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.

Version

v2 · 2026-09-19 · History

Domain-specific #

4871

Domain group

Applied Sciences & Engineering

Origin domain

Computer Science & Software Engineering

Subdomain

Quantum Complexity → Computer Science & Software Engineering

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.

How would you explain it like I'm…

 

No faithful explanation at this level. A kid-level story collapses into 'quantum computers are just faster' or 'try every answer at once'. The concept is actually a specific separation between constant-depth quantum circuits and restricted shallow classical circuits.

Short Quantum Circuit Puzzle

The Hidden linear function problem is a puzzle built from a grid of 0s and 1s. The goal is to find an answer that fits a secret pattern hidden in the puzzle. A quantum circuit with only a few steps can produce valid answers by measuring its qubits. Scientists have shown that ordinary circuits limited to the same small number of steps cannot do it as well. It is a way to prove that very short quantum programs can beat very short ordinary ones on this specific task.

Shallow Quantum Circuit Separation

The Hidden linear function problem (HLF) is given by a binary matrix and vector, which define a quadratic form over 0/1 values; the promise is that a certain hidden linear function matches this form on a special subset. The task is to output a valid hidden linear solution. A constant-depth quantum circuit, often with interactions on a two-dimensional grid, produces measurements whose parities give a valid answer. Restricted shallow classical circuits provably cannot reproduce this output relation efficiently, which gives a shallow-circuit separation. Unlike the Bernstein–Vazirani problem, where the hidden string is accessed through an oracle, HLF instances are given explicitly.

 

The Hidden linear function problem (HLF) is a relation problem over the binary field. A binary matrix and vector specify a quadratic form, and promised structure guarantees that, restricted to an appropriate subspace, the form behaves as a linear function whose description must be output. A constant-depth quantum circuit encodes these phase correlations, and measuring it yields bit strings that constitute valid hidden-linear solutions with the stated probability, possibly after classical postprocessing. The point is a separation: restricted shallow classical circuits, under a stated circuit lower-bound model, cannot reproduce the required input–output relation efficiently. The two-dimensional version constrains interactions to a grid, matching realistic hardware geometry, and underpins shallow-circuit separations. HLF instances are given explicitly, unlike oracle problems such as Bernstein–Vazirani, so the input access model is different. A precise statement fixes the field and dimension, the matrix, vector and promise, the output relation, circuit geometry, depth and gate bounds, measurement and postprocessing, success probability, and the classical model being bounded.

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

1. 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) — outgoing typed relations

• Hidden linear function problem is a kind of Verifier-Prover Asymmetry Prime

  It remains its own entry because its identity is fixed by 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.

Hierarchy path (1) — routes to 1 parentless root

• Hidden linear function problem → is a kind of Verifier-Prover Asymmetry → is a kind of Asymmetry

Neighborhood in Abstraction Space

Hidden linear function problem sits in a crowded region of the domain-specific corpus (18^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Quantum Protocols & Circuit Models (20 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 · 2026-10-25