The accurate estimation of observables is a crucial task in quantum computing. Recent advances have highlighted the need for (a) specialized protocols for qudit-based devices, that include (b) error-aware strategies. Here, we present AQUIRE, the first protocol that can (a) accurately estimate both the mean and the error of an observable on qudit-based quantum computers. AQUIRE achieves this by constructing a Bayesian model to accommodate generalized Pauli operators. It is designed to continuously monitor the estimated average and the associated error of the observable, adjusting the subsequent measurements in real-time. Additionally, AQUIRE is (b) device- and experiment-specific error-aware, and accounts for hardware imperfections and experimental noise during the estimation process. We demonstrate AQUIRE's advantage via numerical simulations and showcase its ability to quantify the noise affecting the estimation by implementing it on a trapped-ion qudit quantum processor. By exploiting general commutation relations and overlap grouping measurements, our protocol is state-of-the-art when restricted to qubit-based quantum computers and extends this advantage to the qudit case.
The accurate estimation of observables is a crucial task in quantum computing. Recent advances have highlighted the need for (a) specialized protocols for qudit-based devices, that include (b) error-aware strategies. Here, we present AQUIRE, the first protocol that can (a) accurately estimate both the mean and the error of an observable on qudit-based quantum computers. AQUIRE achieves this by constructing a Bayesian model to accommodate generalized Pauli operators. It is designed to continuously monitor the estimated average and the associated error of the observable, adjusting the subsequent measurements in real-time. Additionally, AQUIRE is (b) device- and experiment-specific error-aware, and accounts for hardware imperfections and experimental noise during the estimation process. We demonstrate AQUIRE's advantage via numerical simulations and showcase its ability to quantify the noise affecting the estimation by implementing it on a trapped-ion qudit quantum processor. By exploiting general commutation relations and overlap grouping measurements, our protocol is state-of-the-art when restricted to qubit-based quantum computers and extends this advantage to the qudit case.
The accurate estimation of quantum observables is a critical task in science. With progress on the hardware, measuring a quantum system will become increasingly demanding, particularly for variational protocols that require extensive sampling. Here, we introduce a measurement scheme that adaptively modifies the estimator based on previously obtained data. Our algorithm, which we call AEQuO, continuously monitors both the estimated average and the associated error of the considered observable, and determines the next measurement step based on this information. We allow both for overlap and non-bitwise commutation relations in the subsets of Pauli operators that are simultaneously probed, thereby maximizing the amount of gathered information. AEQuO comes in two variants: a greedy bucket-filling algorithm with good performance for small problem instances, and a machine learning-based algorithm with more favorable scaling for larger instances. The measurement configuration determined by these subroutines is further post-processed in order to lower the error on the estimator. We test our protocol on chemistry Hamiltonians, for which AEQuO provides error estimates that improve on all state-of-the-art methods based on various grouping techniques or randomized measurements, thus greatly lowering the toll of measurements in current and future quantum applications.
Measuring the state of quantum computers is a highly non-trivial task, with implications for virtually all quantum algorithms. A promising avenue is multi-copy schemes, where identical copies of a quantum state are measured jointly so that all Pauli operators within the considered observable can be simultaneously assessed. Here, we present a first implementation of such a two-copy scheme in a measurement protocol. Based on Bayesian statistics, it accurately estimates not only the average of the desired observable but also the error en route. This enables an adaptive shot-allocation algorithm that preferentially samples the most uncertain Pauli terms. In regimes with many non-commuting Pauli operators, this “double'' scheme can outperform the state-of-the-art measurement protocol in minimizing total shots for a given precision. We also numerically confirm the finding in previous theoretical works that the two-copy scheme incurs an overhead due to the square-root relationship between the variance of measured quantities and the number of measurement shots.
The pursuit of quantum advantage is driving the co-evolution of quantum processors and classical simulation methods. Despite advances in scale and quality, the accuracy of quantum simulation is ultimately limited by error rates and sampling overheads. Similarly, while classical simulation methods such as Pauli propagation have made remarkable progress, their accuracy is ultimately limited by the exponential growth of operator paths and the truncations needed to control memory and runtime. Here we show that these complementary limitations can be mitigated by embedding Pauli propagation within a hybrid error-mitigation framework that reduces quantum sampling overhead while achieving lower truncation errors with fewer classical resources than traditional Pauli propagation alone. In this framework, a target observable is classically propagated through noise-canceling inverse channels, producing a modified observable that is measured directly on a quantum processor. We prototype two implementations and benchmark their performance numerically on canonical models that challenge traditional Pauli propagation. We also perform experiments on a quantum processor using 56 superconducting qubits, revealing the tradeoffs of their respective truncation strategies. These results illustrate how classical and quantum resources can be orchestrated to extend observable estimation beyond the limits of either approach alone, providing a foundation for quantum-centric supercomputing and future demonstrations of quantum advantage.
Abstract Realisation of experiments even on small and medium-scale quantum computers requires an optimisation of several parameters to achieve high-fidelity operations. As the size of the quantum register increases, the characterisation of quantum states becomes more difficult since the number of parameters to be measured grows as well and finding efficient observables in order to estimate the parameters of the model becomes a crucial task. Here we propose a method relying on application of Bayesian inference that can be used to determine systematic, unknown phase shifts of multi-qubit states. This method offers important advantages as compared to Ramsey-type protocols. First, application of Bayesian inference allows the selection of an adaptive basis for the measurements which yields the optimal amount of information about the phase shifts of the state. Secondly, this method can process the outcomes of different observables at the same time. This leads to a substantial decrease in the resources needed for the estimation of phases, speeding up the state characterisation and optimisation in experimental implementations. The proposed Bayesian inference method can be applied in various physical platforms that are currently used as quantum processors.
Achieving precise control over quantum systems has groundbreaking ap-plications, such as simulating complex quantum systems, or implementing quantum algorithms with an improved efficiency over classical algorithms. Quantum computing has been under development for the last decades, re-sulting in the so-called Noisy Intermediate-Scale Quantum (NISQ) devices with about 50 qubits. While these devices are built with the purpose of achieving practical advantages over classical processors, they are vulnerable to noise sources. In order to account for error sources, one needs to im-plement Quantum Error Correction (QEC) techniques. These techniques protect quantum information by encoding a logical qubit into several phys-ical ones. For the implementation of NISQ devices and QEC techniques, it is vital to keep improving the quantum operations and states involved. Here we develop calibration protocols based on the application of Bayesian inference. We show their advantages compared to the more traditional frequentist approaches, such as their ability to maximise the information gained with each measurement, or their straightforward automation. In our first line of research, we develop a Bayesian protocol for the correction of unwanted phases appearing in the experimental preparation of Steane code states. This protocol requires 13 times less measurements than the frequentist approach. In our second line of research, we develop a proto-col for locking a laser to the qubit transition frequency in a trapped-ion architecture. This calibration is vital, since the correct implementation of single-qubit gates depends on it. In our final line of research, we study the Mølmer-Sørensen entangling gate, used in trapped-ion systems. First we introduce the calibration parameters, and derive a semianalytical model of the effects of one of them, for which there was no previous analytical understanding. Finally, we develop a Bayesian protocol for the calibration of these parameters, which was successfully implemented and tested in an experimental trapped-ion quantum processor.
Quantum computers have enabled solving problems beyond the current machines' capabilities. However, this requires handling noise arising from unwanted interactions in these systems. Several protocols have been proposed to address efficient and accurate quantum noise profiling and mitigation. In this work, we propose a novel protocol that efficiently estimates the average output of a noisy quantum device to be used for quantum noise mitigation. The multi-qubit system average behavior is approximated as a special form of a Pauli Channel where Clifford gates are used to estimate the average output for circuits of different depths. The characterized Pauli channel error rates, and state preparation and measurement errors are then used to construct the outputs for different depths thereby eliminating the need for large simulations and enabling efficient mitigation. We demonstrate the efficiency of the proposed protocol on four IBM Q 5-qubit quantum devices. Our method demonstrates improved accuracy with efficient noise characterization. We report up to 88% and 69% improvement for the proposed approach compared to the unmitigated, and pure measurement error mitigation approaches, respectively.