Arbitrary high-fidelity binomial codes from multiphoton spin-boson interactions

Abstract

Encoding a qubit in the continuous degrees of freedom of a quantum system, such as bosonic modes, is a powerful alternative to modern quantum error correction (QEC). Among the most prominent bosonic QEC codes, binomial codes provide protection against loss and dephasing errors by encoding logical states in superpositions of Fock states with binomially weighted coefficients. While much attention has been given to their error-correcting capabilities and integration into fault-tolerant architectures, efficient methods for generating arbitrary binomial codewords remain scarce. In this work, we propose a scheme for generating these codewords by exploiting nonlinear multiphoton interactions between a continuous-variable bosonic mode (oscillator) and a two-level system (spin/qubit). Our proposed scheme assumes the ability to prepare the oscillator in an arbitrary Fock state and the qubit in an arbitrary superposition of its basis states and access to arbitrarily high multiphoton interactions. To enhance the experimental feasibility of our scheme, we further demonstrate how to reduce the required order parameter of multiphoton interactions by a factor of two for a special class of code states.

Figures (8)

• Figure 1: Schematic of the proposed protocol for synthesizing binomial code states composed of two Fock state superpositions. The protocol begins with state preparation, where the oscillator is initialized in a Fock state $\ket{n_1}$ (blue solid circle) and the qubit in a superposition state $\cos\theta\ket{g}+\sin\theta\ket{e}$ (red arrow). The joint system then undergoes an entangling evolution governed by a nonlinear multiphoton Jaynes–Cummings (MPJC) interaction of order $m$, mediated by a tunable coupling strength $g$. A projective measurement on the qubit at an optimized interaction time $\tau$ collapses the oscillator into a nontrivial superposition of Fock states. The parameters $n_1$, $m$, $\theta$, and $\tau$ uniquely determine the final encoded binomial state. The protocol can be recursively applied to construct more complex binomial states involving additional Fock components (see main text).
• Figure 2: Combinations of $\{\tau,\, \theta\}$ for which the fidelity $F_{2g}(\tau,\,\theta)$ of the postselected oscillator state—obtained by measuring the qubit in the ground state $\ket{g}$ and given in Eq. \ref{['eqn_fid_two_fock_ket_g']}—matches each of the eight target logical binomial states with unit fidelity are shown in panels (a)–(h), respectively. The parameters $\tau$ and $\theta$ were sampled uniformly over $0 \leqslant \tau \leqslant 2\pi$ and $0 \leqslant \theta \leqslant \pi$, using 1001 and 501 points, respectively. All displayed points satisfy a numerical fidelity threshold of $|F_{2g}-1|\leqslant 10^{-4}$, with red points marking those that additionally meet the stricter threshold $|F_{2g}-1|\leqslant 10^{-6}$.
• Figure 3: Combinations of $\{\tau,\, \theta\}$ for which the fidelity $F_{2e}(\tau,\,\theta)$ of the postselected oscillator state—obtained by measuring the qubit in the ground state $\ket{e}$ and given in Eq. \ref{['eqn_fid_two_fock_ket_e']}—matches each of the eight target logical binomial states with unit fidelity are shown in panels (a)–(h), respectively.. All sampling details and fidelity thresholds are identical to those used in Fig. \ref{['fig:two_sup_ket_g']}.
• Figure 4: Combinations of $\{\tau, \theta, \phi\}$ for which the fidelity ($F_{3g_\pm}$ in Eq. \ref{['eqn_fid_three_fock_ket_3g_pm']}) of the postselected oscillator state in Eq. \ref{['eqn_psit_3g_both']} reaches unity (within numerical precision) with each of the four three-component binomial targets $\ket{\bar{0}}_{2,4}= \frac{1}{2\sqrt{2}} \left( \ket{0} + \sqrt{6}\ket{4} + \ket{8} \right)$, $\ket{\bar{0}}_{2,5}=\frac{1}{4} \left( \ket{0} + \sqrt{10}\ket{4} + \sqrt{5}\ket{8} \right)$, $\ket{\bar{1}}_{2,5}=\frac{1}{4}\left(\sqrt{5}\ket{2} + \sqrt{10}\ket{6} + \ket{10}\right)$, and $\ket{\bar{1}}_{2,6}=\frac{1}{4 \sqrt{2}}\left(\sqrt{6}\ket{2} + \sqrt{20}\ket{6} + \sqrt{6}\ket{10}\right)$ are shown. The top row corresponds to the case $n_2 = n_1 + m$ ($F_{3g_+}$), while the bottom row corresponds to $n_2 = n_1 - m$ ($F_{3g_-}$). The parameters were sampled uniformly over $0 \le \tau \le 2\pi$ (1001 points) and $0 \le \theta,\phi \le \pi$ (501 points each). As in Figs. \ref{['fig:two_sup_ket_g']} and \ref{['fig:two_sup']}, all plotted points satisfy $|F - 1| \le 10^{-4}$, with red points indicating those additionally meeting the stricter threshold $|F - 1| \le 10^{-6}$.
• Figure 5: Combinations of $\{\tau, \theta, \phi\}$ for which the fidelity ($F_{2e_\pm}$ in Eq. \ref{['eqn_fid_three_fock_ket_3e_pm']}) of the postselected oscillator state in Eq. \ref{['eqn_psit_3e_both']} reaches unity (within numerical precision) with each of the four three-component binomial targets $\ket{\bar{0}}_{2,4}$, $\ket{\bar{0}}_{2,5}$, $\ket{\bar{1}}_{2,5}$, and $\ket{\bar{1}}_{2,6}$ are shown. As in Fig. \ref{['fig:three_sup_ket_g']}, the top row corresponds to the case $n_2 = n_1 + m$ ($F_{2e_+}$), while the bottom row corresponds to $n_2 = n_1 - m$ ($F_{2e_-}$. All sampling details and fidelity thresholds are identical to those used in Fig. \ref{['fig:three_sup_ket_g']}.