Critical Schrödinger Cat Qubit

Encoding quantum information onto bosonic systems is a promising route to quantum error correction. In a cat code, this encoding relies on the confinement of the dynamics of the system onto the two-dimensional manifold spanned by Schrödinger cats of opposite parity. In dissipative cat qubits, an engineered dissipation scheme combining two-photon drive and two-photon loss has been used to autonomously stabilize this manifold, ensuring passive protection against, e.g., bit-flip errors regardless of their origin. Similarly, in Kerr-cat qubits, where highly performing gates can be engineered, two-photon drive and Kerr nonlinearity cooperate to confine the system to a twofold-degenerate ground-state manifold spanned by cat states of opposite parity. Dissipative, Hamiltonian, and hybrid confinement mechanisms have been investigated at resonance, i.e., for driving frequencies matching that of the cavity. Here, we propose a critical cat code, where both two-photon loss and Kerr nonlinearity are present and the two-photon drive is allowed to be out of resonance. The performance of this code is assessed via the spectral theory of Liouvillians in all configurations ranging from the purely dissipative to the Kerr limit. We show that large detunings and small, but non-negligible, two-photon loss rates are fundamental to achieve optimal performance. We further demonstrate that the competition between nonlinearity and detuning results in a first-order dissipative phase transition, leading to a squeezed vacuum steady state. We show that to achieve the maximal suppression of the logical bit-flip rate requires initializing the system in the metastable state emerging from the first-order transition and we detail a protocol to do so. Efficiently operating over a broad range of detuning values, the critical cat code is particularly resistant to random frequency shifts characterizing multiple-qubit operations, opening avenues for the realization of reliable protocols for scalable and concatenated bosonic qubit architectures.

Popular Summary

In quantum computers lies the promise to vastly outperform state-of-the-art supercomputers by harnessing the laws of quantum mechanics in solving specific tasks. Like classical computers, quantum computers are also affected by random errors, usually handled by storing information redundantly. The problem of scaling up quantum computers while retaining protection against said errors has stood the test of time. By exploiting the redundancy of the infinite-dimensional Hilbert space of the harmonic oscillator, bosonic quantum codes are a resource-efficient solution to this problem.

Among the few examples of viable bosonic codes, the Schrödinger-cat code stands out for its biased response to different types of errors and its experimental feasibility on current superconducting platforms. Prior examinations of this code have been restricted to two limiting cases where simultaneous access to high-performance logical gates and autonomous protection against errors is ultimately unattainable. We propose a novel regime of operation where first-order criticality endows the system with extreme protection from noise processes. We examine this new code, which we dub the “critical Schrödinger-cat code,” under the lens of the spectral theory of Liouvillians, proving that it outperforms its predecessors in its error-correcting capabilities and versatility toward coupled-qubit architectures.

Our analysis suggests that the largely unexplored parameter space of bosonic codes may still offer regions where specific properties of the code are enhanced. A more systematic investigation of the regimes of operation of all bosonic codes may thus lead to their significant improvement and to a competitively efficient design of a fault-tolerant bosonic quantum code architecture.

Figure 1

(a) The even and odd cat states, represented by their Wigner functions, encode the logical states $|0_L⟩$ and $|1_L⟩$. Their superpositions, approximately the two coherent states $|\pm \alpha ⟩$, define the logical states $|{\pm }_L⟩$. (b) The block structure of the Liouvillian arising from its strong ${\mathcal{Z}}_2$ symmetry. Finite values of $\mathrm{\Delta }$ or ${\kappa }_{\varphi }$ act only within the blocks ${\mathcal{L}}_0^{+-}$ and ${\mathcal{L}}_0^{-+}$, so that the four-block-diagonal structure of ${\mathcal{L}}_0$ is preserved. The single-photon loss ${\kappa }_1$, instead, connects different blocks, resulting in a two-block-diagonal structure typical of weakly ${\mathcal{Z}}_2$-symmetric systems.

Figure 2

(a),(b) The bit-flip error rate $\mathrm{\Gamma }$ as a function of the detuning $\mathrm{\Delta }$ and the ratio $U/\eta =\tan (\theta )$ defined in Eq. (20). The dissipative limit corresponds to $\theta =0$ ($U=0$, $\eta =1$). The opposite limit, i.e., the Kerr (or Hamiltonian) limit, is reached for $\theta =\pi /2$ ($U=1$, $\eta =0$). The optimal detuning values ${\mathrm{\Delta }}_{\mathrm{opt}}(G=5,\theta )$ are shown as continuous yellow lines. The dot-dashed line denotes the critical threshold ${\mathrm{\Delta }}_c$ separating the phase with large steady-state photon number from the vacuum one (see Sec. 4). We set the two-photon driving-field amplitude $G=5$ and (a) ${\kappa }_1={10}^{-5}\ll {\kappa }_{\varphi }={10}^{-3}$ and (b) ${\kappa }_1={10}^{-3}\gg {\kappa }_{\varphi }={10}^{-5}$. (c) ${\mathrm{\Delta }}_{\mathrm{opt}}(G,\theta )$ as a function of $G$ for different values of $\theta$ and ${\kappa }_1={10}^{-3}\gg {\kappa }_{\varphi }={10}^{-5}$.

Figure 3

The optimal bit-flip error rate $\mathrm{\Gamma }({\mathrm{\Delta }}_{\mathrm{opt}})$ as a function of the single-photon loss rate ${\kappa }_1$ and of the ratio $U/\eta$ for ${\kappa }_{\varphi }={10}^{-3}$. The value of $\mathrm{\Gamma }({\mathrm{\Delta }}_{\mathrm{opt}})$ is normalized to ${\kappa }_{\varphi }$ to ensure consistency in the limiting values of the displayed quantity for all choices of ${\kappa }_{\varphi }$. The solid line highlights the lowest-$(U/\eta )$ isoline $\mathrm{\Gamma }({\mathrm{\Delta }}_{\mathrm{opt}})/{\kappa }_{\varphi }=5\times {10}^{-5}$. The dashed curves display the same quantity for ${\kappa }_{\varphi }={10}^{-4}$, ${10}^{-5}$, ${10}^{-6}$, and ${10}^{-7}$. The optimal configuration is clearly seen to shift toward the Kerr limit for increasing values of ${\kappa }_{\varphi }$.

Figure 4

The bit-flip error rate $\mathrm{\Gamma }$ as a function of the driving-field amplitude $G$ for different values of $\mathrm{\Delta }$ and $\theta$, for (a) ${\kappa }_1={10}^{-5}\ll {\kappa }_{\varphi }={10}^{-3}$ and (b) ${\kappa }_1={10}^{-3}\gg {\kappa }_{\varphi }={10}^{-5}$. In both panels, we display results for (i) the Kerr limit $U/\eta ={10}^6$, (ii) the dissipative limit $U/\eta =0$, (iii) the optimal configuration obtained by optimizing both $\theta$ and $\mathrm{\Delta }$, and (iv) the result obtained by setting $U/\eta ={10}^6$ and optimizing over $\mathrm{\Delta }$. The curve denoted as (v) is obtained by setting $U/\eta ={10}^2$ in (a) and $U/\eta ={10}^5$ in (b).

Figure 5

The exponential rate $\zeta$ computed as a function of $U/\eta$ for $\mathrm{\Delta }=0$ (dashed) and $\mathrm{\Delta }={\mathrm{\Delta }}_{\mathrm{opt}}$ (solid): (a) ${\kappa }_1={10}^{-5}\ll {\kappa }_{\varphi }={10}^{-3}$; (b) ${\kappa }_1={10}^{-3}\gg {\kappa }_{\varphi }={10}^{-5}$. The dot-dashed curve in (b) shows $\zeta ({\mathrm{\Delta }}_{\mathrm{opt}})$ for ${\mathrm{\Delta }}_{\max }=90$.

Figure 6

(a) The average photon number of the steady and metastable states as defined in the text. The jump discontinuity in the photon number signals the presence of a first-order DPT at ${\mathrm{\Delta }}_c\sim 15$ [58]. (b) The Liouvillian gaps ${\mathrm{\Lambda }}_1^{+}$ and ${\mathrm{\Lambda }}_0^{-}$ as defined in Eq. (12). The vertical dotted lines denote the values of $\mathrm{\Delta }$ selected for further investigation in Fig. 9. (c) Expansion of the state ${\hat{\rho }}_0^{-}(+)\equiv |1_L⟩⟨1_L|$ [cfr Eq. (22)] in the left eigenstates of $\mathcal{L}$. The expansion coefficients are $c_m=⟨{\sigma }_m^{\mu }⟩{\hat{\rho }}_0^{-}(+)$. Each component in the expansion evolves with one specific time scale. For $\mathrm{\Delta }<{\mathrm{\Delta }}_c$, only two components are nonvanishing: the steady state and that corresponding to bit flips. For $\mathrm{\Delta }>{\mathrm{\Delta }}_c$, a single additional component corresponding to the leakage with rate ${\mathrm{\Gamma }}_{\mathrm{leak}}={\mathrm{\Lambda }}_1^{+}$ emerges and the dynamics are that of Eq. (19). We consider $G=5$, $U/\eta ={10}^5$, and ${\kappa }_1={10}^{-3}\gg {10}^{-5}={\kappa }_{\varphi }$.

Figure 7

An illustration of the optimal encoding on the two sides of the phase boundary. For $\mathrm{\Delta }<{\mathrm{\Delta }}_c$, the steady-state manifold coincides with the encoding manifold and the logical code space is characterized by a single phase-flip error rate $\mathrm{\Gamma }$. For $\mathrm{\Delta }>{\mathrm{\Delta }}_c$, the steady state is a squeezed vacuum and is distinct from the optimal encoding manifold characterized by a finite average photon number. In this case, a second time scale emerges, which is characterized by the rate ${\mathrm{\Lambda }}_1^{+}$ at which the logical states decay into the vacuum.

Figure 8

The phase diagrams of (a) the diagonal Liouvillian gap ${\mathrm{\Lambda }}_1^{+}$ and (b) the steady-state photon number as a function of the detuning $\mathrm{\Delta }$ and the ratio $U/\eta$. The yellow line denotes the optimal detuning ${\mathrm{\Delta }}_{\mathrm{opt}}$ that minimizes the bit-flip error rate $\mathrm{\Gamma }$. The circles and triangles indicate the specific cases selected for the discussion of the state-preparation protocol in Sec. 4b and shown in Fig. 9.

Figure 9

The time-dependent simulation of the initialization protocols detailed in the text. The colors and symbols in the legend correspond to those points selected in the phase diagram of Fig. 8. The dashed lines represent the adiabatic ramping of $G(t)=f(t,\tau )G$, where $f(t,\tau )=\tanh (t/\tau )$, with $\tau =50$ and $G=5$. The solid lines describe $\mathrm{\Delta }(t)={\mathrm{\Delta }}_0+f(t,\tau )(\mathrm{\Delta }-{\mathrm{\Delta }}_0)$ for ${\mathrm{\Delta }}_0=-10$ and $U/\eta =5\times {10}^2$. The shaded lines describe the amplitude of the oscillation for the same $\mathrm{\Delta }(t)$ protocol, setting $U/\eta ={10}^5$. (a) The evolution of the photon number during the initialization protocol. The dotted horizontal lines define the desired photon number for the different detuning configurations. (b) The quantity $1-O$ as defined in Eq. (26), describing the similarity between the prepared state and the encoding one. We consider ${\kappa }_1={10}^{-3}\gg {10}^{-5}={\kappa }_{\varphi }$.

Figure 10

The quantity $1-O$, defined in Eq. (27), displayed as a function of the detuning $\mathrm{\Delta }$ and of the angle $\theta$ in Eq. (20) for ${\kappa }_1={10}^{-3}\gg {\kappa }_{\varphi }={10}^{-5}$. The data are obtained by appropriately selecting at each point the correct encoding, either steady-state or metastable. The initial state $|0_L⟩⟨0_L|$ and the final state obtained from the recovery procedure are detailed in the text. The data in the two panels are computed assuming a frequency shift of (a) ${\mathrm{\Delta }}_{\mathrm{err}}=1$ and (b) ${\mathrm{\Delta }}_{\mathrm{err}}=4$. The white dashed lines bound the region wherein $1-O<{10}^{-5}$ in the cases (a) ${\mathrm{\Delta }}_{\mathrm{err}}=-1$ and (b) ${\mathrm{\Delta }}_{\mathrm{err}}=-4$.

Figure 11

(a),(b) The total $Z$ and $X$ error probabilities of the $Z$ gate as defined by Eqs. (28) and (29). (c) Their ratio, the noise bias $\chi =P_Z/P_X$. We consider three different operation regimes: critical ($\mathrm{\Delta }=40$, $U/\eta =400$), Kerr ($\mathrm{\Delta }=0$, $U/\eta =\mathrm{\infty }$), and dissipative ($\mathrm{\Delta }=0$, $U/\eta =0$). The solid and dashed lines correspond to applying the gate at $t=0$ or after an idling time of $t=T$, respectively. We consider the noise configuration ${\kappa }_1={10}^{-5}\ll {\kappa }_{\varphi }={10}^{-3}$.

Figure 12

The Liouvillian spectral analysis for $G=5$ and different values of $\theta$ in Eq. (20). (a) The average occupation of the cavity as a function of the detuning $\mathrm{\Delta }$ for different values of the Kerr nonlinearity $U$ and two-photon loss $\eta$. As $|\alpha (\mathrm{\Delta }=0){|}^2=G/W$, all curves intersect at $\mathrm{\Delta }=0$. The jump in $⟨{\hat{a}}^{†}\hat{a}⟩$ appearing at $\mathrm{\Delta }>0$ becomes sharper for larger values of $\theta$, heralding the emergence of a first-order phase transition. (b) The Liouvillian gap of the $+-$ symmetry sector ${\mathrm{\Lambda }}_0^{+-}=\mathrm{\Gamma }$. Optimal configurations with nonvanishing detuning appear as local minima of the curves and allow us to take advantage of the nonlinearity (see the text). In addition to this general trend, $\mathrm{\Gamma }$ exhibits dips for specific detuning values, namely, $\mathrm{\Delta }=mU$ with $m\in \mathbb{N}$. These dips become more pronounced as the system approaches the Kerr limit, where they have recently been measured [32]. The insets to these plots display the Wigner distribution of $|1_L⟩$ for $U/\eta =5$ and increasing detuning values identified by red markers. (c) In an ideal noiseless subsystem, the eigenoperators ${\hat{\rho }}_0^{\mu \nu }$ obey ${\hat{R}}_{\mu }{\rho }_0^{\mu \nu }{\hat{R}}_{\nu }^{†}\equiv |\mu ⟩⟨\nu |\otimes z_{\mu \nu }$, where $z_{\mu \nu }=z_{{\mu }^{\mathrm{\prime }}{\nu }^{\mathrm{\prime }}}\mathrm{\forall }\mu ,\nu ,{\mu }^{\mathrm{\prime }},{\nu }^{\mathrm{\prime }}$ and ${\hat{R}}_{\mu }$ is the unitary transformation diagonalizing the density matrix ${\hat{\rho }}_0^{\mu \mu }$ [33]. We quantify the discrepancy from this ideal case using the trace distance between the two diagonal blocks $z_{++}$ and $z_{--}$. We show this as a function of $\mathrm{\Delta }$ for $2\theta /\pi =0.01$, $0.96$. The insets show the structure of the full density matrix for the points indicated in the main figure: (i) exemplifies a decoherence-free subspace and (ii) a noiseless subsystem with a mixed $\hat{M}$.

Figure 13

A schematic representation of the states comprising the steady-state encoding defined in Eq. (A7). We show ${\hat{\rho }}_{\pm Z}$ and ${\hat{\rho }}_{\pm X}$. ${\hat{\rho }}_{\pm Y}$ can be obtained by acting on ${\hat{\rho }}_{\pm Z}$ with the rotation ${\hat{X}}_{\phi }=\cos \frac{\phi }{2}({\hat{\rho }}_{+Z}+{\hat{\rho }}_{-Z})+i\sin \frac{\phi }{2}({\hat{\rho }}_{+X}-{\hat{\rho }}_{-X})$ with $\phi =\pi /2$.

Figure 14

A schematic representation of the states comprising the steady-state encoding defined in Eq. (A8).

Figure 15

The same distance quantifier as that used in Fig. 12. Here, it is evaluated between ${\hat{\rho }}_Z$ and ${\hat{\rho }}_X$, following the procedure detailed in the main text. We plot the results as a function of the detuning $\mathrm{\Delta }$ for ${\kappa }_1={10}^{-3}\gg {\kappa }_{\varphi }={10}^{-5}$ and $G=5$. The dips at integer values of detuning are to be imputed to the tunneling-suppression effects of Hamiltonian nature discussed in Ref. [32].

Figure 16

The average photon number of the steady and metastable states as defined in the main text and the Liouvillian gaps ${\mathrm{\Lambda }}_1^{+}$ and ${\mathrm{\Lambda }}_0^{-}$ as defined in Eq. (12) are shown as a function of the detuning $\mathrm{\Delta }$. The shaded blue (orange) area identifies the region hosting a steady-state (metastable) encoding. We consider $G=5$ and ${\kappa }_1={10}^{-3}\gg {\kappa }_{\varphi }={10}^{-5}$.

Figure 17

The same as Fig. 16 but for ${\kappa }_1={10}^{-5}\ll {\kappa }_{\varphi }={10}^{-3}$.

Figure 18

The Liouvillian gaps ${\mathrm{\Lambda }}_1^{+}$ and ${\mathrm{\Lambda }}_0^{-}$ as defined in Eq. (12). The dephasing noise at rate ${\kappa }_{\varphi }$ and the thermal excitations at rate $n_{\mathrm{th}}{\kappa }_1$ are turned on separately, with the single-photon loss at rate $(1+n_{\mathrm{th}}){\kappa }_1$ underlying all of the simulations. The dashed line at $\mathrm{\Lambda }={10}^{-9}$ serves as a guide to the eye to follow the considerations in the text. We set ${\kappa }_1={10}^{-3}$, $n_{\mathrm{th}}=0.06$, and ${\kappa }_{\varphi }={10}^{-5}$. Within the range of experimentally meaningful values of $n_{\mathrm{th}}$, the one selected above allows us to immediately grasp the similar effects of dephasing and thermal noise.

Figure 19

The Liouvillian gaps (a) ${\mathrm{\Lambda }}_0^{-}$ and (b) ${\mathrm{\Lambda }}_1^{+}$ as a function of the detuning $\mathrm{\Delta }$ for different values of ${\lambda }_3$ in the perturbing Hamiltonian ${\hat{H}}_3$ defined in Eq. (B1). We set $U/\eta ={10}^4$ and ${\kappa }_1={10}^{-3}\gg {\kappa }_{\varphi }={10}^{-5}$, with the black curve detailing the unperturbed behavior (${\lambda }_3=0$).

Figure 20

The same as Fig. 19, with the Hamiltonian ${\hat{H}}_4$ defined in Eq. (B2) as the perturbing term.