Exponential-to-polynomial scaling of measurement overhead in circuit knitting via quantum tomography

TL;DR

This paper tackles the exponential measurement overhead in circuit knitting by introducing tomography-assisted, rescaling-free wire cuts that enable polynomial-scaling reconstruction for tree-structured quantum circuits. The authors develop randomized tomography protocols to learn Heisenberg-evolved observables under unknown channels and integrate these into two main approaches for circuit knitting (timelike-cut and processing-based) to achieve polynomial scaling with the number of cuts. They prove substantial improvements for two-layer and multi-layer tree circuits and establish an information-theoretic exponential separation from conventional quasiprobability-based wire-cutting methods. The results offer a scalable pathway for distributing large quantum computations across local devices, with potential impact on near-term quantum computing and distributed quantum architectures.

Abstract

Circuit knitting is a family of techniques that enables large quantum computations on limited-size quantum devices by decomposing a target circuit into smaller subcircuits. However, it typically incurs a measurement overhead exponential in the number of cut locations, and it remains open whether this scaling is fundamentally unavoidable. In conventional circuit-cutting approaches based on the quasiprobability decomposition (QPD), for example, rescaling factors lead to an exponential dependence on the number of cuts. In this work, we show that such an exponential scaling is not universal: it can be circumvented for tree-structured quantum circuits via concatenated quantum tomography protocols. We first consider estimating the expectation value of an observable within additive error $ε$ for a tree-structured circuit with tree depth 1 (two layers), maximum branching factor $R$, and bond dimension at most $d$ on each edge. Our approach uses quantum tomography to construct, for each cut edge, a local decomposition that eliminates the rescaling factors in conventional QPD, instead introducing a controllable bias set by the tomography sample size. After cutting $R$ edges, we show that $\mathcal{O}(d^3R^3\ln(dR)/ε^2)$ total measurements suffice, including tomography cost. Next, we extend the tree-depth-1 case to general trees of depth $L\geq2$, and give an algorithm whose total measurement cost $\tilde{\mathcal{O}}(d^3K^{5}/ε^2)$ scales polynomially with the number of cuts for complete $R$-ary trees. Finally, we perform an information-theoretic analysis to show that, in a comparable tree-depth-1 setting, conventional QPD-based wire-cutting methods require at least $Ω((d+1)^R/ε^2)$ measurements. This exponential separation highlights the significance of tomography-based construction for reducing measurement overhead in hybrid quantum-classical computations.

Figures (8)

• Figure 1: (a) Schematic representation of tree-structured quantum circuits considered in our analysis, characterized by a maximum branching factor $R$, maximum depth $L$, and maximum bond dimension $d$. Each node represents either a quantum state, an observable with operator norm bounded by 1, or a quantum channel. Each edge denotes an identity channel of dimension at most $d$. (b) Comparison of the number of measurements required by existing circuit-knitting methods to achieve an accuracy $\epsilon$ with high probability. In the "No assumption" setting, we consider a general quantum circuit without structural assumptions, and a $d$-dimensional wire cut is applied at $R$ locations. The bound is estimated via Hoeffding’s inequality, as in standard circuit-cutting analyses. The "Two-layer tree circuit" and "$(L+1)$-layer tree circuit" settings consider situations where we cut the tree-structured circuits in (a), with depth $1$ and $L$, respectively, along the red dashed edges.
• Figure 2: (a) An example of a quantum circuit for simulating a clustered quantum system. (b) Conventional circuit-cutting protocol that uses a measure-and-prepare channel to decompose the identity channel for a clustered quantum system. Due to the rescaling factor in the quasi-probability decomposition, the sampling overhead scales exponentially with the number of clustered systems. (c) Our learning-based cluster simulation. (c1) By randomly choosing the input states, we can estimate the Heisenberg-evolved observable with the weighted measurement outcome. (c2) Using the learned information about the observable, we perform circuit cutting with bounded bias and no rescaling. (c3) We apply the procedure in (c2) to multiple clustered systems.
• Figure 3: (a) Characterization of the identity channel $\mathrm{id}$ in terms of the invariance under the Hilbert-Schmidt inner product (Eq. \ref{['eq:identity_equiv_def']}). A linear map $\mathcal{E}$ coincides with $\mathrm{id}$ if and only if it preserves the expectation value of any observable $Y$ for any input $X$. (b) Schematic figure of the map $\mathcal{E}_{Y_2}$ defined in Eq. \ref{['eq:identity_constrained']}. We focus on the decomposition of the map $\mathcal{E}_{Y_2}$ that preserves the expectation value of a fixed observable $Y_2$ for any input $X$.
• Figure 4: (a) Original quantum circuit before decomposition. Here, $\Phi$ denotes a CPTP map, and $O$ is a Hermitian operator satisfying $\|O \|_{\infty}\leq1$. We consider a decomposition of the identity channel enclosed by the blue dashed circle. (b) When the conventional wire cut in Eq. \ref{['eq:QPD_of_timelike_cut']} is applied to the dashed circuit in (a), a rescaling factor $2d-1$ is required, which consequently increases the variance. (c) There exists an MP channel $\mathcal{M}_{\rm id}$ that preserves the unbiasedness of the ideal expectation value without introducing any additional rescaling factor, i.e., without increasing the variance (Theorem \ref{['thm:existence_of_no_cost_cut']}). (d1) When the dashed circle is replaced by an MP channel $\mathcal{M}_{\rm apr}$ constructed from an approximate effective observable $\tilde{O}_{\Phi}$ satisfying $\| \tilde{O}_{\Phi} - O_{\Phi} \|_{\infty}\leq \epsilon$, the bias in the expectation value is bounded by $2\epsilon$, and no additional rescaling factor is required (Theorem \ref{['thm:zero_cost_timelike_cut_with_an_approximated_unitary']}). (d2) When a classical post-processing function $\mathcal{C}_{\rm apr}$ constructed from $\tilde{O}_{\Phi}$ is used, the bias is bounded by $\epsilon$, and the additional rescaling factor is at most $1+\epsilon$ (Theorem \ref{['thm:zero_cost_proccessing']}).
• Figure 5: Tensor-network representation of an $(L,R,d)$-tree quantum circuit. Nodes correspond to a quantum state $\rho$, CPTP maps $\Phi_{(i_1,...,i_l)}$ ($l=1,...,L$), and local observables $O_{(i_1,...,i_L)}$ with $\|O_{(i_1,...,i_L)} \|_{\infty} \leq 1$. Each edge represents the identity channel of dimension at most $d$, except for those connecting a CPTP map (circle node) to an observable (triangle node).

Theorems & Definitions (46)

• proof
• Theorem 1: Existence of a rescaling-free wire cut
• Theorem 2: Approximate rescaling-free wire cut
• Theorem 3
• Theorem 4: Performance guarantee for $\hat{O}_{\Phi,x}$
• Theorem 5: Two-layer tree circuits
• proof : Proof sketch of Theorem \ref{['thm:2-layer']}
• Theorem 6: Multi-layer tree circuits
• Remark 1: $R=1$
• proof : Proof sketch of Theorem \ref{['thm:tree_simulation_general']}