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First-order fast-check channel

PDF p. 14: Approximate Kraus operators before Eq. (S4.1); PDF p. 15: Eqs. (S4.3)–(S4.4)

Statement

Adopt the finite lifted model with blocks $J = 0, \dots, N$ , $N \geq 1$ , its equal-rate downward jumps, and all physical, spectral and choice assumptions inherited by that model and the uniform-decay result. In particular the Axiom of Choice, Countable Choice and Dependent Choice are assumed. The orthonormal magnetic basis is $∣ J, m ⟩$ , with integer $m = - J, \dots, J$ . The Hamiltonian is $H = \sum_{J = 0}^{N} \sum_{m = - J}^{J} [ω_{0} J (J + 1) + Δ (J, m)] ∣ J, m ⟩ ⟨ J, m ∣$ , with real diagonal entries, and the three downward jump matrices are $L_{δ}$ for $δ = - 1, 0, 1$ as specified by the model. Set $V = - \frac{1}{2} \sum_{δ} L_{δ}^{†} L_{δ}$ and $G = - i H + V$ .

For $τ \geq 0$ , define $K_{δ} = \sqrt{τ} L_{δ}$ and $K_{n u l l} = I + τ G$ . Then $K_{n u l l} = I - i τ \sum_{J = 0}^{N} \sum_{m = - J}^{J} [ω_{0} J (J + 1) + Δ (J, m)] ∣ J, m ⟩ ⟨ J, m ∣ - (γ τ / 2) \sum_{J = 1}^{N} \sum_{m = - J}^{J} ∣ J, m ⟩ ⟨ J, m ∣$ . The ground block $J = 0$ has no downward jump.

Let $N_{τ}$ be the finite GKSL channel of the model, and put ${\tilde{N}}_{τ} (X) = K_{n u l l} X K_{n u l l}^{†} + \sum_{δ} K_{δ} X K_{δ}^{†}$ . With matrix spectral norm, induced map norm $∥ Φ ∥ = \sup_{X \neq 0} ∥ Φ (X) ∥ / ∥ X ∥$ , $g = ∥ G ∥$ , $j = \sum_{δ} ∥ L_{δ} ∥^{2}$ , and $r_{τ} = \frac{1}{2} e^{τ g} (τ g)^{2}$ , the explicit bound is $∥ N_{τ} - {\tilde{N}}_{τ} ∥ \leq τ j (e^{2 τ g} - 1) + \frac{1}{2} e^{2 τ g + τ j} (τ j)^{2} + r_{τ} (e^{τ g} + 1 + τ g)$ . Thus the displayed Kraus sum approximates the channel with $O (τ^{2})$ induced-norm error as $τ ↓ 0$ , at fixed model parameters. It need not be trace preserving at finite $τ$ .

The corresponding diagonal formula with sums over all integer $J \geq 0$ is an algebraic identity on finite-support combinations of the labeled basis vectors, using the same formal jump coefficients. It does not assert an infinite-dimensional channel. The bound above depends on the full laboratory-frame $∥ G ∥$ , including the rotor term; the model's short-checking conditions alone do not make it uniform in arbitrarily large $ω_{0} τ$ .

Facts & Assumptions

[F1]

The lifted-model postulate specifies the finite diagonal Hamiltonian, the three downward jumps, their finite GKSL evolution and the inherited physical and choice assumptions. General bounded diagonal lift for fast-checking analysis

[F2]

Under those hypotheses the uniform-decay result gives $\sum_{δ} L_{δ}^{†} L_{δ} = γ \sum_{J = 1}^{N} \sum_{m = - J}^{J} ∣ J, m ⟩ ⟨ J, m ∣$ and says the ground block is annihilated by every downward jump. Its infinite-label identity holds on finite-support vectors. Normalized jumps give a scalar total decay operator

[F3]

The angular-basis definition supplies the orthonormal labeled basis and the magnetic-label ranges. Angular-momentum basis and tensor labels

[F4]

The finite GKSL interface defines the generator and its nonnegative-time channel for finite Hermitian $H$ and a finite jump family. Finite GKSL evolution and the QEC channel interface

[F5]

The finite trajectory expansion gives the no-jump term $E (τ) X E (τ)^{†}$ , with $E (u) = e^{u G}$ , and the one-jump term $\sum_{δ} \int_{0}^{τ} E (τ - s) L_{δ} E (s) X E (s)^{†} L_{δ}^{†} E (τ - s)^{†} d s$ . The sum of the terms with at least two jumps has induced norm at most $e^{2 τ g} \sum_{n = 2}^{\infty} (τ j)^{n} / n!$ . Finite no-jump and ordered-jump expansion

[F6]

The finite exponential Taylor result defines the matrix exponential by its norm-convergent power series and gives $∥ e^{τ G} - I - τ G ∥ \leq r_{τ}$ . Its scalar case also gives the real exponential series. Finite exponential and smooth quotient remainder control

[F7]

The finite matrix algebra with spectral norm is a complete complex C*-algebra. Its defining norm properties include submultiplicativity and the isometry of conjugate transpose: $∥ A B ∥ \leq ∥ A ∥ ∥ B ∥$ and $∥ A^{†} ∥ = ∥ A ∥$ . Here $∥ I ∥ = 1$ follows immediately from the unit-ball formula in positive dimension. Finite matrix algebras with the operator norm are C star algebras C star algebra

[F8]

A continuous curve in a complete normed real space has a Riemann integral, linear in the curve and commuting with bounded real-linear maps, with norm at most interval length times the supremum norm. A complete complex matrix space can be regarded as a real space for this assertion. Riemann integration of continuous curves in a complete normed space, with the fundamental theorem of calculus and coefficient functionals

Proof

Given: The finite lifted model, its inherited hypotheses, and $τ \geq 0$ .

Goal: Calculate the null operator and bound the difference between the exact channel and the displayed first-order Kraus sum.

1.1F1F2F3algebra

By [F2], $V$ is minus $γ / 2$ times the projection onto the excited blocks. Substitute this and the diagonal Hamiltonian from [F1] into $I + τ (- i H + V)$ . The orthonormal basis and ranges in [F3] give the displayed diagonal formula. The same jump formula annihilates the ground block.

1.2F6F7algebra

Use the norm properties in [F7] and the matrix and scalar series in [F6]. For $u \geq 0$ , comparison term by term gives $∥ E (u) ∥ \leq \sum_{n = 0}^{\infty} (u g)^{n} / n! = e^{u g}$ and $∥ E (u) - I ∥ \leq \sum_{n = 1}^{\infty} (u g)^{n} / n! = e^{u g} - 1$ . These comparisons use $∥ G^{n} ∥ \leq g^{n}$ by submultiplicativity. For $u, v \in [0, τ]$ , the factorization of $u^{n} - v^{n}$ gives $∣ u^{n} - v^{n} ∣ \leq n τ^{n - 1} ∣ u - v ∣$ for $n \geq 1$ ; the same series comparison yields $∥ E (u) - E (v) ∥ \leq g e^{τ g} ∣ u - v ∣$ . Thus these matrix curves are norm-continuous.

2.1F6F7step 1.2algebra

Put $B = I + τ G$ and $E = E (τ)$ . By [F6], $∥ E - B ∥ \leq r_{τ}$ , while [F7] gives $∥ B ∥ \leq 1 + τ g$ . The identity $E X E^{†} - B X B^{†} = (E - B) X E^{†} + B X (E - B)^{†}$ , the adjoint norm equality in [F7], and step 1.2 bound the induced norm of the difference of these two maps by $r_{τ} (e^{τ g} + 1 + τ g)$ .

2.2F4F5F7step 1.2algebra

The data satisfy the finite hypotheses of [F4] and [F5]. For $0 \leq s \leq τ$ , put $A_{δ} (s) = E (τ - s) L_{δ} E (s)$ . By [F7] and step 1.2, $∥ A_{δ} (s) ∥ \leq e^{τ g} ∥ L_{δ} ∥$ . Subtract $L_{δ}$ using $A_{δ} (s) - L_{δ} = (E (τ - s) - I) L_{δ} E (s) + L_{δ} (E (s) - I)$ to get $∥ A_{δ} (s) - L_{δ} ∥ \leq [(e^{(τ - s) g} - 1) e^{s g} + e^{s g} - 1] ∥ L_{δ} ∥ = (e^{τ g} - 1) ∥ L_{δ} ∥$ . The identity $A_{δ} (s) X A_{δ} (s)^{†} - L_{δ} X L_{δ}^{†} = (A_{δ} (s) - L_{δ}) X A_{δ} (s)^{†} + L_{δ} X (A_{δ} (s) - L_{δ})^{†}$ therefore bounds its norm by $(e^{2 τ g} - 1) ∥ L_{δ} ∥^{2} ∥ X ∥$ .

3.1F5F6F7F8step 1.2step 2.2algebra

By step 1.2 and [F7], each matrix integrand in step 2.2 is continuous in $s$ . The matrix space is complete by [F7], so [F8] applies to it as a real space. Its norm integral agrees with the entrywise integral in [F5]: for the standard coordinate vectors, $∣ X_{a b} ∣ \leq ∥ X ∥$ follows from $∣ X_{a b} ∣ \leq ∥ X e_{b} ∥_{2} \leq ∥ X ∥$ , making real and imaginary coordinate functionals bounded. Apply [F8] to the difference integrand and sum over the three indices. The one-jump term consequently differs from $τ \sum_{δ} L_{δ} X L_{δ}^{†}$ by at most $τ j (e^{2 τ g} - 1) ∥ X ∥$ . For $x = τ j \geq 0$ , $(k + 2)! \geq 2 k!$ and the scalar series in [F6] give $\sum_{n = 2}^{\infty} x^{n} / n! \leq \frac{1}{2} x^{2} e^{x}$ . Thus the at-least-two-jump tail in [F5] has induced norm at most $\frac{1}{2} e^{2 τ g + τ j} (τ j)^{2}$ . For $τ = 0$ , the integrals and both bounds are zero.

4.1F4F6F7step 2.1step 3.1algebra

Combine the no-jump estimate in step 2.1 and the one-jump and tail estimates in step 3.1 by the triangle inequality, and use $τ L_{δ} X L_{δ}^{†} = K_{δ} X K_{δ}^{†}$ . Taking the supremum over nonzero $X$ yields the displayed bound. For fixed $g, j$ , [F6] gives $e^{2 τ g} - 1 = O (τ)$ and bounded exponential factors near zero, so every term is $O (τ^{2})$ . Finally, $G + G^{†} = - \sum_{δ} L_{δ}^{†} L_{δ}$ gives $K_{n u l l}^{†} K_{n u l l} + \sum_{δ} K_{δ}^{†} K_{δ} = I + τ^{2} G^{†} G$ , which need not equal $I$ at finite $τ$ .

5.1F1F2F3step 1.1step 4.1algebra∎

On a finite-support vector, the diagonal sums act through finitely many basis labels. The finite-support extension in [F2] and the displayed diagonal Hamiltonian therefore give the same formal null-operator identity without an infinite-dimensional evolution claim. The bound in step 4.1 contains $g = ∥ - i H + V ∥$ ; consequently its constants retain the full rotor contribution unless a separate rotating-frame argument is supplied.

Depends on

Used by · 4

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Sources

Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 14, approximate Kraus operators before Eq. (S4.1); PDF p. 15, Eqs. (S4.3)–(S4.4)