{ "id": "1501.06911", "version": "v1", "published": "2015-01-27T21:00:12.000Z", "updated": "2015-01-27T21:00:12.000Z", "title": "Quantum Circuits for Isometries", "authors": [ "Raban Iten", "Roger Colbeck", "Ivan Kukuljan", "Jonathan Home", "Matthias Christandl" ], "comment": "9+8 pages, 3 tables, many circuits", "categories": [ "quant-ph" ], "abstract": "Every quantum operation can be decomposed into a sequence of single-qubit and Controlled-NOT (C-NOT) gates. In many implementations, single-qubit gates are simpler to perform than C-NOTs, and it is hence desirable to minimize the number of C-NOT gates required to implement a circuit. Previous work has looked at C-NOT-efficient synthesis of arbitrary unitaries and state preparation. Here we consider the generalization to arbitrary isometries from m qubits to n qubits. We derive a theoretical lower bound on the number of C-NOT gates required to decompose an isometry for arbitrary m and n, and give an explicit gate decomposition that achieves this bound up to a factor of about two in the leading order. We also perform some bespoke optimizations in the case of small m and n. In addition, we show how to apply our result for isometries to give a decomposition scheme for an arbitrary quantum operation via Stinespring's theorem, and derive a lower bound on the number of C-NOTs in this case too.", "revisions": [ { "version": "v1", "updated": "2015-01-27T21:00:12.000Z" } ], "analyses": { "keywords": [ "quantum circuits", "c-not gates", "explicit gate decomposition", "arbitrary quantum operation", "theoretical lower bound" ], "note": { "typesetting": "TeX", "pages": 8, "language": "en", "license": "arXiv", "status": "editable", "adsabs": "2015arXiv150106911I" } } }