The Quantum Atlas

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.
Stabilizer simulation: Clifford structure allowed exact verification with stabilizer methods beyond statevector limits.
Basis-state truth table: 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.
Sub-block verification: Reusable modules (oracle, QFT block, Trotter step, adder…) were verified independently.
Echo / inverse test: U followed by U† returned the input exactly, catching wiring, phase, and ancilla bugs.
Invariant & structural checks: Scale-independent invariants held: qubit counts, unitarity, symmetry, conservation laws, ancilla cleanup.
Verified by construction: 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.

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Exact & formalFermionic Hamiltonians

Fermi-Hubbard dimer operator (Jordan-Wigner encoded)

The two-site Fermi-Hubbard dimer Hamiltonian, mapped to qubits via the Jordan-Wigner transformation: nearest-neighbor hopping competing with on-site Coulomb repulsion.

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Exact & formalSpin Hamiltonians

Heisenberg XXZ spin-chain operator

The anisotropic Heisenberg (XXZ) spin-chain Hamiltonian: exchange-coupled spins with tunable easy-axis/easy-plane anisotropy Δ, U(1)-symmetric under total-Sz rotation.

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Exact & formalSpin Hamiltonians

Classical Ising Hamiltonian operator

The classical (longitudinal-field) Ising Hamiltonian expressed as a diagonal SparsePauliOp: a foundational Z-only spin model with no quantum superposition dynamics of its own.

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Exact & formalSpin Hamiltonians

Transverse-field Ising model operator

The transverse-field Ising model (TFIM): the standard minimal Hamiltonian exhibiting a quantum (zero-temperature) phase transition driven by competing Z-Z order and X-field disorder.

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Atlas stars stay in this public list. Saving an entry to your workspace starts an unstarred private copy.

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The Quantum Atlas

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.
Stabilizer simulation: Clifford structure allowed exact verification with stabilizer methods beyond statevector limits.
Basis-state truth table: Reversible classical logic was checked exhaustively (or at edge cases) on computational-basis inputs.

Strong empirical

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.
Sub-block verification: Reusable modules (oracle, QFT block, Trotter step, adder…) were verified independently.
Echo / inverse test: U followed by U† returned the input exactly, catching wiring, phase, and ancilla bugs.
Invariant & structural checks: Scale-independent invariants held: qubit counts, unitarity, symmetry, conservation laws, ancilla cleanup.
Verified by construction: 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.

Search the Atlas

Exact & formalFermionic Hamiltonians

Fermi-Hubbard dimer operator (Jordan-Wigner encoded)

The two-site Fermi-Hubbard dimer Hamiltonian, mapped to qubits via the Jordan-Wigner transformation: nearest-neighbor hopping competing with on-site Coulomb repulsion.

View frozen graph

Exact & formalSpin Hamiltonians

Heisenberg XXZ spin-chain operator

The anisotropic Heisenberg (XXZ) spin-chain Hamiltonian: exchange-coupled spins with tunable easy-axis/easy-plane anisotropy Δ, U(1)-symmetric under total-Sz rotation.

View frozen graph

Exact & formalSpin Hamiltonians

Classical Ising Hamiltonian operator

The classical (longitudinal-field) Ising Hamiltonian expressed as a diagonal SparsePauliOp: a foundational Z-only spin model with no quantum superposition dynamics of its own.

View frozen graph

Exact & formalSpin Hamiltonians

Transverse-field Ising model operator

The transverse-field Ising model (TFIM): the standard minimal Hamiltonian exhibiting a quantum (zero-temperature) phase transition driven by competing Z-Z order and X-field disorder.

View frozen graph

Atlas stars stay in this public list. Saving an entry to your workspace starts an unstarred private copy.

The Quantum Atlas

What it is

Technique

Problem domain75 of 285 entries

In a pipeline72 of 285 declare ports · 58 meet another entry

Not a pipeline stage

Order by

Fermi-Hubbard dimer operator (Jordan-Wigner encoded)

Heisenberg XXZ spin-chain operator

Classical Ising Hamiltonian operator

Transverse-field Ising model operator

The Quantum Atlas

What it is

Technique

Problem domain75 of 285 entries

In a pipeline72 of 285 declare ports · 58 meet another entry

Not a pipeline stage

Order by

Fermi-Hubbard dimer operator (Jordan-Wigner encoded)

Heisenberg XXZ spin-chain operator

Classical Ising Hamiltonian operator

Transverse-field Ising model operator