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.
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.
Performing experiments on small-scale quantum computers is certainly a challenging endeavor. Many parameters need to be optimized to achieve high-fidelity operations. This can be done efficiently for operations acting on single qubits, as errors can be fully characterized. For multiqubit operations, though, this is no longer the case, as in the most general case, analyzing the effect of the operation on the system requires a full state tomography for which resources scale exponentially with the system size. Furthermore, in recent experiments, additional electronic levels beyond the two-level system encoding the qubit have been used to enhance the capabilities of quantum-information processors, which additionally increases the number of parameters that need to be controlled. For the optimization of the experimental system for a given task (e.g., a quantum algorithm), one has to find a satisfactory error model and also efficient observables to estimate the parameters of the model. In this manuscript, we demonstrate a method to optimize the encoding procedure for a small quantum error correction code in the presence of unknown but constant phase shifts. The method, which we implement here on a small-scale linear ion-trap quantum computer, is readily applicable to other AMO platforms for quantum-information processing.
Bosonic quantum systems provide a hardware-efficient platform for quantum information processing but remain challenging to characterise due to their large Hilbert space and the high measurement cost of state tomography. Existing approaches estimate the fidelity with respect to a single target state, making them unsuitable for applications in which physically equivalent states differ by phase space translations, rotations, or other transformations. Here, we introduce an adaptive reconstruction technique that estimates the fidelity with respect to a family of bosonic states while reconstructing the underlying Wigner function from a small number of measurements. The method combines a physics-informed parametric model with Bayesian inference, bootstrap, and active learning to iteratively select the most informative phase space sampling points. We implement the approach on a circuit quantum electrodynamics platform and benchmark it on Schrödinger cat states with amplitudes α∈[1,3]. The reconstruction yields reproducible fidelity estimates within a few minutes, remains robust to substantial displacements and rotations in phase space despite using a mismatched prior, and is sensitive to subtle state imperfections. We further compare the adaptive strategy with existing Wigner function sampling protocols experimentally, demonstrating the advantage of adaptive sampling for measurement-efficient fidelity estimation with respect to a family of cat states. Finally, we incorporate the reconstructed fidelity into the figure of merit used in a proof-of-principle closed-loop quantum optimal control experiment, demonstrating the applicability of the method to autonomous optimisation of bosonic quantum states.
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-phase-estimation algorithms are critical subroutines in many applications for quantum computers and in quantum-metrology protocols. These algorithms estimate the unknown strength of a unitary evolution. By using coherence or entanglement to sample the unitary N tot times, the variance of the estimates can scale as O ( 1 / N tot 2 ) , compared to the best “classical” strategy with O ( 1 / N tot ) . The original algorithm for quantum phase estimation cannot be implemented on near-term hardware as it requires large-scale entangled probes and fault-tolerant quantum computing. Therefore, alternative algorithms have been introduced that rely on coherence and statistical inference. These algorithms produce quantum-boosted phase estimates without interprobe entanglement. This family of phase-estimation algorithms have, until now, never exhibited the possibility of achieving optimal scaling O ( 1 / N tot 2 ) . Moreover, previous works have not considered the effect of noise on these algorithms. Here, we present a coherence-based phase-estimation algorithm which can achieve the optimal quadratic scaling in the mean absolute error and the mean squared error. In the presence of noise, our algorithm produces errors that approach the theoretical lower bound. The optimality of our algorithm stems from its adaptive nature: Each step is determined, iteratively, using a Bayesian protocol that analizes the results of previous steps. Published by the American Physical Society 2024
This paper investigates the application of Bayesian inference for optimizing quantum measurement schemes. Traditional quantum measurement strategies often rely on heuristic approaches or specific system constraints, potentially leading to suboptimal outcomes. We propose a framework where the quantum measurement process is modeled as a Bayesian inference problem. This allows us to systematically explore the measurement landscape, incorporating prior knowledge and iteratively refining the measurement strategy based on observed outcomes. The core of this approach lies in formulating the measurement problem as a probabilistic model and utilizing Bayesian updating to determine the optimal measurement basis or parameters. We demonstrate, through theoretical analysis and conceptual design, that Bayesian optimization can lead to improved measurement accuracy and efficiency compared to conventional methods. The key benefits include reduced measurement errors, enhanced signal-to-noise ratios, and potentially the ability to extract more information from quantum systems. This work presents a novel methodology for quantum measurement, bridging the gap between classical Bayesian optimization and the unique properties of quantum mechanics.
As the size of quantum devices continues to grow, the development of scalable methods to characterise and diagnose noise is becoming an increasingly important problem. Recent methods have shown how to efficiently estimate Hamiltonians in principle, but they are poorly conditioned and can only characterize the system up to a scalar factor, making them difficult to use in practice. In this work we present a Bayesian methodology, called Bayesian Hamiltonian Learning (BHL), that addresses both of these issues by making use of any or all, of the following: well-characterised experimental control of Hamiltonian couplings, the preparation of multiple states, and the availability of any prior information for the Hamiltonian. Importantly, BHL can be used online as an adaptive measurement protocol, updating estimates and their corresponding uncertainties as experimental data become available. In addition, we show that multiple input states and control fields enable BHL to reconstruct Hamiltonians that are neither generic nor spatially local. We demonstrate the scalability and accuracy of our method with numerical simulations on up to 100 qubits. These practical results are complemented by several theoretical contributions. We prove that a k-body Hamiltonian H whose correlation matrix has a spectral gap Δ can be estimated to precision ε with only O~(n3k/(εΔ)3/2) measurements. We use two subroutines that may be of independent interest: First, an algorithm to approximate a steady state of H starting from an arbitrary input that converges factorially in the number of samples; and second, an algorithm to estimate the expectation values of m Pauli operators with weight ≤k to precision ε using only O(ε−23klogm) measurements, which quadratically improves a recent result by Cotler and Wilczek.