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The Phase Quantum Walk: A Unified Framework for Graph State Distribution in Quantum Networks

Soumyojyoti Dutta

Abstract

Distributing arbitrary graph states across quantum networks is a central challenge for modular quantum computing and measurement-based quantum communication. I introduce the phase quantum walk (PQW), a discrete-time quantum walk in which the conventional position-permuting shift operator is replaced by a diagonal conditional phase (CZ) gate, enabling distribution of arbitrary graph states, not merely GHZ states, from elementary two-qubit resources. The Byproduct Lemma shows that each walk step teleports edge entanglement with a correctable Pauli byproduct; the Coin Invariance Theorem proves that the optimal fidelity F*(C,E) = F*(H,E) for all unitary coins C and noise channels E, with closed-form expressions F_dep = (1 - 3p/4)^k and F_pd = ((1 + sqrt(1 - p))/2)^k. Analytical correction formulas are derived for tree graphs (general theorem) and ring graphs (C4 case study), with F = 1.0 verified across eight topologies (up to 4096 outcomes). Hardware validation on ibm marrakesh (IBM Heron r2, CZ-native) yields F_cl = 0.924 for |GHZ4> and 0.922 for |L4>, statistically identical, providing the first experimental confirmation that fidelity is independent of graph topology as predicted by the Coin Invariance Theorem.

The Phase Quantum Walk: A Unified Framework for Graph State Distribution in Quantum Networks

Abstract

Distributing arbitrary graph states across quantum networks is a central challenge for modular quantum computing and measurement-based quantum communication. I introduce the phase quantum walk (PQW), a discrete-time quantum walk in which the conventional position-permuting shift operator is replaced by a diagonal conditional phase (CZ) gate, enabling distribution of arbitrary graph states, not merely GHZ states, from elementary two-qubit resources. The Byproduct Lemma shows that each walk step teleports edge entanglement with a correctable Pauli byproduct; the Coin Invariance Theorem proves that the optimal fidelity F*(C,E) = F*(H,E) for all unitary coins C and noise channels E, with closed-form expressions F_dep = (1 - 3p/4)^k and F_pd = ((1 + sqrt(1 - p))/2)^k. Analytical correction formulas are derived for tree graphs (general theorem) and ring graphs (C4 case study), with F = 1.0 verified across eight topologies (up to 4096 outcomes). Hardware validation on ibm marrakesh (IBM Heron r2, CZ-native) yields F_cl = 0.924 for |GHZ4> and 0.922 for |L4>, statistically identical, providing the first experimental confirmation that fidelity is independent of graph topology as predicted by the Coin Invariance Theorem.

Paper Structure

This paper contains 30 sections, 14 theorems, 24 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Graph States and Stabiliser Formalism
  4. Standard DTQW and its Limitations for Graph State Distribution
  5. The Phase Quantum Walk
  6. Is the phase quantum walk a quantum walk?
  7. The $\boldsymbol{\ket{L_4}}$ Distribution Protocol
  8. Setup and Circuit
  9. Protocol
  10. Correction Formula
  11. Resource scaling.
  12. Verification
  13. Coin Invariance
  14. Numerical verification.
  15. Noisy Performance
  16. ...and 15 more sections

Key Result

Proposition 4

$\mathrm{CZ}$ is diagonal, and $\blacktriangleleft$$\blacktriangleleft$

Figures (7)

  • Figure 1: Qiskit circuit for the $\ket{L_4}$ distribution protocol (10 qubits). Barriers separate the four stages: (S1) Hadamard on data qubits, (S2) resource state preparation, (S3) phase walk CZ gates, (S4) Hadamard on resource qubits before measurement.
  • Figure 2: Fidelity with $\ket{L_4}$ for each of the 64 measurement outcomes after applying corrections of Thm. \ref{['thm:correction']}. All 64 bars coincide with $F=1.0$ (red dashed). Equal outcome probabilities ($1/64$ each) confirm uniform distribution.
  • Figure 3: Optimal fidelity $F^*(p)$ for the $\ket{L_4}$ protocol under independent noise per resource qubit: depolarising (red, analytical), phase damping (blue, analytical), amplitude damping (green, numerical). Phase damping is least destructive due to $Z$-transparency of $\mathrm{CZ}$.
  • Figure 4: Comparison of $F^*(p)$ under depolarising noise for Bell (analytical $1-3p/4$), GHZ$_4$ (analytical $(1-3p/4)^2$), and $\ket{L_4}$ (6 resource qubits). All are 4-qubit output states; $\ket{L_4}$ and GHZ$_4$ use identical resources.
  • Figure 5: Heavy-hex qubit connectivity graph of ibm_marrakesh (156 qubits). Blue nodes are operational qubits; purple and white nodes indicate qubits with elevated error rates ($>5\times$ median CZ error). Grey edges mark high-error CZ pairs (worst: Q40--Q41 at $13.9\%$, Q95--Q99 at $10.2\%$). The Qiskit transpiler automatically routes circuits away from these defective pairs. Each qubit connects to at most 3 neighbours, requiring SWAP insertion for non-adjacent qubit interactions.
  • ...and 2 more figures

Theorems & Definitions (34)

  • Definition 1: Graph state hein2004multiparty
  • Definition 2: Phase Quantum Walk
  • Remark 3
  • Proposition 4: Diagonality and $Z$-error transparency
  • proof
  • Proposition 5: Symmetry
  • proof
  • Proposition 6: $X$-basis equivalence
  • proof
  • Proposition 7: Graph state output
  • ...and 24 more