Circuit Knitting Faces Exponential Sampling Overhead Scaling Bounded by Entanglement Cost

TL;DR

This work proves that the sampling overhead of circuit knitting is exponentially lower bounded by the exact entanglement cost of the target bipartite dynamic, even for asymptotic overhead in the parallel cut regime.

Abstract

Circuit knitting, a method for connecting quantum circuits across multiple processors to simulate nonlocal quantum operations, is a promising approach for distributed quantum computing. While various techniques have been developed for circuit knitting, we uncover fundamental limitations to the scalability of this technology. We prove that the sampling overhead of circuit knitting is exponentially lower bounded by the exact entanglement cost of the target bipartite dynamic, even for asymptotic overhead in the parallel cut regime. Specifically, we prove that the regularized sampling overhead assisted with local operations and classical communication (LOCC), of any bipartite quantum channel is lower bounded by the exponential of its exact entanglement cost under separable preserving operations. Furthermore, we show that the regularized sampling overhead for simulating a general bipartite channel via LOCC is lower bounded by $κ$-entanglement and max-Rains information, providing efficiently computable benchmarks. Our work reveals a profound connection between virtual quantum information processing via quasi-probability decomposition and quantum Shannon theory, highlighting the critical role of entanglement in distributed quantum computing.

Figures (5)

• Figure 1: Illustration of the main result: the relationship between the exact (parallel) entanglement cost $E_{C,0}$ of implementing a bipartite quantum channel ${\cal N}$ (left) and the regularized sampling overhead $\gamma^{\infty}$ for simulating ${\cal N}$ via circuit knitting (right). The one-shot exact entanglement cost $E_{C,0}^{(1)}$ measures the minimum entanglement required for Alice and Bob to exactly realize ${\cal N}$. The implementation uses a bipartite channel $\Lambda_{AB \rightarrow A'B'}$ from LOCC, SEP, or PPT operations. The $\gamma$-factor indicates the minimal overhead for simulating ${\cal N}$ with QPD. Our results highlight the limitations of the circuit knitting technique, establishing an exponential lower bound of $\gamma^{\infty}$ relative to $E_{C,0}$ for simulating numerous instances of ${\cal N}$.
• Figure 2: Lower bounds comparison for noisy CNOT operation undertaking the two-qubit depolarizing channel by the noise rate $p$. The orange solid line gives the one-shot exact $\gamma_{{\text{\rm PPT}}}({\cal N})$. The dark and light blue dashed lines with square and circle markers illustrate the lower bounds of regularized PPT $\gamma$-factor by max-Rains information and the max logarithmic negativity, respectively. The shaded area is the possible region for $\gamma_{{\text{\rm PPT}}}^{\infty}({\cal N})$ to lie in.
• Figure 3: Lower bounds comparison for noisy SWAP operation undertaking the single-qubit amplitude damping channel by the damping rate $\gamma$. The orange solid line gives the one-shot exact $\gamma_{{\text{\rm PPT}}}({\cal N})$. The dark and light blue lines illustrate the lower bounds by max-Rains information and the max logarithmic negativity, respectively.
• Figure 4: Lower bounds comparison for noisy control-SWAP operation (CSWAP) undertaking the global depolarizing channel by the noise rate $p$ with respect to different partitioning. The Left and right figures illustrate the bound values of cutting qubit $1$ and $2$ and qubit $2$ and $3$, respectively. The orange line gives the one-shot exact $\gamma_{{\text{\rm PPT}}}({\cal N})$. The dark and light blue lines with square and circle markers illustrate the lower bounds of regularized PPT $\gamma$-factor by max-Rains information and the max logarithmic negativity, respectively. The shaded area indicates that our bounds serve as feasible efficient-computable lower bounds for $\gamma_{{\text{\rm PPT}}}^{\infty}({\cal N})$.
• Figure S1: The strategy of aligning three bipartite noisy quantum gates to parallel via local SWAPs. ${\cal N}_1, {\cal N}_2$ and ${\cal N}_3$ are originally acted on the qubit pairs $(A_3, B_2), (A_3, B_3)$ and $(A_1, B_2)$, respectively.

Theorems & Definitions (22)

• Definition 1
• Theorem 1
• Theorem 2: Kim2021one
• Corollary 3
• Remark 1
• Proposition 1
• Corollary 4
• Lemma 5
• Remark 2
• Theorem 6