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In a recently introduced version of the protocol, the randomization step is performed via unitary circuits of variable depth t<\/mml:mi><\/mml:math>, defining the so-called shallow shadows. For sufficiently large t<\/mml:mi><\/mml:math>, this approach allows one to get around the use of non-local unitaries to probe global properties such as the fidelity with respect to a target state or the purity. Still, shallow shadows involve the inversion of a many-body map, the measurement channel, which requires non-trivial computations in the post-processing step, thus limiting its applicability when the number of qubits N<\/mml:mi><\/mml:math> is large. In this work, we put forward a simple approximate post-processing scheme where the infinite-depth inverse channel is applied to the finite-depth classical shadows and study its performance for fidelity and purity estimation. The scheme allows for different circuit connectivity, as we illustrate for geometrically local circuits in one and two spatial dimensions and geometrically non-local circuits made of two-qubit gates. For the fidelity, we find that the resulting estimator coincides with a known linear cross-entropy, achieving an arbitrary small approximation error &#x03B4;<\/mml:mi><\/mml:math> at depth t<\/mml:mi>=<\/mml:mo>O<\/mml:mi>(<\/mml:mo>log<\/mml:mi>&#x2061;<\/mml:mo>(<\/mml:mo>N<\/mml:mi>\/<\/mml:mo><\/mml:mrow>&#x03B4;<\/mml:mi>)<\/mml:mo>)<\/mml:mo><\/mml:math> (independent of the circuit connectivity). For the purity, we show that the estimator becomes accurate at a depth O<\/mml:mi>(<\/mml:mo>N<\/mml:mi>)<\/mml:mo><\/mml:math>. In addition, at those depths, the variances of both the fidelity and purity estimators display the same scaling with N<\/mml:mi><\/mml:math> as in the case of global random unitaries. We establish these bounds by analytic arguments and extensive numerical computations in several cases of interest. Our work extends the applicability of shallow shadows to large system sizes and general circuit connectivity.<\/jats:p>","DOI":"10.22331\/q-2025-04-08-1698","type":"journal-article","created":{"date-parts":[[2025,4,8]],"date-time":"2025-04-08T15:05:27Z","timestamp":1744124727000},"page":"1698","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":4,"title":["Approximate inverse measurement channel for shallow shadows"],"prefix":"10.22331","volume":"9","author":[{"given":"Riccardo","family":"Cioli","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica e Astronomia, Universit\u00e0 di Bologna and INFN, Sezione di Bologna, via Irnerio 46, I-40126 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Elisa","family":"Ercolessi","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica e Astronomia, Universit\u00e0 di Bologna and INFN, Sezione di Bologna, via Irnerio 46, I-40126 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Matteo","family":"Ippoliti","sequence":"additional","affiliation":[{"name":"Department of Physics, The University of Texas at Austin, Austin, TX 78712, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Xhek","family":"Turkeshi","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Theoretische Physik, Universit\u00e4t zu K\u00f6ln, Z\u00fclpicher Strasse 77a, 50937 K\u00f6ln, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lorenzo","family":"Piroli","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica e Astronomia, Universit\u00e0 di Bologna and INFN, Sezione di Bologna, via Irnerio 46, I-40126 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"9598","published-online":{"date-parts":[[2025,4,8]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Steven T Flammia, David Gross, Yi-Kai Liu, and Jens Eisert. ``Quantum tomography via compressed sensing: error bounds, sample complexity and efficient estimators''. 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