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The universal quantum computation model based on quantum walk by Childs hasopened the door for a new way of studying the limitations and advantages ofquantum computation, as well as for its intermediate-term simulation. In recentyears, the growing interest in noisy intermediate-scale quantum computers(NISQ) has lead to intense efforts being directed at understanding thecomputational advantages of open quantum systems. In this work, we extend thequantum walk model to open noisy systems in order to provide such a tool forthe study of NISQ computers. Our method does not use explicit purification, andallows to ignore the environment degrees of freedom and obtain a much moreefficient implementation (linear rather than exponential in the runtime), whichemploys no ancillas hence provides direct access to the entanglement propertiesof the system. In our scheme, the quantum walk amplitudes represent elements ofthe density matrix rather than the wavefunction of a pure state. Despite thenon-trivial manifestation of the normalization requirement in this setting, wemodel the application of general unitary gates and nonunitary channels, with anexplicit implementation protocol for channels that are commonly used in noisemodels.

Noisy Quantum Computation Modeled by Quantum Walk: Universality without Ancillas