qLDPC surgery, measuring pauli product with overlapping support

qLDPC surgery, measuring pauli product with overlapping support Ask Question Asked 4 days ago Modified yesterday Viewed 61 times 1 I'm trying to implement some of the surgery scheme for measuring pauli product in qldpc codes, using homological measurement as described in https://arxiv.org/abs/2410.02753. There is a few point I would need help on clarifying: If I want to measure a logical Pauli product of two logical operators X 1 ¯ X 2 ¯ 𝑋 1 ¯ 𝑋 2 ¯ that happen to have overlapping supports (some physical qubits are included in both logical operator), should the support Q 𝑄 used for the building procedure be the union of both support or it should be the "xor" of the two support (i.e. exclude element that are in both supports). When measuring mixed joint Pauli products (including X and Z), from my understanding, the construction results in a non-CSS code, is that correct? If yes, this introduce a lot of complexity on syndrome extraction circuit and decoding. Is it possible to measure any such product with a CSS construction? error-correctionqldpc Share Improve this question Follow asked May 12 at 13:28 L.DZ 3051 1 silver badge 6 6 bronze badges Add a comment 1 Answer Sorted by: Highest score (default) Date modified (newest first) Date created (oldest first) 1 I would separate two issues: the Pauli product itself, and the particular surgery construction used to measure it. For the first point, the support should be the support of the resulting Pauli operator after multiplying the two representatives, not the union as a multiset. For example, if both logical operators are represented by X-type physical Paulis, X ¯ 1 = ∏ i∈ S 1 X i , X ¯ 2 = ∏ i∈ S 2 X i , 𝑋 ¯ 1 = ∏ 𝑖 ∈ 𝑆 1 𝑋 𝑖 , 𝑋 ¯ 2 = ∏ 𝑖 ∈ 𝑆 2 𝑋 𝑖 , then X ¯ 1 X ¯ 2 = ∏ i∈ S 1 △ S 2 X i , 𝑋 ¯ 1 𝑋 ¯ 2 = ∏ 𝑖 ∈ 𝑆 1 △ 𝑆 2 𝑋 𝑖 , because on the overlap (S_1\cap S_2) one has (X_iX_i=I). So for two X-type representatives, the relevant support is the symmetric difference, i.e. the XOR of the two supports. More generally, you should first multiply the Pauli words qubit by qubit, modulo the overall phase, and then take the support of the resulting Pauli word. Same Pauli on the same qubit cancels: X i X i = Z i Z i =I. 𝑋 𝑖 𝑋 𝑖 = 𝑍 𝑖 𝑍 𝑖 = 𝐼 . Different Paulis on the same qubit do not cancel; they produce the third Pauli up to phase, e.g. X i Z i =−i Y i , Z i X i =i Y i . 𝑋 𝑖 𝑍 𝑖 = − 𝑖 𝑌 𝑖 , 𝑍 𝑖 𝑋 𝑖 = 𝑖 𝑌 𝑖 . For a commuting logical Pauli product, the total phase can be chosen so that the measured observable is Hermitian with eigenvalues (\pm 1). But the local labels may include (Y)'s if X- and Z-type factors overlap on physical qubits. For the second point: yes, if the measured Pauli word contains both X and Z structure in the same stabilizer/check, then viewed directly it is no longer a CSS check. A CSS stabilizer generator is purely X-type or purely Z-type. A mixed Pauli such as one containing both X and Z labels, or Y labels after multiplication, is not a CSS stabilizer generator in the original CSS splitting. There are a few caveats: If the operator is purely X-type or purely Z-type after simplifying the product, then the corresponding CSS-style measurement is still natural. If the mixed operator can be mapped to a pure X- or pure Z-type operator by local Clifford basis changes, the measured Pauli itself becomes simple, but those same basis changes also transform the surrounding code checks. In general this does not mean the whole intermediate code remains CSS in the original basis. Measuring the X part and the Z part separately can keep the measurements CSS, but it is generally not equivalent to measuring only their product. It reveals more syndrome/logical information. That may be acceptable in some protocols, but not if the goal is a non-demolition measurement of only the joint Pauli product. So my rule of thumb would be: for overlapping support, simplify the Pauli product first; for same-type overlaps this means XOR/symmetric difference, not union; for mixed products, a direct joint measurement is generally non-CSS unless there is additional structure or one accepts measuring extra information through separate CSS measurements. Share Improve this answer Follow answered yesterday Matti Sarjala 5112 2 silver badges 10 10 bronze badges Add a comment Your Answer Sign up or log in Sign up using Google Sign up using Email and Password Post as a guest Name Email Required, but never shown Post Your Answer By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions error-correctionqldpc See similar questions with these tags. 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