aqora / qaoa
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Quantum Approximate Optimization Algorithm (QAOA)
A hybrid quantum-classical variational algorithm for combinatorial optimization on near-term quantum hardware. Use the controls below to configure the problem instance and circuit depth — the notebook updates reactively.Interactive QAOA Explorer
Use the controls below to configure the problem instance, circuit depth, and optimizer settings. The notebook will automatically re-run the full QAOA optimization whenever any parameter changes. Tip: Start with $$ to understand the landscape, then increase depth to observe approximation ratio improvements. For small graphs ($$) the simulation runs in under a second.Configuration
Graph summary: 6 nodes, 6 edges, unweighted
QAOA Solution Summary
BEST BITSTRING
010101
CUT VALUE / MAX CUT
6.00 / 6.00
APPROXIMATION RATIO
75.0%
PARTITION S0
{0, 2, 4}
PARTITION S1
{1, 3, 5}
OPTIMAL PARAMETERS (p=1)
gamma: 2.3562
beta: 1.1781
beta: 1.1781
Expected cut (QAOA): 4.5000 | Depth: p=1 | Qubits: n=6
Depth Sweep: Approximation Ratio vs Circuit Layers
The chart below solves the current problem instance for every $$ and tracks how the approximation ratio and expected cut value evolve with circuit depth. This directly illustrates the depth vs. accuracy trade-off central to NISQ algorithm selection. For most small instances, $$ already reaches near-optimal performance, while $$ may saturate. The gain per additional layer varies by graph topology and weight distribution.Check the box above to compute approximation ratios for p = 1 to 4.
Circuit Export & Hardware Deployment
The statevector simulator scales as $$ in memory and time, capping practical classical simulation at roughly $$ qubits. Real devices today operate on 100+ qubits. This section provides two complementary export paths: Optimised circuit (small graphs): Afterqaoa_solve converges, freeze the $$ parameters and emit OpenQASM 2.0 via qaoa_to_qasm. Each RZZ interaction on an edge $$ decomposes into the hardware-native cx, rz, cx triple:
cx q[u], q[v];
rz( 2 w gamma ) q[v];
cx q[u], q[v];
qaoa_circuit_to_qasm generates a circuit for any graph size using pre-supplied angles — either transferred from a small proxy or taken from the published fixed-angle table (Brandao et al., 2018). Gate count scales as $$, so a 1000-node graph at $$ is produced in milliseconds.
| Loader | One-liner |
|---|---|
| Qiskit | QuantumCircuit.from_qasm_str(qasm) |
| Cirq | cirq.Circuit.from_qasm(qasm) |
| Amazon Braket | Circuit.from_openqasm(qasm) |
Large-circuit configuration
Large-Circuit OpenQASM 2.0 Output
Parameters: Fixed angles (Brandao et al., p=1)
Qubits: 50
Edges: 55
CX gates: 110
RZ gates: 55
RX gates: 50
Depth p: 1
Lines: 327
OPENQASM 2.0; include "qelib1.inc"; qreg q[50]; creg c[50]; // Initialise uniform superposition h q[0]; h q[1]; h q[2]; h q[3]; h q[4]; h q[5]; h q[6]; h q[7]; h q[8]; h q[9]; h q[10]; h q[11]; h q[12]; h q[13]; h q[14]; h q[15]; h q[16]; h q[17]; h q[18]; h q[19]; h q[20]; h q[21]; h q[22]; h q[23]; h q[24]; h q[25]; h q[26]; h q[27]; h q[28]; h q[29]; h q[30]; h q[31]; h q[32]; h q[33]; h q[34]; h q[35]; h q[36]; h q[37]; h q[38]; h q[39]; h q[40]; h q[41]; h q[42]; h q[43]; h q[44]; h q[45]; h q[46]; h q[47]; h q[48]; h q[49]; // Layer 1: phase separator U_C(gamma=0.615500) cx q[0], q[2]; rz(1.231000) q[2]; cx q[0], q[2]; cx q[0], q[10]; rz(1.231000) q[10]; cx q[0], q[10]; cx q[0], q[13]; rz(1.231000) q[13]; cx q[0], q[13]; cx q[0], q[20]; rz(1.231000) q[20]; cx q[0], q[20]; cx q[0], q[42]; rz(1.231000) q[42]; cx q[0], q[42]; cx q[1], q[20]; rz(1.231000) q[20]; cx q[1], q[20]; cx q[1], q[42]; rz(1.231000) q[42]; cx q[1], q[42]; cx q[2], q[6]; rz(1.231000) q[6]; cx q[2], q[6]; cx q[2], q[30]; rz(1.231000) q[30]; cx q[2], q[30]; cx q[2], q[32]; rz(1.231000) q[32]; cx q[2], q[32]; cx q[5], q[40]; rz(1.231000) q[40]; cx q[5], q[40]; cx q[6], q[18]; rz(1.231000) q[18]; cx q[6], q[18]; cx q[6], q[25]; rz(1.231000) q[25]; cx q[6], q[25]; cx q[7], q[10]; rz(1.231000) q[10]; cx q[7], q[10]; cx q[7], q[28]; rz(1.231000) q[28]; cx q[7], q[28]; cx q[8], q[13]; rz(1.231000) q[13]; cx q[8], q[13]; cx q[8], q[39]; rz(1.231000) q[39]; cx q[8], q[39]; cx q[9], q[20]; rz(1.231000) q[20]; cx q[9], q[20]; cx q[9], q[32]; rz(1.231000) q[32]; cx q[9], q[32]; cx q[10], q[47]; rz(1.231000) q[47]; cx q[10], q[47]; cx q[11], q[21]; rz(1.231000) q[21]; cx q[11], q[21]; cx q[11], q[29]; rz(1.231000) q[29]; cx q[11], q[29]; cx q[11], q[32]; rz(1.231000) q[32]; cx q[11], q[32]; cx q[11], q[46]; rz(1.231000) q[46]; cx q[11], q[46]; cx q[12], q[19]; rz(1.231000) q[19]; cx q[12], q[19]; cx q[12], q[34]; rz(1.231000) q[34]; cx q[12], q[34]; cx q[13], q[44]; rz(1.231000) q[44]; cx q[13], q[44]; cx q[14], q[18]; rz(1.231000) q[18]; cx q[14], q[18]; cx q[15], q[23]; rz(1.231000) q[23]; cx q[15], q[23]; cx q[16], q[37]; rz(1.231000) q[37]; cx q[16], q[37]; cx q[17], q[37]; rz(1.231000) q[37]; cx q[17], q[37]; cx q[17], q[49]; rz(1.231000) q[49]; cx q[17], q[49]; cx q[20], q[48]; rz(1.231000) q[48]; cx q[20], q[48]; cx q[21], q[29]; rz(1.231000) q[29]; cx q[21], q[29]; cx q[22], q[27]; rz(1.231000) q[27]; cx q[22], q[27]; cx q[22], q[40]; rz(1.231000) q[40]; cx q[22], q[40]; cx q[23], q[32]; rz(1.231000) q[32]; cx q[23], q[32]; cx q[24], q[33]; rz(1.231000) q[33]; cx q[24], q[33]; cx q[26], q[34]; rz(1.231000) q[34]; cx q[26], q[34]; cx q[26], q[44]; rz(1.231000) q[44]; cx q[26], q[44]; cx q[28], q[30]; rz(1.231000) q[30]; cx q[28], q[30]; cx q[28], q[48]; rz(1.231000) q[48]; cx q[28], q[48]; cx q[29], q[41]; rz(1.231000) q[41]; cx q[29], q[41]; cx q[29], q[48]; rz(1.231000) q[48]; cx q[29], q[48]; cx q[29], q[49]; rz(1.231000) q[49]; cx q[29], q[49]; cx q[30], q[45]; rz(1.231000) q[45]; cx q[30], q[45]; cx q[35], q[41]; rz(1.231000) q[41]; cx q[35], q[41]; cx q[35], q[49]; rz(1.231000) q[49]; cx q[35], q[49]; cx q[36], q[42]; rz(1.231000) q[42]; cx q[36], q[42]; cx q[38], q[45]; rz(1.231000) q[45]; cx q[38], q[45]; cx q[38], q[48]; rz(1.231000) q[48]; cx q[38], q[48]; cx q[39], q[44]; rz(1.231000) q[44]; cx q[39], q[44]; cx q[40], q[45]; rz(1.231000) q[45]; cx q[40], q[45]; cx q[40], q[46]; rz(1.231000) q[46]; cx q[40], q[46]; cx q[41], q[46]; rz(1.231000) q[46]; cx q[41], q[46]; // Layer 1: mixer U_B(beta=0.392700) rx(0.785400) q[0]; rx(0.785400) q[1]; rx(0.785400) q[2]; rx(0.785400) q[3]; rx(0.785400) q[4]; rx(0.785400) q[5]; rx(0.785400) q[6]; rx(0.785400) q[7]; rx(0.785400) q[8]; rx(0.785400) q[9]; rx(0.785400) q[10]; rx(0.785400) q[11]; rx(0.785400) q[12]; rx(0.785400) q[13]; rx(0.785400) q[14]; rx(0.785400) q[15]; rx(0.785400) q[16]; rx(0.785400) q[17]; rx(0.785400) q[18]; rx(0.785400) q[19]; rx(0.785400) q[20]; rx(0.785400) q[21]; rx(0.785400) q[22]; rx(0.785400) q[23]; rx(0.785400) q[24]; rx(0.785400) q[25]; rx(0.785400) q[26]; rx(0.785400) q[27]; rx(0.785400) q[28]; rx(0.785400) q[29]; rx(0.785400) q[30]; rx(0.785400) q[31]; rx(0.785400) q[32]; rx(0.785400) q[33]; rx(0.785400) q[34]; rx(0.785400) q[35]; rx(0.785400) q[36]; rx(0.785400) q[37]; rx(0.785400) q[38]; rx(0.785400) q[39]; rx(0.785400) q[40]; rx(0.785400) q[41]; rx(0.785400) q[42]; rx(0.785400) q[43]; rx(0.785400) q[44]; rx(0.785400) q[45]; rx(0.785400) q[46]; rx(0.785400) q[47]; rx(0.785400) q[48]; rx(0.785400) q[49]; // Measure all qubits measure q[0] -> c[0]; measure q[1] -> c[1]; measure q[2] -> c[2]; measure q[3] -> c[3]; measure q[4] -> c[4]; measure q[5] -> c[5]; measure q[6] -> c[6]; measure q[7] -> c[7]; measure q[8] -> c[8]; measure q[9] -> c[9]; measure q[10] -> c[10]; measure q[11] -> c[11]; measure q[12] -> c[12]; measure q[13] -> c[13]; measure q[14] -> c[14]; measure q[15] -> c[15]; measure q[16] -> c[16]; measure q[17] -> c[17]; measure q[18] -> c[18]; measure q[19] -> c[19]; measure q[20] -> c[20]; measure q[21] -> c[21]; measure q[22] -> c[22]; measure q[23] -> c[23]; measure q[24] -> c[24]; measure q[25] -> c[25]; measure q[26] -> c[26]; measure q[27] -> c[27]; measure q[28] -> c[28]; measure q[29] -> c[29]; measure q[30] -> c[30]; measure q[31] -> c[31]; measure q[32] -> c[32]; measure q[33] -> c[33]; measure q[34] -> c[34]; measure q[35] -> c[35]; measure q[36] -> c[36]; measure q[37] -> c[37]; measure q[38] -> c[38]; measure q[39] -> c[39]; measure q[40] -> c[40]; measure q[41] -> c[41]; measure q[42] -> c[42]; measure q[43] -> c[43]; measure q[44] -> c[44]; measure q[45] -> c[45]; measure q[46] -> c[46]; measure q[47] -> c[47]; measure q[48] -> c[48]; measure q[49] -> c[49];
Generated without simulation. Circuit generation time is O(p x |E|).
Submit to hardware via Qiskit, Cirq, or Amazon Braket.
Optimised circuit export (current explorer result)
OpenQASM 2.0 Output
Lines: 48
CX gates: 12
RZ gates: 6
RX gates: 6
Qubits: 6
Depth p: 1
OPENQASM 2.0; include "qelib1.inc"; qreg q[6]; creg c[6]; // Initialise uniform superposition h q[0]; h q[1]; h q[2]; h q[3]; h q[4]; h q[5]; // Layer 1: phase separator U_C(gamma=2.356194) cx q[0], q[1]; rz(4.712389) q[1]; cx q[0], q[1]; cx q[0], q[5]; rz(4.712389) q[5]; cx q[0], q[5]; cx q[1], q[2]; rz(4.712389) q[2]; cx q[1], q[2]; cx q[2], q[3]; rz(4.712389) q[3]; cx q[2], q[3]; cx q[3], q[4]; rz(4.712389) q[4]; cx q[3], q[4]; cx q[4], q[5]; rz(4.712389) q[5]; cx q[4], q[5]; // Layer 1: mixer U_B(beta=1.178097) rx(2.356194) q[0]; rx(2.356194) q[1]; rx(2.356194) q[2]; rx(2.356194) q[3]; rx(2.356194) q[4]; rx(2.356194) q[5]; // Measure all qubits measure q[0] -> c[0]; measure q[1] -> c[1]; measure q[2] -> c[2]; measure q[3] -> c[3]; measure q[4] -> c[4]; measure q[5] -> c[5];
Paste directly into Qiskit, Cirq, or Amazon Braket. Each RZZ interaction is decomposed
into the cx, rz, cx basis-gate triple required by hardware transpilers.
Reference
What is QAOA?
The Quantum Approximate Optimization Algorithm (QAOA), introduced by Farhi, Goldstone, and Gutmann (2014), is a hybrid quantum-classical variational algorithm designed to find approximate solutions to combinatorial optimization problems. It operates on a parameterized quantum circuit whose depth is controlled by a single integer $$ (the number of layers). QAOA alternates between two unitary operators:- Problem unitary $$: encodes the cost function $$ into the phase of quantum states
- Mixer unitary $$: drives quantum tunneling between candidate solutions
Canonical Problem Instances
Three benchmark problems are included. Each maps naturally to a graph whose MaxCut encodes the optimization objective.| Problem | Graph | Industry relevance |
|---|---|---|
| Unweighted MaxCut | Petersen / cycle / complete | Network partitioning, VLSI layout |
| Weighted MaxCut | Random Erdos-Renyi with capacities | Supply-chain balancing, clustering |
| Portfolio optimization | Asset-correlation graph | Asset allocation, risk parity |
Industry Applications of QAOA / QUBO
Finance
Portfolio optimization
Partition assets into uncorrelated buckets to minimize variance. QUBO encoding maps covariance matrices directly to edge weights.
Logistics
Vehicle routing & scheduling
Assign deliveries or jobs to vehicles / machines minimizing total cost. Reduces to MaxCut or graph coloring on conflict graphs.
Networking
Network partitioning & clustering
Bisect communication networks, VLSI circuits, or social graphs to minimize cross-partition traffic. The canonical MaxCut application.
Machine Learning
Feature selection & clustering
Select maximally uncorrelated features or cluster data points by mapping pairwise similarity to edge weights and solving MaxCut.
Energy
Grid load balancing
Partition power grid nodes into zones that minimize inter-zone power flow, reducing transmission loss and congestion risk.
Pharmaceuticals
Molecular docking & design
Binary encoding of conformer selection and protein-ligand binding site assignment maps naturally to QUBO and QAOA.
All problems above reduce to a Quadratic Unconstrained Binary Optimization (QUBO) instance,
which maps directly to a weighted MaxCut graph and can be solved with the qaoa_solve(graph) function in this notebook.