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Finite JRS leakage amplitude and probability

Statement

For a finite adiabatic braid input with rank- $m$ spectral band and schedule $T > 0$ , define $A_{m} (s) = \frac{1}{T} [\frac{m ∥ H^{'} (0) ∥}{g (0)^{2}} + \frac{m ∥ H^{'} (s) ∥}{g (s)^{2}} + \int_{0}^{s} (\frac{m ∥ H^{''} (r) ∥}{g (r)^{2}} + \frac{7 m \sqrt{m} ∥ H^{'} (r) ∥^{2}}{g (r)^{3}}) d r] .$ Then $∥ U_{T} (s) P (0) U_{T} (s)^{*} - P (s) ∥ \leq A_{m} (s)$ , and every unit initial band vector has leakage probability at most $A_{m} (s)^{2}$ . In particular, $∥ [I - P (1)] U_{T} (1) A_{0} ∥_{o p} \leq A_{m} (1)$ and leakage probability is at most $A_{m} (1)^{2}$ . This result does not prove the separate in-band transport condition or identify band transport with Fibonacci $F / R$ data.

Facts & Assumptions

Given: The finite-dimensional $C^{2}$ Hermitian path, its constant-rank uniformly gapped band, preparation isometry, schedule $T > 0$ , and propagator.

[F1]

The JRS theorem gives the displayed rank-dependent projector bound and transition-probability bound. Finite JRS adiabatic projector bound

[F2]

The preparation isometry maps every unit vector in $C^{m}$ to a unit vector in $ran P (0)$ . Finite adiabatic braid compilation input

Proof

1.1F1F2givenalgebra

The hypotheses of [F1] are exactly the finite-path, $C^{2}$ , constant-rank, uniform-gap and $T > 0$ clauses in the input definition. Substituting its $H, P, g, m, T$ into the JRS expression gives the displayed $A_{m} (s)$ and $∥ U_{T} (s) P (0) U_{T} (s)^{*} - P (s) ∥ \leq A_{m} (s)$ .

2.1F1F2step 1.1algebra∎

Let $v \in C^{m}$ be unit and set $ψ = A_{0} v$ . By [F2], $ψ$ is a unit vector in the initial band, so [F1] gives $∥ [I - P (1)] U_{T} (1) ψ ∥^{2} \leq A_{m} (1)^{2}$ . Taking square roots and then the supremum over unit $v$ gives $∥ [I - P (1)] U_{T} (1) A_{0} ∥_{o p} \leq A_{m} (1)$ . The squared bound is its leakage probability. No statement in [F1] supplies the in-band comparison $ϵ_{t r}$ ; that remains an independent input hypothesis.

Depends on

Used by

Finite braid compilation and adiabatic error budget Lemma

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Sabine Jansen, Mary-Beth Ruskai, and Ruedi Seiler, Bounds for the adiabatic approximation with applications to quantum computation (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms (standard reference, not scraped)