Universal quantum computation in waveguide QED using decoherence free subspaces

The general set-up that we consider is depicted in figure 1(a); namely N TLSs, ${\{| g{\rangle }_{n},| e{\rangle }_{n}\}}_{n=1...N}$, placed at positions z_n and coupled to a 1D field with bosonic annihilation operators a_q. Due to the variety of implementations available nowadays, we will keep the discussion as general as possible without making further assumptions on the nature of the TLS and/or 1D waveguides.

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The composite system is described by the Hamiltonian $H={H}_{0}+{H}_{{\rm{I}}}$, where H₀ is the free term given by ${H}_{0}={H}_{\mathrm{qb}}+{H}_{\mathrm{field}}$, (using ${\rm{\hslash }}=1$)

$$\begin{eqnarray}&&{H}_{\mathrm{qb}}={\omega }_{{\rm{a}}}\displaystyle \sum _{n=1}^{N}{\sigma }_{{ee}}^{n},\ {H}_{\mathrm{field}}=\displaystyle \sum _{q}{\omega }_{q}{a}_{q}^{\dagger }{a}_{q},\end{eqnarray} \tag{ 1 }$$

where ${\omega }_{{\rm{a}}}$ is the TLS energy, ${\sigma }_{{ij}}^{n}=| i{\rangle }_{n}\langle j{| }_{n}$ are atomic operators, and ${\omega }_{q}$ is the energy dispersion relation of the waveguide modes. We consider a dipolar coupling of the form

$$\begin{eqnarray}&&{H}_{{\rm{I}}}=\displaystyle \sum _{n}({\sigma }_{{ge}}^{n}E({z}_{n})+{\rm{h.c.}}),\end{eqnarray} \tag{ 2 }$$

with $E(z)={\sum }_{q}{g}_{q}({a}_{q}{{\rm{e}}}^{{\rm{i}}{qz}}+{a}_{q}^{\dagger }{{\rm{e}}}^{-{\rm{i}}{qz}})$, and g_q the single photon coupling constant. When the system-reservoir coupling is weak (Born approximation) and the characteristic timescale of the 1D-reservoir is much faster that the one of the system (Markov approximation), the evolution of ρ, the reduced density matrix for the atom-like system, can be described by a Markovian master equation of the form ${\rm{d}}\rho /{\rm{d}}t={ \mathcal L }[\rho ]$ [25, 26, 33, 34], with the superoperator

$$\begin{eqnarray}&&{ \mathcal L }[\rho ]=\displaystyle \sum _{n,m}{{\rm{\Gamma }}}_{n,m}({\sigma }_{{ge}}^{n}\rho {\sigma }_{{eg}}^{m}-\rho {\sigma }_{{eg}}^{m}{\sigma }_{{ge}}^{n})+{\rm{h.c.}},\end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{n,m}=\displaystyle \frac{{{\rm{\Gamma }}}_{1{\rm{D}}}}{2}{{\rm{e}}}^{{\rm{i}}q({\omega }_{{\rm{a}}})| {z}_{n}-{z}_{m}| }.\end{eqnarray} \tag{ 4 }$$

${{\rm{\Gamma }}}_{1{\rm{D}}}$ is the rate of decay into waveguide modes, that we will assume to be larger than the rate of spontaneous emission into all other modes, ${{\rm{\Gamma }}}^{*}\ll {{\rm{\Gamma }}}_{1{\rm{D}}}$, as this is the regime we are interested in. Moreover, as the propagation lengths of the waveguide modes for many implementations are long (${L}_{{\rm{p}}}\gg {\lambda }_{{\rm{a}}}$), the atoms can be separated several wavelengths apart and can therefore be individually addressed as depicted in figure 1(a). In particular, we assume to control the TLS state through the Hamiltonian (in the interaction picture with respect to H_qb)

$$\begin{eqnarray}&&{H}_{\mathrm{las}}=\displaystyle \sum _{n}\displaystyle \frac{1}{2}({{\rm{\Omega }}}_{n}{\sigma }_{{ge}}^{n}+{\rm{h.c.}})+{{\rm{\Delta }}}_{n}{\sigma }_{{ee}}^{n},\end{eqnarray} \tag{ 5 }$$

where ${{\rm{\Omega }}}_{n}$ is the amplitude of the coherent driving (that we consider to be resonant, i.e., ${\omega }_{{\rm{L}}}={\omega }_{{\rm{a}}}$) which controls the number of excitations of the system, and ${{\rm{\Delta }}}_{n}$ is a phase shift interaction term. The latter can be obtained, e.g., in atomic systems, by adding an off-resonant driving to another excited state $| {e}^{\prime }\rangle$, as depicted in figure 1(b), which results in an Stark shift ${{\rm{\Delta }}}_{n}=| {{\rm{\Omega }}}^{\prime }{| }^{2}/({\omega }_{{\rm{a}}}-{\omega }_{{\rm{L}}}^{\prime })$. In general, the way of implementing ${{\rm{\Omega }}}_{n}$ and ${{\rm{\Delta }}}_{n}$ will depend on the particular system.

For completeness, it is worth mentioning that Λ systems can also be mapped to effective TLS's by using an off-resonant Raman transition as depicted in figure 1(c). By adiabatically eliminating the excited state $| {e}^{\prime }\rangle$, one can formally project the dynamics to the two metastable states, $\{| g\rangle ,| e\rangle \}$, and find a similar light–matter Hamiltonian as the one of equation (2), with the advantage that the effective TLS defined by $\{| g\rangle ,| e\rangle \}$ will be long-lived as they are encoded in metastable states. For example, by switching both ${{\rm{\Omega }}}_{g}$ and ${{\rm{\Omega }}}_{e}$ at the same time with detuning $\delta (\gg | {{\rm{\Omega }}}_{g}| ,| {{\rm{\Omega }}}_{e}| )$ as depicted in figure 1(c), we can implement a coherent driving term with effective amplitude ${\rm{\Omega }}=\tfrac{{{\rm{\Omega }}}_{g}{{\rm{\Omega }}}_{e}^{*}}{4\delta }$. By switching δ in this case big enough one can neglect spontaneous emission processes as they will be proportional to ${{\rm{\Gamma }}}^{*}\left(\tfrac{| {{\rm{\Omega }}}_{e}{| }^{2}+| {{\rm{\Omega }}}_{g}{| }^{2}}{4{\delta }^{2}}\right)$. Moreover, if we switch only ${{\rm{\Omega }}}_{e}$ and adiabatically eliminate the photonic modes we also obtain an irreversible transition from $| e\rangle \to | g\rangle$, but with a renormalization of the decay rates ${{\rm{\Gamma }}}_{1{\rm{D}}}\to {{\rm{\Gamma }}}_{1{\rm{D}}}| \tfrac{{{\rm{\Omega }}}_{e}}{2\delta }{| }^{2}$ and ${{\rm{\Gamma }}}^{*}\to {{\rm{\Gamma }}}^{*}| \tfrac{{{\rm{\Omega }}}_{e}}{2\delta }{| }^{2}$. Hence, the Purcell factor ${P}_{1{\rm{D}}}={{\rm{\Gamma }}}_{1{\rm{D}}}/{{\rm{\Gamma }}}^{*}$ is unchanged. In that situation our analysis is an alternative implementation to the one developed in [29–31].

In the case of equidistant spacing at positions commensurate with the wavelength of the guided mode, i.e., ${z}_{n}=n2\pi /q({\omega }_{{\rm{a}}})$, the effective interaction induced by the waveguide modes yields a pure Dicke model [35] decay described by

$$\begin{eqnarray}&&{{ \mathcal L }}_{{\rm{D}}}[\rho ]=\displaystyle \frac{{{\rm{\Gamma }}}_{1{\rm{D}}}}{2}({S}_{{ge}}\rho {S}_{{eg}}-{S}_{{eg}}{S}_{{ge}}\rho )+{\rm{h.c.}},\end{eqnarray} \tag{ 6 }$$

where we have introduced the collective spin operator ${S}_{{ge}}={\sum }_{n=1}^{N}{\sigma }_{{ge}}^{n}$. The states satisfying ${S}_{{ge}}| {\rm{\Psi }}\rangle =0$ are decoherence-free with respect to the collective dissipation ${{ \mathcal L }}_{{\rm{D}}}$. These states can be easily described in the collective spin basis $\{| J,{m}_{J},{\alpha }_{J}\rangle \}$, that is the eigenstates of the collective operators ${S}^{2}={\sum }_{i=x,y.z}{S}_{i}^{2}$ and S_z with

$$\begin{eqnarray}&&{S}^{2}| J,{m}_{J},{\alpha }_{J}\rangle =J(J+1)| J,{m}_{J},{\alpha }_{J}\rangle ,\end{eqnarray} \tag{ 7a }$$

$$\begin{eqnarray}&&{S}_{z}| J,{m}_{J},{\alpha }_{J}\rangle ={m}_{J}| J,{m}_{J},{\alpha }_{J}\rangle ,\end{eqnarray} \tag{ 7b }$$

where $J=N/2,N/2-1,\ldots \mathrm{and}{m}_{J}=-J,-J+1,\ldots ,J$. The index ${\alpha }_{J}$ is introduced because the states in the collective spin basis are degenerate, with degeneracy given by: ${\alpha }_{J}=1,\ldots ,\left(\begin{array}{c}N\\ J\end{array}\right)-\left(\begin{array}{c}N\\ J-1\end{array}\right)$. It is easy to observe in this basis that the states $| J,-J,{\alpha }_{J}\rangle$ are decoherence free, and therefore span the DFS.

The DFS has a dimension of $\left(\begin{array}{c}N\\ N/2\end{array}\right)$ (assuming even atom number N), and is composed of all the possible states which are antisymmetric with the permutation of two atoms. Thus, an alternative way of characterizing the DFS is to consider all possible (tensor products) of singlet states

$$\begin{eqnarray}&&| {A}_{m,n}\rangle =(| e{\rangle }_{m}\otimes | g{\rangle }_{n}-| g{\rangle }_{m}\otimes | e{\rangle }_{n})/\sqrt{2},\end{eqnarray} \tag{ 8 }$$

where $m,n$ denote the atomic positions of the pair of atoms that form the singlet. This characterization makes it more difficult to describe an orthonormal basis of the DFS. However, we show in the next section that it is convenient to define our computational subspace.

We are interested in the regime where the collective dissipation induced by ${{ \mathcal L }}_{{\rm{D}}}$, with characteristic timescale ${{\rm{\Gamma }}}_{1{\rm{D}}}^{-1}$, dominates over any possible perturbation of the system, ${{ \mathcal L }}_{\mathrm{pert}}$, with characteristic timescale $\tau \gg {{\rm{\Gamma }}}_{1{\rm{D}}}^{-1}$. Under these assumptions, any state outside of the DFS will only be virtually populated due to the strong dissipation and therefore the dynamics will be restricted to the slow subspace, i.e., the DFS. Mathematically, we formalize this intuitive picture by defining a projection superoperator ${\mathbb{P}}$ (with ${{\mathbb{P}}}^{2}={\mathbb{P}}$) satisfying: ${\mathbb{P}}{{ \mathcal L }}_{{\rm{D}}}=$ ${{ \mathcal L }}_{{\rm{D}}}{\mathbb{P}}=0$ that projects out the fast dynamics yielding only the effective evolution in the slow subspace. It is then possible to integrate out the fast dynamics (see appendix) arriving to an effective master equation given by

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\mathbb{P}}\rho }{\partial t}={{ \mathcal L }}_{\mathrm{eff}}[{\mathbb{P}}\rho ]=\left({\mathbb{P}}{{ \mathcal L }}_{\mathrm{pert}}{\mathbb{P}}+{\mathbb{P}}{{ \mathcal L }}_{\mathrm{pert}}{\mathbb{Q}}\displaystyle \frac{1}{-{{ \mathcal L }}_{{\rm{D}}}}{\mathbb{Q}}{{ \mathcal L }}_{\mathrm{pert}}{\mathbb{P}}+{\rm{O}}({\tau }^{-3}/{{\rm{\Gamma }}}_{1{\rm{D}}}^{2})\right)\rho .\end{eqnarray} \tag{ 9 }$$

This result to first order (left-hand term in the brackets) accounts for the ideal quantum Zeno dynamics [27, 28, 32]. The second order in perturbation theory then yields correction terms mainly coming from slightly populating the (super)radiant states.

In our case, there will be two types of perturbations, namely,

• The Hamiltonian ${{ \mathcal L }}_{\mathrm{pert}}[\cdot ]=-{\rm{i}}[{H}_{\mathrm{las}},\cdot ]$ to control the atomic state. This results to first order in an effective Hamiltonian ${H}_{\mathrm{eff}}={ \mathcal P }{H}_{\mathrm{las}}{ \mathcal P }$ that couples only atomic states within the DFS. Here, we introduced the projection onto the DFS for pure states ${ \mathcal P }={\sum }_{i}| {d}_{i}\rangle \langle {d}_{i}|$, where the states $| {d}_{i}\rangle$ form an orthonormal basis of the DFS. We use this effective laser coupling to control the atomic state of the ensemble. Besides, there is a second order correction resulting from ${H}_{\mathrm{las}}$ that will be relevant for the analysis of the error probability of our proposal as we show in section 4.
• The contribution of the emission of photons to other radiative modes different from the guided mode of the waveguide that we embed into a single decay rate, ${{\rm{\Gamma }}}^{*}$ and describe through the Liouvillian
  $$\begin{eqnarray}&&{{ \mathcal L }}_{\mathrm{pert}}[\rho ]={{ \mathcal L }}_{*}[\rho ]=\displaystyle \sum _{n}\displaystyle \frac{{{\rm{\Gamma }}}^{*}}{2}({\sigma }_{{ge}}^{n}\rho {\sigma }_{{eg}}^{n}-\rho {\sigma }_{{ee}}^{n}+{\rm{h.c.}}).\end{eqnarray} \tag{ 10 }$$
  This contribution is relevant for the error analysis of the gates in section 4.

Before finding the appropriate gates, we first need to define the logical qubits within the DFS. In principle, assuming an even number of atoms N, the dimension of the DFS is $\left(\begin{array}{c}N\\ N/2\end{array}\right)$ and therefore, it is possible to encode ${\mathrm{log}}_{2}\left(\begin{array}{c}N\\ N/2\end{array}\right)$ logical qubits. However, it is more useful to restrict the computational subspace to a smaller set of states in order to achieve universal quantum computation. As the DFS is spanned by all (tensor) products of singlet states over two atoms, it is natural to define the logical qubits as

$$\begin{eqnarray}&&| 0{\rangle }_{j}^{{\rm{L}}}\equiv | {G}_{j,j+1}\rangle =| g{\rangle }_{j}\otimes | g{\rangle }_{j+1},\end{eqnarray} \tag{ 11a }$$

$$\begin{eqnarray}&&| 1{\rangle }_{j}^{{\rm{L}}}\equiv | {A}_{j,j+1}\rangle =(| e{\rangle }_{j}\otimes | g{\rangle }_{j+1}-| g{\rangle }_{j}\otimes | e{\rangle }_{j+1})/\sqrt{2}.\end{eqnarray} \tag{ 11b }$$

It is instructive to consider particular examples to see how the DFS and computational space look, i.e., for the case of N = 2 and N = 4 atoms.

Two atoms: In this case it is easy to plot the complete Hilbert space (including states outside DFS) as it consists only of 4 states as depicted in figure 2(a). The separation into DFS states and non-DFS states is easily done in the familiar singlet-triplet basis. The DFS consists of two states: the one with two atoms in the ground state and the singlet state, i.e., the antisymmetric combination of one single excited state. Thus, we can encode one logical decoherence free qubit. The other two states are superradiant, i.e., they decay with an enhanced decay rate of $2{{\rm{\Gamma }}}_{1{\rm{D}}}$.

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Four atoms: The complete Hilbert space consists of ${2}^{4}=16$ states, and the dimension of the DFS, shown in figure 2(b), is $\left(\begin{array}{c}4\\ 2\end{array}\right)=6$. As aforementioned, we want to use as computational subspace the tensor product of the antisymmetric pairs described in equations (11a) and (11b) , which consists only of ${2}^{2}=4$ states. This is why in this situation we have to distinguish within the DFS between the computational space and the additional states that must be either decoupled or used as auxiliary states. In particular, for N = 4 the auxiliary states are $| {A}_{\mathrm{1,2}}{G}_{\mathrm{3,4}}+{A}_{\mathrm{3,4}}{G}_{\mathrm{1,2}}\rangle /\sqrt{3}$ and $| {A}_{\mathrm{1,3}}{A}_{\mathrm{2,4}}+{A}_{\mathrm{1,4}}{A}_{\mathrm{2,3}}\rangle /\sqrt{2}$.

For $N\gt 4$ atoms: In general (for $N\gt 2$) the dimension of the DFS, $\left(\begin{array}{c}N\\ N/2\end{array}\right)$ (for even N), is larger than the one of the computational subspace, ${2}^{N/2}$. Thus, one can split the projection onto the DFS, ${ \mathcal P }$, into two orthogonal projections, i.e. one into the computational subspace ${{ \mathcal P }}_{\mathrm{CS}}$ and its orthogonal counterpart ${{ \mathcal Q }}_{\mathrm{CS}}$:

$$\begin{eqnarray}&&{ \mathcal P }={{ \mathcal P }}_{\mathrm{CS}}+{{ \mathcal Q }}_{\mathrm{CS}},\end{eqnarray} \tag{ 12a }$$

$$\begin{eqnarray}&&{{ \mathcal P }}_{\mathrm{CS}}=\displaystyle \sum _{j\ \mathrm{odd}}| 0{\rangle }_{j}^{{\rm{L}}}\langle 0| +| 1{\rangle }_{j}^{{\rm{L}}}\langle 1| ,\end{eqnarray} \tag{ 12b }$$

such that the effective Hamiltonian can be written as follows

$$\begin{eqnarray}&&{H}_{\mathrm{eff}}={{ \mathcal P }}_{\mathrm{CS}}{H}_{\mathrm{las}}{{ \mathcal P }}_{\mathrm{CS}}+({{ \mathcal P }}_{\mathrm{CS}}{H}_{\mathrm{las}}{{ \mathcal Q }}_{\mathrm{CS}}+{\rm{h.c.}})+{{ \mathcal Q }}_{\mathrm{CS}}{H}_{\mathrm{las}}{{ \mathcal Q }}_{\mathrm{CS}},\end{eqnarray} \tag{ 13 }$$

which separates the transitions within the computational (auxiliary) subspace ${{ \mathcal P }}_{\mathrm{CS}}H{{ \mathcal P }}_{\mathrm{CS}}$ ${\rm{[}}{{ \mathcal Q }}_{\mathrm{CS}}H{{ \mathcal Q }}_{\mathrm{CS}}{\rm{]}}$ and the coupling between these two subspaces: ${{ \mathcal P }}_{\mathrm{CS}}H{{ \mathcal Q }}_{\mathrm{CS}}$. This separation will be useful to argue that we can make operations within the DFS even in the situations with $N\gt 4$ as we will show afterwards. For a general situation, it is easy to show that by projecting ${H}_{\mathrm{las}}$ into the computational subspace we obtain an effective evolution inside the computational subspace given by:

$$\begin{eqnarray}&&{{ \mathcal P }}_{\mathrm{CS}}{H}_{\mathrm{las}}{{ \mathcal P }}_{\mathrm{CS}}=\displaystyle \sum _{j\ \mathrm{odd}}\left(\displaystyle \frac{{{\rm{\Omega }}}_{j,j+1}^{-}}{\sqrt{2}}| 1{\rangle }_{j}^{{\rm{L}}}\langle 0| +{\rm{h.c.}}\right)+{{\rm{\Delta }}}_{j,j+1}^{+}| 1{\rangle }_{j}^{{\rm{L}}}\langle 1| ,\end{eqnarray} \tag{ 14 }$$

where we used the following notation ${x}_{i,j}^{\pm }=\tfrac{1}{2}({x}_{i}\pm {x}_{j})$ to abbreviate the combination of parameters.

The goal is to find the $\{{{\rm{\Omega }}}_{n},{{\rm{\Delta }}}_{n}\}$ such that they define both the phase and Pauli-X (and Y) gates over the computational subspace.

Two atoms: This is the simplest situation because the size of the computational space is the same as the one of the DFS. In this case (see also [30]), a phase shift gate on the logical qubit ($\alpha | 0{\rangle }^{{\rm{L}}}+\beta | 1{\rangle }^{{\rm{L}}}\to$ $\alpha | 0{\rangle }^{{\rm{L}}}+\beta {{\rm{e}}}^{-{\rm{i}}\phi }| 1{\rangle }^{{\rm{L}}}$) is obtained by applying ${{\rm{\Omega }}}_{12}^{-}=0$, ${{\rm{\Delta }}}_{12}^{+}\ne 0$ for a time $T=\tfrac{\phi }{{{\rm{\Delta }}}_{12}^{+}}$. Pauli-X rotations (plus a phase) are obtained for $0\ne {{\rm{\Omega }}}_{12}^{-}\in {\mathbb{R}},{{\rm{\Delta }}}_{12}^{+}=0$ and time $T=\tfrac{\pi }{\sqrt{2}{{\rm{\Omega }}}_{12}^{-}}$. Note that to avoid errors in both cases, one should also set ${{\rm{\Omega }}}_{12}^{+}={{\rm{\Delta }}}_{12}^{-}=0$ as will be discussed in section 4. The Pauli Y can be obtained as the X just by using ${\rm{i}}{{\rm{\Omega }}}_{12}^{-}\in {\mathbb{R}}$, so that we will not discuss it further.

Four atoms: In this case, the way to do phase gates and rotations is the same as for the two atom case. However, in the case of the rotations, states within the computational subspace couple to auxiliary states for more than two atoms. In particular, the state $| 10{\rangle }^{{\rm{L}}}$ is coupled to the auxiliary state $| {A}_{\mathrm{1,3}}{A}_{\mathrm{2,4}}+{A}_{\mathrm{1,4}}{A}_{\mathrm{2,3}}\rangle /\sqrt{2}$ for ${{\rm{\Omega }}}_{12}^{-}\ne 0$ as shown in figure 2. However, this transition can be made far off-resonance by setting $| {{\rm{\Delta }}}_{34}^{+}| \gg | {{\rm{\Omega }}}_{12}^{-}|$. This results in an additional error rate $\tfrac{| {{\rm{\Omega }}}_{12}^{-}{| }^{2}}{2{{\rm{\Delta }}}_{34}^{+}}$ that will be considered when calculating the fidelity of the operation⁴ .

For $N\gt 4$ atoms: Again in the case of rotations, transitions to states outside the computational subspace in the ideal case (${{\rm{\Omega }}}_{12}^{+}={{\rm{\Delta }}}_{12}^{-}=0$) are possible when ${{\rm{\Omega }}}_{12}^{-}\ne 0$, that is when

$$\begin{eqnarray}&&{{ \mathcal Q }}_{\mathrm{CS}}H{{ \mathcal P }}_{\mathrm{CS}}={{ \mathcal Q }}_{\mathrm{CS}}{H}_{\mathrm{eff}}{{ \mathcal P }}_{\mathrm{CS}}\ne 0,\end{eqnarray} \tag{ 15 }$$

where we use that ${{ \mathcal P }}_{\mathrm{CS}}{ \mathcal P }={{ \mathcal P }}_{\mathrm{CS}}$. However, the transitions to these states can be made far off-resonant by setting

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{n}=0,\mathrm{and}\ | {{\rm{\Omega }}}_{12}^{-}| \ll {{\rm{\Delta }}}_{n}={{\rm{\Delta }}}_{{\rm{D}}}\ll {{\rm{\Gamma }}}_{1{\rm{D}}},n\geqslant 3,\end{eqnarray} \tag{ 16 }$$

because the auxiliary states inside the DFS that the computational subspace couples to extend over more than two atoms⁵ and can therefore be detuned as

$$\begin{eqnarray}&&{{ \mathcal Q }}_{\mathrm{CS}}H{{ \mathcal Q }}_{\mathrm{CS}}\sim {{\rm{\Delta }}}_{D}{{ \mathcal Q }}_{\mathrm{CS}},\end{eqnarray} \tag{ 17 }$$

while keeping the desired transition driven by ${{\rm{\Omega }}}_{12}^{-}$ as resonant. One has to make sure that the Stark-shift introduced by this off-resonant transition is small and possibly correct the detuning that it will induce by choosing appropriately the applied laser frequency ${\omega }_{{\rm{L}}}$.

For universal quantum computation, a controlled two-qubit gate is required. In this case, the minimal system to encode the operation is the N = 4 atom case, where two decoherence-free logical qubits can be obtained.

Four atoms: In order to build the controlled-Z gate, we use one of the auxiliary states, $| {A}_{\mathrm{1,2}}{G}_{\mathrm{3,4}}+{A}_{\mathrm{3,4}}{G}_{\mathrm{1,2}}\rangle /\sqrt{2}$. Now, it is possible to drive only the transition between this state and $| 10{\rangle }^{{\rm{L}}}$ without affecting the other states within the DFS by the choice ${{\rm{\Omega }}}_{n}=0,{{\rm{\Delta }}}_{3}={{\rm{\Delta }}}_{4}=0$ and ${{\rm{\Delta }}}_{1}=-{{\rm{\Delta }}}_{2}\ne 0$. A π-pulse on the state $| 10{\rangle }^{{\rm{L}}}$ leads to a relative phase of −1 on this state, i.e.

$$\begin{eqnarray}&&| 10{\rangle }_{1,3}^{{\rm{L}}}\longrightarrow -| 10{\rangle }_{1,3}^{{\rm{L}}},\end{eqnarray} \tag{ 18 }$$

for $\tfrac{1}{\sqrt{2}}{{\rm{\Delta }}}_{12}^{-}T=2\pi$ without affecting the other states of the computational subspace. Hence, we have defined a a controlled controlled-$(-Z)$ gate which is equivalent up to single qubit unitaries to a CNOT-gate [36].

For $N\gt 4$ atoms: One can restrict the dynamics to the subspace of four atoms in a similar way as for the single-qubit rotations. With the choice of

$$\begin{eqnarray}&&{{\rm{\Omega }}}_{n}=0,\mathrm{and}\ | {{\rm{\Delta }}}_{12}^{-}| \ll {{\rm{\Delta }}}_{n}={{\rm{\Delta }}}_{{\rm{D}}}\ll {{\rm{\Gamma }}}_{1{\rm{D}}},n\geqslant 5,\end{eqnarray} \tag{ 19 }$$

transitions to states over more than four atoms are far off-resonant. As before, this adds an error rate proportional to $\tfrac{| {{\rm{\Delta }}}_{12}^{-}{| }^{2}}{{{\rm{\Delta }}}_{{\rm{D}}}}$ with a proportionality factor depending on the coupling strength after the projection onto the DFS.

For the phase shift gate we must set ${{\rm{\Omega }}}_{n}=0$ for all n and ${{\rm{\Delta }}}_{n}={{\rm{\Delta }}}_{{\rm{D}}}\gg {{\rm{\Delta }}}_{12}^{-}$ for all $n\geqslant 3$ to avoid errors from transitions to auxiliary states. By choosing ${{\rm{\Delta }}}_{1}={{\rm{\Delta }}}_{2}$, i.e., ${{\rm{\Delta }}}_{12}^{-}=0$, no errors (from second order perturbation theory) occur because the computational states do not couple to the radiant ones. However, it is instructive to consider the errors that appear for situations where ${{\rm{\Delta }}}_{12}^{-}\ne 0$, e.g, because of imperfect addressing, as this yields a useful understanding about how to deal with situations where the second order correction cannot be avoided.

Two atoms: For the simplest situation the additional errors due to imperfect addressing, i.e., ${{\rm{\Delta }}}_{12}^{-}\ne 0$, enter at a rate proportional to $| {{\rm{\Delta }}}_{12}^{-}{| }^{2}/{{\rm{\Gamma }}}_{1{\rm{D}}}$ through the same error channel as the spontaneous emission into all other modes with rate ${{\rm{\Gamma }}}^{*}$, that is, via the quantum jump operator $| 0{\rangle }^{{\rm{L}}}\langle 1|$. Then, the infidelity, i.e., $1-F$, for a $\pi /2$ phase shift on the normalized state $\alpha | 0{\rangle }^{{\rm{L}}}+\beta | 1{\rangle }^{{\rm{L}}}$ can be approximated by

$$\begin{eqnarray}&&1-F\approx \displaystyle \frac{| \beta {| }^{2}}{4}\displaystyle \frac{\pi }{{{\rm{\Delta }}}_{12}^{+}}\left({{\rm{\Gamma }}}^{*}+4\displaystyle \frac{| {{\rm{\Delta }}}_{12}^{-}{| }^{2}}{{{\rm{\Gamma }}}_{1{\rm{D}}}}\right).\end{eqnarray} \tag{ 20 }$$

One observes, that in the ideal case, ${{\rm{\Delta }}}_{12}^{-}=0$, the infidelity can be arbitrarily close to 0 for large ${{\rm{\Delta }}}_{12}^{+}$. If ${{\rm{\Delta }}}_{12}^{-}$ is not negligible, the transition strength ${{\rm{\Delta }}}_{12}^{+}$ cannot be chosen arbitrarily large to decrease the infidelity. For example, in the worst case scenario where ${{\rm{\Delta }}}_{12}^{-}={{\rm{\Delta }}}_{12}^{+}$ this results in an optimal infidelity scaling $\tfrac{| \beta {| }^{2}\pi }{2}{P}_{1{\rm{D}}}^{-1/2}$ for ${{\rm{\Delta }}}_{12}^{+}={{\rm{\Delta }}}_{12}^{-}=\tfrac{1}{2}\sqrt{{{\rm{\Gamma }}}^{*}{{\rm{\Gamma }}}_{1{\rm{D}}}}$.

Four atoms: A similar behavior can be obtained by choosing ${{\rm{\Delta }}}_{12}^{-}=0$ such that the infidelity is arbitrarily close to 0 (see figure 3(a)). Slight deviations from this ideal value do not change this behavior drastically (see figure 3(b)). However, when ${{\rm{\Delta }}}_{12}^{-}$ is not negligible, it leads to two types of errors that decrease the fidelity (see figure 3(c)): (i) virtual population of non-DFS states, which leads to an error rate proportional to $| {{\rm{\Delta }}}_{12}^{-}{| }^{2}/{{\rm{\Gamma }}}_{1{\rm{D}}}$ as for two atoms; and (ii) transitions to auxiliary states, in particular $| {A}_{\mathrm{1,2}}{G}_{\mathrm{3,4}}+{A}_{\mathrm{3,4}}{G}_{\mathrm{1,2}}\rangle /\sqrt{2}$. The latter can be made far off-resonance by applying a detuning on the second qubit such that $| {{\rm{\Delta }}}_{12}^{-}| \ll | {{\rm{\Delta }}}_{34}^{+}| \ll {{\rm{\Gamma }}}_{1{\rm{D}}}$. With a large off-resonance ratio ${r}_{{\rm{\Delta }}}=| {{\rm{\Delta }}}_{34}^{+}/{{\rm{\Delta }}}_{12}^{-}| \gg 1$, one still achieves a small infidelity (see figure 3(d)).

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As shown in figures 3(a)–(d), the detuning of the third and fourth atom is important when ${{\rm{\Delta }}}_{12}^{-}$ cannot be neglected. As expected, the infidelity decreases by increasing the off-resonance ratio r_Δ (see figure 4(a)). For large enough ${r}_{{\rm{\Delta }}}$, the infidelity can be analytically approximated by

$$\begin{eqnarray}&&1-F\overset{{r}_{{\rm{\Delta }}}\to \infty }{\longrightarrow }\displaystyle \frac{\pi }{8}\left(\displaystyle \frac{{{\rm{\Gamma }}}^{*}}{| {{\rm{\Delta }}}_{12}^{+}| }+2\displaystyle \frac{| {{\rm{\Delta }}}_{12}^{+}| }{{{\rm{\Gamma }}}_{1{\rm{D}}}}\right).\end{eqnarray} \tag{ 21 }$$

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This leads to a minimal infidelity $\propto \;{P}_{1{\rm{D}}}^{-1/2}$ (see figure 4(b)) for ${{\rm{\Delta }}}_{12}^{+}={{\rm{\Delta }}}_{12}^{-}=\sqrt{{{\rm{\Gamma }}}^{*}{{\rm{\Gamma }}}_{1{\rm{D}}}/2}$.

For rotations around the x-axis, we set ${{\rm{\Delta }}}_{1}={{\rm{\Delta }}}_{2}=0$, and ${{\rm{\Delta }}}_{n}={{\rm{\Delta }}}_{{\rm{D}}}\gg {{\rm{\Omega }}}_{12}^{-}$ for all $n\geqslant 3$ to avoid transitions to auxiliary states. In contrast to the phase shift gate, even in the ideal case, ${{\rm{\Omega }}}_{12}^{+}=0$, errors will occur because ${{\rm{\Omega }}}_{12}^{-}$ couples to state outside the DFS, as shown schematically in figure 2(a) for the two atom case. Moreover, we also include a short discussion on deviations due to imperfect control on ${{\rm{\Delta }}}_{1(2)}\ne 0$ and ${{\rm{\Omega }}}_{12}^{+}$.

Two atoms: Using ${{\rm{\Omega }}}_{12}^{+}=0$, the error rate from deviations from the Zeno Hamiltonian enters in the same way as from the spontaneous emission into all other modes, that is via the quantum jump operator $| 0{\rangle }_{{\rm{L}}}\langle 1|$. The corresponding decay rate is $(| {{\rm{\Delta }}}_{12}^{-}{| }^{2}+| {{\rm{\Omega }}}_{12}^{-}{| }^{2}/2)/{{\rm{\Gamma }}}_{1{\rm{D}}}$. The error from ${{\rm{\Omega }}}_{12}^{+}\ne 0$ enters differently, but can still be included in the estimation of the infidelity. Neglecting the errors from ${{\rm{\Delta }}}_{12}^{+}\ne 0$, the infidelity for a Pauli-X gate ($\tfrac{1}{\sqrt{2}}| {{\rm{\Omega }}}_{12}^{-}| T=\pi /2$) on state $| 1{\rangle }_{{\rm{L}}}$ can be approximated by

$$\begin{eqnarray}&&1-F\approx \displaystyle \frac{1}{2}\displaystyle \frac{\pi }{\sqrt{2}| {{\rm{\Omega }}}_{12}^{-}| }\left({{\rm{\Gamma }}}^{*}+\displaystyle \frac{| {{\rm{\Omega }}}_{12}^{-}{| }^{2}}{2{{\rm{\Gamma }}}_{1{\rm{D}}}}+\displaystyle \frac{| {{\rm{\Delta }}}_{12}^{-}{| }^{2}}{{{\rm{\Gamma }}}_{1{\rm{D}}}}+\displaystyle \frac{| {{\rm{\Omega }}}_{12}^{+}{| }^{2}}{2{{\rm{\Gamma }}}_{1{\rm{D}}}}\right)\equiv {\varepsilon }_{0}.\end{eqnarray} \tag{ 22 }$$

In the ideal case of perfect control of addressing parameters, i.e., ${{\rm{\Delta }}}_{12}^{-}={{\rm{\Omega }}}_{12}^{+}=0$, the minimal value of the infidelity, proportional to ${P}_{1{\rm{D}}}^{-1/2}$, is obtained at $| {{\rm{\Omega }}}_{12}^{-}| =\sqrt{2{{\rm{\Gamma }}}^{*}{{\rm{\Gamma }}}_{1{\rm{D}}}}$. Note, that even for ${{\rm{\Omega }}}_{12}^{-}={{\rm{\Omega }}}_{12}^{+}$ and ${{\rm{\Delta }}}_{12}^{-}=0$, the infidelity is still proportional to ${P}_{1{\rm{D}}}^{-1/2}$.

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Four atoms: In this case, apart from the transitions out of the DFS, ${{\rm{\Omega }}}_{12}^{-}$ also couples states inside the DFS, but out of the computational space (see figure 2) such that we need to detune these processes to achieve the rotations. As already explained in the previous section, this can be done by setting $| {{\rm{\Omega }}}_{12}^{-}| \ll | {{\rm{\Delta }}}_{34}^{+}| \ll {{\rm{\Gamma }}}_{1{\rm{D}}}$. As expected, the infidelity decreases when increasing the off-resonance ratio ${r}_{{\rm{\Omega }}}=| {{\rm{\Delta }}}_{34}^{+}/{{\rm{\Omega }}}_{12}^{-}|$ (see figure 5). For large enough ratios ${r}_{{\rm{\Omega }}}(\gtrsim 4)$, the infidelity can be approximated by

$$\begin{eqnarray}&&1-F\approx {\varepsilon }_{0}+\displaystyle \frac{\alpha }{{r}_{{\rm{\Omega }}}^{2}},\end{eqnarray} \tag{ 23 }$$

where the constant $\alpha ={ \mathcal O }(1)$ can be obtained through a numerical fit. The infidelity of a $\pi /2$-pulse on the state $| 00{\rangle }^{{\rm{L}}}$ is plotted in figure 3(e), whereas the minimal infidelity is shown to scale with ${P}_{1{\rm{D}}}^{-1/2}$ in figure 6.

For the controlled-$(-Z)$ gate, we set ${{\rm{\Omega }}}_{n}=0$ for all $n,{{\rm{\Delta }}}_{12}^{+}=0$ and ${{\rm{\Delta }}}_{n}={{\rm{\Delta }}}_{{\rm{D}}}\gg | {{\rm{\Delta }}}_{12}^{-}|$ for all $n\geqslant 5$. As ${{\rm{\Delta }}}_{12}^{-}$ couples $| 10{\rangle }^{{\rm{L}}}$ and $| 11{\rangle }^{{\rm{L}}}$ also to states outside the DFS, the fidelity shows a similar behavior as the Pauli-X gate (see figure 3(f) for example with N = 4), i.e., there is an optimal ${{\rm{\Delta }}}_{12}^{-}$ that sets the maximum fidelity.

The infidelity can be approximated similarly to equation (20), i.e., after a controlled Pauli-Z gate ($| {{\rm{\Delta }}}_{12}^{-}| T/\sqrt{2}=\pi$) acting on the state $(| 10+11{\rangle }^{{\rm{L}}})/\sqrt{2}$, by

$$\begin{eqnarray}&&1-F\approx \displaystyle \frac{3\pi }{2\sqrt{2}| {{\rm{\Delta }}}_{12}^{-}| }\left({{\rm{\Gamma }}}^{*}+\displaystyle \frac{3}{4}\displaystyle \frac{| {{\rm{\Delta }}}_{12}^{-}| }{{{\rm{\Gamma }}}_{1{\rm{D}}}}\right),\end{eqnarray} \tag{ 24 }$$

which attains its minimal value, $3\pi /\sqrt{2{P}_{1{\rm{D}}}}$, for $| {{\rm{\Delta }}}_{12}^{-}| =\sqrt{4{{\rm{\Gamma }}}^{*}{{\rm{\Gamma }}}_{1{\rm{D}}}/3}$. As for the single qubit gates, the infidelity scales with ${P}_{1{\rm{D}}}^{-1/2}$, shown in blue circles in figure 6.

Summing up, from the explicit analysis with two and four TLS, we have shown both numerically and analytically that both the single-qubit rotations and the controlled (−Z) gate show a scaling of the infidelity as ${P}_{1{\rm{D}}}^{-1/2}$ (see figure 6). Only for small values of the Purcell factor ${P}_{1{\rm{D}}}$ does the minimal infidelity deviate slightly from the theoretical analysis because the hierarchy ${{\rm{\Gamma }}}^{*}\ll {{\rm{\Omega }}}_{12}^{-},{{\rm{\Delta }}}_{12}^{-}\ll {{\rm{\Gamma }}}_{1{\rm{D}}}$ is no longer well satisfied.

Moreover, in the N = 4 case, we also showed how to deal with the errors that come from the larger size of the DFS with respect to the computational one. For single qubit rotations in a system of four emitters, the choice $| {{\rm{\Omega }}}_{12}^{-}| ,| {{\rm{\Delta }}}_{12}^{+}| \ll | {{\rm{\Delta }}}_{34}^{+}| \ll {{\rm{\Gamma }}}_{1{\rm{D}}}$ ensures that the dynamics can be restricted to two atoms. In the extreme case where $| {{\rm{\Delta }}}_{34}^{+}| \gg {{\rm{\Gamma }}}_{1{\rm{D}}}$ the perturbation analysis is no longer valid. However, in this case the levels are so strongly shifted, that they are decoupled from the collective dissipation, so that the system can be described as a system of only two emitters. The same is true, if the emitters can be completely decoupled from the waveguide by other means available in a particular implementation. For more atoms the same arguments hold as the second order correction introduced from deviations from Zeno dynamics satisfies

$$\begin{eqnarray}&&\parallel {\mathbb{P}}{{ \mathcal L }}_{\mathrm{pert}}{\mathbb{Q}}\displaystyle \frac{1}{{{ \mathcal L }}_{{\rm{D}}}}{\mathbb{Q}}{{ \mathcal L }}_{\mathrm{pert}}{\mathbb{P}}\rho \parallel \ll \displaystyle \frac{\parallel {{ \mathcal L }}_{\mathrm{pert}}{\parallel }^{2}}{{{\rm{\Gamma }}}_{1{\rm{D}}}},\end{eqnarray} \tag{ 25 }$$

where ${{ \mathcal L }}_{\mathrm{pert}}$ is the perturbation to the purely collective decay and $\parallel \cdot \parallel$ denotes the maximum norm. This is independent of the atom number N, because $\parallel {{ \mathcal L }}_{\mathrm{pert}}\parallel$ does not increase with the number of atoms for one and two-qubit gates. So there is an upper limit on the second order correction, which leads to the ${P}_{1{\rm{D}}}^{-1/2}$-scaling. Finally, the error rate stemming from spontaneous emission of the logical states $| 1{\rangle }_{{\rm{L}}}=| A\rangle$ is proportional to the number of excited states in the system. Therefore, the gate fidelity does depend on the full state, and can be upper bounded by considering the worst-case state, that is the state with $| 1{\rangle }_{{\rm{L}}}$ in all other computational qubits, which indeed will depend on the atom number.