Every quantum algorithm worth knowing about, written down the same way: what it takes, what it returns, what it costs, and who proved it.
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The defining behavior was checked exactly: a mathematical identity, a full statevector or stabilizer simulation, or an exhaustive basis-state truth table.
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60 public entries
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The two-site Fermi-Hubbard dimer Hamiltonian, mapped to qubits via the Jordan-Wigner transformation: nearest-neighbor hopping competing with on-site Coulomb repulsion.
The minimal-basis (STO-3G) H₂ electronic Hamiltonian after Jordan-Wigner/parity mapping and two-qubit tapering: the canonical small-molecule target for variational quantum eigensolver (VQE) demonstrations.
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.
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.
The Jordan-Wigner-encoded fermionic number operator n = c†c = (I-Z)/2, whose eigenvalues count mode occupation exactly.
Removes a fermion from spin orbital p. Representative form: aₚ.
Generator used in unitary coupled-cluster circuits. Representative form: τ - τ†.
Interacting lattice-boson model. Representative form: H = -tΣ⟨ij⟩(b†ᵢbⱼ+h.c.) + U/2Σᵢnᵢ(nᵢ-1).
Balances locality of parity and occupation updates. Representative form: occupation and parity stored in logarithmic update sets.
Measurement partition for shared basis estimation. Representative form: G_k={P_j : [P_i,P_j]_qw=0}.
Raises energy outside a feasible subspace. Representative form: H_penalty = λ(Ax-b)².
Represents two-electron repulsion in an orbital basis. Representative form: V = 1/2 Σ hₚqrs a†ₚa†_q a_r a_s.
Adds a fermion in spin orbital p subject to antisymmetry. Representative form: a†ₚ.
Penalizes an already found eigenstate. Representative form: H' = H + β|ψ⟩⟨ψ|.
Spatial electron-density observable. Representative form: ρ(r) = Σₚq φ*ₚ(r)φ_q(r)a†ₚa_q.
Observable for molecular polarity and response. Representative form: μ = -Σᵢrᵢ + Σ_AR_AZ_A.
Second-quantized molecular energy operator. Representative form: H = Σ hₚq a†ₚa_q + 1/2 Σ hₚqrs a†ₚa†_q a_r a_s.
Ranks adaptive ansatz generators. Representative form: ∂E/∂θ|₀ = ⟨ψ|[H,A]|ψ⟩.
Correlated lattice-fermion benchmark Hamiltonian. Representative form: H = -tΣ⟨ij⟩σ(c†ᵢσcⱼσ+h.c.) + UΣᵢnᵢ↑nᵢ↓.
Moves amplitude between orbitals or lattice sites. Representative form: Tₚq = a†ₚa_q + a†_q aₚ.
Z2 symmetry used for sectors and tapering. Representative form: Π = (-1)^N.
Eigenstate diagnostic and alternative objective. Representative form: H² - ⟨H⟩².
Isotropic interacting-spin model. Representative form: H = ΣJ(XᵢXⱼ + YᵢYⱼ + ZᵢZⱼ).
Diagonal optimization and spin-model objective. Representative form: H_C = Σᵢ hᵢZᵢ + ΣᵢⱼJᵢⱼZᵢZⱼ.
Maps fermionic antisymmetry into a parity string. Representative form: a†ₚ ↦ (Xₚ-iYₚ)/2 ⊗ Z₀···Zₚ₋₁.
Qubit representation of fermion transport. Representative form: a†ₚa_q+h.c. ↦ Pauli strings with a Z parity chain.
Local qubit representation of orbital occupation. Representative form: nₚ ↦ (I-Zₚ)/2.
Topological superconducting-chain model. Representative form: H = -μΣnᵢ - tΣ(c†ᵢcᵢ₊₁+h.c.) + ΔΣ(cᵢcᵢ₊₁+h.c.).
QAOA objective for graph cuts. Representative form: C = 1/2 Σ(i,j)∈E (I - ZᵢZⱼ).
Measures occupation of one spin orbital. Representative form: nₚ = a†ₚaₚ.
Kinetic, external-potential, or orbital-rotation term. Representative form: O₁ = Σ hₚq a†ₚa_q.
Compact one-body state descriptor and orbital gradient input. Representative form: γₚq = ⟨a†ₚa_q⟩.
Creates and annihilates correlated fermion pairs. Representative form: Δₚq = a†ₚa†_q + a_q aₚ.
Alternative encoding that can expose removable symmetries. Representative form: occupation ↦ cumulative parity bits.
Basic measured term in a qubit Hamiltonian. Representative form: P = P₀ ⊗ ··· ⊗ Pₙ₋₁.
Primitive for Trotter simulation and problem-inspired ansätze. Representative form: U_P(t) = exp(-itP).
Qubit-space anti-Hermitian generator candidates. Representative form: {iP_k}.
Maps binary quadratic objectives into diagonal Pauli form. Representative form: xᵀQx, xᵢ ↦ (I-Zᵢ)/2.
Overlap, fidelity, and penalty observable. Representative form: Π_ref = |φ⟩⟨φ|.
Folded-spectrum objective around target energy ω. Representative form: (H-ωI)².
Collective transverse spin observable. Representative form: Sₓ = 1/2 Σᵢ Xᵢ.
Collective quadrature spin observable. Representative form: Sᵧ = 1/2 Σᵢ Yᵢ.
Collective magnetization observable. Representative form: S_z = 1/2 Σᵢ Zᵢ.
Defines fixed-number sectors and symmetry checks. Representative form: N = Σₚ a†ₚaₚ.
Labels total-spin sectors and spin contamination. Representative form: S² = Sₓ² + Sᵧ² + S_z².
Non-commuting spin-chain ground-state benchmark. Representative form: H = -JΣZᵢZᵢ₊₁ - hΣXᵢ.
Product-formula approximation to Hamiltonian evolution. Representative form: e^{-itΣH_j} ≈ ∏_j e^{-itH_j}.
Electron-electron interaction term. Representative form: O₂ = 1/2 Σ hₚqrs a†ₚa†_q a_r a_s.
Two-body correlation descriptor used in energy and response. Representative form: Γₚqrs = ⟨a†ₚa†_q a_s a_r⟩.
Correlated double-excitation generator set. Representative form: {a†_a a†_b a_j a_i - h.c.}.
Adaptive or fixed single-excitation generator set. Representative form: {a†_a a_i - a†_i a_a}.
Canonical qubit-space objective used by VQE. Representative form: H = Σⱼ cⱼPⱼ.
Excitation-preserving spin-exchange model. Representative form: H = ΣJₓXᵢXⱼ + JᵧYᵢYⱼ.
Anisotropic extension of the Heisenberg model. Representative form: H = Σ(JₓXX + JᵧYY + J_zZZ).
Defines conserved sectors and qubit tapering constraints. Representative form: S = ⊗ᵢ Pᵢ, S²=I, [S,H]=0.
The joint parity operator Z^{⊗n} and its standard non-destructive ancilla-based measurement circuit, the core primitive behind stabilizer syndrome extraction.
A bit-flip operator that makes the relationship between a circuit and its truth table explicit.
A foundational code record that compares quantum protection with the narrower classical repetition-code idea.
A fault-tolerance record for comparing physical error, syndrome extraction, decoder choice, and logical failure.
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.