Steering paths mid-flight for fault-tolerance in measurement-based holonomic gates

TL;DR

A fault-tolerant framework for implementing measurement-based holonomic gates that leverages continuous measurements with real-time feedback is introduced that relaxes the stringent adiabaticity requirement and enables faster implementation of holonomic gates.

Abstract

Continuous measurement-based holonomic quantum computation provides a route to universal logical computation in quantum error correcting codes. We introduce a fault-tolerant framework for implementing measurement-based holonomic gates that leverages continuous measurements with real-time feedback. We show that non-Markovian decoherence is intrinsically suppressed through the quantum Zeno effect, while Markovian errors are identified by the decoding of measurement records to reveal the rotated syndrome subspace populated during the evolution. This information enables steering holonomic paths mid-flight to ensure that the final evolution realizes the target logical gate. We further demonstrate that non-adiabatic effects give rise to measurement-induced errors, and we show that these can also be corrected by an analogous protocol. This approach relaxes the stringent adiabaticity requirement and enables faster implementation of holonomic gates.

Figures (5)

• Figure 1: The horizontal lift as a unique curve in $S_{N, K}(\mathbb{C})$ with the base manifold $G_{N, K}(\mathbb{C})$. The difference between the initial point $V(0)$ and the final point $V(T)$ is the holonomy.
• Figure 2: Average gate fidelity in the presence of static, $1/f$ noise (with different values of $\tau$), and white noise (Markovian errors). Frequent measurements impose a Zeno regime on the system dynamics and suppress the non-Markovian noise, which slows the decay of the average code state fidelity. For a given measurement rate, decreasing the pulse lifetime $\tau$ makes the temporal correlations decay faster so the dynamics are effectively more Markovian. In the limit of purely Markovian noise, frequent measurements cannot suppress the error. The plots represent an ensemble average over 1000 trajectories.
• Figure 3: Real-time error correction by dynamic path modulation. Here, the black dashed lines depict the fibers over the respective subspaces. The blue solid curves depict the actual evolution due to the path $V(t)$, and the dashed blue curves depict an emulated evolution. Due to \ref{['prop:unique_lift']}, the point on $\mathcal{F}$ acquires the same unitary transformation $U(t)$ for the evolution path $V(t)$. Assuming that $U(\tau)\tilde{L}_E(0)$ is the instantaneous error state provides $\tilde{L}_E(0)$, and the path can be modified to $\tilde{V}(t)$; at the end of the evolution with $T=2\pi/\omega$, the emulated holonomy is $G$. Note that the figure depicts a scenario where the instantaneous error state belongs to $\mathcal{H}_E(\tau)$. Analogously, the system can occupy $\mathcal{H}_{EX}(\tau)$ or $\mathcal{H}_X(\tau)$.
• Figure 4: Evolution of the expectation values of the rotated stabilizer generators. (a) The expectation values remain close to $+1$, which indicates that there is no error during the evolution. (b) the error rate is intentionally set to $\gamma=0$, and the rotation rate is chosen such that $\omega/\kappa=0.1$ (corresponding to a jump probability of $\approx 36\%$). We observe a syndrome $1011$ at $t\approx 3$ corresponding to an $X$ error purely due to non-adiabatic rotation of the code space. (c) A different trajectory with $\gamma/\kappa=\mathcal{O}(10^{-4})$, and $\omega/\kappa=\mathcal{O}(10^{-2})$. The syndrome $1100$ corresponds to $\sigma^x_2$ error.
• Figure 5: Average fidelity $\mathbb{E}[\mathcal{F}(\rho(T))]$ as a function of (a) the error rate $\gamma$, where $\rho(t)$ is obtained by numerically integrating \ref{['eq:sme_313']}, (b) the rotation rate $\omega$ (equivalently the final evolution time $T$). When the state jumps to a syndrome space at $\tau$, the path is changed from $V(\tau)$ to $\tilde{V}(\tau)$. The code considered is the $[\![3, 1, 3]\!]$ bit-flip code with $2$ ancilla qubits initialized in $\ket{00}$. The logical operator $H=\otimes_{i=1}^3\sigma_i^z$, the logical rotation angle $\theta=\pi/6$ and $X = \sigma_1^x\sigma_3^z\sigma^x_4\sigma^x_5$. The plots are averaged over an ensemble of $2000$ trajectories.

Theorems & Definitions (10)

• Proposition 1: mhqc
• Lemma 1
• proof
• Theorem 1: mhqc
• Proposition 2
• proof
• Proposition 3
• proof
• Theorem 2
• proof