Fariel Shafee
We study the changes if any of the expectation value of a general observable in a quantum system, the difficulties associated with the detection of these changes, and the possible methods for correcting the system through unitary control to maintain a constant average expectation value of the observable.
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We examine the problem of estimating the expectation values of two observables when we have a finite number of copies of an unknown qubit state. Specifically we examine whether it is better to measure each of the observables separately on different copies or to perform a joint measurement of the observables on each copy. We find that joint measurements can sometimes provide an advantage over separate measurements, but only if we make estimates of an observable based solely on the results of measurements of that observable. If we instead use both sets of results to estimate each observable then…
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.
Quantum information-theoretic approach has been identified as a way to understand the foundations of quantum mechanics as early as 1950 due to Shannon. However there hasn't been enough advancement or rigorous development of the subject. In the following paper we try to find relationship between a general quantum mechanical observable and it von Neumann entropy. We find that the expectation values and the uncertainties of the observables have bounds which depend on the entropy. The results also show that von Neumann entropy is not just the uncertainty of the state but also encompasses the infor…
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…
Wave packet revivals and fractional revivals are striking quantum interference phenomena that can occur under suitable conditions in a system with a nonlinear spectrum. In the framework of a specific model (the propagation of an initially coherent wave packet in a Kerr-like medium), it is shown that distinctive signatures of these revivals and fractional revivals are displayed by the time evolution of the expectation values of physical observables and their powers, i.e., by experimentally measurable quantities. Moreover, different fractional revivals can be selectively identified by examining …
Variational Quantum Generative Modeling by Sampling Expectation Values of Tunable Observables
本文提出了一种通过限制可观测量来提升量子生成模型效率的OT-EVS方法，通过经典阴影测量和对抗训练优化样本复杂度。
Experimental characterizations of a quantum system involve the measurement of expectation values of observables for a preparable state |psi> of the quantum system. Such expectation values can be measured by repeatedly preparing |psi> and coupling the system to an apparatus. For this method, the precision of the measured value scales as 1/sqrt(N) for N repetitions of the experiment. For the problem of estimating the parameter phi in an evolution exp(-i phi H), it is possible to achieve precision 1/N (the quantum metrology limit) provided that sufficient information about H and its spectrum is a…
We propose a single observable to witness the nonzero quantum discord of an unknown quantum state provided that we have four copies of the state. The expectation value of this observable provides a necessary and sufficient condition for nonzero quantum discord in $2\times N$ systems and a necessary condition in higher finite-dimensional bipartite systems. Furthermore, a nontrivial lower bound of quantum discord can be obtained from this expectation value. The proposed observable can be experimentally measured in exactly the same easy manner as that of the entanglement witness, moreover a quant…
Expectation value estimation with parametrized quantum circuits
本文提出利用浅层参数化量子电路分解目标可观测量并应用重要性采样，以优化期望值估计的样本复杂度，并推导了基本下界，数值实验展示了优势。
The inference of a hidden variable's historical value, based on observations before and after the fact, is a controversial subject in quantum mechanics. Here I address the controversy by proposing a formalism that unifies and generalizes some of the previous proposals for the task, including the quantum minimum-mean-square-error estimators proposed by Ohki, the generalized conditional expectation proposed by Accardi and Cecchini, the quantum smoothing theory proposed by Tsang, the optimal observables for parameter estimation proposed by Personick, Belavkin, and Grishanin, and the weak values p…
Benchmarking a randomized quantum algorithm for estimating expectation values of molecular observables
本文对Wang等人的随机采样算法进行基准测试，发现其测量开销过高，需改进测量策略或算法缩放才能用于工业量子化学应用。
Exact quantum noise deconvolution with partial knowledge of noise
提出一种无需完全噪声知识或部分噪声层析成像的量子噪声去卷积技术，适用于任何可逆量子噪声通道，通过经典后处理构建具有完全可校正期望值的可观测量集。
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…
Reconstructing Quantum States and Expectations via Dynamical Tomography
该研究通过动力学层析技术，利用系统演化前的测量来重建量子态和期望，探讨了马尔可夫动力学下的确定性和随机性测试方法，并应用到自旋链和电子-核系统中。
Given a physical quantum system described by a Hilbert H, for any bounded quantum observable (a bounded self-adjoint operator) T it is possible to define several ''hidden observable'' functions f:H->R associated to T and for any quantum mixed state (a density matrix) D it is possible to define several ''hidden mixed states'' (probability measures) m on H associated to D in such a way that the following equality is verified: Trace[ b(T). D] =integral[b(f(psi)).dm(psi) whatever is the continuous function b:R->R. This formula gives a general way to express any expectation value computable in a qu…
Maximum expectation of observables with restricted purity states
针对量子噪声下量子信息处理性能评估难的问题，提出快速数值算法求解受限纯度态下可观测量最大期望值，可用于量子态层析等，效率远高于现有凸优化求解器，并得出含噪系统基态能量更高的洞见。
We show that the vast majority of all pure states featuring a common expectation value of some generic observable at a given time will yield very similar expectation values of the same observable at any later time. This is meant to apply to Schroedinger type dynamics in high dimensional Hilbert spaces. As a consequence individual dynamics of expectation values are then typically well described by the ensemble average. Our approach is based on the Hilbert space average method. We support the analytical investigations with numerics obtained by exact diagonalization of the full time-dependent Sch…
Conditional expectation values of quantum mechanical observables reflect unique non-classical correlations, and are generally sensitive to decoherence. We consider the circumstances under which such sensitivity to decoherence is removed, namely, when the measurement process is subjected to conservation laws. Specifically, we address systems with additive conserved quantities and identify sufficient conditions for the system state such that its coherence plays no role in the conditional expectation values of observables that commute with the conserved quantity. We discuss our findings for a spe…
I propose a scheme for reconstructing the weak value of an observable without the need for weak measurements. The post-selection in weak measurements is replaced by an initial projector measurement. The observable can be measured using any form of interaction, including projective measurements. The reconstruction is effected by measuring the change in the expectation value of the observable due to the projector measurement. The weak value may take nonclassical values if the projector measurement disturbs the expectation value of the observable.
We propose a new quantum computational way of obtaining a ground-state energy and expectation values of observables of interacting Hamiltonians. It is based on the combination of the adiabatic quantum evolution to project a ground state of a non-interacting Hamiltonian onto a ground state of an interacting Hamiltonian and the phase estimation algorithm to retrieve the ground-state energy. The expectation value of an observable for the ground state is obtained with the help of Hellmann-Feynman theorem. As an illustration of our method, we consider a displaced harmonic oscillator, a quartic anha…
We consider the quantum expectation value \mathcal{A}=\<ψ|A|ψ\> of an observable A over the state |ψ\> . We derive the exact probability distribution of \mathcal{A} seen as a random variable when |ψ\> varies over the set of all pure states equipped with the Haar-induced measure. The probability density is obtained with elementary means by computing its characteristic function, both for non-degenerate and degenerate observables. To illustrate our results we compare the exact predictions for few concrete examples with the concentration bounds obtained using Levy's lemma. Finally we comment on th…
An entanglement witness is an observable with the property that a negative expectation value signals the presence of entanglement. The question arises how a witness can be improved if the expectation value of a second observable is known, and methods for doing this have recently been discussed as so-called ultrafine entanglement witnesses. We present several results on the characterization of entanglement given the expectation values of two observables. First, we explain that this problem can naturally be tackled with the method of the Legendre transformation, leading even to a quantification …
A Margolus-Levitin speed limit for observables: mean energy bounds expectation-value change quadratically
本文提出可观测量的Margolus-Levitin二次方速度极限，填补了平均能量在可观测量期望值变化约束中的空白，完善了量子速度极限框架，在自主量子时钟中有应用价值。
New directions in dynamical expectation estimation
本文提出一种联合优化状态与可观测值的扫描算法，用于动力学期望值估计，在30量子比特随机电路上误差比变分态压缩低两到三个数量级。
We introduce an approximate description of an $N$-qubit state, which contains sufficient information to estimate the expectation value of any observable with precision independent of $N$. We show, in fact, that the error in the estimation of the observables' expectation values decreases as the inverse of the square root of the number of the system's identical preparations and increases, at most, linearly in a suitably defined, $N$-independent, seminorm of the observables. Building the approximate description of the $N$-qubit state only requires repetitions of single-qubit rotations followed by…
The quantum-mechanical state vector is not directly observable even though it is the fundamental variable that appears in Schrodinger's equation. In conventional time-dependent perturbation theory, the state vector must be calculated before the experimentally-observable expectation values of relevant operators can be computed. We discuss an alternative form of time-dependent perturbation theory in which the observable expectation values are calculated directly and expressed in the form of nested commutators. This result is consistent with the fact that the commutation relations determine the p…
Many quantum algorithms involve the evaluation of expectation values. Optimal strategies for estimating a single expectation value are known, requiring a number of state preparations that scales with the target error $\varepsilon$ as $\mathcal{O}(1/\varepsilon)$. In this paper, we address the task of estimating the expectation values of $M$ different observables, each to within additive error $\varepsilon$, with the same $1/\varepsilon$ dependence. We describe an approach that leverages Gilyén et al.'s quantum gradient estimation algorithm to achieve $\mathcal{O}(\sqrt{M}/\varepsilon)$ scaling…
A quantum expectation identity: Applications to statistical mechanics
本文推导一个量子统计力学期望恒等式，统一联系含连续参数的可观测量，并证明多个经典定理是其特例，进而由最大熵原理给出广义形式。
Measuring Incompatible Observables with Quantum Neural Networks
利用量子神经网络同时测量不相容可观测量的期望值，通过多输出结构实现资源效率提升。
Quantum observables over time for information recovery
本文提出量子随时间观测（QOOT）概念，用于联合描述两个时间点的观测量，并基于不可行定理给出其定义条件，进而构造恢复映射以在无噪声期望值估计中实现优于概率误差消除的采样开销。
Fault-tolerant quantum computation of molecular observables
针对分子基态外可观测量期望值估算研究不足的问题，提出std-EVE和QSP-EVE两种量子算法，经严格误差分析与资源估算，QSP-EVE可大幅降低门数量与比特宽度，为相关领域首个同类研究。
We introduce a procedure based on quantum expectation values of measurement observables to characterize quantum coherence. Our measure allows one to quantify coherence without having to perform tomography of the quantum state, and can be directly calculated from measurement expectation values. This definition of coherence allows the decomposition into contributions corresponding to the non-classical correlations between the subsystems and localized on each subsystem. The method can also be applied to cases where the full set of measurement operators is unavailable. An estimator using the trunc…
What is special about the Kirkwood-Dirac distributions? Only they produce natural conditional expectations
本文证明，在所有与可观测量 $\hat A$ 和 $\hat B$ 的 Born 概率兼容的准概率表示中，只有 Kirkwood-Dirac 表示能提供与量子条件期望一致的条件期望概念。
Observable-targeted variational quantum simulation of Hamiltonian dynamics
针对仅需少数可观测量期望值的场景，提出一种目标导向的变分量子模拟方法，通过精确误差恒等式选择可观测量并更新电路参数，在不增加测量成本下将模拟时间延长至4.2倍。
It is shown that the nature of quantum states that emerge from decoherence is such that one can {\em measure} the expectation value of any observable of the system in a single measurement. This can be done even when such pointer states are a priori unknown. The possibility of measuring the expectation value of any observable, without any prior knowledge of the state, points to the objective existence of such states.
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。
Hall's recent derivation of an exact uncertainty relation [Phys. Rev. A64, 052103 (2001)] is revisited. It is found that the Bayes estimator of an observable between pre- and postselection equals the real part of the weak value. The quadratic loss function equals the expectation of the squared imaginary part of the weak value.
Measurements on current quantum processors are subject to hardware imperfections that lead to readout errors. These errors manifest themselves as a bias in quantum expectation values. Here, we propose a very simple method that forces the bias in the expectation value to appear as a multiplicative factor that can be measured directly and removed at the cost of an increase in the sampling complexity for the observable. The method assumes no specific form of the noise, but only requires that the noise is `weak' to avoid excessive sampling overhead. We provide bounds relating the error in the expe…
Measuring expectation values of observables is an essential ingredient in variational quantum algorithms. A practical obstacle is the necessity of a large number of measurements for statistical convergence to meet requirements of precision, such as chemical accuracy in the application to quantum chemistry computations. Here we propose an algorithm to estimate the expectation value based on its approximate expression as a weighted sum of classically-tractable matrix elements with some modulation, where the weight and modulation factors are evaluated by sampling appropriately prepared quantum st…
Quantum mechanics is challenging even for advanced undergraduate and graduate students. In the Schrödinger representation, the wave function evolves in time according to the time dependent Schrödinger equation. The time dependence of the wave function gives rise to time dependence of the expectation value of observables. We have been exploring the difficulties that advanced undergraduate and graduate students have with time dependence of expectation values in quantum mechanics. We have developed and administered conceptual free response and multiple-choice questions to students to investigate …
Stronger-than-quantum correlations
在对Bell不等式进行初等推导后，讨论了二值可观量的几种期望函数形式。特别强调的是假设强于量子的期望函数，从而产生最大的违反贝尔不等式。
Partial order and topology of Hermitian matrices and quantum Choquet integrals for density matrices with given expectation values
研究 Hermitian 矩阵的 Löwner 偏序与对应拓扑，并基于容量、累积量和 Möbius 变换构建量子 Choquet 积分，用于求解给定多个非对易可观测量期望值的密度矩阵。
Estimating properties of a quantum state by importance-sampled operator shadows
提出一种名为算子影子的方法，通过基于 Pauli 分解的重要性采样估计未知量子态的可观测量期望值，在估计局部 Hamiltonian 能量等局部可观测量的线性组合时，样本复杂度在所有参数上均优于经典影子技术。
Towards enhancing quantum expectation estimation of matrices through partial Pauli decomposition techniques and post-processing
本文提出部分泡利分解方法，将估计任意 $n$ 量子比特矩阵期望值所需量子电路数从 $4^n$ 降至最多 $2^n$，并通过测量计数后处理同时获取多个矩阵元素，有望加速变分量子算法。
Generalized geometric speed limits for quantum observables
本文利用量子信息几何推导可观测量期望值变化率的广义量子速度极限，在维数不低于3时收紧现有界，并通过超导 qutrit 实验验证。
An arbitrary operator corresponding to a physical observable cannot be measured in a single measurement on currently available quantum hardware. To obtain the expectation value of the observable, one needs to partition its operator to measurable fragments. However, the observable and its fragments generally do not share any eigenstates, and thus the number of measurements needed to obtain the expectation value of the observable can grow rapidly even when the wavefunction prepared is close to an eigenstate of the observable. We provide a unified Lie algebraic framework for developing efficient …
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…
Reducing the number of measurements required to estimate the expectation value of an observable is crucial for the variational quantum eigensolver to become competitive with state-of-the-art classical algorithms. To measure complicated observables such as a molecular electronic Hamiltonian, one of the common strategies is to partition the observable into linear combinations (fragments) of mutually commutative Pauli products. The total number of measurements for obtaining the expectation value is then proportional to the sum of variances of individual fragments. We propose a method that lowers …
No-go result for quantum postselection measurements of rank-m degenerate subspace
本文证明了当联合系统-设备可观测量具有秩-m简并性时，后选择测量无法改变其条件期望值，并据此表明量子测量的误差-扰动不确定性在后选择下保持不变。
The translation of Grover's search algorithm from its standard version, designed for implementation on a single quantum system amenable to projective measurements, into one suitable for an ensemble of quantum computers, whose outputs are expectation values of observables, is described in detail. A filtering scheme, which effectively determines expectation values on a limited portion of the quantum state, is presented and used to locate a single item for searches involving more than one marked item. A truncated version of Grover's algorithm, requiring fewer steps than the translated standard ve…
Quantum Advantage with Adaptive Shallow Circuits
该研究证明测量反馈可让自适应浅电路实现量子优势，固定相干深度下增加反馈结果能扩大可访问函数类，多项式反馈可将素域离散对数问题编码为单量子比特期望值，经典估计该期望值是困难的。
An extension of the conditional expectations (those under a given subalgebra of events and not the simple ones under a single event) from the classical to the quantum case is presented. In the classical case, the conditional expectations always exist; in the quantum case, however, they exist only if a certain weak compatibility criterion is satisfied. This compatibility criterion was introduced among others in a recent paper by the author. Then, state-independent conditional expectations and quantum Markov processes are studied. A classical Markov process is a probability measure, together wit…
The solutions of the time independent Schrodinger equation for non-Hermitian (NH) Hamiltonians have been extensively studied and calculated in many different fields of physics by using L^2 methods that originally have been developed for the calculations of bound states. The existing non-Hermitian formalism breaks down when dealing with wavepackets(WP). An open question is how time dependent expectation values can be calculated when the Hamiltonian is NH ? Using the F-product formalism, which was recently proposed, [J. Phys. Chem., 107, 7181 (2003)] we calculate the time dependent expectation v…
Quantum tracking control aims to identify applied fields to steer the expectation values of particular observables along desired paths in time. The associated temporal fields can be identified by inverting the underlying dynamical equations for the observables. However, fields found in this manner are often plagued by undesirable singularities. In this paper we consider a planar molecular rotor, and derive singularity-free tracking expressions for the fields that steer the expectation of the orientation of the rotor along desired trajectories in time. Simulations are presented that utilize two…
We introduce an approximate description of an $N$-qubit state, which contains sufficient information to estimate the expectation value of any observable to a precision that is upper bounded by the ratio of a suitably-defined seminorm of the observable to the square root of the number of the system's identical preparations $M$, with no explicit dependence on $N$. We describe an operational procedure for constructing the approximate description of the state that requires, besides the quantum state preparation, only single-qubit rotations followed by single-qubit measurements. We show that follow…
For a general multipartite quantum state, we formulate a locally checkable condition, under which the expectation values of certain nonlocal observables are completely determined by the expectation values of some local observables. The condition is satisfied by ground states of gapped quantum many-body systems in two spatial dimensions, assuming a widely conjectured form of area law is correct. Its implications on the studies of gapped quantum many-body systems, quantum state tomography, and quantum state verification are discussed. These results are based on a partial characterization of stat…
We present a framework to control and track the observables of a general solid state system driven by an incident laser field. The main result is a non-linear equation of motion for tracking an observable, together with a constraint on the size of expectations which may be reproduced via tracking. Among other applications, this model provides a potential route to the design of laser fields which cause photo-induced superconductivity in materials above their critical temperature. As a first test, the strategy is used to make the expectation value of the current conform to an arbitrary function …
Circuit-Efficient Randomized Quantum Simulation of Non-Unitary Dynamics with Observable-Driven and Symmetry-Aware Designs
提出random-LCHS框架，通过随机化编译高效模拟非幺正动力学，含通用、可观测量驱动及对称性感知三种变体，降低电路资源与样本复杂度，并经数值验证误差界更紧。
Chaos in classical systems has been studied in plenty over many years. Although the search for chaos in quantum systems has been an area of prominent research over the last few decades, the detailed analysis of many inherently chaotic quantum systems based on expectation values of dynamical variables has not been reported in the literature. In this paper, we extend the study of dynamical behaviour using expectation values of variables to a $q$-deformed harmonic oscillator. The system is found to be periodic, quasi-periodic or chaotic depending on the values of the deformation parameter $q$ and…
In operator algebra theory, a conditional expectation is usually assumed to be a projection map onto a sub-algebra. In the paper, a further type of conditional expectation and an extension of the Lueders - von Neumann measurement to observables with continuous spectra are considered; both are defined for a single operator and become a projection map only if they exist for all operators. Criteria for the existence of the different types of conditional expectation and of the extension of the Lueders - von Neumann measurement are presented, and the question whether they coincide is studied. All t…
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