Manuscript Oai283-A@1 · Oai283-A.tex · theorem · 6fb75895

Theorem: Finite-confidence channel interval

Operational certification of the logical channel › A complete finite-shot certificate · lines 944–989 · proof included or linked

Oai283-A@1#thm:tomography, Oai283-A@1#eq:confidence

Under Specification spec:statistics, put

e = b + \frac{2 lo g ( 2 D ^{2} / α )}{N}, a = \frac{1}{2} ∥ ω - ω_{U} ∥_{1} .

With probability at least $1 - α$ ,

max {0, a - D e /2} \leq d_{⋄} (Φ, Ad_{U}) \leq min {1, d a + d D e /2} .

If the calibration contract itself holds only on an event of probability at least $1 - β$ , the unconditional coverage is at least $1 - α - β$ .

Proof. For a variable in $[- 1, 1]$ , Hoeffding's bound gives

P {∣ r_{P} - E r_{P} ∣ > v} \leq 2 e^{- N v^{2} /2} .

A union bound, followed by the bias allowance, gives simultaneous coefficient errors at most $e$ .

Pauli orthogonality is $Tr (P P^{'}) = D δ_{P P^{'}}$ . Therefore

∥ ω - ω ∥_{2}^{2} = \frac{1}{D} P \sum ∣ r_{P} - r_{P} ∣^{2} \leq D e^{2},

and

∥ ω - ω ∥_{1} \leq D ∥ ω - ω ∥_{2} \leq D e .

Apply the triangle inequality and Lemma lem:choi. The last coverage statement is another union bound.

Displayed formulas in this passage

Oai283-A@1#eq:confidence · line 953
$max {0, a - D e /2} \leq d_{⋄} (Φ, Ad_{U}) \leq min {1, d a + d D e /2} .$
unlabeled · line 947
$e = b + \frac{2 lo g ( 2 D ^{2} / α )}{N}, a = \frac{1}{2} ∥ ω - ω_{U} ∥_{1} .$
unlabeled · line 967
$P {∣ r_{P} - E r_{P} ∣ > v} \leq 2 e^{- N v^{2} /2} .$
unlabeled · line 976
$∥ ω - ω ∥_{2}^{2} = \frac{1}{D} P \sum ∣ r_{P} - r_{P} ∣^{2} \leq D e^{2},$
unlabeled · line 982
$∥ ω - ω ∥_{1} \leq D ∥ ω - ω ∥_{2} \leq D e .$