[Paper Review] QuDot Nets: Quantum Computers and Bayesian Networks

Perry Sakkaris|arXiv (Cornell University)|Jul 26, 2016

Bayesian Modeling and Causal Inference3 citations

TL;DR

This paper introduces QuDot Nets, a graph-theoretic framework that models quantum computation using Bayesian Networks, representing qubit states as nodes and quantum gates as edge-weight transformations. It achieves linear scaling for single-qubit gates and measurements, demonstrating efficient simulation of the Quantum Fourier Transform and suggesting a path toward classical implementation of quantum computation if path explosion in coherent gates can be controlled.

ABSTRACT

We present a new implementation of quantum computation that treats quantum computers as a special type of Bayesian Network called a QuDot Net. QuDot Nets allow for the efficient representation of some qubit systems. Single qubit quantum gates can be implemented as edge transformations on QuDot Nets. The X, H, R(k), M and SWAP gates are discussed in detail and results show linear scaling as the number of qubits are increased. We show that measurement and semi-quantum control gates can be efficiently implemented using QuDot Nets and present results from a QuDot Net implementation of the terminal Quantum Fourier Transform. We show how QuDot Nets can implement coherent control gates using multi-digraphs by labelling parallel edges. Lastly, we discuss implications to quantum foundations if a classical implementation of quantum computation is realized.

Motivation & Objective

• To develop a classical, graph-based representation of quantum computation that avoids the exponential resource cost of amplitude vector representations.
• To demonstrate that single-qubit quantum gates and measurements can be implemented in linear time using edge-weight transformations on a directed, weighted multi-digraph.
• To explore whether coherent two-qubit control gates can be efficiently represented and computed within this framework, addressing the key challenge of exponential path growth.
• To investigate the foundational implications if quantum computation can be efficiently simulated on classical hardware, particularly regarding BPP = BQP and Einstein’s view that 'God does not play dice'.

Proposed method

• Represent n-qubit states as a directed, acyclic, weighted multi-digraph (QuDot Net), where nodes represent qubit states |0⟩ and |1⟩, and edges represent tensor products with complex amplitudes.
• Implement single-qubit gates (X, H, R(k), M, SWAP) as transformations of edge weights, enabling linear-time gate application instead of O(2^n × 2^n) matrix operations.
• Model measurement as traversal of all possible paths through the graph, where the product of edge weights along a path gives the amplitude of the corresponding computational basis state.
• Use labeled parallel edges (multi-digraph structure) to represent coherent two-qubit control gates, enabling representation of entanglement and unitary evolution.
• Apply the framework to simulate the Quantum Fourier Transform, showing consistent results with theoretical expectations across multiple test states.
• Analyze entanglement as a graph coloring problem and explore the possibility of path merging to control exponential growth in computational complexity.

Experimental results

Research questions

• RQ1 Can quantum computation be efficiently simulated on classical hardware using a graph-theoretic model based on Bayesian Networks?
• RQ2 Can the exponential growth of computational paths in QuDot Nets be controlled or approximated using known techniques from NP-complete graph problems?
• RQ3 Does the QuDot Net framework enable a classical implementation of BQP, implying BPP = BQP and supporting Einstein’s view that quantum mechanics is fundamentally deterministic?
• RQ4 Can coherent two-qubit control gates be efficiently represented and computed in QuDot Nets without exponential blowup in the number of edges?
• RQ5 Can the framework provide new insights into the foundational question of whether quantum randomness is intrinsic or reducible to classical probabilistic computation?

Key findings

• QuDot Nets represent n-qubit states using O(n) nodes and edges, achieving linear scaling compared to the O(2^n) resources required by standard amplitude vector representations.
• Single-qubit gates (X, H, R(k), M, SWAP) are implemented via edge-weight transformations, enabling linear-time computation instead of matrix multiplication.
• The Quantum Fourier Transform was successfully simulated using QuDot Nets, with measurement probabilities matching theoretical predictions (e.g., |qft12⟩ showed four outcomes each with ~0.25 probability).
• Measurement outcomes correspond to all possible traversals of the graph, with amplitude products matching quantum mechanical probabilities, confirming consistency with standard quantum mechanics.
• Coherent control gates require multi-digraphs with labeled parallel edges, but exponential path growth remains a challenge, with path merging efficiency still unresolved.
• If path explosion can be controlled in bounded-error probabilistic polynomial time, this would support BPP = BQP, implying quantum randomness could be de-randomized and confirming Einstein’s belief that nature does not play dice.