Hidden linear function problem¶
A search problem that asks for a vector satisfying a linear relation hidden by phase data encoded in an explicitly supplied quadratic-form instance.
Core Idea¶
The two-dimensional version constrains interactions to a grid and is used for shallow-circuit separations; oracle Bernstein-Vazirani and explicit HLF instances have different input access. A binary matrix and vector define phase correlations, a constant-depth quantum circuit samples measurements whose parity reveals a valid hidden linear solution while restricted shallow classical circuits cannot reproduce the relation efficiently. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.
Scope of Application¶
Hidden linear function problem belongs to quantum complexity and is useful where the analyst can specify the typed quantum complexity carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the binary field and dimension, matrix vector and promised quadratic form, required output relation, quantum circuit geometry depth and gate bounds, measurement and postprocessing, correctness probability and stated classical circuit lower-bound model are explicit. The scope is broad within that domain but bounded by the need for the binary field and dimension, matrix vector and promised quadratic form, required output relation, quantum circuit geometry depth and gate bounds, measurement and postprocessing, correctness probability and stated classical circuit lower-bound model are explicit.
Clarity¶
The abstraction clarifies a crowded vocabulary by making the binary field and dimension, matrix vector and promised quadratic form, required output relation, quantum circuit geometry depth and gate bounds, measurement and postprocessing, correctness probability and stated classical circuit lower-bound model are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.
Manages Complexity¶
Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Hidden linear function problem. Hidden linear function problem compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.
Abstract Reasoning¶
- Identify the carrier. State what the elements, states, objects, or observations are: the typed quantum complexity carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the binary field and dimension, matrix vector and promised quadratic form, required output relation, quantum circuit geometry depth and gate bounds, measurement and postprocessing, correctness probability and stated classical circuit lower-bound model are explicit independently of one notation or implementation.
Knowledge Transfer¶
Knowledge transfers strongly among subfields of quantum complexity because they reuse the typed quantum complexity carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, A binary matrix and vector define phase correlations, a constant-depth quantum circuit samples measurements whose parity reveals a valid hidden linear solution while restricted shallow classical circuits cannot reproduce the relation efficiently., and type the carrier, state every parameter and convention in the definition, test that the binary field and dimension, matrix vector and promised quadratic form, required output relation, quantum circuit geometry depth and gate bounds, measurement and postprocessing, correctness probability and stated classical circuit lower-bound model are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.
Relationships to Other Abstractions¶
Current abstraction Hidden linear function problem Domain-specific
Parents (1) — more general patterns this builds on
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Hidden linear function problem is a kind of Verifier-Prover Asymmetry Prime
The proposed strict upward parent is
prime:verifier_prover_asymmetry.
Hierarchy path (1) — routes to 1 parentless root
- Hidden linear function problem → Verifier-Prover Asymmetry → Asymmetry
Neighborhood in Abstraction Space¶
Hidden linear function problem sits in a crowded region of the domain-specific corpus (32nd percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.
Family — Quantum Information & State Structure (41 abstractions)
Nearest neighbors
- Quantum circuit — 0.92
- Quantum key distribution — 0.91
- Graph state — 0.90
- Greenberger–Horne–Zeilinger state — 0.90
- AC0 — 0.90
Computed from structural-signature embeddings · 2026-09-08