Performance and structure of single-mode bosonic codes

The early Gottesman, Kitaev, and Preskill (GKP) proposal for encoding a qubit in an oscillator has recently been followed by cat- and binomial-code proposals. Numerically optimized codes have also been proposed, and we introduce codes of this type here. These codes have yet to be compared using the same error model; we provide such a comparison by determining the entanglement fidelity of all codes with respect to the bosonic pure-loss channel (i.e., photon loss) after the optimal recovery operation. We then compare achievable communication rates of the combined encoding-error-recovery channel by calculating the channel's hashing bound for each code. Cat and binomial codes perform similarly, with binomial codes outperforming cat codes at small loss rates. Despite not being designed to protect against the pure-loss channel, GKP codes significantly outperform all other codes for most values of the loss rate. We show that the performance of GKP and some binomial codes increases monotonically with increasing average photon number of the codes. In order to corroborate our numerical evidence of the cat-binomial-GKP order of performance occurring at small loss rates, we analytically evaluate the quantum error-correction conditions of those codes. For GKP codes, we find an essential singularity in the entanglement fidelity in the limit of vanishing loss rate. In addition to comparing the codes, we draw parallels between binomial codes and discrete-variable systems. First, we characterize one- and two-mode binomial as well as multiqubit permutation-invariant codes in terms of spin-coherent states. Such a characterization allows us to introduce check operators and error-correction procedures for binomial codes. Second, we introduce a generalization of spin-coherent states, extending our characterization to qudit binomial codes and yielding a multiqudit code.

Figure 1

Wigner function plots for maximally mixed logical states $\frac{1}{2}{P}_{\mathtt{code}}$ for $\mathtt{code}$ being $\mathtt{cat}$ (5.1), $\mathtt{bin}$ (6.1), $\mathtt{num}$ (Appendix pp2), $\mathtt{gkps}$ (7.7b), and $\mathtt{gkp}$ (7.8), evaluated for given values of the respective parameters of the codes. On the axes, $Q=\frac{1}{2}\langle \widehat{a}+{\widehat{a}}^{†}\rangle$ and $I=\frac{i}{2}\langle {\widehat{a}}^{†}-\widehat{a}\rangle$; color scales are not the same for all plots.

Figure 2

Channel fidelity $F_{\mathcal{E}}$ (1.11) given an optimal recovery operation and optimized over all instances of each code given an occupation number constraint ${\overline{n}}_{\mathtt{code}}\le$ 2 (a), 5 (b), or 10 (c). The dotted diagonal line, drawn for reference, is the optimal $F_{\mathcal{E}}$ for single-rail encoding [6] (whose logical states are the Fock states $|0\rangle ,|1\rangle$). While $\mathtt{gkp}$ codes perform worse than the other codes for sufficiently small $\gamma$ (see insets), they outperform all other codes as $\gamma$ is increased despite not being designed to protect against the pure-loss channel. We were not able to obtain significantly better $\mathtt{num}$ codes with ${\overline{n}}_{\mathtt{num}}>5$ due to a large set of parameters to be optimized over, so red curves in panels (b) and (c) are identical. Parameters for all of the codes used are in Table 4.

Figure 3

Hashing bound $D_{\mathcal{E}}$ (3.3) of the codes which optimize $F_{\mathcal{E}}$ for a given $\gamma$ and given constraints ${\overline{n}}_{\mathtt{code}}\le$ 2 (a), 5 (b), or 10 (c). The boundary of the gray region is $Q_{\overline{n}=2,5,10}$ (3.1), the capacity of ${\mathcal{N}}_{\gamma }$ given the energy constraint of $\overline{n}$. The gray line is the unconstrained capacity $Q_{\overline{n}\to \infty }$ (3.2). The thin dotted diagonal line is $D_{\mathcal{E}}$ for the single-rail encoding, which has the highest $D_{\mathcal{E}}$ at large $\gamma$. Recall that $\mathtt{cat}\text{and}\mathtt{bin}$ codes include the single-rail encoding [i.e., this encoding is also $\mathtt{cat}(\alpha =S=0)$ and $\mathtt{bin}(N=S=0)$], so $\mathtt{bin}\text{and}\mathtt{cat}$ eventually jump to match the dotted line. We also show $D_{\mathcal{E}}$ for the four-qubit $\mathtt{leung}$ code (3.5) versus qubit amplitude damping (i.e., the pure-loss channel restricted to the first two Fock states of four modes). There are no $\mathtt{num}$ codes with ${\overline{n}}_{\mathtt{num}}\ge 5$, so red curves in panels (b) and (c) are identical.

Figure 4

Channel fidelity $F_{\mathcal{E}}(\gamma =0.095)$ (1.11) vs mean occupation number $\overline{n}$ (2.1) for $\mathtt{cat}$ and $\mathtt{bin}$ at spacing (a) $S=3$ and (b) $S=4$. Note that $\mathtt{cat}\left(S\right)$ depends continuously on $\overline{n}$ and $\mathtt{bin}$ exists only at discrete values. For a given $S$, $\mathtt{cat}$ codes perform best at specific $\overline{n}$, corresponding to “sweet-spot” values of $\alpha ={\alpha }_{★}$. However, due to the oscillatory nature of the errors in $\mathtt{cat}(S\gtrsim 3)$, the codes also develop troughs in $F_{\mathcal{E}}$. On the other hand, $\mathtt{bin}\left(S\right)$ performs similar to $\mathtt{cat}\left(S\right)$ at small and large $\overline{n}$ but does not suffer from troughs at intermediate $\overline{n}$. (c) Plots of $\mathtt{cat}$ and $\mathtt{bin}$ for $S\in \{1,2,3,4,5\}$ along with $\mathtt{gkps}$, which turns out to outperform both $\mathtt{cat}$ and $\mathtt{bin}$ and increase monotonically with $\overline{n}$. Similar trends are observed for smaller $\gamma$.

Figure 5

Uncorrectable parts of the QEC matrices, ${\epsilon }^{\mathtt{code}}-c^{\mathtt{code}}$ (see Sec. 4), for $\mathtt{code}$ being (a) $\mathtt{cat}$ (5.1), (b) $\mathtt{bin}$ (6.1), (c) $\mathtt{gkps}$ (7.8), and (d) $\mathtt{gkp}$ (1.9) with parameters such that ${\overline{n}}_{\mathtt{code}}\approx 6$ for all codes and given $\gamma =0.095$. Recall that ${\epsilon }^{\mathtt{code}}$ is a block matrix consisting of $2\times 2$ matrices ${\epsilon }_{\ell {\ell }^{\prime }}^{\mathtt{code}}$ that quantify the overlap between error subspaces $E_{\ell }{P}_{\mathtt{code}}$ and $E_{{\ell }^{\prime }}{P}_{\mathtt{code}}$. These $2\times 2$ matrices are delineated by dotted lines. The four entries in ${\epsilon }_{\ell {\ell }^{\prime }}^{\mathtt{code}}$ are presented as colored squares; note that ${\epsilon }_{\ell {\ell }^{\prime }}^{\mathtt{code}}$ has no imaginary part (all $E_{\ell }{P}_{\mathtt{code}}$ are real). From Fig. 4, we see that $F_{\mathcal{E}}\left(\mathtt{cat}\right)<F_{\mathcal{E}}\left(\mathtt{bin}\right)<F_{\mathcal{E}}\left(\mathtt{gkps}\right)$ at ${\overline{n}}_{\mathtt{code}}\approx 6$, and the above QEC plots nicely corroborate that order of performance. Since $\mathtt{cat}$ and $\mathtt{bin}$ have spacing $S=3$, there are no off-diagonal errors for $\ell \le 3$ (inside the red square). However, both codes suffer from distortion on the diagonal portions of ${\epsilon }_{\ell {\ell }^{\prime }}^{\mathtt{code}}$, with $\mathtt{bin}$ suffering noticeably less at this particular $\gamma$. On the other hand, $\mathtt{gkps}$ (square lattice) and $\mathtt{gkp}$ (shifted hexagonal lattice) barely suffer from any noticeable errors. Since $\mathtt{gkps}$ codes have spacing $S=1$ due to a symmetry of their underlying lattice, ${\epsilon }_{\ell {\ell }^{\prime }}$ does not contain off-diagonal errors for $\ell \le 1$ (more generally, for $\ell -{\ell }^{\prime }$ odd). Shifted $\mathtt{gkp}$ codes do not have that symmetry, but perform comparably.

Figure 6

${\text{Log}}_{10}$ plot of $1-F_{\mathcal{E}}(\gamma =0.095)$ (1.11) vs (a) $\mathtt{cat}$ code parameters $S$ and $\alpha$ and (b) $\mathtt{bin}$ code parameters $S$ and $N$ (cf. Fig. 1 in Ref. [23]). For a given $S$, $\mathtt{cat}$ achieves the best performance (purple) at multiple $S$-dependent sweet spots ${\alpha }_{★}$. While $F_{\mathcal{E}}\left({\alpha }_{★}\right)$ for $\mathtt{cat}\left(S\right)$ does not increase with increasing $S$ for the values we have sampled, performance of $\mathtt{bin}(N,S\approx 2N)$ clearly does (cyan). There is thus reason to believe that $\mathtt{bin}$ outperforms $\mathtt{cat}$ when no energy constraints are imposed.

Figure 7

(a) Table listing the basis elements used to express binomial ($\mathtt{bin}$) (6.1), permutation-invariant ($\mathtt{perm}$) (6.14), and two-mode binomial codes ($\mathtt{bin}2$) (6.16) (with $m\in \{0,1,\cdots ,2J\}$). The coefficients next to these basis states are those of spin-coherent states $|\frac{\pi }{2},\pi \mu {\rangle }_J$ (6.10) of a spin $J$. (b) Plots of overlaps ${|}_J{\langle \theta ,\varphi |\psi \rangle |}^2$ vs $\theta ,\varphi$ given a spin state $|\psi \rangle$. The first (second, third) column shows normalized states ${\left\{{\left(J_z\right)}^p\right|\frac{\pi }{2},\pi \mu \rangle }_{J=4}{\}}_{\mu =0}^1$ for $p\in \{0,2,4\}$. These plots show that powers of the “error” $J_z$ cause $|\frac{\pi }{2},0{\rangle }_J$ and $|\frac{\pi }{2},1{\rangle }_J$ to approach each other in the Bloch sphere and eventually overlap at the two poles. The dashed arcs connecting $\theta =\pm \frac{\pi }{2J}p$ serve as a guide to the eye. Since $J_z$ is mapped to $\widehat{n}-J$ under the Holstein-Primakoff transformation, this provides an interpretation of dephasing errors for $\mathtt{bin}$ codes, which are correctable as long as $p<J=\frac{1}{2}(N+1)$. Here, $N=7$ and one can see from the third column that ${\left(J_z\right)}^4{|\frac{\pi }{2},0\rangle }_J$ and ${\left(J_z\right)}^4{|\frac{\pi }{2},1\rangle }_J$ overlap.

Figure 8

Wigner function sketches of the two $Z_{\mathtt{gkps}}$-, $X_{\mathtt{gkps}}$-, $Y_{\mathtt{gkps}}$-logical states. Comparing to the fourth panel in Fig. 1, which shows that the code lattice is square, here we see that the lattices formed by the logical states may be square or rectangular, depending on which logical operator is considered. The unit cell of the state lattices (7.6) is marked by “$\times$” in the two leftmost panels; the remaining dots appear as a result of the coherences between different coherent states.

Figure 9

Contour plot of $F_{\mathcal{E}}$ vs dimensionless Kerr parameter $Kt$ and damping parameter $\chi \equiv \kappa t$ for $\mathtt{cat}$, $\mathtt{bin}$, $\mathtt{num}$, and $\mathtt{gkps}$ picked such that they all have $\overline{n}\approx 2$. Here, $K$ is the strength of the Kerr Hamiltonian (8.1), $\kappa$ is the cavity decay rate, and $t$ is time. Starting with a fixed $\chi$ and tracking increasing $Kt$, we see that $F_{\mathcal{E}}$ quickly decreases to a constant for all codes considered, implying a potentially universal failure of error correction when $Kt$ is large. However, $F_{\mathcal{E}}$ is minimal at high-symmetry points of the codes (dashed red lines) and maximal in between. For example, $\frac{1}{2}{P}_{\mathtt{num}}$ for the $\mathtt{num}$ code is threefold symmetric (see Fig. 1), and $F_{\mathcal{E}}\left(\mathtt{num}\right)$ is minimal at $Kt$, being the first few multiples of $2\pi /3$.

Figure 10

Channel fidelity for $\mathtt{cat}$, $\mathtt{bin}$, $\mathtt{num}$, and $\mathtt{gkps}$ picked such that they all have $\overline{n}\approx 2$, given (a) the pure-loss channel $\mathcal{N}$ (1.4) and (b) the pure-loss channel with the capability of knowing the number of photons lost, modeled by the quantum instrument $\tilde{\mathcal{N}}$ (8.4). (c) Uncorrectable parts of the QEC matrices, ${\epsilon }^{\mathtt{code}}-c^{\mathtt{code}}$ (see Sec. 4), for the four codes (cf. Fig. 5). Since uncorrectable parts in the diagonal blocks ${\epsilon }_{\ell \ell }^{\mathtt{code}}$ are all that matter in recovering from $\tilde{\mathcal{N}}$, we see that $\mathtt{cat}$, $\mathtt{bin}$, and $\mathtt{num}$ outperform $\mathtt{gkp}$ at this $\overline{n}$.

Figure 11

Overlap ${|}_J{\langle \theta ,\varphi |\frac{\pi }{2},\frac{2\pi }{d}\mu \rangle }_{J,d}{|}^2$ vs $\theta ,\varphi$ (in radians) for qudit states $|\frac{\pi }{2},\frac{2\pi }{d}\mu {\rangle }_{J,d}$ (C1) with $\mu =0$ and $d,N$ picked such that the spin $J=\frac{1}{2}(d-1)(N+1)=12$. These states resemble spin-squeezed states and characterize the qu$d$it $\mathtt{bin}$ and $\mathtt{bin}\mathtt{2}$ codes (see Appendix pp3). For fixed $J$, the degree of squeezing increases with $d$.