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Clifford+T Exact and Approximate Synthesis — Examples

1 · Prerequisites

2 · Summary

The worked example verifies the exact period of T and proves that the stated Rz target is not an exact one-qubit Clifford+T matrix word by comparing its cyclotomic field degree with the dyadic eighth-cyclotomic entry field.

3 · Logical flowchart

4 · Definitions, theorems and proofs

None yet.

5 · Examples, counterexamples and false statements

ExampleConstruction: AI-adaptedVerification: AI-adaptedOpen item page →

Exact T period and a non-exact Rz phase

Example

Let $T = diag (1, ω)$ with $ω = e^{i π / 4}$ , and let $ζ_{64} = e^{2 π i / 64}$ . The gate $T$ has exact period eight, while $R_{z} (π / 16)$ is not an exact one-qubit Clifford+T word.

Facts & Assumptions

Given: $ω = e^{i π / 4}$ , $ζ_{64} = e^{i π / 32}$ , and $R_{z} (θ) = diag (e^{- i θ / 2}, e^{i θ / 2})$ .

[F1]

Clifford+T circuit entries lie in the dyadic cyclotomic ring: every exact one-qubit Clifford+T word has entries in $D = Z [ζ_{8}, 1 / 2] \subseteq Q (ζ_{8})$ .

[F2]

$Φ_{n}$ is irreducible in $Q [t]$ for every $n \geq 1$ : the cyclotomic polynomial $Φ_{n}$ is irreducible over $Q$ .

[F3]

The recursion defines a unique monic $Φ_{n} \in Z [t]$ , of degree $φ (n)$ : $\deg Φ_{n} = φ (n)$ .

Verification

1.1givenalgebra

Since $ω^{8} = 1$ , direct matrix powering gives $T^{8} = diag (1, ω^{8}) = I$ . No smaller positive power is the identity because $ω$ has order eight.

2.1F1F2F3givenalgebra∎

The lower-right entry of $R_{z} (π / 16)$ is $e^{i π / 32} = ζ_{64}$ . If an exact H/T word equaled this rotation, [F1] would put $ζ_{64}$ in $Q (ζ_{8})$ . But $ζ_{8} = ζ_{64}^{8}$ , so $Q (ζ_{8}) \subseteq Q (ζ_{64})$ ; containing $ζ_{64}$ in the smaller field would force equality. By [F2]–[F3], these fields have degrees $φ (8) = 4$ and $φ (64) = 32$ respectively: among residues modulo $2^{m}$ , exactly the $2^{m - 1}$ odd residues are coprime. The unequal degrees contradict equality, so no exact H/T word implements this rotation.

Remarks

The example uses the necessary direction of the ring characterization only; it does not rely on the converse theorem or on an approximation algorithm.