Degeneracy-Preserving Quantum Nondemolition Measurement of Parity-Type Observables for Cat Qubits

A central requirement for any quantum error correction scheme is the ability to perform quantum nondemolition measurements of an error syndrome, corresponding to a special symmetry property of the encoding scheme. It is in particular important that such a measurement does not introduce extra error mechanisms, not included in the error model of the correction scheme. In this Letter, we ensure such a robustness by designing an interaction with a measurement device that preserves the degeneracy of the measured observable. More precisely, we propose a scheme to perform continuous and quantum nondemolition measurement of photon-number parity in a microwave cavity. This corresponds to the error syndrome in a class of error correcting codes called the cat codes, which have recently proven to be efficient and versatile for quantum information processing. In our design, we exploit the strongly nonlinear Hamiltonian of a high-impedance Josephson circuit, coupling a high-$Q$ cavity storage cavity mode to a low-$Q$ readout one. By driving the readout resonator at its resonance, the phase of the reflected or transmitted signal carries directly exploitable information on parity-type observables for encoded cat qubits of the high-$Q$ mode.

Figure 1

(a) Eigenvalues of ${\mathbf{H}}^{\mathrm{RWA}}/E_J$ for ${\phi }_a=4$. The changing signs around the Fock state $|4⟩$ explain why under two or four-photon process, the Hamiltonian acts as a parity Hamiltonian for a coherence state $|\alpha ⟩$ with $|\alpha |={\phi }_a/2=2$. Although the parity operator $\cos (\pi {\mathbf{a}}^{†}\mathbf{a})$ requires also its eigenvalues to have the same module, this sign alternance is sufficient for having a parity Hamiltonian under two-photon loss. (b) Nonvanishing matrix elements of projected Hamiltonian for the two-photon driven dissipation, $c_{\alpha }^{\pm }=⟨{\mathcal{C}}_{\alpha }^{\pm }|{\mathbf{H}}^{\mathrm{RWA}}|{\mathcal{C}}_{\alpha }^{\pm }⟩$ as a function of ${\phi }_a$ ($\alpha$ being set to 2). As shown in the inset, $c_{\alpha }^{\pm }$ take opposite values for $3<{\phi }_a<5$, indicating that the projected Hamiltonian acts as the ${\mathit{\sigma }}_Z$ Pauli operator in the logical basis $|{\mathcal{C}}_{\alpha }^{\pm }⟩$. (c) Nonvanishing matrix elements of projected Hamiltonian for the four-photon driven dissipation, $c_{\alpha }^{jj}=⟨{\mathcal{C}}_{\alpha }^{(j\text{mod}4)}|{\mathbf{H}}^{\mathrm{RWA}}|{\mathcal{C}}_{\alpha }^{(j\text{mod}4)}⟩$ as a function of ${\phi }_a$ ($\alpha$ being set to 5). We note that for $9<{\phi }_a<12$, the Hamiltonian is degenerate in each parity subspace. (d) Effect of the amplitude $|\alpha |$ on the parity-subspace degeneracy for the four-photon process. Fixing ${\phi }_a=2\alpha$, we observe that for $\alpha <4$, we deal with a nondegenerate Hamiltonian [hence, the choice of $\alpha =5$ in (c)]. The inset illustrates that while the parity Hamiltonian strength ${\mathrm{\Delta }}_{\text{parity}}=\sqrt{(c_{\alpha }^{00}-c_{\alpha }^{11}{)}^2+(c_{\alpha }^{22}-c_{\alpha }^{33}{)}^2}$ decreases in $1/|\alpha |$, the parity subspace nondeegeneracy ${\delta }_{\text{parity}}=\sqrt{(c_{\alpha }^{00}-c_{\alpha }^{22}{)}^2+(c_{\alpha }^{11}-c_{\alpha }^{33}{)}^2}$ decreases exponentially in $|\alpha {|}^2$.

Figure 2

(a) Schematic of a realization of a single-mode continuous parity measurement in the presence of two-photon driven dissipation. Similarly to [30], on one side we mediate a two-photon dissipation of the storage high-$Q$ cavity mode and on the other, we couple to a low-$Q$ readout mode through a high-impedance Josephson circuit. (b) Electrical circuit equivalent, without the two-photon driven dissipation. The cavity (blue) and readout (green) modes are modeled by LC oscillators and are capacitively coupled to a high impedance Josephson circuit mode. This Josephson mode consists of a large superinductance, formed from an array of large Josephson junctions (as in the fluxonium [14]), in series with a nonlinear circuit element, depicted as a crosshatched box. (c) Schematic design of the joint-parity measurements between two high-$Q$ cavity modes under two-photon driven dissipation, inspired from [6] and based on an extension of (a).

Figure 3

The left axis (black straight curve) illustrates the measurement rate in the single-mode case (4) in units of $E_J/\hslash$ and renormalized by the parameters of the readout mode. We observe an optimal measurement rate around ${\phi }_a=2\alpha$. The right axis (colored dashed curves) illustrate the efficiency of the measurement limited by the higher-order Zeno effects [simulating (5)]. Here, we fix ${\phi }_c=0.1$ and the number of readout photons $n_c=1$, and we plot ${\eta }_{1\mathrm{ph}}={\mathrm{\Gamma }}_m^{1\mathrm{ph}}/({\mathrm{\Gamma }}_m^{1\mathrm{ph}}+{\mathrm{\Gamma }}_Z)$. This efficiency achieves a local optimum near ${\phi }_a=2\alpha$ corresponding also to the optimum measurement rate ${\mathrm{\Gamma }}_m$. This higher-order effect is suppressed by decreasing the Zeno parameter ${\epsilon }_{\text{Zeno}}=E_J/\hslash {\kappa }_{2\mathrm{ph}}$.