Explore quantum algorithms and circuits, with resource estimates, verification evidence, and sources.
◈Verification legend
Every record is classified by how it was verified. The badge shows the strongest tier of evidence; the chips list each method that applies.
≡Exact & formal
The defining behavior was checked exactly: a mathematical identity, a full statevector or stabilizer simulation, or an exhaustive basis-state truth table.
Direct mathematics
An analytic identity or closed-form derivation proves the defining property (e.g. H² = I, Y = iXZ).
Unitary / matrix equivalence
The circuit's unitary was computed and compared against the reference matrix.
Exact statevector simulation
A full statevector simulation reproduced the exact expected state or distribution.
Reversible classical logic was checked exhaustively (or at edge cases) on computational-basis inputs.
✓Strong empirical
The design was verified by construction plus measured evidence: statistical re-execution, small-instance analytic agreement, sub-block, echo, or invariant checks. Scale-specific bugs can still survive.
Statistical re-execution
Measured counts from independent executions agreed within statistical tolerance (e.g. TVD bounds).
Small-instance agreement
The same generator matched analytic results at tractable sizes; large instances are inferred.
The circuit follows a specification whose correctness argument is standard; the construction was audited against it.
◐Attested & literature
The record rests on external authority: peer-reviewed papers, standard textbooks, expert review, or evidence carried over from related verified entries. Nothing here was re-executed by this catalog.
Peer-reviewed paper
The record's claims trace to one or more peer-reviewed publications cited on the entry.
Textbook / standard citation
The behavior is standard material (e.g. Nielsen & Chuang, OpenQASM spec) and is cross-referenced, not re-derived.
Expert review
A named human reviewer with domain expertise checked the record.
Tangential evidence
Correctness is supported indirectly through related verified entries (e.g. a gate verified inside a verified algorithm).
○Automated & unreviewed
Only automated (LLM-assisted) review or an unreviewed community submission backs this record so far. Treat it as a starting point, not evidence.
LLM-assisted review
An LLM checked the record for internal consistency. Useful screening, not evidence of correctness.
Community submission
Submitted by the community and not yet through the review pipeline.
Let G be a finitely generated Abelian group and let H be a subgroup of G such that G/H is finite, and let f be a function on G with the promise that f(g1) = f(g2) if and only if g1 and g2 lie in the same coset of H. The task is to find H, that is, a set of generators for H, by making queries to f. Mosca and Ekert work order finding and period finding as G = Z with hidden subgroup rZ, Simon's problem as G = Z_2^l, and the discrete logarithm as G = Z_r × Z_r.
Find a better parameterized ansatz for the quantum approximate optimization algorithm (QAOA) applied to combinatorial optimization problems such as Max-Cut, where the standard, fixed-form QAOA ansatz is not known to be optimal and no systematic method exists for improving on it.
Give polynomial-time quantum algorithms for three average-case lattice problems in parameter regimes where none was known: the short integer solution problem under the infinity norm, the learning-with-errors problem when the input is supplied as LWE-like quantum states rather than classical samples, and the extrapolated dihedral coset problem.
Backtracking is the general classical technique for exploiting problem structure in constraint satisfaction: explore a tree of partial assignments and prune the branches a predicate rules out. The question is whether an arbitrary backtracking algorithm — any predicate and any branching heuristic — can be sped up quantumly, rather than replaced by brute-force search over the whole assignment space.