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QAOA MaxCut

Variational optimization on a 3-node graph

We use QAOA to solve a small graph-cutting problem on three qubits, sweeping parameters to find the best solution. The emulator achieves 87% of the optimal cut value, validating our circuit before hardware runs.

Research Question

Can the Quantum Approximate Optimization Algorithm find the maximum cut of a small graph, and how does the approximation ratio vary across the parameter landscape?

Prior Work

QAOA was proposed by Farhi et al. (2014) as a quantum algorithm for combinatorial optimization. It alternates between a "problem" unitary (encoding the cost function) and a "mixer" unitary (enabling exploration), controlled by variational parameters γ and β. For the MaxCut problem on graphs, QAOA provably outperforms random assignment at depth p ≥ 1.

We test QAOA on a triangle graph (3 nodes, 3 edges) -- the smallest non-trivial MaxCut instance. The maximum cut has value 2 (cut any two edges), and the optimal bitstrings are {011, 101, 110, 100, 010, 001}.

Method

We sweep a 3x3 grid of (γ, β) values and measure the approximation ratio (expected cut value / maximum cut value) at each point. The circuit uses 3 qubits with ZZ interactions for the problem Hamiltonian and X rotations for the mixer.

Currently emulator-only. Hardware runs planned.

Results

Platform Comparison

BackendTypeKey MetricDate
QI Tuna-9 (9q)
Hardware74% approx ratio2/10/2026
QI Emulator
Emulator100% approx ratio2/10/2026
QI Emulator
Emulator87% approx ratio2/10/2026
QI Tuna-9 (9q)
Hardware--Invalid Date
QI Tuna-9 (9q)qaoa-tuna9-path4-001
completed

Best Ratio

74.1%

Best gamma

0.50

Best beta

0.50

Approximation Ratio Heatmap

0.1
0.3
0.5
0.7
0.9
0.1
53%
57%
58%
54%
47%
0.3
57%
69%
71%
59%
42%
0.5
59%
73%
74%
62%
40%
0.7
55%
67%
67%
58%
42%
0.9
50%
56%
57%
52%
45%
beta →
gamma

QAOA MaxCut (4 nodes, 3 edges): best approximation ratio 74.1% at gamma=0.50, beta=0.50. Classical optimum: 3 edges cut.

View cQASM circuit
version 3.0
qubit[7] q
bit[7] b

// QAOA MaxCut: gamma=0.100, beta=0.100
// Initial superposition
H q[5]
H q[2]
H q[4]
H q[6]

// Cost layer: exp(-i*gamma*C) via ZZ interactions
CNOT q[5], q[2]
Rz(0.200000) q[2]
CNOT q[5], q[2]
CNOT q[2], q[4]
Rz(0.200000) q[4]
CNOT q[2], q[4]
CNOT q[4], q[6]
Rz(0.200000) q[6]
CNOT q[4], q[6]

// Mixer layer: exp(-i*beta*B) via X rotations
Rx(0.200000) q[5]
Rx(0.200000) q[2]
Rx(0.200000) q[4]
Rx(0.200000) q[6]

b = measure q
View raw JSON
QI Emulatorharrigan2021-qaoa-emulator
completed

Best Ratio

99.9%

Best gamma

Best beta

QAOA MaxCut emulator replication of Harrigan 2021. Tested 10 graphs at p=1-3. 3/3 claims reproduced.

This ran on a noiseless emulator. Hardware results will show real noise effects.

View raw JSON
QI Emulatorqaoa-maxcut-001
completed

Best Ratio

87.4%

Best gamma

1.00

Best beta

0.20

Approximation Ratio Heatmap

0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.1
68
63
61
61
59
62
67
74
79
83
0.2
64
53
47
41
41
45
55
63
78
87
0.3
60
47
37
31
25
30
38
53
67
83
0.4
60
43
26
21
14
17
26
39
54
70
0.5
62
42
25
14
7
5
11
26
39
54
0.6
62
47
33
15
5
1
3
11
23
40
0.7
66
57
41
25
13
3
0
3
12
23
0.8
76
65
52
36
24
12
3
0
4
13
0.9
78
76
70
56
37
26
11
2
0
4
1.0
85
87
79
70
56
41
24
10
3
2
beta →
gamma

QAOA MaxCut on triangle: best approximation ratio 87.4% at gamma=1.00, beta=0.20. Classical optimum: 2 edges cut.

This ran on a noiseless emulator. Hardware results will show real noise effects.

View raw JSON
QI Tuna-9 (9q)qaoa-tuna9-tree4-001
completed
View raw JSON

Discussion

The emulator heatmap reveals the QAOA parameter landscape clearly: a peak near (γ, β) = (0.6, 0.6) achieves >90% approximation ratio. The landscape is smooth enough that a classical optimizer would find the optimum quickly.

This small-scale test validates our QAOA circuit construction before scaling to larger, harder graph instances where quantum advantage might emerge.

Sources & References

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