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
These links open the shared core's formal companion panel. Each card distinguishes admitted fixed-reference views from pending formalization. Preserve the exact core contract and split; a formal view is not another core.
A single core of a finite real tensor train is a real matrix whose rows are indexed by the left bond and whose columns are indexed by a composite label: an ordered pair of an emitted bit (two values) and a right-bond ind…
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During backward canonicalization of a finite real tensor train, a residual matrix R (left over from factorizing a later core) must be absorbed into the right bond of the preceding core A, where A emits one bit of two pos…
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A stencil coefficient is written in a small symbolic language whose expressions are built from rational constants, named symbols, and the operations addition, multiplication, and negation. Given an environment that assig…
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A block-encoding unitary represents an operator that acts on the 2^n states of an n-qubit system register, using a auxiliary qubits. (a) What is the dimension of the square matrix representing the full unitary, and why? …
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You are given r real column vectors in R^N that are orthonormal (each has unit length and distinct ones are orthogonal), presented as the columns of an N-by-r matrix V satisfying V-transpose times V equals the r-by-r ide…
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Suppose r orthonormal real columns are placed at r distinct positions among the N columns of an orthogonal matrix, and suppose r < N so at least one column position is left free. The naive completion to an orthogonal mat…
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In a finite complex matrix backend, acting with an operator on a computational-basis state (a vector with a single 1 at one position and 0 elsewhere) returns the corresponding column of the operator. Using only this fact…
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Two proof styles are commonly used to certify that a flattened signal-system unitary block-encodes a target operator on the system register at a fixed 'clean' signal value. Style A proves a pointwise equality: the matrix…
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Let V be a real N-by-r matrix whose columns are orthonormal, expressed by the identity V-transpose times V equals the r-by-r identity. (a) Prove the dimensional obstruction r <= N. (b) Suppose further that the strict ine…
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When embedding a right-canonical tensor-train core into a larger fixed physical register by zero-padding its outputs, one claims the active clean-input columns remain orthonormal and that right-multiplication of the boun…
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The special-orthogonal completion result guarantees that prescribed orthonormal columns can be extended to a determinant-+1 orthogonal matrix WHENEVER the number of prescribed columns is strictly less than the ambient di…
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A finite real tensor train is right-canonicalized by sweeping from the terminal bond backward, replacing each core by one with orthonormal rows while absorbing a residual into the preceding core. The active left bond dim…
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