How many gates are in the $ k $-th level of the single qubit Clifford hierarchy?

How many gates are in the k 𝑘 -th level of the single qubit Clifford hierarchy? Ask Question Asked 3 years, 3 months ago Modified today Viewed 141 times 2 Define the single qubit Clifford hierarchy recursively by C 1 :=<iX,iZ> 𝐶 1 :=< 𝑖 𝑋 , 𝑖 𝑍 > (the determinaint 1 subgroup of the Pauli group) as well as C k :={V∈SU(2):Vp V −1 for all p∈ C k−1 } 𝐶 𝑘 := { 𝑉 ∈ 𝑆 𝑈 ( 2 ) : 𝑉 𝑝 𝑉 − 1 for all  𝑝 ∈ 𝐶 𝑘 − 1 } How many elements are in C k 𝐶 𝑘 ? I know C 1 =<iX,iZ> 𝐶 1 =< 𝑖 𝑋 , 𝑖 𝑍 > has size 8 8 . And C 2 =<iH, ζ 8 ¯ ¯ ¯ ¯ P> 𝐶 2 =< 𝑖 𝐻 , 𝜁 8 ¯ 𝑃 > has 48 48 elements. How many elements does C 3 𝐶 3 have? Here X 𝑋 is pauli X 𝑋 , Z 𝑍 is Pauli Z 𝑍 , H 𝐻 is Hadamard, and P=diag(1,i) 𝑃 = d i a g ( 1 , 𝑖 ) is the phase gate. clifford-groupclifford-hierarchy Share Improve this question Follow edited Aug 16, 2024 at 13:58 Frederik vom Ende 4,3413 3 gold badges 15 15 silver badges 51 51 bronze badges asked Dec 1, 2022 at 18:21 Ian Gershon Teixeira 5,5187 7 silver badges 35 35 bronze badges 1 As I indicated in my other answer, it should be possible to have an elementary counting argument for the third level. For higher levels, you may use that all elements in the Clifford hierarchy for a single qubit are semi-Clifford, so you can take the (known) order of the diagonal Clifford hierarchy, multiply that with the squared order of the Clifford group and mod out the stabilizer. I think this should give you the right cardinality –  Markus Heinrich Commented Dec 2, 2022 at 12:04 +1 that sounds very promising. I'll try to work out some counting with semi Clifford for some small cases for n=1 𝑛 = 1 and n=2 𝑛 = 2 qubits. So is counting the number of elements pretty hopeless for n≥3,k≥3 𝑛 ≥ 3 , 𝑘 ≥ 3 since no one knows much about the structure of the hierarchy for those values? –  Ian Gershon Teixeira Commented Dec 2, 2022 at 22:19 Yeah, I think there's not much hope to count them, we don't know enough. –  Markus Heinrich Commented Dec 3, 2022 at 18:49 Add a comment 1 Answer Sorted by: Highest score (default) Date modified (newest first) Date created (oldest first) 0 In this paper, they affirm (Eq. 57) that "the number of gates in C k 𝐶 𝑘 , up to phase, is exactly |[Ck]|= d 2 𝑑 2 if k = 1, d 3 ( d 2 −1)( d k−1 + d k−2 −d) 𝑑 3 ( 𝑑 2 − 1 ) ( 𝑑 𝑘 − 1 + 𝑑 𝑘 − 2 − 𝑑 ) if k ≥ 2, where d is the dimension. Share Improve this answer Follow answered 4 hours ago user38248 535 5 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 clifford-groupclifford-hierarchy See similar questions with these tags. 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