Random unitary matrices sampled from the uniform Haar ensemble have a number of important applications both in cryptography and in the simulation of a variety of fundamental physical systems. Since the Haar ensemble is very expensive to sample, pseudorandom ensembles in the form of t-designs are frequently used as an efficient substitute, and are sufficient for most applications. We investigate t-designs generated using a measurement-based approach on superconducting quantum computers. In particular, we implemented an exact single-qubit 3-design on IBM quantum processors by performing measurements on a 6-qubit graph state. By analysing channel tomography results, we were able to show that the ensemble of unitaries realised was a 1-design, but not a 2-design or a 3-design under the test conditions set, which we show to be a result of depolarising noise during the measurement-based process. We obtained improved results for the 2-design test by implementing an approximate 2-design, in which measurements were performed on a smaller 5-qubit graph state, but the test still did not pass for all states. This suggests that the practical realisation of measurement-based t-designs on superconducting quantum computers will require further work on the reduction of depolarising noise in these devices.

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.

Quantum computers have the potential to outperform classical computers in a range of computational tasks, such as prime factorisation and unstructured searching. However, real-world quantum computers are subject to noise. Quantifying noise is of vital importance, since it is often the dominant factor preventing the successful realisation of advanced quantum computations. Here we propose and demonstrate an interleaved randomised benchmarking protocol for measurement-based quantum computers that can be used to estimate the fidelity of any single-qubit measurement-based gate. We tested the protocol on IBM superconducting quantum processors by estimating the fidelity of the Hadamard and T gates — a universal single-qubit gate set. Measurements were performed on entangled cluster states of up to 31 qubits. Our estimated gate fidelities show good agreement with those calculated from quantum process tomography. By artificially increasing noise, we were able to show that our protocol detects large noise variations in different implementations of a gate.

Quantum computers have the potential to outperform classical computers at certain computational tasks, such as prime factorisation and unstructured searching. However, experimental realisations of quantum computers are subject to noise. Quantifying the noise is of fundamental importance, since noise is often the dominant factor preventing the successful realisation of advanced quantum computations. Here we propose an interleaved randomised benchmarking protocol for measurement-based quantum computers, in which any single-qubit measurement-based 2-design can be used to estimate the fidelity of any single-qubit measurement-based gate. We test our protocol by using a weak approximate measurement-based 2-design to estimate the fidelity of the Hadamard gate and the T gate (a universal single-qubit set) on IBM superconducting quantum computers. To this end, single-qubit measurements were performed on entangled linear cluster states of up to 31 qubits. Our estimated gate fidelities show good agreement with gate fidelities calculated from process tomography results. Furthermore, by artificially increasing noise in the measurement-based gates, we were able to show that our protocol is able to detect large noise variations in different measurement-based implementations of a gate.