In this report we study certification of quantum measurements, which can be viewed as the extension of quantum hypotheses testing. This extension involves also the study of the input state and the measurement procedure. Here, we will be interested in two-point (binary) certification scheme in which the null and alternative hypotheses are single element sets. Our goal is to minimize the probability of the type II error given some fixed statistical significance. In this report, we begin with studying the two-point certification of pure quantum states and unitary channels to later use them to prove our main result, which is the certification of von Neumann measurements in single-shot and parallel scenarios. From our main result follow the conditions when two pure states, unitary operations and von Neumann measurements cannot be distinguished perfectly but still can be certified with a given statistical significance. Moreover, we show the connection between the certification of quantum channels or von Neumann measurements and the notion of q-numerical range.
In this work, we analyze the local certification of unitary quantum channels, which is a natural extension of quantum hypothesis testing. A particular case of a quantum channel operating on two systems corresponding to product states at the input, is considered. The goal is to minimize the probability of the type II error, given a specified maximum probability of the type I error, considering assistance through entanglement with auxiliary systems. Our result indicates connection of the local certification problem with a product numerical range of unitary matrices. We show that the optimal local strategy does not need usage of auxiliary systems and requires only single round of one-way classical communication. Moreover, we compare local and global certification strategies and show that typically local strategies are optimal, yet in some extremal cases, where global strategies make no errors, local ones may fail miserably. Finally, some application for local certification of von Neumann measurements are discussed as well.
A central task in quantum information science is state certification: testing whether an unknown state is
ε1-close to a fixed target state, or
ε2-far. Recent work has shown that surprisingly simple measurement protocols--comprising only single-qubit measurements--suffice to certify arbitrary
n-qubit states [Huang, Preskill, Soleimanifar '25; Gupta, He, O'Donnell '25]. However, these certification protocols are not robust: rather than allowing constant
ε1, they can only positively certify states within
ε1=O(1/n) trace distance of the target. In many experimental settings, the appropriate error tolerance is constant as the system size grows, so this lack of robustness renders existing tests inapplicable at scale, no matter how many times the test is repeated.
Here we present robust certification protocols based on few-qubit measurements that apply to all but a
O(2−n)-fraction of pure target states. Our first protocol achieves constant robustness, i.e.
ε1=Θ(1), using a single
O(logn)-qubit measurement along with single-qubit measurements in the
Z or
X basis on the other qubits. As a corollary of its robustness, this protocol also achieves constant (in
n) copy complexity, which is optimal. Our second protocol uses exclusively single-qubit measurements and is nearly robust:
ε1=Ω(1/logn). Our tests are based on a new uncertainty principle for conditional fidelities, which may be of independent interest.
Abstract In this report, we explore benchmarking gate-based quantum devices based on the certification of quantum measurements, which can be viewed as an extension of quantum hypothesis testing. Here, we are interested in a binary certification scheme in which the null and alternative hypotheses are qubit von Neumann measurements. The certification procedure can be used to evaluate the accuracy of Noisy Intermediate-Scale Quantum devices. To support this methodology, we contribute benchmarks to PyQBench, an open-source Python library for benchmarking gate-based quantum computers. We provide a user-friendly command-line interface for certifying measurements from the Fourier family. For more advanced scenarios, PyQBench can also be used as a Python library, allowing users to define custom measurement schemes and error models. The latest version of PyQBench is designed to support IBM Q devices through integration with the Qiskit library, enabling seamless benchmarking on real quantum hardware. By providing flexible benchmarking tools, PyQBench addresses the critical need for reliable performance metrics in the NISQ era, contributing to the development of error mitigation techniques and the verification of quantum measurement fidelity.
Certification of quantum channels is based on quantum hypothesis testing and involves also preparation of an input state and choosing the final measurement. This work primarily focuses on the scenario when the false negative error cannot occur, even if it leads to the growth of the probability of false positive error. We establish a condition when it is possible to exclude false negative error after a finite number of queries to the quantum channel in parallel, and we provide an upper bound on the number of queries. On top of that, we found a class of channels which allow for excluding false negative error after a finite number of queries in parallel, but cannot be distinguished unambiguously. Moreover, it will be proved that parallel certification scheme is always sufficient, however the number of steps may be decreased by the use of adaptive scheme. Finally, we consider examples of certification of various classes of quantum channels and measurements.
For any pair of quantum states (the hypotheses), the task of binary quantum hypotheses testing is to derive the tradeoff relation between the probability p01 of rejecting the null hypothesis and p10 of accepting the alternative hypothesis. The case when both hypotheses are explicitly given was solved in the pioneering work by Helstrom. Here, instead, for any given null hypothesis as a pure state, we consider the worst-case alternative hypothesis that maximizes p10 under a constraint on the distinguishability of such hypotheses. Additionally, we restrict the optimization to separable measurements, in order to describe tests that are performed locally. The case p01=0 has been recently studied under the name of ``quantum state verification''. We show that the problem can be cast as a semi-definite program (SDP). Then we study in detail the two-qubit case. A comprehensive study in parameter space is done by solving the SDP numerically. We also obtain analytical solutions in the case of commuting hypotheses, and in the case where the two hypotheses can be orthogonal (in the latter case, we prove that the restriction to separable measurements generically prevents perfect distinguishability). In regards to quantum state verification, our work shows the existence of more efficient strategies for noisy measurement scenarios.