A Multilayer Network Approach to Quantum Computing

Abstract

The circuit model of quantum computation is reformulated as a multilayer network theory [3] called a Quantum Multiverse Network (QuMvN). The QuMvN formulation allows us to interpret the quantum wave function as a combination of ergodic Markov Chains where each Markov Chain occupies a different layer in the QuMvN structure. Layers of a QuMvN are separable components of the corresponding wave function. Single qubit measurement is defined as a state transition of the Markov Chain that emits either a $0$ or $1$ making each layer of the QuMvN a Discrete Information Source. A message is equivalent to a possible measurement outcome and the message length is the number of qubits. Therefore, the quantum wave function can be treated as a combination of multiple discrete information sources analogous to what Shannon called a "mixed" information source [18]. We show the QuMvN model has significant advantages in the classical simulation of some quantum circuits by implementing quantum gates as edge transformations on the QuMvNs. We implement a quantum virtual machine capable of simulating quantum circuits using the QuMvN model and use our implementation to classically simulate Shor's Algorithm [19]. We present results from multiple simulations of Shor's Algorithm culminating in a $70$ qubit simulation of Shor's Algorithm on a commodity cloud server with $96$ CPUS and $624$GB of RAM. Lastly, the source of quantum speedups is discussed in the context of layers in the QuMvN framework and how randomized algorithms can push the quantum supremacy boundary.