Investigating the effect of noise channels on the quality of unitary t-designs

Abstract

Unitary t-designs have a wide variety of applications in quantum information theory, such as quantum data encryption and randomised benchmarking. However, experimental realisations of t-designs are subject to noise. Here we investigate the effect of noise channels on the quality of single-qubit t-designs. The noise channels we study are bit flips, phase flips, bit and phase flips, phase damping, amplitude damping and depolarising noise. We consider two noise models: the first has noise applied before the t-design unitary operations, while the second has noise applied after the unitary operations. We show that for single qubits the 1-design is completely unaffected by an arbitrary noise channel, for both noise models, while numeric results obtained for the 2-, 3-, 4and 5-design suggest that a 2t-design is significantly more sensitive to noise than a (2t − 1)-design and that, with the exception of amplitude damping, a (2t + 1)-design is as sensitive to noise as a 2t-design. Numeric results also reveal substantial variations in sensitivity to noise throughout the Bloch sphere. In particular, t-designs appear to be most sensitive to noise when acting on pure states and least sensitive to noise for the maximally mixed state. For depolarising noise, we show that our two noise models are equivalent, and for the other noise channels, numeric results suggest that applying a noise channel after the unitary operations effectively transforms the channel into a depolarising channel, an effect exploited in randomised benchmarking with 2-designs.

Measurement-based interleaved randomised benchmarking using IBM processors

“…Randomised benchmarking relies heavily on a unitary 2-design's ability to transform an arbitrary noise channel into a depolarising channel [29,30,38], a property which was studied extensively in our previous work [71]. In interleaved randomised benchmarking, two experiments are performed, namely an experiment to determine the reference depolarising noise parameter p ref and an experiment to determine the interleaved depolarising noise parameter p int .…”

Measurement-based interleaved randomised benchmarking using IBM processors

“…Randomised benchmarking relies heavily on a unitary 2-design's ability to transform an arbitrary noise channel into a depolarising channel [29,30,38], a property which was studied extensively in our previous work [69]. In interleaved randomised benchmarking, two experiments are performed, namely an experiment to determine the reference depolarising noise parameter p ref and an experiment to determine the interleaved depolarising noise parameter p int .…”