On the freedom in representing quantum operations

Abstract

We discuss the effects of a gauge freedom in representing quantum information processing devices, and its implications for characterizing these devices. We demonstrate with experimentally relevant examples that there exists equally valid descriptions of the same experiment which distribute errors differently among objects in a gate-set, leading to different error rates. Consequently, it can be misleading to attach a concrete operational meaning to figures of merit for individual gate-set elements. We propose an alternative operational figure of merit for a gate-set, the mean variation error, and a protocol for measuring this figure.

Gate Set Tomography

“…15 form a continuous group, so the standard representation of gate sets described above contains entire "orbits" of equivalent gate sets. This degeneracy, referred to generically as "gauge freedom", is a perennial irritant and obstacle to characterization of quantum processors [19,44,94,127]. More importantly, it shows that a gate set is not just the sum of its parts, and that tomography on a gate set is not just tomography on its components.…”

“…The most powerful and comprehensive technique for this task is currently gate set tomography (GST). Developed around 2012-13, GST has been used in a variety of experiments [14,19,42,72,82,85,128,153,167,176], discussed in the literature [25,49,87,93,104,124,130,137,139,142,146,170], and implemented in a large open-source software project [120,121]. But no comprehensive explanation of the theory behind GST has appeared in the literature.…”

Compressive gate set tomography

“…So far we have not made explicit what 'finding a device description' actually means. What is well studied in the GST and RB literature [6,12,14,[68][69][70], is that without additional prior assumptions, there is a freedom in representing a device in the gate set model. In particular, this freedom needs to be considered when defining a metric for gate sets [68] w.r.t.…”

“…1 Meaningful distance measures for gate sets should have the same gauge freedom as the GST data. The problem of finding gauge invariant distance has been studied by Lin et al [68]. For individual gate sequences, any measure that compares only the ideal and observed outcome probabilities is naturally gauge invariant.…”

Modeling Quantum Devices and the Reconstruction of Physics in Practical Systems

“…Given that unique models are rare, developing quantum technologies independent of the gauge freedom is important, e.g. quantum error mitigation based on the gate set tomography [26] and gauge-invariant measure of the gate error [27]. A ray is a set of vectors in the Hilbert space representing the same physical state, i.e.…”