Theory of Quantum System Certification

The precise control of complex quantum systems promises numerous technological applications including digital quantum computing. The complexity of such devices renders the certification of their correct functioning a challenge. To address this challenge, numerous methods were developed in the last decade. In this tutorial, we explain prominent protocols for certifying the physical layer of quantum devices described by quantum states and processes. Such protocols are particularly important in the development of near-term devices. Specifically, we discuss methods of direct quantum-state certification, direct-fidelity estimation, shadow-fidelity estimation, direct quantum-process certification, randomized benchmarking, and cross-entropy benchmarking. Moreover, we provide an introduction to powerful mathematical methods, which are widely used in quantum-information theory, in order to derive theoretical guarantees for the protocols.

Popular Summary

The precise control of complex quantum systems promises numerous technological applications. Prominently quantum-computing devices are envisioned to efficiently solve problems that are intractable on classical computers. Potential applications include the simulation of complex molecules and difficult optimization problems. To achieve these goals, the novel computing devices need to be highly complex quantum systems. Furthermore, to ensure their correct functioning, tools for their certification are needed. In the high-complexity regime, standard certification techniques such as comparing their behavior to classical simulations are not feasible.

To address the challenge of certification in regimes of high complexity, numerous methods were developed in the last decade. In this tutorial, we provide a self-contained introduction to the field of quantum-system certification and discuss prominent protocols for the certification of quantum states and quantum-gate implementations. Such protocols are particularly important in the development of near-term devices.

Figure 1

Left: the theoretical description of protocols makes use of the distinction into the device, measurement apparatus, and classical processor. Right: a complex quantum device comprises multiple abstraction layers. Different protocols aim at certifying the functioning of the device on different layers. NISQ devices are not expected to feature a powerful logical-gate layer. Instead, applications are directly tailored to the physical-gate layer.

Figure 2

The task of quantum-state certification is to detect when a state preparation $\tilde{\rho }$ is not close to a chosen target state $\rho$, i.e., when $dist(\rho ,\tilde{\rho })>ϵ$.

Figure 3

The (upper) tail of a random variable $X$ is the probability of $X$ being greater than some threshold $t$. This probability is given by the corresponding area under the graph of the probability density function (PDF) of $X$.

Figure 4

Illustration of a prototypical RB protocol. After the preparation of an initial state, one applies a random sequence of unitaries $g=(g_1,\dots ,g_m)$ succeeded by an inversion gate and final measurement of $M$. This experiment is repeated for different sequences and different sequence lengths $m$. In the classical postprocessing, the decay parameter of resulting empirical estimates for different sequence lengths $m$ are extracted and reported as the RB parameters.