2}$ for the $| 1\rangle \leftrightarrow | L\rangle$ process or ${J}_{10}(t)=| t{| }^{3}| \exp [\tfrac{-{\rm{i}}{a}2\sqrt{2}| t| }{{\hslash }{v}_{t1}}]+\exp \left[\tfrac{{\rm{i}}{a}2\sqrt{2}| t| }{{\hslash }{v}_{t1}}\right]-2{| }^{2}$ for the $| 1\rangle \leftrightarrow | 0\rangle$ process, see appendix F. Here, v_t1 is the speed of sound for transverse acoustic phonons, which dominate over longitudinal phonons for the parameters used. The origins of J₁₀(t) and J_1L(t) are easy to understand: $| t{| }^{2}$ corresponds to the phonon density of states, and another factor of $| t|$ comes from the form of the deformation potential, while the exponential terms reflect the phase differences between the different matrix elements, which arise from the dot separations. Figure 2 shows that the calculated Γ_Lar is well described by the function ${A}_{1}{J}_{1L}(t)+{C}_{1}{J}_{10}(t)$, with fitting constants A₁ and C₁ given in the caption to figure 2. The t-dependence of Γ_Ram fits well to the same functional form, as shown in figure 2, even though its dependence on $| t|$ is more complicated.

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It is possible to tune the phonon-induced decoherence rates by modifying geometrical device parameters. The dependence of Γ_Lar and Γ_Ram on interdot distance is shown in figure 3. Here, we tune a while holding the tunnel coupling t = −8.2 μeV constant; experimentally this can be accomplished by varying gate voltages. We take l = 18.5 nm for all points here, to consider solely effects of changing a. Figure 3 shows that the rates grow with a as a linear combination of power laws a² and a⁴, which is justified by expanding the functions J_1L and J₁₀ in small values of their exponential arguments. For Γ_Ram this expansion is not strictly valid, however we find that a fit with the same power laws seems to work. The physics underlying the growth of Γ_Lar and Γ_Ram with a is that the decoherence is dominated by phonons with wavelengths larger than the triple dot size; as the QD separations are increased, the phonon causes larger shifts within the triple QD, and consequently more decoherence.

The temperature dependence of the decoherence rates is shown in figure 4. Both rates grow very slowly with temperature up until about 0.1 K, particularly Γ_Ram, and then increase linearly with T at higher temperatures. The linear dependence on temperature appears when ${\hslash }{\omega }_{10},{\hslash }{\omega }_{1L}\ll {k}_{B}T$, causing the Bose–Einstein distribution to reduce to a classical distribution. The transition occurs at lower temperatures for Γ_Lar than for Γ_Ram, because the Larmor pulse sequence involves smaller energy splittings than the Ramsey sequence.

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One of the main questions to be answered in this work is whether phonon-mediated noise presents a challenge for charge quadrupole qubits, and how phonon-induced decoherence compares to decoherence from charge noise, which was studied in [31, 32]. Similar to [32], we assume the charge noise to be quasistatic, with fluctuations of the dipolar detuning parameter that follow a Gaussian distribution with a standard deviation of 1 μeV [58], as typical for experiments. A composite $Z(\pi )X(3\pi )Z(-\pi )$ pulse sequence was implemented, to eliminate the effects of leakage to leading order, and the average gate fidelity was calculated as described in [32]. The pulse sequence used in these simulations is shown in figure 6, with additional details given in appendix G. For the phonon noise, we use the same pulse sequence. However, we modified the Ramsey sequence in figure 5 by setting t = 0 during the Z rotation, as proposed in [32]. This has the benefit of suppressing the phonon-induced tunneling during Ramsey oscillations, so that decoherence occurs only during Larmor precession. To compute the average gate fidelity for phonon-induced noise during the free evolution portion of the Larmor sequence, F_ph, we average over initial states using the standard definition ${F}_{{\rm{ph}}}=1/4{\sum }_{\xi }{\rm{Tr}}\{\sqrt{\sqrt{{\rho }_{\xi }^{{\rm{id}}}}{\rho }_{\xi }\sqrt{{\rho }_{\xi }^{{\rm{id}}}}}\}$. Here, ξ are indices denoting the initial states that form a regular tetrahedron on the Bloch sphere [59], and ${\rho }_{\xi }^{{\rm{id}}}$ is a density matrix for the ideal system, i.e. without phonons.

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Results of our calculations are shown in figure 6 for decoherence from both phonons and charge noise. For phonons, the infidelity increases with $| t|$, for the same reason as in figure 2. For charge noise, the fidelity shows the opposite trend, due to the suppression of leakage for large $| t|$, and the broadening of the charge noise sweet spot, which is used to suppress decoherence in the quadrupole qubit [31]. As a result, an optimal working point emerges near t ≃ 17 μeV, corresponding to a maximum fidelity >99.99%.

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In this work, we have considered phonon-induced decoherence of a charge quadrupole qubit to determine its optimal working regime. We find that decoherence rates increase quickly with energy level splitting, due to the strong dependence of the phonon density of states and electron–phonon interaction matrix elements on it. This suggests that large tunnel couplings and quadrupolar detunings should be avoided during qubit operation. However, decoherence caused by charge noise is found to decrease rapidly as the tunnel coupling is increased, due to the suppression of leakage and broadening of the sweet spot. Hence an optimal working point emerges for the tunnel coupling, which we estimate to be t ≃ 17 μeV for typical devices, corresponding to average gate fidelities greater than 99.99%, with X gate periods of 0.13 ns and Z gate periods of 0.055 ns. We also find that smaller dot separations tend to reduce the phonon-induced decoherence; a similar trend was previously observed for the coupling to charge noise [31].

Our calculations here focus on the effects of phonons and charge noise within the three-level system $\{| 1\rangle ,| 0\rangle ,| L\rangle \}$, and we do not explicitly consider other possible sources of decoherence. For example, we do not consider the presence of non-qubit excited states (other than $| L\rangle$), which can provide additional leakage channels when their energy splittings are comparable to the qubit energy. In Si heterostructures, these can include both excited orbital and valley states [60]. To quantify such effects, we note that the orbital excitation energies of the dots considered in this work is > 0.83 meV, while the reported values of valley splittings in similar QDs can be ∼0.12 meV in Si/SiGe heterostructures [61, 62] and 0.3–0.8 meV in Si/SiO₂ heterostructures [63, 64]. In comparison, for the phonon-induced decoherence the largest energy splitting considered here was ∼0.1 meV, while at the optimal working point shown in figure 6, the largest energy splitting was 0.048 meV. For the temperatures usually used for such devices (<100 mK), the dominant phonon-induced decoherence processes are $| 1\rangle \to | 0\rangle$ and $| 1\rangle \to | L\rangle$, accompanied by the emission of a phonon. Compared to these, the transition processes involving absorption of a phonon, $| 0\rangle \to | 1\rangle$ and $| 0\rangle \to | L\rangle$, are strongly suppressed because their transition rates are proportional to the Bose–Einstein distribution function, ${n}_{{\rm{BE}}}=1/(\exp [{\hslash }{\omega }_{{\bf{q}}s}/({k}_{B}T)]-1)$, while emission rates are proportional to ${n}_{{\rm{BE}}}+1$, which is exponentially larger than n_BE. For more details about transition rates see appendix F. Therefore, transitions to even higher energy states are strongly suppressed and can be neglected. However, in some situations, the excited states could lie closer to the qubit subspace, in which case the absorption of a phonon could cause leakage that suppresses the qubit fidelity.

Another source of infidelity not considered in this paper is the possible population of highly excited states due to the non-adiabatic nature of sudden gate pulses. However, infidelity arising from this source can be suppressed greatly by application of shaped pulses, as discussed in [65–67].

In conclusion, we have studied the effects of phonons on the coherence of a charge quadrupole qubit. Our results suggest that phonons can be a dominating source of decoherence in some regimes and should be taken into account if a qubit with this architecture is realized experimentally. We show, however, that there are also regimes, where the combined effects of phonons and quasistatic charge noise lead to qubit infidelities less than 0.01%.

We thank Mark Eriksson, Joydip Ghosh, and Zhenyi Qi for helpful discussions. This work was supported in part by ARO (W911NF-15-1-0248, W911NF-17-1-0274) and the Vannevar Bush Faculty Fellowship program sponsored by the Basic Research Office of the Assistant Secretary of Defense for Research and Engineering and funded by the Office of Naval Research through Grant No. N00014-15-1-0029. The views and conclusions contained here are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the Army Research Office (ARO), or the US Government.

The full wave function of an electron in a triple QD can be represented as a product of a component in the direction of heterostructure growth (z) and a lateral component (x − y), because the confinement potential can be approximated as a sum of z and x − y terms. The lateral components of the wave functions ψ(x, y) are discussed in the main text. Here we summarize our treatment of the z component φ(z). Taking into account both subband and valley physics, a convenient approximate expression for the wave function of an electron in the direction of quantum well confinement is (see e.g., [43, 68])

$\begin{eqnarray}&&\varphi (z)=\displaystyle \frac{\sqrt{2}{\rm{i}}\sin {k}_{0}z}{{\pi }^{1/4}\sqrt{{d}(1-{{\rm{e}}}^{-{k}_{0}^{2}{{d}}^{2}})}}\exp \left[-\displaystyle \frac{{z}^{2}}{2{{d}}^{2}}\right],\end{eqnarray} \tag{ A.1 }$

where k₀ is a wave vector corresponding to the 'valley' minima of the conduction band in Si, ${k}_{0}\approx 0.82\times 2\pi /{a}_{0}$, with the length of the Si cubic cell a₀ = 0.54 nm. The parameter d defines the width of the quantum confinement. As d is typically < 10 nm, the wavelengths of the relevant phonons are much larger than d. Therefore the effect of phonons in the z direction is approximately a constant shift of atoms, and the contribution from the z dimension to the electron–phonon interaction matrix element $\int | \varphi {| }^{2}{{\rm{e}}}^{{{\rm{i}}{q}}_{z}z}{\rm{d}}{z}\approx {{\rm{e}}}^{{{\rm{i}}{q}}_{z}{z}_{0}}\int | \varphi {| }^{2}{\rm{d}}{z}={{\rm{e}}}^{{{\rm{i}}{q}}_{z}{z}_{0}}$, where z₀ is a point near the wave function maximum. Consequently, the contribution from the z dimension to the decoherence rates is roughly the same for any approximation of φ. Therefore in our calculations we further simplify φ as follows:

$\begin{eqnarray}&&\varphi (z)\approx \displaystyle \frac{1}{{\pi }^{1/4}\sqrt{d}}\exp \left[-\displaystyle \frac{{z}^{2}}{2{d}^{2}}\right].\end{eqnarray} \tag{ A.2 }$

Through numerical simulations we have verified that this simplification does not noticeably affect the results obtained in our Fermi Golden rule calculations, and therefore we believe that it also does not affect the results of the Bloch–Redfield formalism.

As described in section 2, the wave functions of an electron in a triple QD Φ_L, Φ_C, and Φ_R are constructed from the harmonic oscillators ground states ψ_L, ψ_C, and ψ_R using a well known procedure described in [47, 48]. We first define the overlap integral $S=\langle {\psi }_{L}| {\psi }_{C}\rangle =\langle {\psi }_{R}| {\psi }_{C}\rangle$. Taking into account that $\langle {\psi }_{L}| {\psi }_{R}\rangle \approx 0$ due to the large distance between left and right dots, we express the electron wave functions in terms of linear combinations of ψ_L, ψ_C, and ψ_R as follows

$\begin{eqnarray}&&{{\rm{\Phi }}}_{L,R}=\displaystyle \frac{{\psi }_{L,R}-{g}_{1}{\psi }_{C}}{{g}_{\mathrm{norm},1}},\end{eqnarray} \tag{ B.1 }$

$\begin{eqnarray}&&{{\rm{\Phi }}}_{C}=\displaystyle \frac{{\psi }_{C}-{g}_{2}{\psi }_{L}-{g}_{2}{\psi }_{R}}{{g}_{\mathrm{norm},2}},\end{eqnarray} \tag{ B.2 }$

and then determine g₁, g₂, ${g}_{\mathrm{norm},1}$, and ${g}_{\mathrm{norm},2}$ by solving the conditions for orthogonalization and normalization of Φ_L, Φ_C, and Φ_R, which are

$\begin{eqnarray}&&\langle {{\rm{\Phi }}}_{L}| {{\rm{\Phi }}}_{R}\rangle =\langle {{\rm{\Phi }}}_{L}| {{\rm{\Phi }}}_{C}\rangle =\langle {{\rm{\Phi }}}_{R}| {{\rm{\Phi }}}_{C}\rangle =0,\end{eqnarray} \tag{ B.3 }$

$\begin{eqnarray}&&\langle {{\rm{\Phi }}}_{L}| {{\rm{\Phi }}}_{L}\rangle =\langle {{\rm{\Phi }}}_{R}| {{\rm{\Phi }}}_{R}\rangle =\langle {{\rm{\Phi }}}_{C}| {{\rm{\Phi }}}_{C}\rangle =1.\end{eqnarray} \tag{ B.4 }$

These conditions yield two solutions. The first solution, g₁ = 0 and g₂ = S, corresponds to localized basis states for the left and right dots, and a delocalized basis state for the center dot. The second solution, which is used in this work, is given by ${g}_{1}=2S$ and ${g}_{2}=g=S/(4{S}^{2}-1)$ (for this solution, ${g}_{\mathrm{norm},1}=1$ and ${g}_{{\rm{norm}},2}=\sqrt{1-4{gS}+2{g}^{2}}$). The latter appears more physical than the first solution because it treats the three dots on an equal footing by introducing a small amount of delocalization in each case. It also appears more analogous to the basis states used in the analysis of double QDs [47]. It would be interesting to compare how these different solutions affect our final results; however, we have not done so here.

From equation (5) in the main text, the definition of tunnel coupling is $t=\langle {{\rm{\Phi }}}_{R}| {H}_{0}| {{\rm{\Phi }}}_{C}\rangle =\langle {{\rm{\Phi }}}_{L}| {H}_{0}| {{\rm{\Phi }}}_{C}\rangle$. We evaluate t by considering the tunneling of a particle in a double-well potential given by $\tfrac{{{\hslash }}^{2}}{2{m}_{\mathrm{eff}}{l}^{4}{a}^{2}}{\left({x}^{2}-\tfrac{{a}^{2}}{4}\right)}^{2}$. This potential includes only two quantum wells, but this is sufficient for our purposes because the third dot is far away, and its effect is exponentially suppressed. To evaluate t we use the wave functions constructed similarly to Φ_R and Φ_C, namely[47, 48]

$\begin{eqnarray}&&{\tilde{{\rm{\Phi }}}}_{L,R}=\displaystyle \frac{{\psi }_{L,R}^{(a/2)}-\tilde{g}{\psi }_{R,L}^{(a/2)}}{\sqrt{1-2\tilde{g}\tilde{S}+{\tilde{g}}^{2}}},\end{eqnarray} \tag{ C.1 }$

$\begin{eqnarray}&&\tilde{S}=\langle {\psi }_{L}^{(a/2)}| {\psi }_{R}^{(a/2)}\rangle ,\end{eqnarray} \tag{ C.2 }$

$\begin{eqnarray}&&\tilde{g}=\displaystyle \frac{1-\sqrt{1-{\tilde{S}}^{2}}}{\tilde{S}},\end{eqnarray} \tag{ C.3 }$

where ${\psi }_{L,R}^{(a/2)}$ are the same as for the x-components of ${\psi }_{R,L}$ in the main text, but shifted along x by a/2, rather than a. Consequently, the formula for the tunnel coupling becomes $\langle {\tilde{{\rm{\Phi }}}}_{L}| -\tfrac{{{\hslash }}^{2}{{\rm{\nabla }}}^{2}}{2{m}_{\mathrm{eff}}}+\tfrac{{{\hslash }}^{2}}{2{m}_{\mathrm{eff}}{l}^{4}{a}^{2}}{\left({x}^{2}-\tfrac{{a}^{2}}{4}\right)}^{2}| {\tilde{{\rm{\Phi }}}}_{R}\rangle$. Then the expression for t is

$\begin{eqnarray}&&t=-\displaystyle \frac{3{{\hslash }}^{2}({a}^{2}+4{l}^{2})}{64{m}_{\mathrm{eff}}{l}^{4}\sinh \tfrac{{a}^{2}}{4{l}^{2}}}.\end{eqnarray} \tag{ C.4 }$

The rapid growth of t with l arises because $t\propto {\sinh }^{-1}\tfrac{{a}^{2}}{4{l}^{2}}$.

We note that as $| {{\rm{\Phi }}}_{L}\rangle$ and $| {{\rm{\Phi }}}_{R}\rangle$ contain ψ_C, there is the tunneling term between them $\langle {{\rm{\Phi }}}_{L}| {H}_{0}| {{\rm{\Phi }}}_{R}\rangle \lesssim S| t| \sim {S}^{2}$. In the parameter regime we consider here, its effect on the spectrum is not significant and will not lead to a noticeable change in phonon-induced decoherence processes.

In this appendix we present a detailed expression for the electron–phonon Hamiltonian in bulk Si; it is given by [50]

$\begin{eqnarray}&&{H}_{{\rm{el}}-{\rm{ph}}}={{\rm{\Xi }}}_{d}{\rm{Tr}}{\boldsymbol{\varepsilon }}+{{\rm{\Xi }}}_{u}{\varepsilon }_{{zz}},\end{eqnarray} \tag{ D.1 }$

where ${\boldsymbol{\varepsilon }}$ is a strain tensor defined as [45, 51]

$\begin{eqnarray}&&{\varepsilon }_{{ij}}=\displaystyle \frac{1}{2}\left(\displaystyle \frac{\partial {u}_{i}}{\partial {r}_{j}}+\displaystyle \frac{\partial {u}_{j}}{\partial {r}_{i}}\right),\end{eqnarray} \tag{ D.2 }$

$\begin{eqnarray}&&{\boldsymbol{u}}=\displaystyle \sum _{{\boldsymbol{q}},s}\sqrt{\displaystyle \frac{\hslash }{2{\rho }_{{\rm{Si}}}{{Vqv}}_{s}}}{{\boldsymbol{e}}}_{{\boldsymbol{q}}s}({b}_{{\boldsymbol{q}},s}{\mp }_{s}{b}_{-{\boldsymbol{q}},s}^{\dagger }){{\rm{e}}}^{{\rm{i}}{\boldsymbol{q}}\cdot {\boldsymbol{r}}},\end{eqnarray} \tag{ D.3 }$

where Ξ_d and Ξ_u are deformation potential constants, ${\boldsymbol{u}}$ is a displacement operator, $s\in \{l,{t}_{1},{t}_{2}\}$ (corresponding to one longitudinal and two transverse acoustic modes), v_s is the speed of sound for each of these modes, ρ_Si is the mass density of bulk Si, V is the volume we consider, ${b}_{{\boldsymbol{q}},s}$ and ${b}_{{\boldsymbol{q}},s}^{\dagger }$ are annihilation and creation operators for phonons of mode s and wave vector ${\boldsymbol{q}}$. The polarization vectors are chosen as follows: ${{\boldsymbol{e}}}_{{\boldsymbol{q}},l}={\boldsymbol{q}}/q$, ${{\boldsymbol{e}}}_{-{\boldsymbol{q}},{t}_{1}}=-{{\boldsymbol{e}}}_{{\boldsymbol{q}},{t}_{1}}$ and ${{\boldsymbol{e}}}_{-{\boldsymbol{q}},{t}_{2}}={{\boldsymbol{e}}}_{{\boldsymbol{q}},{t}_{2}}$ leading to the definition ${\mp }_{l}=-$, ${\mp }_{{t}_{1}}=-$, and ${\mp }_{{t}_{2}}=+$. The explicit expressions for polarizations are

$\begin{eqnarray}&&{{\boldsymbol{e}}}_{{\boldsymbol{q}},l}=\left(\begin{array}{c}\cos {\phi }_{{\boldsymbol{q}}}\sin {\theta }_{{\boldsymbol{q}}}\\ \sin {\phi }_{{\boldsymbol{q}}}\sin {\theta }_{{\boldsymbol{q}}}\\ \cos {\theta }_{{\boldsymbol{q}}}\ \ \ \ \end{array}\right),\end{eqnarray} \tag{ D.4 }$

$\begin{eqnarray}&&{{\boldsymbol{e}}}_{{\boldsymbol{q}},{t}_{1}}=\left(\begin{array}{c}\sin {\phi }_{{\boldsymbol{q}}}\\ -\cos {\phi }_{{\boldsymbol{q}}}\\ 0\ \ \ \ \ \end{array}\right),\end{eqnarray} \tag{ D.5 }$

$\begin{eqnarray}&&{{\boldsymbol{e}}}_{{\boldsymbol{q}},{t}_{2}}=\left(\begin{array}{c}\cos {\phi }_{{\boldsymbol{q}}}\cos {\theta }_{{\boldsymbol{q}}}\\ \sin {\phi }_{{\boldsymbol{q}}}\cos {\theta }_{{\boldsymbol{q}}}\\ -\sin {\theta }_{{\boldsymbol{q}}}\ \ \ \ \end{array}\right),\end{eqnarray} \tag{ D.6 }$

where $0\leqslant {\phi }_{{\boldsymbol{q}}}\lt 2\pi$ and $0\leqslant {\theta }_{{\boldsymbol{q}}}\lt \pi$ are angles describing the phonon wave vector ${\boldsymbol{q}}$ in spherical coordinates ${\boldsymbol{q}}=\left(\begin{array}{c}q\cos {\phi }_{{\boldsymbol{q}}}\sin {\theta }_{{\boldsymbol{q}}}\\ q\sin {\phi }_{{\boldsymbol{q}}}\sin {\theta }_{{\boldsymbol{q}}}\\ q\cos {\theta }_{{\boldsymbol{q}}}\ \ \ \ \end{array}\right)$. Writing equation (D.1) in a more explicit form, we get

$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{H}_{{\rm{el}}-{\rm{ph}}} & = & {\rm{i}}{{\rm{\Xi }}}_{d}\displaystyle \sum _{{\boldsymbol{q}}}\sqrt{\displaystyle \frac{\hslash q}{2{\rho }_{{\rm{Si}}}{{Vv}}_{l}}}({b}_{{\boldsymbol{q}},l}-{b}_{-{\boldsymbol{q}},l}^{\dagger }){{\rm{e}}}^{{\rm{i}}{\boldsymbol{q}}\cdot {\boldsymbol{r}}}\\ & & +{\rm{i}}{{\rm{\Xi }}}_{u}\displaystyle \sum _{{\boldsymbol{q}},s}\sqrt{\displaystyle \frac{\hslash }{2{\rho }_{{\rm{Si}}}{{Vqv}}_{s}}}{q}_{z}{e}_{{\boldsymbol{q}}s}^{z}({b}_{{\boldsymbol{q}},s}{\mp }_{s}{b}_{-{\boldsymbol{q}},s}^{\dagger }){{\rm{e}}}^{{\rm{i}}{\boldsymbol{q}}\cdot {\boldsymbol{r}}}.\end{array}\end{array}\end{eqnarray} \tag{ D.7 }$

As the ${\tilde{H}}_{{\rm{el}}-{\rm{ph}}}={U}_{d}^{-1}{H}_{{\rm{el}}-{\rm{ph}}}{U}_{d}$ matrix elements will be important below, particularly $\langle L| {\tilde{H}}_{{\rm{el}}-{\rm{ph}}}| 1\rangle$ and $\langle 1| {\tilde{H}}_{{\rm{el}}-{\rm{ph}}}| 0\rangle$, we present their explicit forms here:

$\begin{eqnarray}&&\begin{array}{l}\langle L| {\tilde{H}}_{{\rm{el}}-{\rm{ph}}}| 1\rangle \\ \quad =\,\left[2({P}_{{RR}}-{P}_{{LL}})-\displaystyle \frac{1}{t}({\epsilon }_{q}+\sqrt{8{t}^{2}+{\epsilon }_{q}^{2}})({P}_{{LC}}-{P}_{{CR}})\right]\sqrt{\displaystyle \frac{\sqrt{8{t}^{2}+{\epsilon }_{q}^{2}}-{\epsilon }_{q}}{2\sqrt{8{t}^{2}+{\epsilon }_{q}^{2}}}},\end{array}\end{eqnarray} \tag{ D.8 }$

$\begin{eqnarray}&&\begin{array}{l}\langle 0| {\tilde{H}}_{{\rm{el}}-{\rm{ph}}}| 1\rangle \\ \quad =\,\displaystyle \frac{\sqrt{\sqrt{8{t}^{2}+{\epsilon }_{q}^{2}}-{\epsilon }_{q}}}{4t\sqrt{8{t}^{2}+{\epsilon }_{q}^{2}}\sqrt{\sqrt{8{t}^{2}+{\epsilon }_{q}^{2}}+{\epsilon }_{q}}}[t({\epsilon }_{q}+\sqrt{8{t}^{2}+{\epsilon }_{q}^{2}})({P}_{{LL}}+{P}_{{RR}}-2{P}_{{CC}})\\ \quad +\,4{t}^{2}({P}_{{CR}}^{\dagger }+{P}_{{LC}}-{P}_{{LC}}^{\dagger }-{P}_{{CR}})+({P}_{{CR}}^{\dagger }+{P}_{{LC}}){\epsilon }_{q}({\epsilon }_{q}+\sqrt{8{t}^{2}+{\epsilon }_{q}^{2}})].\end{array}\end{eqnarray} \tag{ D.9 }$

Here we derive the equation for the reduced density matrix that describes our three-level system, which we use to obtain decoherence rates from the decay of the occupation of the system's states. We follow the standard procedure for deriving Bloch equations [39, 54, 69].

The Hamiltonian in our problem is given by

$\begin{eqnarray}&&H={H}_{q}+{H}_{{\rm{ph}}}+{H}_{q-{\rm{ph}}},\end{eqnarray} \tag{ E.1 }$

where H_q is the Hamiltonian of a qubit (including the leakage state in our case), H_ph describes the Hamiltonian of the phonons and ${H}_{q-{\rm{ph}}}$ is an interaction between them. The von Neumann equation for the full density matrix in the Schrödinger representation is given by

$\begin{eqnarray}&&\dot{\tilde{\rho }}={\rm{i}}[\tilde{\rho },H],\end{eqnarray} \tag{ E.2 }$

where we set ℏ = 1 for brevity. In the interaction representation defined as ${\tilde{\rho }}_{{\rm{int}}}={{\rm{e}}}^{{\rm{i}}({H}_{q}+{H}_{{\rm{ph}}})t}\tilde{\rho }{{\rm{e}}}^{-{\rm{i}}({H}_{q}+{H}_{{\rm{ph}}})t}$, the von Neumann equation becomes

$\begin{eqnarray}&&{\dot{\tilde{\rho }}}_{{\rm{int}}}={\rm{i}}[{\tilde{\rho }}_{{\rm{int}}},{H}_{q-{\rm{ph}}}^{{\rm{int}}}],\end{eqnarray} \tag{ E.3 }$

and integration yields

$\begin{eqnarray}&&{\tilde{\rho }}_{{\rm{int}}}(t)={\rm{i}}{\int }_{0}^{t}{\rm{d}}\tau [{\tilde{\rho }}_{{\rm{int}}}(\tau ),{H}_{q-{\rm{ph}}}^{{\rm{int}}}(\tau )]+{\tilde{\rho }}_{{\rm{int}}}(0).\end{eqnarray} \tag{ E.4 }$

Substituting into equation (E.3) yields

$\begin{eqnarray}&&{\dot{\tilde{\rho }}}_{{\rm{int}}}(t)=-{\int }_{0}^{t}{\rm{d}}\tau [[{\tilde{\rho }}_{{\rm{int}}}(\tau ),{H}_{q-{\rm{ph}}}^{{\rm{int}}}(\tau )],{H}_{q-{\rm{ph}}}^{{\rm{int}}}(t)]+{\rm{i}}[{\tilde{\rho }}_{{\rm{int}}}(0),{H}_{q-{\rm{ph}}}^{{\rm{int}}}(t)].\end{eqnarray} \tag{ E.5 }$

As we are mainly interested in the three-level system of the qubit with the leakage state, we trace over phonon degrees of freedom to obtain the reduced density matrix:

$\begin{eqnarray}\begin{array}{rcl}{\dot{\tilde{\rho }}}_{{\rm{int}}}^{{\rm{red}}}(t) & = & -{\displaystyle \int }_{0}^{t}{\rm{d}}\tau {{\rm{Tr}}}_{{\rm{ph}}}\{[[{\tilde{\rho }}_{{\rm{int}}}(\tau ),{H}_{q-{\rm{ph}}}^{{\rm{int}}}(\tau )],{H}_{q-{\rm{ph}}}^{{\rm{int}}}(t)]\}\\ & & +{\rm{i}}{{\rm{Tr}}}_{{\rm{ph}}}\{[{\tilde{\rho }}_{{\rm{int}}}(0),{H}_{q-{\rm{ph}}}^{{\rm{int}}}(t)]\}.\end{array}\end{eqnarray} \tag{ E.6 }$

We then apply the Born approximation: ${\tilde{\rho }}_{{\rm{int}}}(t)={\rho }_{{\rm{int}}}^{{\rm{ph}}}(0)\otimes {\rho }_{{\rm{int}}}^{q}(t)$ and the Markov approximation ${\tilde{\rho }}_{{\rm{int}}}(\tau )\to {\tilde{\rho }}_{{\rm{int}}}(t)$ inside the integral. Here we note that ${{\rm{Tr}}}_{{\rm{ph}}}\{[{\rho }_{{\rm{int}}}^{{\rm{ph}}}(0)\otimes {\rho }_{{\rm{int}}}^{q}(0),{H}_{q-{\rm{ph}}}^{{\rm{int}}}(t)]\}=0$ because ${{\rm{Tr}}}_{{\rm{ph}}}\{{\rho }_{{\rm{int}}}^{{\rm{ph}}}{b}_{k}\}=0$, and similarly for ${b}_{k}^{\dagger }$. Therefore only the first term remains in equation (E.6), which after the Born–Markov approximations reads

$\begin{eqnarray}&&{\dot{\rho }}_{{\rm{int}}}^{q}(t)=-{\int }_{0}^{t}{\rm{d}}\tau {{\rm{Tr}}}_{{\rm{ph}}}\{[[{\rho }_{{\rm{int}}}^{{\rm{ph}}}\otimes {\rho }_{{\rm{int}}}^{q}(t),{H}_{q-{\rm{ph}}}^{{\rm{int}}}(\tau )],{H}_{q-{\rm{ph}}}^{{\rm{int}}}(t)]\}.\end{eqnarray} \tag{ E.7 }$

We expand the commutator to obtain

$\begin{eqnarray}\begin{array}{rcl}{\dot{\rho }}_{{\rm{int}}}^{q}(t) & = & -{\displaystyle \int }_{0}^{t}{\rm{d}}\tau {{\rm{Tr}}}_{{\rm{ph}}}\{{\rho }_{{\rm{int}}}^{{\rm{ph}}}{\rho }_{{\rm{int}}}^{q}(t){H}_{q-{\rm{ph}}}^{{\rm{int}}}(\tau ){H}_{q-{\rm{ph}}}^{{\rm{int}}}(t)-{H}_{q-{\rm{ph}}}^{{\rm{int}}}(\tau ){\rho }_{{\rm{int}}}^{{\rm{ph}}}{\rho }_{{\rm{int}}}^{q}(t){H}_{q-{\rm{ph}}}^{{\rm{int}}}(t)\\ & & -{H}_{q-{\rm{ph}}}^{{\rm{int}}}(t){\rho }_{{\rm{int}}}^{{\rm{ph}}}{\rho }_{{\rm{int}}}^{q}(t){H}_{q-{\rm{ph}}}^{{\rm{int}}}(\tau )+{H}_{q-{\rm{ph}}}^{{\rm{int}}}(t){H}_{q-{\rm{ph}}}^{{\rm{int}}}(\tau ){\rho }_{{\rm{int}}}^{{\rm{ph}}}{\rho }_{{\rm{int}}}^{q}(t)\}.\end{array}\end{eqnarray} \tag{ E.8 }$

We then project this equation onto the qubit-leakage basis (dropping the notation 'int' for brevity), yielding:

$\begin{eqnarray}\begin{array}{rcl}{\dot{\rho }}_{{nm}}^{q}(t) & = & -{\displaystyle \int }_{0}^{t}{\rm{d}}\tau {{\rm{Tr}}}_{{\rm{ph}}}\{{\rho }^{{\rm{ph}}}{\rho }_{{nk}}^{q}(t){H}_{q-{\rm{ph}}}^{{kj}}(\tau ){H}_{q-{\rm{ph}}}^{{jm}}(t)-{H}_{q-{\rm{ph}}}^{{nk}}(\tau ){\rho }^{{\rm{ph}}}{\rho }_{{kj}}^{q}(t){H}_{q-{\rm{ph}}}^{{jm}}(t)\\ & & -{H}_{q-{\rm{ph}}}^{{nk}}(t){\rho }^{{\rm{ph}}}{\rho }_{{kj}}^{q}(t){H}_{q-{\rm{ph}}}^{{jm}}(\tau )+{H}_{q-{\rm{ph}}}^{{nk}}(t){H}_{q-{\rm{ph}}}^{{kj}}(\tau ){\rho }^{{\rm{ph}}}{\rho }_{{jm}}^{q}(t)\},\end{array}\end{eqnarray} \tag{ E.9 }$

where the convention of summing over repeated indices is used. We also switch to a different interaction picture in the following way:

$\begin{eqnarray}\begin{array}{rcl}{H}_{q-{\rm{ph}}}^{{nk}} & = & \langle n| {{\rm{e}}}^{{\rm{i}}({H}_{q}+{H}_{{\rm{ph}}})t}{H}_{q-{\rm{ph}}}{{\rm{e}}}^{-{\rm{i}}({H}_{q}+{H}_{{\rm{ph}}})t}| k\rangle \\ & = & {{\rm{e}}}^{{\rm{i}}{\omega }_{{nk}}t}{{\rm{e}}}^{{{\rm{i}}{H}}_{{\rm{ph}}}t}\langle n| {H}_{q-{\rm{ph}}}| k\rangle {{\rm{e}}}^{-{{\rm{i}}{H}}_{{\rm{ph}}}t}={{\rm{e}}}^{{\rm{i}}{\omega }_{{nk}}t}{\bar{H}}_{q-{\rm{ph}}}^{{nk}},\end{array}\end{eqnarray} \tag{ E.10 }$

where the bar denotes the interaction representation with respect to the phonon Hamiltonian only, and ${\omega }_{{nk}}={E}_{n}-{E}_{k}$, where E_n and E_k are the energies of states n and k, respectively. We apply such representation to all the Hamiltonian terms in equation (E.9), obtaining

$\begin{eqnarray}\begin{array}{rcl}{\dot{\rho }}_{{nm}}^{q}(t) & = & -{\displaystyle \int }_{0}^{t}{\rm{d}}\tau {{\rm{Tr}}}_{{\rm{ph}}}\{{\rho }^{{\rm{ph}}}{\rho }_{{nk}}^{q}(t){\bar{H}}_{q-{\rm{ph}}}^{{kj}}(\tau ){\bar{H}}_{q-{\rm{ph}}}^{{jm}}(t){{\rm{e}}}^{{\rm{i}}{\omega }_{{kj}}\tau +{\rm{i}}{\omega }_{{jm}}t}\\ & & -{\bar{H}}_{q-{\rm{ph}}}^{{nk}}(\tau ){\rho }^{{\rm{ph}}}{\rho }_{{kj}}^{q}(t){\bar{H}}_{q-{\rm{ph}}}^{{jm}}(t){{\rm{e}}}^{{\rm{i}}{\omega }_{{nk}}\tau +{\rm{i}}{\omega }_{{jm}}t}\\ & & -{\bar{H}}_{q-{\rm{ph}}}^{{nk}}(t){\rho }^{{\rm{ph}}}{\rho }_{{kj}}^{q}(t){\bar{H}}_{q-{\rm{ph}}}^{{jm}}(\tau ){{\rm{e}}}^{{\rm{i}}{\omega }_{{nk}}t+{\rm{i}}{\omega }_{{jm}}\tau }\\ & & +{\bar{H}}_{q-{\rm{ph}}}^{{nk}}(t){\bar{H}}_{q-{\rm{ph}}}^{{kj}}(\tau ){\rho }^{{\rm{ph}}}{\rho }_{{jm}}^{q}(t){{\rm{e}}}^{{\rm{i}}{\omega }_{{nk}}t+{\rm{i}}{\omega }_{{kj}}\tau }\}.\end{array}\end{eqnarray} \tag{ E.11 }$

We make the substitution $\tau =t-\tau ^{\prime}$, obtaining

$\begin{eqnarray}&&\begin{array}{l}{\dot{\rho }}_{{nm}}^{q}(t)\,=\\ \quad -\,{\displaystyle \int }_{0}^{t}{\rm{d}}\tau ^{\prime} {{\rm{Tr}}}_{{\rm{ph}}}\{{\bar{H}}_{q-{\rm{ph}}}^{{kj}}(-\tau ^{\prime} ){\bar{H}}_{q-{\rm{ph}}}^{{jm}}(0){\rho }^{{\rm{ph}}}{\rho }_{{nk}}^{q}(t){{\rm{e}}}^{{\rm{i}}({\omega }_{{kj}}+{\omega }_{{jm}})t-{\rm{i}}{\omega }_{{kj}}\tau ^{\prime} }\\ \quad -\,{\bar{H}}_{q-{\rm{ph}}}^{{jm}}(\tau ^{\prime} ){\bar{H}}_{q-{\rm{ph}}}^{{nk}}(0){\rho }^{{\rm{ph}}}{\rho }_{{kj}}^{q}(t){{\rm{e}}}^{{\rm{i}}({\omega }_{{nk}}+{\omega }_{{jm}})t-{\rm{i}}{\omega }_{{nk}}\tau ^{\prime} }\\ \quad -\,{\bar{H}}_{q-{\rm{ph}}}^{{jm}}(-\tau ^{\prime} ){\bar{H}}_{q-{\rm{ph}}}^{{nk}}(0){\rho }^{{\rm{ph}}}{\rho }_{{kj}}^{q}(t){{\rm{e}}}^{{\rm{i}}({\omega }_{{nk}}+{\omega }_{{jm}})t-{\rm{i}}{\omega }_{{jm}}\tau ^{\prime} }\\ \quad +\,{\bar{H}}_{q-{\rm{ph}}}^{{nk}}(\tau ^{\prime} ){\bar{H}}_{q-{\rm{ph}}}^{{kj}}(0){\rho }^{{\rm{ph}}}{\rho }_{{jm}}^{q}(t){{\rm{e}}}^{{\rm{i}}({\omega }_{{nk}}+{\omega }_{{kj}})t-{\rm{i}}{\omega }_{{kj}}\tau ^{\prime} }\}.\end{array}\end{eqnarray} \tag{ E.12 }$

Defining

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{jmnk}}^{+}={\int }_{0}^{t}{\rm{d}}\tau ^{\prime} {{\rm{Tr}}}_{{\rm{ph}}}\{{\bar{H}}_{q-{\rm{ph}}}^{{jm}}(\tau ^{\prime} ){\bar{H}}_{q-{\rm{ph}}}^{{nk}}(0){\rho }^{{\rm{ph}}}\}{{\rm{e}}}^{-{\rm{i}}{\omega }_{{nk}}\tau ^{\prime} },\end{eqnarray} \tag{ E.13 }$

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{jmnk}}^{-}={\int }_{0}^{t}{\rm{d}}\tau ^{\prime} {{\rm{Tr}}}_{{\rm{ph}}}\{{\bar{H}}_{q-{\rm{ph}}}^{{jm}}(-\tau ^{\prime} ){\bar{H}}_{q-{\rm{ph}}}^{{nk}}(0){\rho }^{{\rm{ph}}}\}{{\rm{e}}}^{-{\rm{i}}{\omega }_{{jm}}\tau ^{\prime} },\end{eqnarray} \tag{ E.14 }$

our equation takes the form

$\begin{eqnarray}\begin{array}{rcl}{\dot{\rho }}_{{nm}}^{q}(t) & = & -{{\rm{\Gamma }}}_{{kjjm}}^{-}{\rho }_{{nk}}^{q}(t){{\rm{e}}}^{{\rm{i}}({\omega }_{{kj}}+{\omega }_{{jm}})t}+{{\rm{\Gamma }}}_{{jmnk}}^{+}{\rho }_{{kj}}^{q}(t){{\rm{e}}}^{{\rm{i}}({\omega }_{{nk}}+{\omega }_{{jm}})t}\\ & & +{{\rm{\Gamma }}}_{{jmnk}}^{-}{\rho }_{{kj}}^{q}(t){{\rm{e}}}^{{\rm{i}}({\omega }_{{nk}}+{\omega }_{{jm}})t}-{{\rm{\Gamma }}}_{{nkkj}}^{+}{\rho }_{{jm}}^{q}(t){{\rm{e}}}^{{\rm{i}}({\omega }_{{nk}}+{\omega }_{{kj}})t}.\end{array}\end{eqnarray} \tag{ E.15 }$

Note, that we omitted the superscript 'int' here to simplify the notation; however the reduced density matrix is in interaction representation as shown above in equation (E.3). To transform back to the Schrödinger representation we use

$\begin{eqnarray}&&{\rho }_{{nm}}^{{\rm{int}},q}(t)={{\rm{e}}}^{{\rm{i}}{\omega }_{{nm}}t}{\rho }_{{nm}}^{q}(t),\end{eqnarray} \tag{ E.16 }$

$\begin{eqnarray}&&{\partial }_{t}{\rho }_{{nm}}^{{\rm{int}},q}(t)={{\rm{e}}}^{{\rm{i}}{\omega }_{{nm}}t}({\rm{i}}{\omega }_{{nm}}+{\partial }_{t}){\rho }_{{nm}}^{q}(t).\end{eqnarray} \tag{ E.17 }$

Finally, in the Schrödinger representation, we obtain

$\begin{eqnarray}&&{\dot{\rho }}_{{nm}}^{q}(t)=-{\rm{i}}{\omega }_{{nm}}{\rho }_{{nm}}^{q}(t)-{{\rm{\Gamma }}}_{{kjjm}}^{-}{\rho }_{{nk}}^{q}(t)+({{\rm{\Gamma }}}_{{jmnk}}^{+}+{{\rm{\Gamma }}}_{{jmnk}}^{-}){\rho }_{{kj}}^{q}(t)-{{\rm{\Gamma }}}_{{nkkj}}^{+}{\rho }_{{jm}}^{q}(t).\ \ \ \ \ \ \ \ \end{eqnarray} \tag{ E.18 }$

Here, the initial conditions are defined by the desired pulse sequence, as discussed in the main text and figure 1(d). By computing the density matrix as a function of time for a particular initial condition, we can determine the decoherence rates by fitting the decay of the measured state to the exponential form ${{\rm{e}}}^{-{\rm{\Gamma }}t}$.

In this section we use Fermi Golden rule to calculate the transition rates of the relaxation processes between different states. These rates have similar form to equations (E.13) and (E.14), which helps us to better understand the behavior of results obtained from the Bloch–Redfield formalism. We assume that the initial state is one of the qubit states.

The Fermi Golden rule expression for the transition rate from initial state $| i\rangle$ to final state $| f\rangle$ is

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{if}}=\displaystyle \frac{2\pi }{{\hslash }}\displaystyle \sum _{{\boldsymbol{q}},s}| \langle f| {\tilde{H}}_{{\rm{el}}-{\rm{ph}}}^{{\boldsymbol{q}},s}| i\rangle {| }^{2}\delta ({E}_{i}-{E}_{f}\pm {\hslash }{\omega }_{{\boldsymbol{q}},s}),\end{eqnarray} \tag{ F.1 }$

where − corresponds to absorption and + corresponds to emission of a phonon, ${\tilde{H}}_{{\rm{el}}-{\rm{ph}}}={\sum }_{{\boldsymbol{q}},s}{\tilde{H}}_{{\rm{el}}-{\rm{ph}}}^{{\boldsymbol{q}},s}$, and the initial and final states $| i\rangle$ and $| f\rangle$, respectively, describe both the electron and the phonon bath. Here, the initial and final energies of the electron are E_i and E_f, and the frequency ${\omega }_{{\boldsymbol{q}},s}$ corresponds to a phonon that is emitted or absorbed.

The full expressions for the rates are lengthy, therefore we do not include them all here. However we provide one of the shortest of the expressions as an example:

$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{{\rm{\Gamma }}}_{1L} & = & \displaystyle \sum _{s}\displaystyle \int \displaystyle \frac{{\rm{d}}{\theta }_{{\boldsymbol{q}}}{\rm{d}}{\phi }_{{\boldsymbol{q}}}\sin {\theta }_{{\boldsymbol{q}}}}{{(2\pi )}^{2}{\hslash }^{2}{v}_{s}}\\ & & \times | [[{p}_{{CR},s}(q,{\phi }_{{\boldsymbol{q}}},{\theta }_{{\boldsymbol{q}}}){\pm }_{s}{p}_{{LC},s}^{\ast }(q,{\phi }_{{\boldsymbol{q}}}+\pi ,\pi -{\theta }_{{\boldsymbol{q}}})][\sqrt{8{t}^{2}+{\epsilon }_{q}^{2}}+{\epsilon }_{q}]\\ & & {+2t[{p}_{{RR},s}(q,{\phi }_{{\boldsymbol{q}}},{\theta }_{{\boldsymbol{q}}})-{p}_{{LL},s}(q,{\phi }_{{\boldsymbol{q}}},{\theta }_{{\boldsymbol{q}}})]]/[4\sqrt{2}t{(8{t}^{2}+{\epsilon }_{q}^{2})}^{1/4}]| }_{q={q}_{1L}}^{2}\\ & & \times \sqrt{\sqrt{8{t}^{2}+{\epsilon }_{q}^{2}}-{\epsilon }_{q}}{\left[\displaystyle \frac{1}{2\hslash {v}_{s}}(\sqrt{8{t}^{2}+{\epsilon }_{q}^{2}}+{\epsilon }_{q})\right]}^{2}\left[{n}_{{\rm{BE}}}\left(\displaystyle \frac{1}{2}[\sqrt{8{t}^{2}+{\epsilon }_{q}^{2}}+{\epsilon }_{q}]\right)+1\right],\end{array}\end{array}\end{eqnarray} \tag{ F.2 }$

where ${q}_{1L}=\tfrac{\sqrt{8{t}^{2}+{\epsilon }_{q}^{2}}+{\epsilon }_{q}}{2{\hslash }{v}_{s}}$, n_BE is the Bose–Einstein distribution function, and ${p}_{{ij},s}$ is defined in terms of the electron–phonon matrix elements as follows:

$\begin{eqnarray}\begin{array}{c}\begin{array}{rcl}{P}_{{ij}} & = & \displaystyle \sum _{{\boldsymbol{q}}}{p}_{{ij},l}(q,{\phi }_{{\boldsymbol{q}}},{\theta }_{{\boldsymbol{q}}})({b}_{{\boldsymbol{q}},l}-{b}_{-{\boldsymbol{q}},l}^{\dagger })+{p}_{{ij},{t}_{1}}(q,{\phi }_{{\boldsymbol{q}}},{\theta }_{{\boldsymbol{q}}})({b}_{{\boldsymbol{q}},{t}_{1}}-{b}_{-{\boldsymbol{q}},{t}_{1}}^{\dagger })\\ & & +{p}_{{ij},{t}_{2}}(q,{\phi }_{{\boldsymbol{q}}},{\theta }_{{\boldsymbol{q}}})({b}_{{\boldsymbol{q}},{t}_{2}}+{b}_{-{\boldsymbol{q}},{t}_{2}}^{\dagger }).\end{array}\end{array}\end{eqnarray} \tag{ F.3 }$

The relaxation rates are calculated for the following set of Si material parameters: deformation potential constants Ξ_d = 5 eV, Ξ_u = 8.77 eV, longitudinal speed of sound ${v}_{l}=9\times {10}^{3}$ m s⁻¹, transverse speed of sound ${v}_{{t}_{1}}={v}_{{t}_{2}}=5.4\times {10}^{3}$ m s⁻¹, and mass density ${\rho }_{{\rm{Si}}}=2.33\ {\rm{g}}\,{\mathrm{cm}}^{-3}$. The quantum confinement in the z direction is characterized by d = 2 nm, as defined in equation (A.2).

Since we consider the low temperature T = 0.05 K, as consistent with QD experiments performed in a dilution refrigerator, the phonon absorption rates are much slower than emission rates. Mathematically this follows from the fact that emission rates have a n_BE + 1 term, as in the last bracket of equation (F.2), while absorption rates only have n_BE. Therefore, since n_BE becomes very small for low temperatures, emission rates dominate over absorption. These are given by Γ₁₀ between the qubit states $| 1\rangle$ and $| 0\rangle$, and ${{\rm{\Gamma }}}_{1L}$ between the states $| 1\rangle$ and $| L\rangle$, as depicted in figure F1.

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When _q = 0 the main contribution to ${{\rm{\Gamma }}}_{1L}$ in the parameters ranges we consider, comes from the difference ${P}_{{RR}}-{P}_{{LL}}$, appearing in equation (D.8). Similarly, for Γ₁₀, the main contribution comes from the combination ${P}_{{LL}}+{P}_{{RR}}-2{P}_{{CC}}$, appearing in equation (D.9), for nearly all the parameter ranges considered here. The rates ${{\rm{\Gamma }}}_{1L}$ and Γ₁₀ for _q = 0 can be approximated as follows

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{1L}\propto {J}_{1L}=| t{| }^{3}{\left|\exp \left(-\displaystyle \frac{{\rm{i}}\sqrt{2}{ta}}{{\hslash }{v}_{{t}_{1}}}\right)-\exp \left(\displaystyle \frac{{\rm{i}}\sqrt{2}{ta}}{{\hslash }{v}_{{t}_{1}}}\right)\right|}^{2},\end{eqnarray} \tag{ F.4 }$

$\begin{eqnarray}&&{{\rm{\Gamma }}}_{10}\propto {J}_{10}=| t{| }^{3}{\left|\exp \left(-\displaystyle \frac{{\rm{i}}2\sqrt{2}{ta}}{{\hslash }{v}_{{t}_{1}}}\right)+\exp \left(\displaystyle \frac{{\rm{i}}2\sqrt{2}{ta}}{{\hslash }{v}_{{t}_{1}}}\right)-2\right|}^{2}.\end{eqnarray} \tag{ F.5 }$

Here, a factor of $| t{| }^{2}$ comes from the phonon density of states, another factor of $| t|$ comes from the form of the electron–phonon interaction and the exponential terms describe the phase shifts between matrix elements P_LL, P_CC, and P_RR due to the interdot separation a. The transverse phonons dominate for these parameters, therefore the approximation with only ${v}_{{t}_{1}}$ works well. Following the fact that the relaxation processes $| 1\rangle \leftrightarrow | L\rangle$ and $| 1\rangle \leftrightarrow | 0\rangle$ are dominating in the decoherence rates for Larmor and Ramsey pulse sequences, we used J_1L and J₁₀ to fit the results for the dependences of the decoherence rates on $| t|$ and a for figures 2 and 3.

As the dots are gate-defined, the size of each dot can be varied. Here, the transverse size of the dots is characterized by l. Since the experimentally relevant quantity is the tunnel coupling, here we present the dependence of Γ₁₀ and ${{\rm{\Gamma }}}_{1L}$ on $| t|$, whereas t is calculated using geometrical parameters a and l of the triple QD, see figure F2. When $\sqrt{2}{ta}/({\hslash }{v}_{{t}_{1}})\lt 1$, we can expand the exponents in equation (F.4), obtaining the power law $| t{| }^{5}$, whereas for equation (F.5), when $2\sqrt{2}{ta}/({\hslash }{v}_{{t}_{1}})\lt 1$, we get the power law $| t{| }^{7}$.

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Another geometrical parameter we consider is the interdot separation a. We plot the dependence of ${{\rm{\Gamma }}}_{1L}$ and Γ₁₀ on a in figure F3 for _q = 0 and _q = 30 μeV. We see that Γ₁₀ and ${{\rm{\Gamma }}}_{1L}$ are both more than ≃10 times smaller for _q = 0 than for _q = 30 μeV. In the former case we find that the rates depend on interdot distance a as ${{\rm{\Gamma }}}_{1L}\propto {a}^{2}$ and ${{\rm{\Gamma }}}_{10}\propto {a}^{4}$, with very good precision for all a in ${{\rm{\Gamma }}}_{1L}$ and for a > 80 nm in Γ₁₀. This behavior arises from the exponential terms in equations (F.4) and (F.5), which can be expanded to give the power laws a² and a⁴ in the leading order. The behavior of Γ₁₀ for a < 80 nm can be explained as follows. The overlap between wave functions becomes strong enough to cause the off-diagonal matrix elements P_LC, P_CR, ${P}_{{LC}}^{\dagger }$, and ${P}_{{CR}}^{\dagger }$ to contribute significantly to the relaxation rate, therefore modifying the power law. Note that when _q = 30 μeV, the arguments of the exponential factors in equations (F.4) and (F.5) are not small for part of the parameter range, so the expansion is no longer valid.

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The increase of the relaxation rates with a can be explained physically as follows. Phonons induce transitions by perturbing the triple dot potential, as schematically depicted in figure 1(c). Since the phonons responsible for the transitions are long-wavelength, if the triple QD has small dimensions, it experiences the phonon as an approximately uniform shift in energy. If the dots are further apart, the same phonon produces larger shifts between the dots, appearing in the ${H}_{{\rm{el}}-{\rm{ph}}}$ matrix elements, which consequently produces more relaxation.

The dependence of ${{\rm{\Gamma }}}_{1L}$ and Γ₁₀ on quadrupolar detuning is determined by the energy level spacing between the states we consider. As figure F4 shows, the rate Γ₁₀ is at a minimum when the energy splitting between $| 0\rangle$ and $| 1\rangle$ is at a minimum (_q = 0) and exhibits symmetric behavior about _q = 0, as depicted in the inset. In contrast, the energy splitting between $| 1\rangle$ and $| L\rangle$ is asymmetric along the _q axis, yielding an asymmetric dependence of ${{\rm{\Gamma }}}_{1L}$ on _q. We also present in figures F5 and F6 the contour plots for the dependences of Γ₁₀ and ${{\rm{\Gamma }}}_{1L}$ on $| t|$ and _q.

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In this section we present the model we used to calculate quasistatic charge noise-induced infidelity for the charge quadrupole qubit. To find an optimal working point for the charge quadrupole qubit, we considered pulse sequences that reduce leakage effects caused by charge noise, as described in [32]. Following the procedure from [32], we first define

$\begin{eqnarray}{H}_{z}=\displaystyle \frac{{\epsilon }_{q}}{2}\left(\begin{array}{ccc}1 & \ 0 & \ 0\\ 0 & -1 & \ 0\\ 0 & \ 0 & -1\end{array}\right),{H}_{x}=\sqrt{2}t\left(\begin{array}{ccc}0 & 1 & 0\\ 1 & 0 & 0\\ 0 & 0 & 0\end{array}\right),{H}_{\mathrm{leak}}={\epsilon }_{d}\left(\begin{array}{ccc}0 & 0 & 0\\ 0 & 0 & 1\\ 0 & 1 & 0\end{array}\right)\end{eqnarray} \tag{ G.1 }$

in the basis

$\begin{eqnarray}&&| \tilde{C}\rangle =| {{\rm{\Phi }}}_{C}\rangle ,| \tilde{E}\rangle =\displaystyle \frac{| {{\rm{\Phi }}}_{L}\rangle +| {{\rm{\Phi }}}_{R}\rangle }{\sqrt{2}},| \tilde{L}\rangle =\displaystyle \frac{| {{\rm{\Phi }}}_{L}\rangle -| {{\rm{\Phi }}}_{R}\rangle }{\sqrt{2}},\end{eqnarray} \tag{ G.2 }$

where $| \tilde{C}\rangle$ and $| \tilde{E}\rangle$ correspond to logical states and $| \tilde{L}\rangle$ is a leakage state. Here _d represents the charge noise on the gates; we take it to be _d = 1 μeV [58]. The operators that generate Z and X rotations, including charge noise, are defined as follows

$\begin{eqnarray}&&{U}_{z}({\epsilon }_{q},{\epsilon }_{d},\phi )=\exp [-{\rm{i}}[{H}_{z}({\epsilon }_{q})+{H}_{\mathrm{leak}}({\epsilon }_{d})]\phi /{\epsilon }_{q}],\end{eqnarray} \tag{ G.3 }$

$\begin{eqnarray}&&{U}_{x}(t,{\epsilon }_{d},\theta )=\exp [-{\rm{i}}[{H}_{x}(t)+{H}_{\mathrm{leak}}({\epsilon }_{d})]\theta /(2\sqrt{2}t)],\end{eqnarray} \tag{ G.4 }$

for the rotation angles ϕ and θ. We define gate times using ϕ and θ as ${\tau }_{z}=\phi {\hslash }/{\epsilon }_{q}$ and ${\tau }_{x}=\theta {\hslash }/(2\sqrt{2}t)$, which correspond to the case of absence of charge noise. From equation (G.3) follows that during Z rotation we set t to zero.

To construct an effective, leakage-suppressed X(π) rotation, we apply the pulse sequence

$\begin{eqnarray}&&{R}_{\phi ,\theta }={U}_{z}\left({\epsilon }_{q},{\epsilon }_{d},\displaystyle \frac{\phi }{2}\right){U}_{x}(t,{\epsilon }_{d},\theta ){U}_{z}\left(-{\epsilon }_{q},{\epsilon }_{d},-\displaystyle \frac{\phi }{2}\right),\end{eqnarray} \tag{ G.5 }$

and take ϕ = 2π, θ = 3π, and ${\epsilon }_{q}=-t\phi \cot \left(\tfrac{\theta }{4}\right)/\sqrt{2}$, which is equivalent to $Z(\pi )X(3\pi )Z(-\pi )$ in the absence of charge noise. We then define the operation W to be the projection of ${R}_{2\pi ,3\pi }$ onto the 2D logical space [70]. The definition of the average gate fidelity is then

$\begin{eqnarray}&&F=\displaystyle \frac{\mathrm{Tr}({{WW}}^{\dagger })+| \mathrm{Tr}({W}_{\mathrm{target}}^{\dagger }W){| }^{2}}{D(D+1)},\end{eqnarray} \tag{ G.6 }$

where the desired gate operation, W_target = X(π), is in 2D logical space, and D = 2.