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.
13 public entries
Atlas stars stay in this public list. Saving an entry to your workspace starts an unstarred private copy.
Given oracle access to a weighted graph on N vertices of maximum degree d whose edge weights are conductances, so that an edge of weight w carries resistance 1/w, estimate the effective resistance between a chosen pair of vertices s and t to within a factor of 1 + ε. Wang poses it as ENA-ER: for a network with |V| = N, deg(G) ≤ d, edge conductances normalized to 1 ≤ w_e ≤ c for every edge — equivalently edge resistances in [1/c, 1] — and spectral gap λ2(L_G) ≥ λ > 0, estimate R_eff(s, t) up to multiplicative error ε, succeeding with probability at least 2/3, given a procedure Pv that on input a vertex index and a number k returns the k-th edge incident to that vertex and a procedure Pe that on input an edge index returns that edge's two endpoints and its weight, both of which Wang assumes can be implemented in time poly(log(N)).
effective resistanceelectrical networkquantum walk
Adapt the Harrow-Hassidim-Lloyd (HHL) quantum linear-systems algorithm, most of whose components current noisy quantum hardware cannot reach, into a form that near-term devices can actually execute, and demonstrate it on an application.
Given a first-order linear differential equation d/dt x = A(t)x + b(t) with N-dimensional vectors x and b and an N×N matrix A, and given an initial condition x(0), produce the solution x(t) at a later time t to precision ε, in the sense that the normalized vector x(t)/‖x(t)‖ returned is at distance at most ε from the exact solution.
Encode a linear kinetic plasma problem — modeling electrostatic linear waves, driven by a spatially localized external current, in a one-dimensional Maxwellian electron plasma — into a quantum circuit that solves the resulting linear system.
kinetic plasmavlasov-ampere systemquantum signal processing
Given a system of nonlinear ordinary differential equations, in the primary algorithm's case a dissipative quadratic n-dimensional system, produce the solution at a chosen evolution time T to error ε, encoded in the amplitudes of a quantum state.
Given quantum access to a historical record of asset returns, determine the optimal risk-return tradeoff curve of a portfolio and provide a way to sample from the optimal portfolio.
Solve the transport equation — for variable grid sizes and discrete particle velocities, in two and three spatial dimensions — on a fault-tolerant universal quantum computer, including the reflection of particles at the walls, edges and corners of obstacles.
transport equationfault-tolerant quantum computingcnot reduction
Given a function f and a positive semi-definite matrix A whose eigenvalues are λⱼ, estimate the spectral sum Tr[f(A)] = Σⱼ f(λⱼ), a family whose typical examples the paper gives as the von Neumann entropy, the trace of A⁻¹, the log-determinant and the Schatten p-norm, the last of which it says does not require the matrix to be positive semi-definite.
Given query access to the entries of a d × d Hermitian matrix A, output a classical description of a good approximation of its top eigenvector, the eigenvector belonging to the largest eigenvalue.
Simulate the wave equation under Dirichlet and Neumann boundary conditions on a quantum computer, using Hamiltonian simulation and quantum linear system algorithms as subroutines.