Bias-preserving gates with stabilized cat qubits

Abstract

The code capacity threshold for error correction using biased-noise qubits is known to be higher than with qubits without such structured noise. However, realistic circuit-level noise severely restricts these improvements. This is because gate operations, such as a controlled-NOT (CX) gate, which do not commute with the dominant error, unbias the noise channel. Here, we overcome the challenge of implementing a bias-preserving CX gate using biased-noise stabilized cat qubits in driven nonlinear oscillators. This continuous-variable gate relies on nontrivial phase space topology of the cat states. Furthermore, by following a scheme for concatenated error correction, we show that the availability of bias-preserving CX gates with moderately sized cats improves a rigorous lower bound on the fault-tolerant threshold by a factor of two and decreases the overhead in logical Clifford operations by a factor of five. Our results open a path toward high-threshold, low-overhead, fault-tolerant codes tailored to biased-noise cat qubits.

Fig. 1. Cat qubit in parametrically driven Kerr nonlinear oscillator.

(A) Bloch sphere of the cat qubit. The figure also shows cartoons of the Wigner functions corresponding to the eigenstates of $\widehat{X},\widehat{Y}$, and $\widehat{Z}$ Pauli operators. (B) Eigenspectrum of the two-photon, driven, nonlinear oscillator in the rotating frame. The Hamiltonian in the rotating wave approximation is given in Eq. 5. The cat states $\mid {\mathcal{C}}_{\mathrm{\alpha }{e}^{i\mathrm{\varphi }}}^{\pm }\mathrm{\rangle }$ with $\mathrm{\alpha }=\sqrt{P/K}$ are exactly degenerate. The eigenspectrum can be divided into an even- and odd-parity manifold. The cat subspace, highlighted in green, is separated from the first excited state by an energy gap ∣∆ω_gap∣ ∼ 4Kα². In the rotating frame, the excited states appear at a lower energy. This is because the Kerr nonlinearity is negative and implies that transitions out of the cat manifold occur at a lower frequency compared to transitions within the cat subspace. The energy difference between the first n ∼ α²/4 pairs of excited states (highlighted in orange) $\mid {\mathrm{\psi }}_{\mathcal{E},\mathcal{N}}^{\pm }\rangle$ decreases exponentially with P or equivalently with α². These excited state pairs are consequently referred to as quasi-degenerate states. In the limit of large α, the first n excited states are approximately given by $\mid {\mathrm{\psi }}_{\mathcal{E},\mathcal{N}}^{\pm }\mathrm{\rangle }=\left(D\right(\mathrm{\alpha })\pm D(-\mathrm{\alpha }\left)\right)\mid n\mathrm{\rangle }$ when n is even and $\mid {\mathrm{\psi }}_{\mathcal{E},\mathcal{N}}^{\pm }\rangle =(D(\mathrm{\alpha })\mp D(-\mathrm{\alpha }))\mid n\rangle$ when n is odd (20). Here, $D(\pm \mathrm{\alpha })=\text{exp}(\pm \mathrm{\alpha }{\hat{a}}^{\mathrm{†}}\mp \mathrm{\alpha }\hat{a})$ is the displacement operator, and ∣n〉 is the n-photon Fock state.

Fig. 2. Noise channel of the cat qubit in the presence of white thermal noise.

(A) Addition of single photon at frequency ω_r + Δω_gap excites $\mid {\mathrm{\psi }}_0\mathrm{\rangle }=x_{+}|{\mathcal{C}}_{\mathrm{\alpha }}^{+}\mathrm{\rangle }+x_{-}|{\mathcal{C}}_{\mathrm{\alpha }}^{-}\mathrm{\rangle }$ to $x_{+}|{\mathrm{\psi }}_{\mathcal{E},1}^{-}\mathrm{\rangle }+x_{-}|{\mathrm{\psi }}_{\mathcal{E},1}^{+}\mathrm{\rangle }$. The state evolves freely for time τ, during which $|{\mathrm{\psi }}_{\mathcal{E},1}^{\mp }\mathrm{\rangle }$ acquire phases $\sim \mathrm{\tau }{E}_{\mathcal{E},1}^{\mp }$. After loss of two photons, the final state is $x_{+}{e}^{-i\mathrm{\tau }{E}_{\mathcal{E},1}^{-}}|{\mathcal{C}}_{\mathrm{\alpha }}^{-}\mathrm{\rangle }+x_{-}{e}^{-i\mathrm{\tau }{E}_{\mathcal{E},1}^{+}}|{\mathcal{C}}_{\mathrm{\alpha }}^{+}\mathrm{\rangle }\equiv \hat{Z}{e}^{i(E_{\mathcal{E},1}^{-}-E_{\mathcal{E},1}^{+})\mathrm{\tau }\hat{X}/2}\mid {\mathrm{\psi }}_0\mathrm{\rangle }$. Therefore, the autonomous correction of leakage leads to both dephasing and nondephasing error. However, $E_{\mathcal{E},1}^{-}-E_{\mathcal{E},1}^{+}$ decreases exponentially with α², and hence, the nondephasing error is exponentially suppressed. (B) Natural logarithm of the coefficients of the error channel (Eq. 18) at t = 50/K with n_th = 0.01, κ = K/400, and κ_2ph = K/10. As expected, the amount of non-phase errors decreases exponentially with α². (C) Natural logarithm of the amount of leakage in the presence of white thermal noise without two-photon dissipation (red solid line in the left panel) and with it (blue solid line in the right panel). As expected, the two-photon dissipation autonomously corrects for leakage. The dashed black lines show the leakage predicted by the theoretical expressions for the rates of out-of-subspace excitations (∼κn_th) and correction due to single-photon loss ∼κ(1 + n_th) and two-photon loss ∼4κ_2phα². These expressions are only approximations, which become more and more exact as α increases. The figure confirms that the numerically estimated leakage converges to the theoretically predicted value for large α.

Fig. 3. Pauli transfer matrix of the CX gate.

The transfer matrix is obtained by simulating the Hamiltonian in Eq. 26 with α = β = 2, ϕ(t) = πt/T, and T = 10/K. The infidelity of this CX operation with respect to an ideal two-level CX is 9.3 × 10⁻⁷ and results from nonadiabatic transitions due to finite KT.

Fig. 4. Error correction and CX¯cat gadgets.

(A) Each blue shaded block is an error correction gadget for a repetition code with n = 3. The black and green lines indicate code and ancilla qubits, respectively. The green triangles facing the left and right represent preparation and measurement of the ancilla, respectively. In the naive scheme, (n − 1) stabilizer generators for the repetition code are measured using CX gates between pairs of data qubits and ancilla. Transversal CX gates between error-corrected code blocks (shown in the red shaded region) implement a ${\overline{\text{CX}}}_{\text{cat}}$ operation. The code words are further error-corrected at the output. (B and C) Logical error rate for ${\overline{\text{CX}}}_{\text{cat}}$ given in Eq. 38 (solid blue line) and from (8) (solid red line) for different bias η. The black line with slope = 1 is shown for reference. (D and E) Overhead of the ${\overline{\text{CX}}}_{\text{cat}}$ gadget (blue line) for a target logical error rate of 0.67 × 10⁻³ (8, 28). The overhead for the gadget proposed in (8) for the same target error rate is shown in red.

Fig. 5. Schematic for possible realization of the bias-preserving CX gate with superconducting circuits.

Here, the Kerr nonlinear oscillators (of frequencies ω_t and ω_c) are implemented with superconducting nonlinear asymmetric inductive elements or SNAILs (38, 39). A SNAIL can be biased with an external magnetic field so that it has both three- and four-wave mixing capabilities. It can therefore be used to implement the two-photon driven Kerr nonlinear oscillator and realize a cat qubit with biased-noise channel (19). The Hamiltonian in Eq. 26 can be simplified as $\hat{H}=-K{\hat{a}}_{\mathcal{C}}^{\mathrm{†}2}{\hat{a}}_{\mathcal{C}}^2-K{\hat{a}}_{\mathrm{t}}^{\mathrm{†}2}{\hat{a}}_{\mathrm{t}}^2+K{\mathrm{\beta }}^2({\hat{a}}_{\mathcal{C}}^{\mathrm{†}2}+\mathrm{h}.\mathcal{C}.)+K{\mathrm{\alpha }}^2\text{cos}(\mathrm{\varphi }(t))(e^{i\mathrm{\varphi }(t)}{\hat{a}}_{\mathrm{t}}^{\mathrm{†}2}+\mathrm{h}.\mathcal{C}.)-(K{\mathrm{\alpha }}^2\text{sin}(\mathrm{\varphi }(t))/\mathrm{\beta })(i{e}^{i\mathrm{\varphi }(t)}{\hat{a}}_{\mathrm{t}}^{\mathrm{†}2}{\hat{a}}_{\mathcal{C}}+\mathrm{h}.\mathcal{C}.)+(K{\mathrm{\alpha }}^4/2\mathrm{\beta })\text{sin}(2\mathrm{\varphi }(t))(i{\hat{a}}_{\mathcal{C}}^{\mathrm{†}}+\mathrm{h}.\mathcal{C}.)-(K{\mathrm{\alpha }}^4\stackrel{2}{\text{sin}}(\mathrm{\varphi }(t))/{\mathrm{\beta }}^2){\hat{a}}_{\mathcal{C}}^{\mathrm{†}}{\hat{a}}_{\mathcal{C}}^{\mathrm{†}}-\dot{\mathrm{\varphi }}(t){\hat{a}}_{\mathrm{t}}^{\mathrm{†}}{\hat{a}}_{\mathrm{t}}/2+(\dot{\mathrm{\varphi }}(t)/4\mathrm{\beta }){\hat{a}}_{\mathrm{t}}^{\mathrm{†}}{\hat{a}}_{\mathrm{t}}({\hat{a}}_{\mathcal{C}}^{\mathrm{†}}+\mathrm{h}.\mathcal{C}.)$. By expressing the Hamiltonian in this form, the drives required to realize the Hamiltonian become immediately clear. First, a drive to the control cavity (fixed amplitude and phase) centered at 2ω_c is required for the two-photon term driving the control cavity via three-wave mixing. Next, a drive to the target cavity with time-dependent amplitude at 2ω_t results in the two-photon term driving the target cavity via three-wave mixing. An additional drive 2ω_t − ω_c (time-dependent amplitude and phase) is applied to the target cavity to realize the coupling terms $\propto {\widehat{a}}_{\mathrm{t}}^2{\widehat{a}}_{\mathcal{C}}$ in Eq. 26. A drive applied directly to the control cavity centered at ω_c with time-dependent phase and amplitude realizes the single-photon drive to the control cavity. A final drive to the target cavity at ω_c with time-dependent amplitude and phase realizes the last term in the Hamiltonian (40).