distributed-iqft-communication

name	distributed-iqft-communication
description	Communication-efficient distributed Inverse Quantum Fourier Transform (iQFT) using communication horizon pruning to reduce inter-node quantum communication from O(P^2) to O(P). Use when: distributed quantum computing, iQFT across quantum networks, quantum communication optimization, distributed Shor algorithm, scalable QFT implementation, or quantum network resource allocation.

Communication-Efficient Distributed Inverse Quantum Fourier Transform

Distributed iQFT implementation over quantum networks with communication horizon pruning to reduce inter-node communication complexity from O(P^2) to O(P) (arXiv: 2605.10710).

Core Problem

Standard iQFT requires O(n^2) gates for n qubits. When qubits are distributed across P nodes (Q qubits each, n = P*Q), non-local controlled-phase rotations between distant nodes incur expensive inter-node quantum communication (e.g., teleportation). This scales as O(P^2) entanglement resources.

Communication Horizon Strategy

Controlled-phase rotation angles in iQFT decrease exponentially with qubit distance: CR_k has angle 2pi/2^k. Remote controlled-phase gates with small angles contribute negligibly to the final state and can be safely pruned.

Mathematical Foundation

The iQFT on n qubits applies:

|j> -> 1/sqrt(2^n) sum_{k=0}^{2^n-1} exp(2pi*i*j*k/2^n) |k>

Circuit decomposition uses Hadamard + controlled-phase gates:

H on qubit i, then CR_k from qubit j to i for all j > i, k = j - i + 1

For controlled-phase gate CR_k, the rotation angle is 2pi/2^k. When k is large:

• Phase rotation becomes negligible
• Gate impact on fidelity decays exponentially
• Can be safely omitted with bounded error

Pruning Threshold

Define communication horizon h:

h = ceil(-log2(epsilon / (n * sqrt(2))))

For target error epsilon, prune all inter-node CR gates where the phase angle < epsilon/(n*sqrt(2)). This bounds the worst-case error.

Distributed iQFT Algorithm

Step 1: Partition Qubits

• n qubits partitioned across P nodes
• Each node hosts Q = n/P qubits (assume divisible for simplicity)
• Logical qubit i maps to node floor(i/Q), local index i mod Q

Step 2: Local iQFT

• Each node performs local iQFT on its Q qubits
• Only involves intra-node gates (no communication)
• Applies H + local CR gates within each node

Step 3: Inter-node Controlled-Phase Gates

• For each pair of qubits (i, j) on different nodes:
  • Apply CR_{|i-j|+1} gate
  • Requires quantum communication (teleportation or entanglement swapping)
  • Total: O(P^2 * Q^2) inter-node gates without pruning

Step 4: Communication Horizon Pruning

• Define horizon h based on acceptable error epsilon
• Prune CR gates where controlled-phase angle < threshold
• Remote qubits beyond horizon h are skipped
• Remaining gates: O(P * min(h, n)) per node
• Global complexity: O(P * min(h, n)) = O(P) when h is constant

Complexity Analysis

Metric	Standard	With Horizon
Gates	O(n^2)	O(n * min(h, n))
Inter-node	O(P^2 * Q^2)	O(P * min(h*Q, n))
Entanglement/node	O(P*Q)	O(min(h*Q, n))
Global comm	O(P^2)	O(P) (h constant)

Error Bounds

Total error from pruning is bounded by:

||U - U_pruned|| <= n * sqrt(2) * max_theta

Where max_theta is the largest pruned rotation angle. For horizon h:

max_theta = 2pi / 2^h

Choose h to satisfy desired fidelity threshold.

Implementation Pattern

def distributed_iqft(n_qubits, p_nodes, epsilon=1e-6):
    q_per_node = n_qubits // p_nodes
    h = compute_horizon(epsilon, n_qubits)
    
    for node in range(p_nodes):
        # Local iQFT
        local_iqft(node, q_per_node)
        
        # Inter-node gates with horizon
        for qubit_i in node_qubits(node):
            for qubit_j in all_qubits():
                if same_node(qubit_i, qubit_j):
                    continue
                distance = abs(qubit_i - qubit_j) + 1
                if distance > h:
                    continue  # Prune: negligible contribution
                apply_controlled_phase(qubit_i, qubit_j, distance)
    
    return circuit

When to Use

• Distributed quantum computing architectures
• Quantum networks with multiple QPUs
• Scalable Shor algorithm implementation
• Resource-constrained quantum communication
• Large-scale quantum Fourier transform
• Cross-node quantum algorithm compilation

Key Advantages

1. Linear scaling: Communication O(P) instead of O(P^2)
2. Bounded error: Fidelity guarantee from pruning threshold
3. Hardware-agnostic: Works with any distributed quantum architecture
4. Tunable: Trade accuracy for communication cost via horizon parameter
5. Composable: iQFT is a building block for Shor, HHL, and other algorithms

Related Algorithms

• Distributed Shor: iQFT is the final step; communication savings apply
• Distributed HHL: Uses QFT subroutine for phase estimation
• Quantum phase estimation: Core application of QFT/iQFT
• Quantum teleportation: Mechanism for inter-node gate execution

References

• arXiv: 2605.10710 - Communication-Efficient Distributed Inverse Quantum Fourier Transform
• Authors: F. Javier Cardama, Jorge Vazquez-Perez, Tomas F. Pena, Andres Gomez