distribution. In order to reduce the number of TLS-S<sub>qb</sub><sup>Δ</sup> rounds, we propose to use the SB<sub>+q</sub> sequence with the Kraus operators J<sub>SBq</sub><sup>±</sup> = e<sup>±iδ<sub>q</sub>/2</sup> P<sub>b</sub> J<sub>SBq</sub><sup>±</sup> (η<sub>q</sub>), in addition right before the BS<sub>+q</sub> rounds. The freedom to choose the value δ<sub>q</sub> much larger and indepen- dently of η<sub>q</sub> is exploited, allowing large feedback displacements without measuring strongly the ancilla p<sub>b</sub>-envelope position as it is the case in the BS<sub>+q</sub> sequence.

It should be strongly emphasized, that this sequence is introduced for a faster
correction of errors in the $q_b$-distribution, at the expense of perturbing slightly the en-
velope information contained in the ancilla $p_b$-distribution. In figure 4.12, we depict
the impact of the additional $SB_{+q}$ rounds (for $\delta_q = \eta_q$) on the average photon number
of the ancilla and target mode in steady state, that signifies the information loss of
the target envelope position. In numerical simulations, we find the optimum of $\eta_q$ in
the $SB_{+q}$ sequence (for minimal perturbation of the envelope position information)
corresponds to the value used in the succeeding $BS_{+q}$ rounds, that sets the $q_b$-peak
variance for the next error correction cycle. In summary, the error correction scheme
is modified by applying a first batch of $M$ $SB_{+q}$ rounds with large feedback displace-
ments, followed by $N-M$ $BS_{+q}$ rounds, after the TLS-S<sub>p<sub>b</sub></sub><sup>Δ</sup> readout.

<figure><img src="image_36.png" /><figcaption>Figure 4.12: Analysis on the impact of additional SB+q rounds on the steady state envelope size. Numerical simulations of the error correction scheme in figure 4.7 for M SB+q rounds, where the parameters of the sequence are chosen to be δq = ηq = 0.06 followed by a fixed number of 6 BSBq rounds with ηq = 0.06. The average photon number in steady state is plotted versus the number of SB+q rounds M.</figcaption></figure>

The optimal value of the feedback displacement $\delta_q(m)$ at each round $m$ are found
by an optimization algorithm (BFGS-algorithm, scipy library [40]), in which the differ-
ent $A_k$ moments are used as the objective to be minimized. The number of rounds $M$
are chosen such that residual TLS-induced errors during the $SB_{+p}$ rounds are bal-
anced with the intrinsic oscillator and TLS errors that accumulate during the $SB_{+q}$
and $BS_{+q}$ rounds. Moreover, $M$ can not be taken too large, as the measurement
back-action slightly perturbs the envelope information and prevents an efficient sta-
bilization of the target envelope (see figure 4.12).

<figure><img src="image_37.png" /><figcaption>Figure 4.13: Numerical simulations on the optimized ancilla error suppression for a fixed number of $N = 12/16$ (left/right column) $\mathcal{R}_q$ rounds for a TLS with $T_1 = 200\mu s$, $T_\phi = 300\mu s$. The $\mathcal{R}_q$ rounds are partitioned in $M$ $SB_{+q}$ and $N-M = 6$ $BS_{+q}$ rounds (separated by the vertical dashed line). The top row shows the optimized feedback values with respect to the moments $A_0$ and $A_4$ for each round $n$. For clarity, the control parameter $\eta_q = 0.12$ during the $BS_{+q}$ rounds is kept constant. The lower row depicts the behaviour of the first $A_k$ moments over the rounds $n$, for the two different feedback strategies $A_0/A_4$, that are distinguished in dotted/dashed line-styles.</figcaption></figure>

In the end of this chapter, we turn to the numerical simulations of the error-
correction circuit in presence of TLS errors ($\sigma_x$, $\sigma_z$) and photon loss. Since a full
sweep over all free parameters is numerically exhaustive, we fix the number of $N \mathcal{R}_{qb}$ rounds and use the optimal values for the ancilla preparation optimized with regard to a respective $A_k$ as an input. For illustration, we give an example in figure 4.13 for a fixed number of $M = 6/10$ $SB+q$ rounds with varying $\delta_q(m)$ and $N-M = 6$ $BS+q$ rounds with fixed value of $\eta_q = 0.12$, to show the optimal feedback values and the suppression of the first $A_k$ values. The first $M$ rounds with large feedback displacements allow a rapid correction of large shift errors along $q_b$ (reducing $A_0$) and the successive $BS+q$ rounds extract the envelope information and simultaneously further correct the ancilla peaks in $q_b$. More qualitatively, the $SB+q$ rounds suppress $A_0$ faster at the cost of increasing the higher $A_k$ moments temporarily. The successive $BS+q$ rounds with small $\eta_q$ values reduce the latter contributions, to minimize the propagation of errors. In the end, the strength $\eta_q$ in the $BS+q$ sequence, that is kept fixed for clarity in these curves, can also be decreasingly varied (in a small interval, for the optimal extraction of the envelope information) to improve the preparation of ancilla-$q_b$ distribution even further.

In figure 4.14, we depict the lifetime of the logical qubit, as a function of different feedback strategies that have been individually optimized on the different $A_k$ values. An approximate error model is used to include the impact of TLS errors during the $\mathcal{R}_b$ rounds, with which we update the density matrix of the composite system $\rho_{a,b}$. Considering a $\mathcal{R}_b$ round on the ancilla system only (for instance for state preparation), for a single TLS error that happens with probability $p_l$, the ancilla state results in $\rho'_b = p_l\rho_b + (1-p_l)\rho_b^{err}$. We then consider such an event happening when the target and ancilla state are entangled and that the composite state evolves as

$$ \rho'_{a,b} = p_l \rho_{a,b} + (1-p_l) \mathrm{Tr}_b(\rho_{a,b}) \otimes \rho_b^{\mathrm{err}} = p_l \rho_{a,b} + \mathrm{Tr}_b(\rho_{a,b}) \otimes (\rho'_b - p_l \rho_b). \quad (4.25) $$

where we take $\mathrm{Tr}_a(\rho_{a,b})$ as an input for $\rho_b$ and $\rho'_b$ is computed in numerical simulations for the respective $\mathcal{R}_b$ round that is used. This is under the approximation of small residual entanglement shared (at the end of the cycle $C_r$) between the two modes to justify the partial trace over the mode a. For more rapid simulations, we further approximate $\mathrm{Tr}_b(\rho_{a,b})$ by the initial ancilla input state at the start of the two-mode simulations, such that $\rho_b^{err}$ can be pre-computed. The specific error rates in these simulations were: $T_{int} = 5\mu s$, $T_{CD} = 1\mu s$, $T_{osc} = 2ms$, $T_1 = 100\mu s$, $T_\phi = 200\mu s$ and a Fock space truncation value of $N = 120$.

We demonstrate an substantial increase with respect to the bare lifetime of the harmonic oscillators and of the TLS for this set of values. In this parameter regime the feedback strategy for high values of $k$ in $A_k$ is optimal, since the large feedback
<figure><img src="image_38.png" /><figcaption>Figure 4.14: Numerical simulation results on the logical qubit lifetime of the target oscillator as a function of the feedback strategies that are optimized on the respective $A_k$ values. (Left) The explicit values of the feedback displacements for $M = 10 \text{ SB}+q$ rounds $\delta_q(m)$ and for the $L = N - M = 6 \text{ SB}+q$ rounds $\eta_q(l)$ are plotted on the left. The dashed vertical line separates the two batches. (Right) Extracted logical qubit lifetime is plotted versus the different feedback strategies that have been optimized on the respective $A_k$ values.</figcaption></figure>

strengths for $k$ being small are too large for that given round number $M$ to reinitialize the narrow ancilla $q_b$-peak distribution.
**5** | Conclusion

In this thesis, we addressed the issue of error propagation in GKP error-correction
schemes and presented a hardware-efficient proposal to robustly suppress logical er-
rors that are induced by the noise of the ancillary system. To this date, experiments
that demonstrated state-preparation and correction were based on a two-level-system
ancilla [5],[6],[7],[24]. We have extensively reviewed and presented the strengths and
roadblocks of this approach in chapter 2. In particular, bit-flips of the TLS during the
instrumental conditional displacement gates entail uncorrectable errors.

This problem can be circumvented by error-correction with a bosonic ancilla pre-
pared in a GKP-state. However, in previously proposed schemes [41], [9], the ancilla
preparation has been mostly omitted. In this thesis, we showed that by exploiting the
liberty to couple to a single quadrature of the ancilla in the Steane error-correction
circuit and tailoring the ancilla preparation accordingly, the logical errors induced by
noise propagation from the TLS can be robustly suppressed. In chapter 3, we demon-
strated that for infinite energy states both TLS errors and quadrature noise on the
target and ancillary oscillators, the logical qubit coherence time can be extended by
more than an order of magnitude beyond that of the embedding hardware for values
within reach of state-of-the-art superconducting circuit experiments.

In chapter 4, we adapted this protocol to finite energy states. In that case, the fi-
nite squeezing of the ancilla will contribute to uncorrectable errors in the target state.
However, we focused on the limit of a very asymmetric ancilla state, which coincides
with a squeezed cat state. We demonstrated that a readout of the target stabilizer
information encoded in the ancilla via a TLS outperforms a homodyne readout, even
for unit efficiency. Further we presented an optimized readout sequence of the target
error-syndromes that is merged with the asymmetric ancilla preparation. This allows
a significant reduction of the error-correction cycle duration which is crucial to effi-
ciently correct errors induced by intrinsic noise of the target oscillator. For hardware
parameters that are realistic in the domain of superconducting circuits, we demon-
strate that the logical qubit lifetime surpasses substantially the break-even point.
Our proposal is tailored to minimal hardware requirements and control techniques
for near-term experiments. Except for the tuneable quadrature-quadrature gate, it
only requires the conditional displacement gates that have already been demonstrated
in the framework of trapped ions [5] and superconducting circuits [7],[24]. As pro-
posed in [21] the quadrature-quadrature gate can be engineered through single-mode
squeezing and Beam-splitter operations, which have been individually demonstrated.
Moreover, this gate is a matter of active experimental research and are central to
implement logical gates in the GKP code.
# A | Appendix to chapter 4

## 1 Gaussian comb formalism for error correction with homodyne detection readout

A general Gaussian comb distribution

$$P(q) = \sum_t e^{-\frac{(q-\epsilon)^2}{\Delta^2}} e^{-\frac{(q-n2\alpha-\phi)^2}{\sigma^2}} := G[\Delta^2, \sigma^2, 2\alpha, \epsilon, \phi], \quad (A.1)$$

is characterized by 5 quantities, the peak/envelope variance $\sigma^2/\Delta^2$ of the wave-function, the peak/envelope shift error $\phi/\epsilon$ and the lattice spacing $\alpha$. In this representation the two logical target states are identified by $\epsilon = 0, \phi = 0$ ($|+Z_L\rangle$) and $\epsilon = 0, \phi = \alpha$ ($|-Z_L\rangle$) respectively.

The product of two Gaussian combs, with equal lattice spacing $2\alpha$, can be expressed as a sum of Gaussian combs

$$G[\Delta^2, \sigma^2, 2\alpha, \epsilon, \phi] G[d^2, s^2, 2\alpha, e, f] = e^{-\frac{(\epsilon-\epsilon)^2}{\Delta^2+d^2}} \sum_k e^{-\frac{((\phi-f)-2k\alpha)^2}{\sigma^2+s^2}} G[\tilde{\Delta}^2, \tilde{\sigma}^2, 2\alpha, \tilde{\epsilon}, \tilde{\phi} - 2k\alpha \frac{\sigma^2}{\sigma^2+s^2}] \quad (A.2)$$

$$\text{with } \tilde{\Delta}^2 = \frac{\Delta^2 d^2}{\Delta^2+d^2}, \quad \tilde{\sigma}^2 = \frac{\sigma^2 s^2}{\sigma^2+s^2}, \quad \tilde{\epsilon} = \frac{e\Delta^2+\epsilon d^2}{\Delta^2+d^2}, \quad \tilde{\phi} = \frac{f\sigma^2+\phi s^2}{\sigma^2+s^2}.$$

In order to describe the dynamics of a single error-correction cycle in terms of the target probability distribution, $P'(q_a) = M_r^2 P(q_a)$, the operator $M_r$

$$M_r \propto \sum_n e^{-\frac{1}{2d^2}(\mathbf{q}_a - \mathbf{r})} e^{-\frac{1}{2s^2}(\mathbf{q}_a - n\alpha - \mathbf{r})^2}. \quad (A.3)$$
is represented in the Gaussian comb representation by rewriting it in terms of the period $2\alpha$

$$G[2d^2, 2s^2, \alpha, r, r] = G[2d^2, 2s^2, 2\alpha, r, r] + G[2d^2, 2s^2, 2\alpha, r, r + \alpha]. \quad (A.4)$$

By using equation A.2, the expression for $M_r^2$ can be derived

$$M_r^2 \hat{=} \sum_{\lambda_i, \lambda_j \in \{0,1\}} G[2d^2, 2s^2, 2\alpha, r, r + \lambda_i \alpha] G[2d^2, 2s^2, 2\alpha, r, r + \lambda_j \alpha] \\ = \sum_{\lambda_i, \lambda_j \in \{0,1\}} \left( \sum_k e^{-\frac{1}{4s^2}((\lambda_i - \lambda_j)\alpha - 2k\alpha)^2} G[d^2, s^2, 2\alpha, r, r + \frac{(\lambda_i - \lambda_j)\alpha}{2} - k\alpha] \right). \quad (A.5)$$

This expression can be rewritten in a compact form as the sum over four Gaussian combs centered at $\{0, \frac{\alpha}{2}, \alpha, \frac{3\alpha}{2}\}$,

$$M_r^2 \hat{=} \sum_{\lambda_i, \lambda_j \in \{0,1\}} e^{-\frac{(2k\alpha - \lambda_i\alpha)^2}{4s^2}} G[d^2, s^2, \alpha, r, r + \lambda_j\alpha + \frac{\lambda_i\alpha}{2}} \quad (A.6)$$

We consider the dynamics of one $q$-error correction cycle, the product of the $q$-probability distribution of a general target state and the expression of $M_r^2$ in equation A.6. For clarity, the multiplication with a single Gaussian comb in the $M_r^2$ expression indexed by $\lambda = \{0, 1, 2, 3\}$ is computed below

$$G[\Delta^2, \sigma^2, 2\alpha, \epsilon, \phi] G[d^2, s^2, 2\alpha, r, r + \alpha \frac{\lambda}{2}] \\ = e^{-\frac{(\epsilon - r)^2}{\Delta^2 + d^2}} \sum_k e^{-\frac{(2k\alpha - (\phi - (r + \alpha \frac{\lambda}{2})))^2}{\sigma^2 + s^2}} G[\tilde{\Delta}^2, \tilde{\sigma}^2, 2\alpha, \tilde{\epsilon}, \tilde{\phi} - 2k\alpha \frac{\sigma^2}{\sigma^2 + s^2}] \quad (A.7)$$

$$\text{with } \tilde{\Delta}^2 = \frac{\Delta^2 d^2}{\Delta^2 + d^2}, \quad \tilde{\sigma}^2 = \frac{\sigma^2 s^2}{\sigma^2 + s^2}, \quad \tilde{\epsilon} = \frac{r \Delta^2 + \epsilon d^2}{\Delta^2 + d^2}, \quad \tilde{\phi} = \frac{(r + \alpha \frac{\lambda}{2}) \sigma^2 + \phi s^2}{\sigma^2 + s^2}$$

This formalism allows to describe the dynamics by solely updating the variances $\Delta^2$ and $\sigma^2$ and the distributions of $\epsilon$ and $\phi$.

General initial distributions for the peak and envelope shift errors $\epsilon$ and $\phi$ are considered
$$P(\phi) = \int d\phi_0 P(\phi_0) \delta(\phi - \phi_0) \qquad (A.8)$$

$$P(\epsilon) = \int d\epsilon_0 P(\epsilon_0) \delta(\epsilon - \epsilon_0)$$

A single peak in $\phi$ and $\epsilon$, transforms according to equation A.7 as

$$\delta(\phi - \phi_0) \delta(\epsilon - \epsilon_0) \rightarrow e^{-\frac{(\epsilon_0 - r)^2}{\Delta^2 + d^2}} \sum_k e^{-\frac{(2k\alpha - \phi_0 + r + k\alpha \frac{\lambda}{2})^2}{\sigma^2 + s^2}} \delta(\phi - \tilde{\phi} - 2k\alpha \frac{\sigma^2}{\sigma^2 + s^2}) \delta(\epsilon - \tilde{\epsilon}). \quad (A.9)$$

Subsequent to the quadrature readout, a corrective feedback proportional to the outcome $r$, to re-center the peak $\gamma_{\phi}r$ and envelope $\gamma_{\epsilon}r$ position is applied

$$e^{-\frac{(\epsilon_0 - r)^2}{\Delta^2 + d^2}} \sum_k e^{-\frac{(2k\alpha - \phi_0 - r - k\alpha \frac{\lambda}{2})^2}{\sigma^2 + s^2}} \delta\left(\phi - \frac{(r + k\alpha \frac{\lambda}{2})(1 + \gamma_{\phi})\sigma^2 + \phi s^2}{\sigma^2 + s^2} - 2k\alpha \frac{\sigma^2}{\sigma^2 + s^2}\right) \delta\left(\epsilon - \frac{r(1 + \gamma_{\epsilon})\Delta^2 + \epsilon d^2}{\Delta^2 + d^2}\right) \quad (A.10)$$

Further, the expression is integrated over all possible outcomes $r$, resulting in

$$e^{-\frac{1}{\Delta^2}(\epsilon - \eta_{\epsilon}\epsilon_0)^2} \sum_k e^{-\frac{1}{\sigma^2}(\phi - \eta_{\phi}\phi_0 - 2k\alpha\omega_{\phi})^2} \quad (A.11)$$

with $\hat{\Delta}^2 = \frac{\Delta^4(1+\gamma_{\epsilon})^2}{\Delta^2+d^2}$, $\hat{\sigma}^2 = \frac{\sigma^4(1+\gamma_{\sigma})^2}{\sigma^2+d^2}$, $\eta_{\phi} = \frac{\sigma^2(1+\gamma_{\phi})+s^2}{\sigma^2+d^2}$, $\eta_{\epsilon} = \frac{\Delta^2(1+\gamma_{\epsilon})+d^2}{\Delta^2+d^2}$ and $\omega_{\phi} = \frac{\sigma^2\gamma_{\phi}}{\sigma^2+d^2}$.

Therefore, the probability distributions in $\phi$ and $\epsilon$ evolve as

$$P(\phi) = \int d\phi_0 P(\phi_0) \sum_k e^{-\frac{1}{\sigma^2}(\phi - \eta_{\phi}\phi_0 - 2k\alpha\omega_{\phi})^2} \quad (A.12)$$

$$P(\epsilon) = \int d\epsilon_0 P(\epsilon_0) e^{-\frac{1}{\Delta^2}(\epsilon - \eta_{\epsilon}\epsilon_0)^2}$$

The error correction cycle in $p$, realized through $e^{i\theta_r r_a q_b}$, represents for the q-probability distribution a convolution with the q-distribution of the ancilla. Solely, the variances of both the peaks and envelope are updated accordingly

$$\tilde{\Delta}^2 = \Delta^2 + d_2^2 \qquad \tilde{\sigma}^2 = \sigma^2 + s_2^2 \qquad (A.13)$$
with $s_2^2/d_2^2$ being the ancilla q-peak/envelope variance.
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# RÉSUMÉ

Cette thèse porte sur le code GKP qui permet la correction d'erreur générique d'un qubit codé dans un oscillateur harmonique. Des expériences récentes ont démontré la stabilisation de la variété de code basée sur les interactions de Rabi avec un système auxiliaire à deux niveaux. Cependant, ces schémas souffrent de retournements logiques non corrigibles déclenchés par des erreurs de relaxation ancilla pendant l'interaction. Cette thèse propose un protocole pour stabiliser le code GKP dans un mode cible en cartographiant ses syndromes d'erreur à un mode GKP auxiliaire via une interaction quadrature-quadrature. Contrairement aux schémas précédemment proposés, le couplage à une seule quadrature auxiliaire permet d'adapter l'état auxiliaire et sa préparation en conséquence pour assurer une forte suppression des erreurs de rétro-propagation vers le mode cible. Les informations sur le syndrome d'erreur sont récupérées et l'auxiliaire réinitialisé efficacement en utilisant des techniques similaires démontrées dans les récentes expériences GKP. Pour des paramètres système réalistes, les simulations numériques confirment la suppression robuste des erreurs logiques induites par les auxiliaires et montrent une amélioration de la durée de vie du qubit logique d'un ordre de grandeur au-delà du seuil de rentabilité.

# MOTS CLÉS

Correction d'erreur quantique, Circuits supracondacteur, Codes bosoniques

# ABSTRACT

This thesis focuses on the GKP code that allows for generic error-correction of a qubit encoded in a harmonic oscillator. Recent experiments have demonstrated the stabilization of the code manifold based on Rabi interactions with an ancillary two-level system. However, these schemes suffer from uncorrectable logical flips triggered by ancilla relaxation errors during the interaction. This thesis, proposes a protocol to stabilize the GKP code in a target mode by mapping its error syndromes to an ancillary GKP mode via a quadrature-quadrature interaction. In contrast to previously proposed schemes, coupling to solely one ancilla quadrature allows tailoring the ancilla state and its preparation accordingly to ensure a strong suppression of back-propagating errors to the target mode. The error-syndrome information is retrieved and the ancilla efficiently re-initialized using similar techniques demonstrated in the recent GKP experiments. For realistic system parameters, numerical simulations confirm the robust suppression of ancilla induced logical errors and show an enhancement of the logical qubit lifetime by an order of magnitude beyond the break-even point.

# KEYWORDS

Quantum error correction, Superconducting circuits, Bosonic codes