Remote Entanglement in Lattice Surgery: To Distill, or Not to Distill

TL;DR

The fidelity crossover point separating the two regimes is identified and it is shown that choosing the right strategy can reduce resource overhead by up to two orders of magnitude at low fidelities and up to 68% at high fidelities.

Abstract

Distributed quantum computing can potentially address the scalability challenge by networking processors through photon-mediated remote entanglement. Prior approaches assumed that remote Bell pairs require distillation, resulting in substantial overhead, to achieve sufficiently high fidelity before use. However, recent results show that lattice-surgery operations at logical qubit boundaries tolerate significantly higher error rates than previously assumed. We quantify the resource trade-offs between distillation overhead and surface-code distance requirements under realistic constraints including probabilistic entanglement generation and memory decoherence. We identify the fidelity crossover point separating the two regimes and show that choosing the right strategy can reduce resource overhead by up to two orders of magnitude at low fidelities and up to 68% at high fidelities. We briefly describe the application of these methods to ion-trap and neutral-atom platforms. These results provide co-design principles for optimizing photonic interconnects and fault-tolerant architectures in distributed quantum computers.

Figures (8)

• Figure 1: Remote lattice surgery for distributed architecture. (a) Logical level: Modules A and B are physically separated and perform remote operations on encoded logical qubits through communication qubits (green) via Pauli product measurements. (b) Lattice surgery level: $Z \otimes Z$ measurements between logical qubits C$_A$ and C$_B$ are implemented via merge-split of the logical qubit boundaries. The merge operation requires $d$ syndrome measurement rounds, where $d$ is the surface code distance, consuming $\mathcal{O}(d)$ Bell pairs per round. The split operation is instantaneous as it coincides with the standard syndrome measurement cycle. (c) Physical level: Top illustrates the merge operation between two surface code patches encoding logical qubits q$_1$ (left) and q$_2$ (right). Data qubits are shown as white circles and stabilizer plaquettes as colored regions. The light yellow circles and faded areas represent ancilla qubits and additional stabilizers generated during merging. Blue dashed lines indicate the physical module boundaries. Bottom shows the gate teleportation circuit with Bell pair. The Bell pairs connect the interface regions to enable the remote CNOT operations (highlighted in red in the top panel).
• Figure 2: Comparison of remote entanglement utilization schemes. (a) Direct use of raw Bell pairs. Heralded entanglement is generated via optical links between communication qubits in Module A and B, with link error $p_{\mathrm{raw}}$. (b) Distillation-based approach. $n_{\mathrm{pairs}}$ raw Bell pairs are consumed by an entanglement distillation protocol $D$ to produce a single high-fidelity Bell pair with error $p_{\mathrm{eff}} < p_{\mathrm{raw}}$.
• Figure 3: Required surface-code distance $\bm{d}$ for remote lattice surgery versus raw Bell-pair fidelity $\bm{F_0}$, at target logical error rates (a) $\bm{p_L \le 10^{-3}}$, (b) $\bm{10^{-6}}$, (c) $\bm{10^{-9}}$, (d) $\bm{10^{-12}}$. The no-distillation baseline (black solid) is compared with three entanglement distillation protocols from Ref. Krastanov2019optimized: double-select (blue solid, $3{\to}1$, 3 ops), EXPEDIENT (orange dashed, $5{\to}1$, 6 ops), and STRINGENT (green dash-dot, $13{\to}1$, 18 ops). Grey-shaded regions indicate the minimum achievable distance at perfect input fidelity ($F_0 = 100\%$), where only local errors contribute. The vertical dashed line in each panel marks the fidelity at which no distillation first achieves a shorter total operation time than every distillation protocol (see Sec. \ref{['subsubsec:time']}); its color matches the last protocol overtaken (double-select, blue). Distance--fidelity model from Eq. (\ref{['eq:bell-logical-error']}); error parameters in Table \ref{['tab:noise_parameters']}.
• Figure 4: Raw Bell-pair consumption per QEC cycle between two logical qubits via remote lattice surgery versus raw Bell-pair fidelity ($\bm{F_0}$) assuming no decay/decoherence at target logical error rates (a) $\bm{p_L \le 10^{-3}}$, (b) $\bm{10^{-6}}$, (c) $\bm{10^{-9}}$, (d) $\bm{10^{-12}}$. Curves compare the no-distillation case (black) with three entanglement distillation protocols from Ref. Krastanov2019optimized: double-select $3{\to}1$ (blue), EXPEDIENT $5{\to}1$ (orange), and STRINGENT $13{\to}1$ (green). Cost quantifies total raw Bell-pair consumption, incorporating surface-code distance scaling ($\propto d_s^2$) and distillation overhead. Colored background regions indicate the resource-optimal strategy for different input fidelity regimes. The vertical dashed line marks the fidelity above which no distillation is optimal; its colour (blue) indicates the last protocol overtaken (double-select in all cases). No distillation is optimal for $F_0 \gtrsim 97.07\%$ (a), $95.44\%$ (b), $95.38\%$ (c), and $95.50\%$ (d). The greatest no-distillation advantage is a $68\%$ overhead reduction ($F_0 = 98.64\%$, $p_L = 10^{-3}$: 45 vs 140 pairs/cycle, both $d_s = 5$). Shaded bands show Monte Carlo uncertainty from $10^3$ runs at 150 fidelity values spanning $F_0 \in [90.0\%, 99.0\%]$. Distillation parameters are taken from Ref. Krastanov2019optimized; distance--fidelity model is from Eq. (\ref{['eq:bell-logical-error']}); error rates are in Table \ref{['tab:noise_parameters']}.
• Figure 5: Bell-pair scheduling schemes for remote lattice surgery above and below the on-the-fly threshold $\lambda_{\mathrm{th}}$ (Eq. \ref{['eq:on_the_fly_condition']}). (a) On-the-fly regime ($\lambda \geq \lambda_{\mathrm{th}}$): the generation rate is sufficient to supply each syndrome round on demand, so Bell pairs are consumed immediately with worst-case storage time bounded by $T^{\mathrm{round}}$ (Eq. \ref{['eq:fidelity_decay_bound']}). (b) Below threshold ($\lambda < \lambda_{\mathrm{th}}$), two strategies arise. Strategy 1 (round-by-round): because generation cannot keep pace, collecting $n^{\mathrm{round}}$ pairs per round requires a finite accumulation window; pairs generated early in the window must wait in memory until the round begins, degrading their fidelity (Eq. \ref{['eq:fidelity_S1']}), and data qubits likewise accumulate idle error during the wait (Eq. \ref{['eq:idle_error']}). Strategy 2 (pre-buffered): all $N_{\mathrm{QEC}}$ pairs are collected before any syndrome extraction begins; the $\tilde{d}_s$ rounds then proceed back to back, eliminating data-qubit idle error but exposing the earliest pairs to decay over the full accumulation time $T_{\mathrm{QEC}}$ (Eq. \ref{['eq:fidelity_S2']}). In both strategies the required code distance $\tilde{d}_s$ and the per-round consumption $n^{\mathrm{round}}$ are mutually dependent, necessitating the self-consistent iteration of Eq. \ref{['eq:self_consistent_ds']} (Algorithm \ref{['alg:self_consistent']}).