Stabilizers may be poor bounds for fidelities

TL;DR

The paper shows that stabilizer expectation values (SEVs) for GKP codes do not reliably bound fidelity to ideal GKP states; it constructs states with SEVs near unity yet arbitrarily low fidelity, proving an upper bound F ≤ ((s_q+1)/2)((s_p+1)/2). Through Gaussian and rectangle approximations, the authors demonstrate that even normalizable, physically realizable states can spoof SEVs while remaining far from ideal GKP states. Consequently, SEVs can only rule out poor states, not certify high-quality encodings, with implications for state characterization and fault-tolerant quantum computation using GKP codes. The work emphasizes the need for fidelity-based or distance metrics in certification and provides a general framework applicable to various lattice spacings and multimode implementations.

Abstract

The defining feature of ideal Gottesman-Kitaev-Preskill (GKP) states is that they are unchanged by stabilizers, which allow them to detect and correct for common errors without destroying the quantum information encoded in the states. Given this property, can one use the amount to which a state is unchanged by the stabilizers as a proxy for the quality of a GKP state? This is shown to hold in the opposite manner to which it is routinely assumed, because in fact the fidelity a state has to an ideal GKP state is only upper bounded by the stabilizer expectation values. This means that, for qubits encoded in harmonic oscillators via the GKP code, a good stabilizer expectation value does not guarantee proximity to an ideal GKP state in terms of any distance based on fidelity.

Figures (6)

• Figure 1: Exemplary state formed from a superposition of GKP states with equal values of $k=0$. These are equivalent to the base state $\psi(x)$ repeating with period $2\sqrt{\pi}$ and thus have position stabilizer satisfying $s_q=1$. The sharpness of the peaks relate to the quality of the GKP state and to the momentum stabilizer.
• Figure 2: Approximate GKP state formed from superpositions of states with Gaussians probability distributions in position space with variances $V=1/10$. The heights are all equal but the pattern only repeats $2N+1$ times, here with $N=3$, so that the overall state is normalizable.
• Figure 3: Comparison of a GKP state to a state with large stabilizer expectation values (SEVs) yet small fidelity to the former; this discrepancy increases between the top plot ($V=1/1000$, $\epsilon=1/5$) and the bottom plot ($V=1/10000$, $\epsilon=1/10$). The GKP state is the blue, single-peaked function that repeats every $2\sqrt{\pi}$ in position, while the spoofing state is the orange, triple-peaked function with spacing $\epsilon$ that then repeats every $2\sqrt{\pi}$. With these parameters, the error is on the order of $\exp(-\epsilon^2/8V)$, which is less than $1\%$ for the top plot and less than $10^{-5}$ for the bottom. Since the spoofing state is narrow in position, it has large momentum SEV; since the spoofing state repeats in position, it has large position SEV; but, since the spoofing state is never as tall as the GKP state, it has low fidelity. Here the triple peaks imply $M=1$ and maximal fidelity $F=1/3$ even though $s_q,s_p\approx 1$.
• Figure 4: Adding vectors whose lengths sum to unity leads to different possible combinations of maximal vector length and length of summed vector. The black circle has radius $1/2$. Top: black solid vector has length $F=11/20>1/2$ and can be added to any one of the other dashed vectors of length $1-F$ to get a new vector with a range of possible lengths. Progressing from topmost (red) to bottommost (orange), the sum ranges from close to unity to close to $2F-1$. Bottom: black solid vector has length $F=9/20<1/2$ and can be added to any symmetric pair of dashed and dotted vectors of length $(1-F)/2$ to get a new vector with any possible length from zero to unity. The symmetric pairs are chosen to be reflections of each other about the black solid line, ranging from topmost (red) whose sum is close to 1 to bottommost (orange) whose sum is close to $2F-1$. It may seem that this does not pass through zero because $2F-1>0$ when $F<1/2$, but indeed the total vector length is not monotonic for the bottom figure because the summed vector goes from pointing up to pointing down.
• Figure 5: Bounds between GKP state fidelity $F$ and momentum SEV $s_p$ for the rectangle approximation to GKP states. With small $N^\prime$ (poor GKP approximations; dashed curves), a SEV above a certain threshold gives an upper and lower bound for the GKP-state fidelity (the shaded region between two dashed lines for a given value of $s_p$). With large $N^\prime$ (good GKP approximations; solid curves), a SEV only gives an upper bound for the GKP-state fidelity (the fidelity can be anything to the left of the diagonal line that runs from the bottom-left to the top-right of the figure). The means that measuring $s_p$ cannot certify a good value of $F$ but can rule out large values of $F$ when $s_p$ is small. In the limit of true GKP states ($N^\prime\to\infty$), the filled region is exactly a triangle that includes the entire top left of the plot.