metatron_qso

One qubit with P(1) = p.

The type of the None singleton.

The type of the None singleton.

P(objective = 1) in Q^k A|0⟩ for ascending powers k.

Transition amplitude to the output occupation output.

Probability of output for indistinguishable photons.

Probability of output for distinguishable photons.

Exact output distribution: (patterns, probabilities) with one pattern per row.

shots samples as an integer matrix (one output occupation per row). Indistinguishable photons need at most one photon per input mode.

ρ ↦ (1−p)ρ + p/3 (XρX + YρY + ZρZ).

Two-qubit depolarising channel ρ ↦ (1−p)ρ + p/15 Σ_{P≠II} PρP.

Amplitude damping with decay probability gamma.

Phase damping: coherences shrink by √(1−lam).

Bit flip with probability p.

Phase flip with probability p.

Relaxation towards |0⟩ for duration time (t2 ≤ 2·t1).

Channel from explicit 2×2 or 4×4 Kraus operators (checked for Σ K†K = I).

Pauli channel from probabilities of non-identity Pauli strings, e.g. {"ZZ": 0.02, "XI": 0.001} (first letter on the first qubit).

Unitary channel ρ ↦ UρU† (coherent error) from a 2×2 or 4×4 unitary.

Coherent rotation exp(−iθP/2) about a Pauli string ("X", "ZZ", …).

Probabilities p_P (base-4 index, first qubit most significant) if this is a Pauli channel, else None.

The Pauli-twirled channel.

then ∘ self: this channel followed by then on the same qubits.

Kraus operators as complex matrices.

Hadamard.

Pauli X.

Pauli Y.

Pauli Z.

Phase gate S = diag(1, i).

S†.

T = diag(1, e^{iπ/4}).

T†.

√X.

(√X)†.

RX(θ) = exp(−iθX/2).

RY(θ) = exp(−iθY/2).

RZ(θ) = exp(−iθZ/2).

Phase gate P(λ) = diag(1, e^{iλ}).

Generic single-qubit gate U(θ, φ, λ) (OpenQASM convention).

RXX(θ) = exp(−iθ X⊗X/2).

RYY(θ) = exp(−iθ Y⊗Y/2).

RZZ(θ) = exp(−iθ Z⊗Z/2).

exp(−iθP/2) for a Pauli string label such as "X0 Y1 Z3".

Controlled X (CNOT).

Controlled Y.

Controlled Z.

Controlled Hadamard.

SWAP.

Toffoli.

Fredkin (controlled SWAP).

X on target controlled by all controls.

Multi-controlled Z on qubits (symmetric in its qubits).

Controlled RX.

Controlled RY.

Controlled RZ.

Controlled phase.

Custom unitary on 1–6 targets with optional controls.

matrix is 2^k × 2^k (checked for unitarity); targets[0] is the most significant bit of its row/column index.

Fresh symbolic parameter.

n fresh symbolic parameters.

Gate counts by name; controls are prefixed with c (ccx, cry…).

Instructions as (name, targets, controls, params) tuples; params are floats or Parameters.

Adjoint circuit (gates reversed and inverted).

Copy with every parameter replaced by its value for theta.

Append the gates of other (on the first other.n_qubits qubits).

Append other with its qubit k mapped to qubits[k].

OpenQASM 2.0 text with parameters theta bound.

OpenQASM 3.0 text with parameters theta bound.

Parse an OpenQASM 2.0 program; measurements are dropped. Use Program.from_qasm for OpenQASM 3 and for programs with mid-circuit measurements, resets or conditions.

The type of the None singleton.

snapshots random-basis snapshots of a state vector.

snapshots snapshots of circuit(theta), measured exactly, with a noise model or on a device (see estimate).

Mean over the snapshots with its standard error.

Median of the means of groups blocks of snapshots.

The arcs (tail, head) in the order of the state vector.

Walker at node with coin state coin (one amplitude per arc leaving the node), by default the uniform superposition.

Uniform superposition over all arcs.

One step: coin, then shift.

steps steps from psi0.

Probability of each node (total weight of the arcs leaving it).

Node probabilities after steps steps from start (uniform coin).

Search from the uniform superposition: {"steps", "success_probability", "history"} with the step of the highest success probability within max_steps steps. The probability oscillates; about (π/2)·√(N/2) steps reach the first peak on the hypercube.

Dimension of the walk (twice the number of edges).

Node probabilities at time t for a walker started at start.

exp(−iHt)|ψ₀⟩ for an arbitrary (not necessarily normalised) vector.

Infinite-time-averaged distribution (degenerate eigenspaces grouped).

Dimension of the smallest invariant subspace containing the start node: the number of distinct eigenvalues the walker sees.

Exact time average of the node probabilities over [0, t_max].

Mixing time (Aharonov et al. 2001): smallest T on the grid t_max·k/steps from which on the time-averaged distribution stays within total-variation distance epsilon of the limit, up to t_max (None if the bound fails at t_max).

Dense propagator U(t) = exp(−iHt).

Eigenvalues of H (ascending), or None for Krylov-only walks.

|0…0⟩⟨0…0|.

Maximally mixed state I/2ⁿ.

|ψ⟩⟨ψ|.

Run circuit(theta) on |0…0⟩⟨0…0|, inserting noise after every gate if given, or with the noise of device.

Apply circuit(theta) (with optional noise or device) in place.

Apply channel to qubits (one or two) in place.

Diagonal of ρ.

Tr ρ² (1 for pure states, 2⁻ⁿ for the maximally mixed state).

Tr ρ.

Tr(ρH).

⟨ψ|ρ|ψ⟩.

Dense copy of ρ.

Measure all qubits shots times with optional readout flips (P(1|0), P(0|1)): {bitstring: count}.

Measure all qubits with per-qubit readout errors [(P(1|0), P(0|1)), …] (qubit 0 first).

The type of the None singleton.

Node populations at time t ≥ 0 for a walker started at start.

Evolve the density matrix rho0 (n × n) for time t ≥ 0.

Parse a JSON device description.

The same noise as a NoiseModel.

The description as JSON.

Set T1, T2 (≤ 2·T1) and readout errors (P(1|0), P(0|1)) of qubit q; values left out follow the default qubit.

Set the properties of every qubit not set with with_qubit.

Append a gate rule. gate ("cx", "rz", …), qubits (controls first) and num_qubits select gates (None: any); channels act after the gate (one-qubit channels on each qubit, two-qubit channels on consecutive pairs).

Readout errors (P(1|0), P(0|1)) of qubits 0..n.

Two-sided normal confidence interval (low, high) at level.

(coefficient, [(mode, is_creation), …]) pairs.

The type of the None singleton.

Canonical form: creators first, indices descending.

The type of the None singleton.

Map to qubits: mapping is "jordan_wigner", "parity" or "bravyi_kitaev"; n_modes defaults to num_modes.

Graph from (u, v) or (u, v, w) edges; weights (one per edge) may be given for unweighted edges.

Graph from a NetworkX graph (or anything with nodes() and edges(data=True)); nodes are numbered in the order of nodes(), weights are read from the edge attribute weight (default 1). NetworkX itself is not needed.

Graph from an edge list: one u v [weight] per line, nodes from 0, # or % comments, # nodes N for the number of nodes.

The graph as an edge list (see from_edge_list).

Graph from a GraphML document (nodes, edges and a weight key; node ids are numbered in order of appearance).

The graph as a GraphML document.

Add the edge {u, v} with weight weight (finite, non-zero).

Graph from a specification string such as "metatron", "fruit-of-life", "cycle:8", "grid:3x4", "platonic:cube" or "random-regular:12,3,7" (same syntax as the qso CLI).

Metatron's Cube: the complete graph K13 on the 13 Fruit-of-Life centres.

Fruit-of-Life contact graph (18 edges, symmetry group D6).

Lines of Metatron's Cube of length ≤ max_length (circle diameters).

Complete graph K_n.

Cycle C_n (n ≥ 3).

Path P_n.

Star: node 0 joined to nodes 1..n.

Complete bipartite graph K_{a,b}.

d-dimensional hypercube Q_d (2^d nodes, d ≤ 20).

The Petersen graph.

The Heawood graph, the smallest cubic graph of girth 6.

rows × cols grid; a torus if periodic (each side ≥ 3).

Edge graph of a Platonic solid: tetrahedron, cube, octahedron, dodecahedron or icosahedron.

Random d-regular simple graph (n·d even, d < n).

Erdős–Rényi graph G(n, p).

Johnson graph J(n, k): k-subsets of n elements, adjacent when they share k − 1 elements.

Kneser graph K(n, k): k-subsets, adjacent when disjoint (K(5, 2) is the Petersen graph).

Hamming graph H(d, q): words of length d over q letters, adjacent when they differ in one letter (H(d, 2) is the hypercube).

Rook's graph K_m □ K_n (m × n board, adjacent in a row or column).

Paley graph of prime order q ≡ 1 (mod 4).

Circulant graph: node i joined to i ± s (mod n) for every jump s.

Cayley graph of Z_2^d: node x joined to x XOR g for every generator g.

The Shrikhande graph: strongly regular (16, 6, 2, 2) like the 4 × 4 rook's graph, but not isomorphic to it.

Cartesian product: node (g, h) has index g·|H| + h.

(n, k, λ, μ) if the graph is strongly regular, else None.

The graph with node u renamed perm[u].

Canonical labelling: labels[u] is the canonical index of node u; isomorphic graphs get identical canonical forms.

The graph relabelled canonically.

Whether the graphs are isomorphic (edge weights included).

An isomorphism to other (node u maps to iso[u]), or None.

Node degrees.

Sorted neighbours of u as (node, weight) pairs.

Dense weighted adjacency matrix.

Dense Laplacian L = D − A.

Eigenvalues of the adjacency matrix (ascending).

Eigenvalues of the Laplacian (ascending).

Structural summary as a dict.

Order of the automorphism group (None if it exceeds 2^128).

log10 of the automorphism group order (always finite).

Orbits of the automorphism group (each sorted).

Strong generating set of the automorphism group; permutation p maps node u to p[u].

The type of the None singleton.

Largest hop distance (None if disconnected).

Length of the shortest cycle (None for a forest).

Number of triangles.

The type of the None singleton.

Edges (u, v, weight) with u < v, in insertion order.

Node coordinates as an (n, 3) array, or None.

Train on graphs with 0/1 labels (L-BFGS, restarts seeded starts, best loss kept).

Normalised expected cut ⟨C_G⟩ / W(G), with the trained angles or with angles = [γ…, β…].

p(y = 1 | G).

Predicted class (p ≥ ½).

Fraction of correctly classified graphs.

Trained parameters [γ…, β…, a, b] (empty before fit).

Position distribution after steps steps from the origin; entry k is position k − steps. The default coin state (|L⟩ + i|R⟩)/√2 gives the symmetric Hadamard walk.

Expected cut weight ⟨C⟩.

⟨C⟩ and its exact gradient.

Statistics of the QAOA state at params.

QAOA state at params.

Maximise ⟨C⟩ from restarts seeded starts (initial is used for the first one).

All optimal cuts as basis indices (bit q = side of node q).

Weight of the cut encoded by basis index b.

Exact maximum cut weight.

Parse the text of an FCIDUMP file.

Read an FCIDUMP file.

The type of the None singleton.

The type of the None singleton.

The type of the None singleton.

The type of the None singleton.

Qubit Hamiltonian (constant term = core energy).

The type of the None singleton.

X gates preparing the Hartree–Fock determinant.

The lowest n_roots full-CI energies (core energy included).

Depolarising noise: p1 after one-qubit gates, two-qubit depolarising p2 after two-qubit gates.

Value for the parameter vector theta.

Sum of (coefficient, "Pauli string") terms.

Terms as (coefficient, string) pairs ("I" for the identity).

⟨ψ|H|ψ⟩.

Exact ground-state energy on n_qubits (default: min_qubits).

Ground-state energy and state (Lanczos) on n_qubits.

Dense 2ⁿ × 2ⁿ matrix (n ≤ 14; default min_qubits).

Smallest register containing every term.

Coefficient of the identity term.

Parse an OpenQASM 2 or 3 program (the version is read from its header).

The program of circuit with parameters theta bound, measuring every qubit into the bit of the same index if measure.

Exact probabilities of the classical records, {bitstring: p} (records with probability below 1e-15 are left out). Mid-circuit measurements are followed branch by branch.

shots (at most 10⁷) seeded samples of the classical record, {bitstring: count}.

The gates as a Circuit; a ValueError unless the program is unitary.

OpenQASM 3.0 text of the program.

OpenQASM 2.0 text of the program; a ValueError if it needs OpenQASM 3 (for example a condition on part of a register).

Qubit registers as {"name", "offset", "size"} dictionaries.

Whether the program has only gates.

Bit registers as {"name", "offset", "size"} dictionaries.

The parametrised circuit [γ…, β…] (apply to initial_state).

The type of the None singleton.

(⟨C⟩, gradient) at [γ…, β…].

The type of the None singleton.

Minimise ⟨C⟩ from seeded random starts.

(coefficient, label) pairs.

The type of the None singleton.

The operator as a PauliSum; fails if it is not Hermitian.

The type of the None singleton.

[self, other].

Sum of the absolute coefficients.

From (i, j, value) entries.

f(x) for a bit mask or a 0/1 sequence.

The Ising form as a dict with h, j (list of (i, j, J)) and offset.

The diagonal cost Hamiltonian.

(minimum, [bit masks]) by enumeration (at most 30 variables).

Coarsest equitable partition refining the node colouring colours.

Coarsest equitable partition in which each of nodes is a cell of its own (the reduction for a walker started at, or a search for, those nodes).

The partition cell_of (node → cell label), checked to be equitable.

Quotient from cell sizes and the matrix of weights between cells (sizes[i]·matrix[i][j] = sizes[j]·matrix[j][i] is checked).

K_n seen from one node: the node and the n − 1 others.

Complete multipartite graph seen from a node of the first part.

Hamming graph H(d, q) by distance from a word (d + 1 cells).

Hypercube Q_d by Hamming weight (d + 1 cells, any d up to about 1000).

Johnson graph J(n, k) by distance from a k-subset.

Weighted degree of the nodes of each cell.

Adjacency matrix on the normalised cell states (symmetric).

Laplacian on the normalised cell states (symmetric).

The uniform superposition over all nodes in the cell basis.

Per-node probabilities from per-cell ones (explicit graphs only).

Weight from any node of cell i into cell j.

Cell of every node (partitions of an explicit graph only).

Number of nodes of the graph (a float: implicit families can exceed every integer type).

Number of nodes in each cell.

⟨C|L⁺|C⟩: the Childs–Goldstone rate for one marked node of a vertex-transitive graph, from the reduced Laplacian.

Optimise the hopping rate and the measurement time together. t_max=None uses π√N.

Probability of finding a node of a marked cell at time t.

Maximise the success probability over [0, t_max].

Probability of each cell at time t for the walk started at start.

⟨C_to| exp(−iHt) |C_from⟩ (the node amplitude for single-node cells).

exp(−iHt) applied to amplitudes in the cell basis.

Limiting (infinite-time-averaged) probability of each cell.

Exact time average of the cell probabilities over [0, t_max].

Eigenvalues of the reduced Hamiltonian (ascending).

Tensored calibration from known per-qubit errors [(P(1|0), P(0|1)), …] (qubit 0 first).

Full calibration from the confusion matrix A[measured, prepared] (columns are probability distributions).

Measure a tensored calibration (|0…0⟩ and |1…1⟩, shots each).

Measure a full calibration (all 2ⁿ basis states, shots each).

Per-qubit errors (P(1|0), P(0|1)).

Quasi-probabilities {bitstring: q} from measured counts.

Childs–Goldstone critical hopping rate (1/N) Σ_{λ>0} 1/λ over the Laplacian spectrum.

Optimise the hopping rate (around the critical rate) and the measurement time together. t_max=None uses π√N. On complete graphs this recovers the exact optimum γ = 1/N with success 1.

Probability of measuring a marked node at time t.

Maximise the success probability over [0, t_max] (grid scan plus golden-section refinement).

Transfer from the single-node cell from_cell to cell to_cell of a quotient (for example Quotient.hypercube(30), cells 0 → 30).

⟨v| exp(−iHt) |u⟩.

|⟨v| exp(−iHt) |u⟩|².

Highest fidelity over [0, t_max], as {"time", "fidelity"}.

Earliest time of perfect state transfer, or None.

Signs σ_θ with E_θ e_v = σ_θ E_θ e_u on the support, if the nodes are strongly cospectral.

Distinct eigenvalues of H in the support of u.

|0…0⟩.

Uniform superposition |+⟩^⊗n.

Computational basis state |index⟩.

State from amplitudes (length a power of two; normalised).

Haar-random state.

Probabilities of all basis states.

Measure all qubits shots times: {bitstring: count} with qubit n−1 first.

⟨ψ|H|ψ⟩.

⟨self|other⟩.

|⟨self|other⟩|².

Von Neumann entropy (bits) of qubits against the rest.

Reduced density matrix of qubits (qubits[j] is bit j of the row/column index).

Apply circuit(theta) in place; returns the state.

The type of the None singleton.

Amplitudes as a complex128 array (a copy).

The start state |π⟩ in the arc basis.

One step W' = (S·R)².

‖W'^t|π⟩ − |π⟩‖² for t = 0, …, steps.

Quantum hitting time, or None if not reached within max_steps.

Stationary probability of the marked set.

Number of arcs the walk acts on.

Average of ⟨ψ|H|ψ⟩ over the trajectories, with its standard error.

Averages of several observables over the same trajectories.

Measure all qubits shots times with the device's readout errors.

Average of |ψ⟩⟨ψ| over the trajectories (up to 14 qubits).

Final state of trajectory k.

Train on samples xs (shape (m, n_features)) and 0/1 labels ys with L-BFGS from a seeded start.

p(y=1|x): a float for one sample, an array for a 2-D batch.

Predicted class (p ≥ ½): a bool for one sample, an array for a batch.

Fraction of correctly classified samples.

Trained parameters (empty before fit).