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

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.