QuantumComputingBench · graph-grounded exercise book
Start with dimensions, inner products, isometries and explicit conventions. The same foundations support state preparation, block encoding and later quantum-information arguments.
Back to Shared mathematical foundations
Work in an exact-real word-operation model that keeps eight separate running counters: field operations, square roots, inverse-cosine calls, other trigonometric calls, exact comparisons, stored-word reads, stored-word wr…
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Continue in the eight-counter exact-real operation model where stored matrices are vectors of N stored rows and replacing one stored row is charged as a full persistent copy costing N reads and N writes, never an in-plac…
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In the eight-counter exact-real model, a column sweep eliminates entries below a pivot by performing a chronological chain of elimination steps. Each step first computes an elimination angle and the two rotation coeffici…
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A rectangular elimination records a chronological list of two-row rotation steps. Rather than forming the accumulated orthogonal transform by multiplying out dense plane-rotation matrices, a stored replay starts from a m…
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A full stored rectangular compilation of an N-by-M matrix produces a reduced residual, a chronological list of rotation steps, and a fully accumulated N-by-N transform. In the eight-counter exact-real model its per-count…
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Let A be a stored real m-by-n matrix, with m at most n. Its wide thin-LQ routine transposes A, runs the full stored rectangular compiler on the transpose, extracts R as an m-by-m matrix from the residual, and extracts Q …
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In the eight-counter exact-real model, a bounded sum over an index set of size k is evaluated by a monadic fold: starting from the pure value zero (no charge), each of the k summands is produced and then added to the run…
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A stored matrix-product chain over n+1 cores, each of bond size D, is closed into a scalar-boundary chain in two conceptual halves. First the tail is processed recursively: at the last position the terminal core is contr…
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A stored tensor-train norm is computed by streaming local Gram environments from right to left. At the empty tail the environment is a materialized identity of bond size r. At a non-empty node carrying a core of left bon…
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Given the stored right-to-left Gram-environment computation of a scalar-boundary tensor-train chain (which returns a 1-by-1 stored environment), the stored norm is obtained by reading the single scalar entry of that envi…
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Two notions of total cost appear in the stored exact-real model. One, call it the ordinary total, is the sum of a cost function over the finite type of all operation tags. The other, call it the enumerated total, is the …
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A stored Hermite raw-source assembly builds a scalar-boundary tensor-train chain for a state-preparation amplitude. One single run of the assembly returns the chain together with a work ledger carrying both the eight ord…
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A primitive quantum circuit is a chronological list of gates, each gate acting on a set of touched wires. A per-wire scheduling fold tracks, for every wire, the earliest layer at which its next gate can be placed: proces…
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A stored N-by-M matrix is built by materializing an entry supplier: for each of the N*M entries the supplier is run to produce a value, and the construction charges, on top of the supplier's own counted work, two reads a…
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The elimination-angle routine in the exact-real model computes a norm from two inputs x and y: it squares both, adds them, takes a square root, and tests whether that root is zero. If the root is zero it returns the angl…
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In the stored exact-real model the identity matrix of size N is not a primitive constant but is materialized: each of the N*N entries is produced by an exact comparison deciding whether its row and column indices are equ…
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When a stored sweep concatenates its chronological step logs, it uses a persistent list append that copies the spine of the left list while sharing the right list unchanged; the append is charged exactly the length of th…
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A stored thin-LQ dispatcher takes an m-by-n stored matrix and charges one exact comparison to decide the shape. When m is at most n (the wide case) it transposes, runs the full stored rectangular compilation, and extract…
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Consider the ACTUAL stored Hermite raw-source producer raw(k,n,L), for natural k,n and legal positive real scale L>0, not an arbitrary manually populated cache. Keep its original base ledger n+2 exponential calls and one…
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A stored matrix-product chain is assembled from n+1 stored cores, each of bond size D, closed against left and right boundary vectors of length D to form a scalar-boundary chain. It is proved that the maximum bond of the…
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