Testing the context-independence of quantum gates

The actual gate performed on, say, a qubit in a quantum computer may depend, not just on the actual laser pulses and voltages we programmed to implement the gate, but on its {\em context} as well. For example, it may depend on what gate has just been applied to the same qubit, or on how much a long series of previous laser pulses has been heating up the qubit's environment. This paper analyzes several tests to detect such context-dependent errors (which include various types of non-Markovian errors). A key feature of these tests is that they are robust against both state preparation and measurement (SPAM) errors and gate-dependent errors. Since context-dependent errors are expected to be small in practice, it becomes important to carefully analyze the effects of statistical fluctuations and so we investigate the power and precision of our tests as functions of the number of repetitions and the length of the sequences of gates. From our tests an important quantity emerges: the logarithm of the determinant (log-det) of a probability (relative frequency) matrix $\mathcal{P}.$ For this reason, we derive the probability distribution of the log-det estimates which we then use to examine the performance of our tests for various single- and two-qubit sets of measurements and initial states. Finally, we emphasize the connection between the log-det and the degree of reversibility (the unitarity) of a context-independent operation.
Comment: To appear on QIC Vol.20 No.15&16 pp1304-1353