AI for Quantum Circuit Optimization

AI for Quantum Circuit Optimization

Abstract

Quantum algorithms must be compiled to hardware-native gate sets, mapped to physical qubit topologies, and optimized for minimal depth. Classical compilers use heuristics that often produce suboptimal results. This document explores how machine learning—neural compilers, reinforcement learning for gate synthesis, and graph neural networks for qubit mapping—can dramatically improve circuit optimization.

1. The Compilation Challenge

1.1 From Algorithm to Hardware

┌─────────────────────────────────────────────────────────────────┐
│                    Compilation Pipeline                          │
├─────────────────────────────────────────────────────────────────┤
│                                                                  │
│  Input: High-Level Circuit                                      │
│  ┌───────────────────────────────────────────────────────┐     │
│  │  H(0), CNOT(0,3), Rz(0.5, 2), Toffoli(0,1,2)          │     │
│  └───────────────────────────────────────────────────────┘     │
│                           ↓                                      │
│  Stage 1: Gate Decomposition                                    │
│  ┌───────────────────────────────────────────────────────┐     │
│  │  Toffoli → 6 CNOT + 9 single-qubit gates               │     │
│  │  Arbitrary U → native gates (√X, Rz, CNOT)             │     │
│  └───────────────────────────────────────────────────────┘     │
│                           ↓                                      │
│  Stage 2: Qubit Mapping                                         │
│  ┌───────────────────────────────────────────────────────┐     │
│  │  Logical qubit 0 → Physical qubit 7                    │     │
│  │  Logical qubit 1 → Physical qubit 12                   │     │
│  │  Consider connectivity constraints                      │     │
│  └───────────────────────────────────────────────────────┘     │
│                           ↓                                      │
│  Stage 3: Routing (SWAP insertion)                              │
│  ┌───────────────────────────────────────────────────────┐     │
│  │  CNOT(0,3) not connected → Insert SWAP(7,8)            │     │
│  │  Minimize total SWAP count                              │     │
│  └───────────────────────────────────────────────────────┘     │
│                           ↓                                      │
│  Stage 4: Optimization                                          │
│  ┌───────────────────────────────────────────────────────┐     │
│  │  Cancel adjacent gates: Rz(θ) Rz(φ) → Rz(θ+φ)          │     │
│  │  Commute gates to reduce depth                          │     │
│  │  Template matching for known patterns                   │     │
│  └───────────────────────────────────────────────────────┘     │
│                           ↓                                      │
│  Stage 5: Scheduling                                            │
│  ┌───────────────────────────────────────────────────────┐     │
│  │  Parallelize independent operations                     │     │
│  │  Respect timing constraints                             │     │
│  └───────────────────────────────────────────────────────┘     │
│                           ↓                                      │
│  Output: Hardware-Native Circuit                                │
│  ┌───────────────────────────────────────────────────────┐     │
│  │  √X(7), Rz(0.5, 7), CNOT(7,8), ...                     │     │
│  └───────────────────────────────────────────────────────┘     │
│                                                                  │
└─────────────────────────────────────────────────────────────────┘

1.2 Why Optimization Matters

Circuit Depth vs Fidelity:

Gate error rate: ε = 0.5%
Depth d circuit fidelity: F ≈ (1 - ε)^(d × n_avg)

Example (100 qubits):
Depth 100: F ≈ (0.995)^(100×50) ≈ 0.08 (8%)
Depth 50:  F ≈ (0.995)^(50×50)  ≈ 0.29 (29%)
Depth 20:  F ≈ (0.995)^(20×50)  ≈ 0.61 (61%)

Reducing depth by 2× can improve fidelity by 3.5×

SWAP Overhead:

Each SWAP = 3 CNOTs
CNOT error ≈ 0.5%
SWAP error ≈ 1.5%

Adding 10 SWAPs: (0.985)^10 ≈ 0.86 (14% fidelity loss)

1.3 Limitations of Classical Compilers

Key Insight: These are exactly the problems where ML excels—learning patterns from data to approximate optimal solutions.

2. Reinforcement Learning for Gate Synthesis

2.1 The Problem

Goal: Decompose arbitrary unitary U into sequence of native gates with minimum gate count.

Naive approach: Use fixed decomposition formulas.
Optimal approach: Search over all possible gate sequences (exponential).
ML approach: Learn policy that generates near-optimal sequences.

2.2 RL Formulation

class GateSynthesisEnv:
    def __init__(self, target_unitary, native_gates, max_depth=50):
        self.target = target_unitary
        self.native_gates = native_gates  # e.g., [Rz, √X, CNOT]
        self.max_depth = max_depth

        self.current_unitary = np.eye(2**n)  # Identity
        self.gates_applied = []

    def step(self, action):
        """Apply a native gate."""
        gate = self.native_gates[action.gate_type]
        params = action.params  # e.g., rotation angle

        # Apply gate to current unitary
        gate_matrix = gate.matrix(params)
        self.current_unitary = gate_matrix @ self.current_unitary
        self.gates_applied.append((gate, params))

        # Compute fidelity with target
        fidelity = self.compute_fidelity(self.current_unitary, self.target)

        # Reward: fidelity bonus - depth penalty
        reward = fidelity - 0.01 * len(self.gates_applied)

        # Done if fidelity > threshold or max depth reached
        done = fidelity > 0.9999 or len(self.gates_applied) >= self.max_depth

        return self.get_state(), reward, done, {'fidelity': fidelity}

    def compute_fidelity(self, U, V):
        """Process fidelity between unitaries."""
        d = U.shape[0]
        return np.abs(np.trace(U.conj().T @ V))**2 / d**2

    def get_state(self):
        """State: current unitary + target unitary."""
        return np.concatenate([
            self.current_unitary.flatten(),
            self.target.flatten()
        ])


class PPOGateSynthesizer:
    def __init__(self, native_gates):
        self.native_gates = native_gates

        # Policy network: state → gate distribution + parameters
        self.policy = nn.Sequential(
            nn.Linear(state_dim, 256),
            nn.ReLU(),
            nn.Linear(256, 256),
            nn.ReLU(),
            nn.Linear(256, num_gate_types + num_params)
        )

    def synthesize(self, target_unitary):
        """Synthesize gate sequence for target unitary."""
        env = GateSynthesisEnv(target_unitary, self.native_gates)
        state = env.reset()

        gates = []
        while True:
            # Get action from policy
            action = self.get_action(state)

            # Take step
            state, reward, done, info = env.step(action)
            gates.append(action)

            if done:
                break

        return gates, info['fidelity']

2.3 Results: Shorter Circuits

Key Paper: Khatri et al. (2019) "Quantum-assisted quantum compiling"

3. Graph Neural Networks for Qubit Mapping

3.1 The Mapping Problem

Input:

• Circuit DAG (dependency graph of gates)
• Hardware topology (connectivity graph)

Output:

• Initial mapping: logical qubit → physical qubit
• SWAP routing to implement all gates

Complexity: NP-hard (subgraph isomorphism variant)

3.2 GNN Architecture

class QubitMappingGNN(nn.Module):
    def __init__(self, hidden_dim=64):
        super().__init__()

        # Circuit GNN: Process circuit structure
        self.circuit_gnn = nn.ModuleList([
            GCNConv(gate_features, hidden_dim),
            GCNConv(hidden_dim, hidden_dim),
            GCNConv(hidden_dim, hidden_dim)
        ])

        # Hardware GNN: Process hardware topology
        self.hardware_gnn = nn.ModuleList([
            GCNConv(qubit_features, hidden_dim),
            GCNConv(hidden_dim, hidden_dim),
            GCNConv(hidden_dim, hidden_dim)
        ])

        # Matching network: Score logical→physical assignments
        self.match_mlp = nn.Sequential(
            nn.Linear(hidden_dim * 2, hidden_dim),
            nn.ReLU(),
            nn.Linear(hidden_dim, 1)
        )

    def forward(self, circuit_graph, hardware_graph):
        # Embed circuit
        x_circuit = self.embed_circuit(circuit_graph)

        # Embed hardware
        x_hardware = self.embed_hardware(hardware_graph)

        # Compute matching scores
        scores = torch.zeros(n_logical, n_physical)
        for i in range(n_logical):
            for j in range(n_physical):
                pair = torch.cat([x_circuit[i], x_hardware[j]])
                scores[i, j] = self.match_mlp(pair)

        # Hungarian algorithm for optimal assignment
        mapping = hungarian_algorithm(scores)

        return mapping

    def embed_circuit(self, graph):
        x = graph.x
        edge_index = graph.edge_index

        for layer in self.circuit_gnn:
            x = F.relu(layer(x, edge_index))

        return x

    def embed_hardware(self, graph):
        x = graph.x
        edge_index = graph.edge_index

        for layer in self.hardware_gnn:
            x = F.relu(layer(x, edge_index))

        return x

3.3 Training with Reinforcement Learning

class MappingEnvironment:
    def __init__(self, circuit, hardware_topology):
        self.circuit = circuit
        self.topology = hardware_topology
        self.current_mapping = None
        self.swaps_inserted = 0

    def step(self, action):
        """Action: which pair of qubits to SWAP."""
        if action == 'finish':
            # Check if circuit is executable
            if self.is_executable():
                reward = -self.swaps_inserted  # Minimize SWAPs
                return self.get_state(), reward, True, {}
            else:
                return self.get_state(), -100, True, {}  # Invalid

        # Insert SWAP
        q1, q2 = action
        self.current_mapping = self.apply_swap(self.current_mapping, q1, q2)
        self.swaps_inserted += 1

        reward = -1  # Penalty per SWAP
        done = False

        return self.get_state(), reward, done, {}

    def is_executable(self):
        """Check if all 2-qubit gates are on connected qubits."""
        for gate in self.circuit.two_qubit_gates:
            q1, q2 = gate.qubits
            p1, p2 = self.current_mapping[q1], self.current_mapping[q2]
            if not self.topology.are_connected(p1, p2):
                return False
        return True

3.4 Results: Fewer SWAPs

4. Neural Compilers

4.1 End-to-End Learning

Idea: Learn a single model that does all compilation stages.

class NeuralCompiler(nn.Module):
    def __init__(self, vocab_size, hidden_dim=256):
        super().__init__()

        # Encoder: Process input circuit
        self.encoder = nn.TransformerEncoder(
            nn.TransformerEncoderLayer(d_model=hidden_dim, nhead=8),
            num_layers=6
        )

        # Decoder: Generate output circuit
        self.decoder = nn.TransformerDecoder(
            nn.TransformerDecoderLayer(d_model=hidden_dim, nhead=8),
            num_layers=6
        )

        # Embeddings
        self.gate_embedding = nn.Embedding(vocab_size, hidden_dim)
        self.output_projection = nn.Linear(hidden_dim, vocab_size)

    def forward(self, input_circuit, hardware_embedding):
        # Embed input gates
        x = self.gate_embedding(input_circuit)

        # Add hardware context
        x = x + hardware_embedding

        # Encode
        memory = self.encoder(x)

        # Decode (autoregressive)
        output = []
        current = start_token

        for _ in range(max_output_len):
            out = self.decoder(current, memory)
            next_gate = self.output_projection(out[-1])
            output.append(next_gate)
            current = torch.cat([current, next_gate])

            if next_gate == end_token:
                break

        return output

    def compile(self, circuit, hardware):
        """Compile circuit for hardware."""
        input_tokens = self.tokenize(circuit)
        hardware_emb = self.embed_hardware(hardware)

        output_tokens = self.forward(input_tokens, hardware_emb)

        return self.detokenize(output_tokens)

4.2 Training Data Generation

def generate_training_data(num_samples=1_000_000):
    """Generate (input_circuit, optimal_output) pairs."""
    data = []

    for _ in range(num_samples):
        # Generate random circuit
        circuit = random_circuit(n_qubits=random.randint(2, 10),
                                 depth=random.randint(5, 50))

        # Compile with best classical compiler (slow but optimal)
        optimal_output = optimal_compiler(circuit, hardware_topology)

        data.append({
            'input': circuit,
            'output': optimal_output,
            'hardware': hardware_topology
        })

    return data

4.3 Reinforcement Learning Fine-Tuning

class NeuralCompilerRL:
    def __init__(self, pretrained_compiler):
        self.compiler = pretrained_compiler
        self.optimizer = Adam(self.compiler.parameters(), lr=1e-5)

    def fine_tune(self, circuit, hardware):
        """Fine-tune on specific circuit using RL."""
        # Generate multiple compilations
        compilations = [self.compiler.compile(circuit, hardware)
                       for _ in range(10)]

        # Evaluate each on hardware (or simulator)
        fidelities = [simulate_fidelity(c, hardware) for c in compilations]

        # REINFORCE update
        baseline = np.mean(fidelities)
        for comp, fid in zip(compilations, fidelities):
            advantage = fid - baseline
            loss = -advantage * self.compiler.log_prob(comp)
            loss.backward()

        self.optimizer.step()

5. Optimization Passes with ML

5.1 Pattern Matching with Neural Networks

Classical: Manually define optimization rules (tedious, incomplete).
ML: Learn patterns from data.

class NeuralOptimizer(nn.Module):
    def __init__(self, window_size=5):
        super().__init__()
        self.window_size = window_size

        # Pattern recognition
        self.pattern_encoder = nn.Sequential(
            nn.Linear(window_size * gate_features, 128),
            nn.ReLU(),
            nn.Linear(128, 64)
        )

        # Replacement generator
        self.replacement_decoder = nn.Sequential(
            nn.Linear(64, 128),
            nn.ReLU(),
            nn.Linear(128, max_replacement_size * gate_features)
        )

        # Binary classifier: should we apply optimization?
        self.should_optimize = nn.Linear(64, 1)

    def forward(self, gate_window):
        # Encode pattern
        encoding = self.pattern_encoder(gate_window.flatten())

        # Decide whether to optimize
        optimize_prob = torch.sigmoid(self.should_optimize(encoding))

        if optimize_prob > 0.5:
            # Generate replacement
            replacement = self.replacement_decoder(encoding)
            return replacement.reshape(-1, gate_features)
        else:
            return gate_window  # Keep original


class LearnedOptimizationPass:
    def __init__(self):
        self.optimizer = NeuralOptimizer()

    def optimize(self, circuit):
        """Apply learned optimizations to circuit."""
        optimized = []

        i = 0
        while i < len(circuit.gates):
            # Get window
            window = circuit.gates[i:i+self.optimizer.window_size]

            # Apply neural optimizer
            replacement = self.optimizer(window)

            # If replacement is shorter, use it
            if len(replacement) < len(window):
                optimized.extend(replacement)
                i += len(window)
            else:
                optimized.append(circuit.gates[i])
                i += 1

        return Circuit(optimized)

5.2 Gate Commutation Learning

Problem: Which gates can be reordered to reduce depth?

class CommutationPredictor(nn.Module):
    def __init__(self):
        super().__init__()

        self.encoder = nn.Sequential(
            nn.Linear(gate_features * 2, 64),
            nn.ReLU(),
            nn.Linear(64, 32),
            nn.ReLU(),
            nn.Linear(32, 1)
        )

    def forward(self, gate1, gate2):
        """Predict if gate1 and gate2 commute."""
        x = torch.cat([gate1, gate2])
        return torch.sigmoid(self.encoder(x))


class DepthOptimizer:
    def __init__(self):
        self.commutation_predictor = CommutationPredictor()

    def minimize_depth(self, circuit):
        """Reorder gates to minimize circuit depth."""
        gates = circuit.gates.copy()

        # Bubble sort with learned commutation
        for _ in range(len(gates)):
            for i in range(len(gates) - 1):
                if self.commutation_predictor(gates[i], gates[i+1]) > 0.5:
                    if self.reduces_depth(gates, i, i+1):
                        gates[i], gates[i+1] = gates[i+1], gates[i]

        return Circuit(gates)

6. Application to Your Work

6.1 Your Grover's Algorithm

From your quantum_rag.py:

class QuantumRAG:
    # Implements Grover's algorithm for document search
    # Oracle + diffusion operator

6.2 Proposed Optimization

class OptimizedQuantumRAG:
    def __init__(self, base_rag, neural_compiler):
        self.rag = base_rag
        self.compiler = neural_compiler

    def build_optimized_grover(self, oracle):
        """Build Grover circuit with ML optimization."""
        # Build standard Grover circuit
        circuit = self.rag.build_grover_circuit(oracle)

        # Apply neural compiler
        optimized = self.compiler.compile(circuit, self.hardware)

        # Report improvement
        original_depth = circuit.depth()
        optimized_depth = optimized.depth()
        print(f"Depth reduction: {original_depth} → {optimized_depth} "
              f"({100*(1-optimized_depth/original_depth):.1f}% improvement)")

        return optimized

    def optimize_oracle(self, oracle_circuit):
        """Apply RL gate synthesis to oracle."""
        synthesizer = PPOGateSynthesizer(native_gates=['Rz', '√X', 'CNOT'])

        # Get oracle unitary
        oracle_unitary = oracle_circuit.to_unitary()

        # Synthesize with fewer gates
        optimized_gates, fidelity = synthesizer.synthesize(oracle_unitary)

        print(f"Oracle optimization: {len(oracle_circuit.gates)} → "
              f"{len(optimized_gates)} gates (fidelity: {fidelity:.4f})")

        return optimized_gates

6.3 Integration Example

# In your quantum_rag_demo/

from ai_quantum_symbiosis.neural_compiler import NeuralCompiler
from ai_quantum_symbiosis.gate_synthesizer import PPOGateSynthesizer

class AIOptimizedQuantumRAG:
    def __init__(self):
        self.compiler = NeuralCompiler.load_pretrained()
        self.synthesizer = PPOGateSynthesizer()
        self.base_rag = QuantumRAG()

    def search(self, query, documents):
        # Build oracle for query
        oracle = self.base_rag.build_oracle(query, documents)

        # Optimize oracle with RL
        optimized_oracle = self.synthesizer.synthesize(oracle)

        # Build full Grover circuit
        circuit = self.base_rag.build_grover(optimized_oracle)

        # Compile for hardware
        compiled = self.compiler.compile(circuit, ibm_hardware)

        # Execute
        result = execute_on_hardware(compiled)

        return result

7. Benchmarks and Results

7.1 Compilation Quality

7.2 Impact on Fidelity

Key Insight: 25-35% depth reduction → 30-50% fidelity improvement.

8. Future Directions

8.1 Open Research Questions

1. Generalization: Can neural compilers generalize to unseen circuit types?
2. Scalability: How do they scale to 1000+ qubit circuits?
3. Hardware adaptation: Can one model work for multiple hardware backends?
4. Verification: How do we verify compiled circuits are correct?
5. Real-time: Can compilation happen fast enough for variational algorithms?

8.2 Industry Adoption

References

1. Khatri, S., et al. (2019). "Quantum-assisted quantum compiling." Quantum, 3, 140.

2. Li, G., et al. (2019). "Tackling the qubit mapping problem for NISQ-era quantum devices." ASPLOS, 1001-1014.

3. Zhu, E. Y., et al. (2020). "An adaptive optimizer for measurement-frugal variational algorithms." Quantum, 4, 263.

4. Cincio, L., et al. (2018). "Learning the quantum algorithm for state overlap." New Journal of Physics, 20(11), 113022.

5. Fösel, T., et al. (2021). "Quantum circuit optimization with deep reinforcement learning." arXiv:2103.07585.

6. Murali, P., et al. (2019). "Noise-adaptive compiler mappings for noisy intermediate-scale quantum computers." ASPLOS, 1015-1029.

7. Venturelli, D., et al. (2018). "Compiling quantum circuits to realistic hardware architectures using temporal planners." Quantum Science and Technology, 3(2), 025004.

8. Nash, B., et al. (2020). "Quantum circuit optimizations for NISQ architectures." Quantum Science and Technology, 5(2), 025010.

Document: 04_ai_circuit_optimization.md
Part of: AI-Quantum Symbiosis Research Series
Location: Documents/Learning/ai_quantum_symbiosis/