Giacomo Mauro D'Ariano, Vittorio Giovannetti, Paolo Perinotti
We consider the problem of estimating the ensemble average of an observable on an ensemble of equally prepared identical quantum systems. We show that, among all kinds of measurements performed jointly on the copies, the optimal unbiased estimation is achieved by the usual procedure that consists in performing independent measurements of the observable on each system and averaging the measurement outcomes.
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Quantum estimation theory provides optimal observations for various estimation problems for unknown parameters in the state of the system under investigation. However, the theory has been developed under the assumption that every observable is available for experimenters. Here, we generalize the theory to problems in which the experimenter can use only locally accessible observables. For such problems, we establish a Cram{é}r-Rao type inequality by obtaining an explicit form of the Fisher information as a reciprocal lower bound for the mean square errors of estimations by locally accessible ob…
We study the optimal way to estimate the quantum expectation value of a physical observable when a finite number of copies of a quantum pure state are presented. The optimal estimation is determined by minimizing the squared error averaged over all pure states distributed in a unitary invariant way. We find that the optimal estimation is "biased", though the optimal measurement is given by successive projective measurements of the observable. The optimal estimate is not the sample average of observed data, but the arithmetic average of observed and "default nonobserved" data, with the latter c…
We investigate the optimal estimation of quantum expectation value of a physical observable, which minimizes a mean error with respect to general measure of deviation, when a finite number of copies of a pure state are prepared. If pure sates are uniformly distributed, the minimum value of mean error for any measure of deviation is achieved by projective measurement on each copy.
Any method for estimating the ensemble average of arbitrary operator (observables or not, including the density matrix) relates the quantity of interest to a complete set of observables, i.e. a quorum}. This corresponds to an expansion on an irreducible set of operators in the Liouville space. We give two general characterizations of these sets. All the known unbiased reconstruction techniques, i.e. ``quantum tomographies'', can be described in this framework. New operatorial resolutions are given that can be used to implement novel reconstruction schemes.
High-Precision Observable Estimation with Single Qubit Quantum Memory
针对多量子比特可观测量估计中散粒噪声累积问题，提出利用单量子比特量子存储器累积测量数据，以更少交互次数实现高精度估计，所需测量次数随泡利字符串数按N^(2/3)规模减少。
We present observable upper bounds of squared concurrence, which are the dual inequalities of the observable lower bounds introduced in [F. Mintert and A. Buchleitner, Phys. Rev. Lett. 98, 140505 (2007)] and [L. Aolita, A. Buchleitner and F. Mintert, Phys. Rev. A 78, 022308 (2008)]. These bounds can be used to estimate entanglement for arbitrary experimental unknown finite-dimensional states by few experimental measurements on a twofold copy $ρ\otimesρ$ of the mixed states. Furthermore, the degree of mixing for a mixed state and some properties of the linear entropy also have certain relations…
We introduce the quantum indirect estimation theory, which provides a general framework to address the problem of which ensemble averages can be estimated by means of an available set of measuring apparatuses, e. g. estimate the ensemble average of an observable by measuring other observable. A main ingredient in this approach is that of informationally complete (infocomplete in short) measurements, which allow to estimate the ensemble average of any arbitrary system operator, as for quantum tomography. This naturally leads to the more stringent concept of AB-informationally complete measureme…
Enhanced observable estimation through classical optimization of informationally over-complete measurement data -- beyond classical shadows
本文提出在信息过完备测量完成后优化对偶POVM算子，以在经典阴影需指数级开销时实现零方差可观测量估计，并在随机泡利测量量子模拟中显著降低统计误差。
Propagation of error is a widely used estimation tool in experiments, where the estimation precision of the parameter depends on the fluctuation of the physical observable. Thus which observable is chosen will greatly affect the estimation sensitivity. Here we study the optimal observable for the ultimate sensitivity bounded by the quantum Cramér-Rao theorem in parameter estimation. By invoking the Schrödinger-Robertson uncertainty relation, we derive the necessary and sufficient condition for the optimal observables saturating the ultimate sensitivity for single parameter estimate. By applyin…
We discuss how continuous probing of a quantum system allows estimation of unknown classical parameters embodied in the Hamiltonian of the system. We generalize the stochastic master equation associated with continuous observation processes to a Bayesian filter equation for the probability distribution of the desired parameters, and we illustrate its application by estimating the direction of a magnetic field. In our example, the field causes a ground state spin precession in a two-level atom which is detected by the polarization rotation of off-resonant optical probes, interacting with the at…
Various forms of optimality for quantum observables described as normalized positive operator valued measures (POVMs) are studied in this paper. We give characterizations for observables that determine the values of the measured quantity with probabilistic certainty or a state of the system before or after the measurement. We investigate observables which are free from noise caused by classical post-processing, mixing, or pre-processing of quantum nature. Especially, a complete characterization of pre-processing and post-processing clean observables is given, and necessary and sufficient condi…
Verifiable blind observable estimation
本文提出VBOE协议，通过高效构建SDOE资源，实现对量子-经典混合算法的严格可组合验证，解决可观测量估计的验证难题。
An Error-aware and Adaptive Method for the Estimation of Quantum Observables on Qudit-Based Quantum Computers
本文提出AQUIRE方法，通过构建贝叶斯模型准确估计量子比特量子计算机上可观测量的均值和误差，并通过数值模拟和离子阱量子处理器实验验证其有效性。
Unbiased observable estimation with approximate channels in fault-tolerant quantum computation
本研究提出了一种利用噪声通道分解来纠正可观测量估计偏移的方法，通过模拟ising哈密顿量的时间动态验证了其在容错量子计算中的有效性。
Four common optimality criteria for measurements are formulated using relations in the set of observables, and their connections are clarified. As case studies, 1-0 observables, localization observables, and photon counting observables are considered.
Comprehensive Study on Heisenberg-limited Quantum Algorithms for Multiple Observables Estimation
本文提出了一种适应性量子梯度估计算法的通用方案，并分析了量子振幅估计算法在估计费米子k-RDMs中的性能。
We present observable lower and upper bounds for the entanglement of formation (EOF) and quantum discord (QD), which facilitates estimates of EOF and QD for arbitrary experimental unknown states in finite-dimensional bipartite systems. These bounds can be easily obtained by a few experimental measurements on a twofold copy $\varrho\otimes\varrho$ of the mixed states. Based on our results, we use the experimental measurement data of the real experiment given by Schmid \textit{et al.} [Phys. Rev. Lett. \textbf{101}, 260505 (2008)] to obtain the lower and upper bounds of EOF and QD for the experi…
Low variance estimations of many observables with tensor networks and informationally-complete measurements
本文提出用张量网络后处理信息完备测量，以低统计误差、无偏地估计多个可观测量，并在自旋和化学系统中验证其误差较经典阴影低数个数量级。
Heisenberg's uncertainty relations for measurement quantify how well we can jointly measure two complementary observables and have attracted much experimental and theoretical attention recently. Here we provide an exact tradeoff between the worst-case errors in measuring jointly two observables of a qubit, i.e., all the allowed and forbidden pairs of errors, especially asymmetric ones, are exactly pinpointed. For each pair of optimal errors we provide an optimal joint measurement that is realizable without introducing any ancilla and entanglement. Possible experimental implementations are disc…
Quantum memory assisted observable estimation
针对 Pauli 字符串测量中散粒噪声累积的问题，提出 CPS 方法，利用单量子比特量子存储器相干累积信息，使给定方差所需测量次数随 Pauli 字符串数量线性减少。
Recent works identified resolution limits for the distance between incoherent point sources. However, it remains unclear how to choose suitable observables and estimators to reach these limits in practical situations. Here, we show how estimators saturating the Cramér-Rao bound for the distance between two thermal point sources can be constructed using an optimally designed observable in the presence of practical imperfections, such as misalignment, crosstalk and detector noise.
By invoking quantum estimation theory we formulate bounds of errors in quantum measurement for arbitrary quantum states and observables in a finite-dimensional Hilbert space. We prove that the measurement errors of two observables satisfy Heisenberg's uncertainty relation, find the attainable bound, and provide a strategy to achieve it.
Direct estimation of arbitrary observables of an oscillator
本文提出OREO协议，通过将振子可观测量的期望值映射到辅助量子比特上，实现高效测量相空间四阶矩和非高斯性等级，为连续变量量子信息处理提供新的测量和控制方法。
Machine learning of measurement schemes for efficient quantum observable estimation
针对量子可观测量估计这一关键且受限的任务，提出用机器学习技术直接从可观测量学习测量方案的框架，具体实现C-LBCS，经数值验证其在分子系统上优于现有方法，为大量子系统可观测量估计开辟新途径。
Quantum advantages for syndrome-aware noisy logical observable estimation
本文基于量子估计理论，提出一个信息论框架，量化错误综合征在噪声逻辑可观测量估计中的效用，并区分经典和量子协议，证明经典协议中综合征信息最多可将有效逻辑错误率降低两倍，而量子协议中错误率随编码块数指数衰减。
Quantum state estimation is an important task of many quantum information protocols. We consider two families of unitary evolution operators, one with a one-parameter and the other with a two-parameter, which enable the estimation of a single spin component and all spin components, respectively, of a two-level quantum system. To evaluate the tomographic performance, we use the quantum tomographic transfer function (qTTF), which is calculated as the average over all pure states of the trace of the inverse of the Fisher information matrix. Our goal is to optimize the qTTF for both estimation mod…
Parameter estimation with limited access of measurements
针对最优测量不可访问时的量子参数估计问题，本文提出受限测量理论框架，通过优化可观测对象并利用最小欧氏距离刻画其与最优测量的接近程度，提升估计精度。
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-…
We show that the entanglement witnesses based on local orthogonal observables which are introduced in [S. Yu and N.-L. Liu, Phys. Rev. Lett. 95, 150504 (2005)] and [O. Gühne, M. Mechler, G. Tóth and P. Adam, Phys. Rev. A 74, 010301 (2006)] in linear and nonlinear forms can be optimized, respectively. As applications, we calculate the optimal nonlinear witnesses of pure bipartite states and show a lower bound on the I-concurrence of bipartite higher dimensional systems with our method.
Simultaneous Measurement of Multiple Incompatible Observables and Tradeoff in Multiparameter Quantum Estimation
针对多不相容可观测量单测量实现的误差研究不足问题，提出任意有限可观测量单测量实现框架与半定规划紧致误差界，并经超导量子处理器实验验证，为量子信息多非对易可观测量任务优化提供支撑。
Experimental determination of entanglement is important not only to characterize the state and use it in quantum information, but also in understanding complicated phenomena such as phase transitions. In this paper we show that in many cases, it is possible to determine entanglement of a two qubit state, as represented by concurrence, with a few observables, most of which are local. In particular, rank 1 and rank 2 states need exclusively measurement of local observables while rank 3 states need measurement of just one correlation observable in addition to local observables. Only the rank 4 st…
Entanglement does not correspond to any observable and its evaluation always corresponds to an estimation procedure where the amount of entanglement is inferred from the measurements of one or more proper observables. Here we address optimal estimation of entanglement in the framework of local quantum estimation theory and derive the optimal observable in terms of the symmetric logarithmic derivative. We evaluate the quantum Fisher information and, in turn, the ultimate bound to precision for several families of bipartite states, either for qubits or continuous variable systems, and for differ…
The uncertainty relations, pioneered by Werner Heisenberg nearly 90 years ago, set a fundamental limitation on the joint measurability of complementary observables. This limitation has long been a subject of debate, which has been reignited recently due to new proposed forms of measurement uncertainty relations. The present work is associated with a new error trade-off relation for compatible observables approximating two incompatible observables, in keeping with the spirit of Heisenberg's original ideas of 1927. We report the first \textsl{direct} test and confirmation of the tight bounds pre…
General noise-resilient quantum amplitude estimation
本文提出一种通用抗噪量子振幅估计算法，可在噪声随电路深度变化的情况下增强振幅与可观测量的估计精度，并通过数值分析和理论分析验证其鲁棒性与计算准确性。
Heisenberg-limited adaptive gradient estimation for multiple observables
本文提出一种自适应量子算法，以海森堡极限精度同时估计多个可观测量的期望值，查询复杂度为 $\mathcal{O}(\varepsilon^{-1}\sqrt{M}\log M)$，空间开销与精度无关，并避免大规模电路构建中的数值不稳定问题。
Improving shadow estimation with locally-optimal dual frames
提出局部最优阴影方法，通过关联经典阴影实现单比特测量下任意可观测量的无偏估计，显著降低估计误差。
We propose a quantum algorithm to compute low-energy expectation values of a quantum Hamiltonian by sampling a partition function associated with the average energy of that Hamiltonian. For any given quantum circuit-Hamiltonian pair, there is an associated average energy. The sampling is done through an accept/reject Metropolis-style algorithm on the quantum gates of the circuit itself. Observables calculated under the canonical ensemble from these samples of circuits are extrapolated from higher-energies to the ground state.
We investigate the role of symmetric quantum cloning machines (QCMs) in quantifying the mutual incompatibility of quantum observables. Specifically, we identify a cloning-based incompatibility measure whereby the incompatibility of a set of observables maybe quantified in terms of how well a uniform ensemble of their eigenstates can be cloned via a symmetric QCM. We show that this new incompatibility measure $\mathcal{Q}_{c}$ is {\it faithful} since it vanishes only for commuting observables. We prove an upper bound for $\mathcal{Q}_{c}$ for any set of observables in a finite-dimensional syste…
Reducing quantum resources for observable estimation with window-assisted coherent QPE
本文研究窗口辅助的相干 QPE 作为子例程估计可观测量的量子任务，通过加窗和改进块编码技术，将 Toffoli 门数量减少两个数量级以上，并提供了窗口函数改善复合算法误差的模型。
Joint or simultaneous measurements of non-commuting quantum observables are possible at the cost of increased unsharpness or measurement uncertainty. Many different criteria exist for defining what an "optimal" joint measurement is, with corresponding different tradeoff relations for the measurements. Understanding the limitations of such measurements is of fundamental interest and relevant for quantum technology. Here, we experimentally test a tradeoff relation for the sharpness of qubit measurements, a relation which refers directly to the form of the measurement operators, rather than to er…
We introduce a technique to estimate error-mitigated expectation values on noisy quantum computers. Our technique performs shadow tomography on a logical state to produce a memory-efficient classical reconstruction of the noisy density matrix. Using efficient classical post-processing, one can mitigate errors by projecting a general nonlinear function of the noisy density matrix into the codespace. The subspace expansion and virtual distillation can be viewed as special cases of the new framekwork. We show our method is favorable in the quantum and classical resources overhead. Relative to sub…
Faster Quantum Algorithm for Multiple Observables Estimation in Fermionic Problems
本文提出一种通用自适应QGE算法框架及其两种变体，实现费米系统集体性质估计的高效计算，相比现有算法在有限粒子数系统中提供二次加速。
We study the joint estimation of multiple Rabi frequencies in a multilevel system. By analytically identifying the optimal probe state and the optimal measurement, we obtain the highest precision limit for the joint estimation of multiple Rabi frequencies. We show that the joint estimation does not always outperform the separate estimation. In different regimes the joint estimation can either outperform or underperform the separate estimation. We further show that if additional adaptive quantum controls are allowed then the advantage of the joint estimation over the separate estimation can be …
Toward Efficient End-to-End Quantum Elliptic PDE Solvers: a Multilevel Correction Algorithm for Direct Observable Estimation
针对量子线性系统算法在椭圆PDE求解中可观测量估计的读出瓶颈，提出基于多级有限元离散和Schur补分解的多级校正框架，消除多项式网格依赖开销，实现海森堡标度或标准蒙特卡洛标度。
We study the issue of simultaneous estimation of several phase shifts induced by commuting operators on a quantum state. We derive the optimal positive operator-valued measure corresponding to the multiple-phase estimation. In particular, we discuss the explicit case of the optimal detection of double phase for a system of identical qutrits and generalise these results to optimal multiple phase detection for d-dimensional quantum states.
We establish a connection between optimal quantum cloning and optimal state estimation for d-dimensional quantum systems. In this way we derive an upper limit on the fidelity of state estimation for d-dimensional pure quantum states and, furthermore, for generalized inputs supported on the symmetric subspace.
We identify optimal measurement strategies for phase estimation in different scenarios. For pure states of a single qubit, we show that optimal measurements form a broad set parametrized with a continuous variable. When the state is mixed this set reduces to merely two possible measurements. For two qubits, we focus on the symmetric Werner state. We find an optimal measurement and show that estimation from the population imbalance is optimal only if the state is pure. Finally, for a pure state of $N$ qubits, we finds under which conditions the estimation from the full $N$-body correlation and …
We address the problem of optimal estimation of the relative phase for two-dimensional quantum systems in mixed states. In particular, we derive the optimal measurement procedures for an arbitrary number of qubits prepared in the same mixed state.
We present the optimal estimation of an unknown squeezing transformation of the radiation field. The optimal estimation is unbiased and is obtained by properly considering the degeneracy of the squeezing operator. For coherent input states, the r.m.s. of the estimation scales as $(2\sqrt {\bar n})^{-1}$ versus the average photon number $\bar n$, while it can be enhanced to $(2\bar n)^{-1}$ by using displaced squeezed states.
We investigate simultaneous estimation of multi-parameter quantum estimation with time-dependent Hamiltonians. We analytically obtain the maximal quantum Fisher information matrix for two-parameter in time-dependent three-level systems. The optimal coherent control scheme is proposed to increase the estimation precisions. In a example of a spin-1 particle in a uniformly rotating magnetic field, the optimal coherent Hamiltonians for different parameters can be chosen to be completely same. However, in general, the optimal coherent Hamiltonians for different parameters are incompatibility. In th…
We describe algorithms, and experimental strategies, for the Pareto optimal control problem of simultaneously driving an arbitrary number of quantum observable expectation values to their respective extrema. Conventional quantum optimal control strategies are less effective at sampling points on the Pareto frontier of multiobservable control landscapes than they are at locating optimal solutions to single observable control problems. The present algorithms facilitate multiobservable optimization by following direct paths to the Pareto front, and are capable of continuously tracing the front on…
We consider the problem of jointly estimating expectation values of many Pauli observables, a crucial subroutine in variational quantum algorithms. Starting with randomized measurements, we propose an efficient derandomization procedure that iteratively replaces random single-qubit measurements with fixed Pauli measurements; the resulting deterministic measurement procedure is guaranteed to perform at least as well as the randomized one. In particular, for estimating any $L$ low-weight Pauli observables, a deterministic measurement on only of order $\log(L)$ copies of a quantum state suffices.…
We address quantum estimation in situations where one has at disposal data from the measurement of an incomplete set of observables and some a priori information on the state itself. By expressing the a priori information in terms of a bias toward a given state the problem may be faced by minimizing the quantum relative entropy (Kullback entropy) with the constraint of reproducing the data. We exploit the resulting minimum Kullback entropy principle for the estimation of a quantum state from the measurement of a single observable, either from the sole mean value or from the complete probabilit…
We present a versatile inequality of uncertainty relations which are useful when one approximates an observable and/or estimates a physical parameter based on the measurement of another observable. It is shown that the optimal choice for proxy functions used for the approximation is given by Aharonov's weak value, which also determines the classical Fisher information in parameter estimation, turning our inequality into the genuine Cram{é}r-Rao inequality. Since the standard form of the uncertainty relation arises as a special case of our inequality, and since the parameter estimation is avail…
We explore the task of optimal quantum channel identification, and in particular the estimation of a general one parameter quantum process. We derive new characterizations of optimality and apply the results to several examples including the qubit depolarizing channel and the harmonic oscillator damping channel. We also discuss the geometry of the problem and illustrate the usefulness of using entanglement in process estimation.
We address the problem of the optimal quantum estimation of the coupling parameter of a bilinear interaction, such as the transmittivity of a beam splitter or the internal phase-shift of an interferometer. The optimal measurement scheme exhibits Heisenberg scaling of the measurement precision versus the total energy.
We show that the stochastic Schrodinger equation for the filtered state of a system, with linear free dynamics, undergoing continual non-demolition measurement or either position or momentum, or both together, can be solved explicitly within a class of Gaussian states which we call extended coherent states. The asymptotic limit yields a class of relaxed states which we describe explicitly. Bellman's principle is then applied directly to optimal feedback control of such dynamical systems and the Hamilton Jacobi Bellman equation for the minimum cost is derived. The situation of quadratic perform…
We study various optimality criteria for quantum observables. Observables are represented as covariant positive operator valued measures and we consider the case when the symmetry group is compact. Phase observables are examined as an example.
We analyze the statistics of observables in continuous variable quantum teleportation in the formalism of the characteristic function. We derive expressions for average values of output state observables in particular cumulants which are additive in terms of the input state and the resource of teleportation. Working with Squeezed Bell-like states, which may be optimized in a free parameter for better teleportation performance we discuss the relation between resources optimal for fidelity and for different observable averages. We obtain the values of the free parameter which optimize the centra…
We present the experimental realization of the optimal estimation protocol for a Pauli noisy channel. The method is based on the generation of 2-qubit Bell states and the introduction of quantum noise in a controlled way on one of the state subsystems. The efficiency of the optimal estimation, achieved by a Bell measurement, is shown to outperform quantum process tomography.
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