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HHL capped postselection amplification cost

Statement

For the OP5 HHL attempt with $p \geq 1 / (16 κ^{2})$ and failure target $0 < ρ < 1$ , the capped schedule uses at most $2 R K = O (κ (1 + \log (1 / ρ)))$ calls in total to $V$ and $V^{†}$ , where $K = ⌈ 4 κ ⌉$ and $R = ⌈ 16 \ln (1 / ρ) ⌉ + 16$ . Thus it uses $O (κ (1 + \log (1 / ρ)))$ calls to the supplied preparation form (the tagged $B_{A}$ in dilation mode) and separately to its inverse, multiplied by their actual supplied costs.

Facts & Assumptions

Given: OP5 access costs and the lower success bound.

[F1]

Finite capped amplification succeeds with a supplied lower bound: each round uses at most $K$ calls to $V$ and at most $K$ calls to $V^{†}$ , over $R$ rounds.

[F2]

HHL finite sine-window phase-estimation algorithm: each $V$ contains one call to the supplied preparation form and each $V^{†}$ one call to its inverse. In rectangular mode this form is the tagged $B_{A}$ from the Definition, and its controlled/gate cost is the actual declared cost for that access.

[F3]

Circuit-family and computation cost model: sequential costs add and every inverse or controlled access has its own declared cost.

Proof

1.1F1algebra

By [F1], the total number of preparation and inverse-preparation calls is bounded separately by $R K$ each. Since $K = ⌈ 4 κ ⌉ \leq 5 κ$ for $κ \geq 1$ and $R \leq 16 \ln (1 / ρ) + 17$ , each count is $O (κ (1 + \log (1 / ρ)))$ with an absolute constant.

2.1F2F3step 1.1algebra∎

Each preparation call invokes the supplied preparation form once, and each inverse preparation invokes its actual inverse once by [F2]. Their circuit/query/gate costs therefore multiply the two counts separately and add by [F3]. No call to a controlled form, simulator, reflection, or verifier is included unless it appears in the declared circuit.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

40 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

Aram W. Harrow, Avinatan Hassidim, and Seth Lloyd, Quantum algorithm for linear systems of equations (standard reference, not scraped)
Andrew M. Childs, Robin Kothari, and Rolando D. Somma, Quantum algorithm for systems of linear equations with exponentially improved dependence on precision (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms (standard reference, not scraped)