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arXiv:quant-ph/0501140 · Journal of Computational and Theoretical Nanoscience

Simulation of Quantum Computation: A Deterministic Event-Based Approach

Kristel F. L. Michielsen, Koen De Raedt, Hans A. De Raedt

Published Jun 1, 2005• 33 citations
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Abstract

We demonstrate that locally connected networks of machines that have primitive learning capabilities can be used to perform a deterministic, event-based simulation of quantum computation. We present simulation results for basic quantum operations such as the Hadamard and the controlled-NOT gate, and for seven-qubit quantum networks that implement Shor's numbering factoring algorithm.

DOI
10.1166/jctn.2005.106

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We demonstrate that locally connected networks of machines that have primitive learning capabilities can be used to perform a deterministic, event-based simulation of quantum computation. We present simulation results for basic quantum operations such as the Hadamard and the controlled-NOT gate, and for seven-qubit quantum networks that implement Shor's numbering factoring algorithm.
We propose and analyse simple deterministic algorithms that can be used to construct machines that have primitive learning capabilities. We demonstrate that locally connected networks of these machines can be used to perform blind classification on an event-by-event basis, without storing the information of the individual events. We also demonstrate that properly designed networks of these machines exhibit behavior that is usually only attributed to quantum systems. We present networks that simulate quantum interference on an event-by-event basis. In particular we show that by using simple geometry and the learning capabilities of the machines it is possible to simulate single-photon interference in a Mach–Zehnder interferometer. The interference pattern generated by the network of deterministic learning machines is in perfect agreement with the quantum theoretical result for the single-photon Mach–Zehnder interferometer. To illustrate that networks of these machines are indeed capable of simulating quantum interference we simulate, event-by-event, a setup involving two chained Mach–Zehnder interferometers, and demonstrate that also in this case the simulation results agree with quantum theory.
We study a reduced quantum circuit computation paradigm in which the only allowable gates either permute the computational basis states or else apply a ``global Hadamard operation'', \emph{i.e.} apply a Hadamard operation to every qubit simultaneously. In this model, we discuss complexity bounds (lower-bounding the number of global Hadamard operations) for common quantum algorithms~: we illustrate upper bounds for Shor's Algorithm, and prove lower bounds for Grover's Algorithm. We also use our formalism to display a gate that is neither quantum-universal nor classically simulable, on the assumption that Integer Factoring is not in \textbf{BPP}.
Quantum computing is based on the exploitation of quantum effects to perform some important computations more efficiently than ever possible on classical computers. Well known quantum algorithms are Shors polynomial time factoring algorithm and Grovers data base search scheme. Their potential is due to the parallelism of a quantum computation increasing exponentially with the size of the computing device. IBM has succeeded in building a first working quantum computing device with up to 7 qubits based on NMR technology, however, the construction of large multi-qubit systems appears to become extremely difficult due to the loss of coherence of the quantum states. Like the design process of modern microprocessors which is carried out by means of sophisticated software, modelling and simulation of the entire QC device is even more relevant. Simulation is required in order to understand the functioning of quantum algorithms, to construct novel quantum algorithms, to write quantum programs, to teach quantum computing, to construct real quantum computers, to understand and fight de-coherence and to understand and control the influence of the environment. Since quantum parallelism increases exponentially with the size of the quantum computer, parallel supercomputers are required for realistic simulations. I intend to give an introduction into the simulation of quantum computers using conventional computers. The simulation is based on a generic physical model of quantum computer hardware. Issues like parallel gate level simulators, modelling of QCs, simulators for physical quantum computers are addressed.
A medium-scale quantum computer with full universal quantum computing capability is necessary for various practical aims and testing applications. Here we report a 34-qubit quantum virtual machine (QtVM) based on a medium server. Our QtVM can run quantum assembly language with graphic interfaces. The QtVM is implemented with single qubit rotation gate, single to multiple controlled NOT gates to realize the universal quantum computation. Remarkably, it can realize a series of basic functions, such as, the "if" conditional programming language based on single-shot projective measurement results, "for" iteration programming language, build in arithmetic calculation. The measurement can be single-shot and arbitrary number of multi-shot types. In addition, there is in principle no limitation on number of logic gates implemented for quantum computation. By using QtVM, we demonstrate the simulation of dynamical quantum phase transition of transverse field Ising model by quantum circuits, where 34 qubits with one million gates are realized. We also show the realization of programmable Shor algorithm for factoring 15 and 35.