01 From graphs to quantum operators
Max-Cut: Find the best partition.
Choose a graph. Max-Cut maximizes the number of edges crossing the two node groups.
Problem input: nodes and edges
OPTIONAL · CLASSICAL DEMONSTRATION Classical partition demo (manual; separate from QAOA)
Toggle node groups to see how a cut is scored. These demo colors, bitstrings, and reference optima are separate from QAOA inputs and measured outputs.
MANUAL PARTITION
The leftmost bit is q₀; node i corresponds to qubit i.
CLASSICAL REFERENCE Enumerate all classical solutions (reference)
For small-scale teaching and numerical checks only. QAOA does not require all solutions to be listed in advance; enumeration is a reference, not an algorithm output.
02 Optimize parameters · Sample · Read the partition
Optimize parameters. Sample a cut.
Improve the full-circuit objective, then measure a batch of candidate partitions.
1. Optimize parameters
Shared by the original and reduced circuitsHow the objective is evaluated
ALGORITHM OUTPUT · MEASURED CANDIDATES
2. Sample a partition
Measure the full circuit. The highest-cut candidate actually observed in this batch is shown first.
READY WHEN YOU ARE
No measured output yet
Select Sample to reveal an observed bitstring, its cut value, and its partition.
All candidates and batch statistics
QUANTUM EXPLORERExplore the circuit and quantum state
Step through the circuit to connect cost phases, interference, and measurement probabilities.
Follow the steps from state preparation to measurement. Every step uses the current problem and parameters.
Standard QAOA circuit
Click a circuit layer to inspect the state after it.
Quantum-state probabilities (theory)
Computational basis |z⟩Preview the selected basis partition (theory)
This preview shows the partition represented by the selected basis state. It does not indicate that the candidate was measured and does not change the measured results above.
ADVANCED Amplitudes, complex plane, and local Bloch vectors
Complex amplitude ψz
The squared distance from the origin gives probability; the angle gives phase.
Local Bloch vectors
Computed from each qubit's reduced density matrix. The drawing shows the x–z projection; the coordinate labels include y. Local vectors cannot fully describe entanglement or the joint state.
Full statevector
ADVANCED Parameter landscape and slices
Hold the other layers fixed and scan γ and β for the selected layer. Click the heatmap to use those parameters.
03 Equivalence-Preserving Qubit Efficient QAOA
From identifying Q to an equivalent reduction.
Inspect candidate conserved quantities, locate the initial-state sector, build the mapping and reduced operators, then evolve, decode, and verify.
EQE METHOD · VISUAL WALKTHROUGH
Define the problem and Hamiltonians
The shared graph defines the cost operator H_C; the transverse-field mixer H_M determines how quantum amplitudes can mix.
The shared Max-Cut graph
Use the nodes and edges from Problem Setup, without supplying a partition as the answer.
PROBLEM → OPERATORS
Encode the graph as two evolution operators
Before
After the selected flips
Test the specified candidate family
Check both commutators
Here Q = X₀X₁⋯Xₙ₋₁ has only two eigenvalues, +1 and −1. The geometry illustrates a direct-sum decomposition; area does not represent dimension or probability.
Start with the full Hilbert space H. Then reveal the Q eigenspaces, locate the initial state, and trace evolution within the same eigenspace.
Embed a reduced basis state in the original symmetric sector
See how each operator maps to the reduced space
Independent initial-state explorer. Standard QAOA and measured batches stay unchanged.
INITIAL STATE → INVARIANT EVOLUTION
Choose a sector, then follow the state.
ADVANCED EXPLORATION General Q · coefficients · checks · circuit
Actual Q=X⊗n has two eigenspaces. The general-Q view is a separate illustration.
Independent reference components in H₊ and H₋; not QAOA output or a path crossing sectors.
Basis coefficients and isometry
Complementary states
Q eigenstate
Reduced coordinates
All complementary pairs
Percentages are actual weights; bars use a relative scale.Why the sector is conserved
Reduced Hamiltonians and circuit
Explorer Hamiltonians
Explorer reduced circuit
Same γ, β · independently evolvedReconstruct the original space and check the full result
04 Independent simulation · Full reconstruction · Numerical checks
Same result. What changes in resources?
Compare independently simulated states at the same parameters, then inspect the qubit and gate tradeoffs.
Compare measurement distributions in the same space
Compare probabilities in the same n-qubit computational basis, after mapping the reduced state back through V.
Resource counts and assumptions
Resource comparison
Inspect the synchronized circuits
Synchronized circuit evolution
Original QAOA
Original Hilbert spaceEQE-QAOA
Q = +1 effective subspaceInspect all equivalence metrics
All original and reconstructed probabilities+
SCIENTIFIC CHECKS Mathematical and numerical checks
Checks of operators, isometry, states, and probability equivalence.