David Bulger
Classically, determining the gradient of a black-box function f:R^p->R requires p+1 evaluations. Using the quantum Fourier transform, two evaluations suffice. This is based on the approximate local periodicity of exp(2*pi*i*f(x)). It is shown that sufficiently precise machine arithmetic results in gradient estimates of any required accuracy.
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Quantum Amplitude Estimation in Gradient-Based Stochastic Optimization
本文证明量子振幅估计算法在基于梯度的随机优化中相比蒙特卡洛方法可获得二次加速，并强调量子相位估计的集中保证是实现该优势的关键。
Many optimization methods for training variational quantum algorithms are based on estimating gradients of the cost function. Due to the statistical nature of quantum measurements, this estimation requires many circuit evaluations, which is a crucial bottleneck of the whole approach. We propose a new gradient estimation method to mitigate this measurement challenge and reduce the required measurement rounds. Within a Bayesian framework and based on the generalized parameter shift rule, we use prior information about the circuit to find an estimation strategy that minimizes expected statistical…
We show that the quantum Cramér-Rao bound on the precision of measurements of the optical phase gradient, or the wavefront tilt, with a beam of finite width is consistent with the Heisenberg uncertainty principle for a single-photon state, and is a factor of 2 better for the two-photon state that is maximally entangled. This fundamental bound governs a trade-off between quantum sensitivity and spatial resolution. Precision bounds based on a structured configuration using binary projective measurements implemented by an image-inversion interferometer, are higher, and the two-photon factor of 2 …
Gradient-based numerical methods are ubiquitous in optimization techniques frequently applied in industry to solve practical problems. Often times, evaluating the objective function is a complicated process, so estimating the gradient of a function with as few function evaluations as possible is a natural problem. We investigate whether quantum computers can perform $\ell^{\infty}$-approximate gradient estimation of multivariate functions $f : \mathbb{R}^d \to \mathbb{R}$ with fewer function evaluations than classically. Following previous work by Jordan [Jor05] and Gilyén et al. [GAW18], we p…
We develop a general theory to estimate magnetic field gradients in quantum metrology. We consider a system of $N$ particles distributed on a line whose internal degrees of freedom interact with a magnetic field. Usually gradient estimation is based on precise measurements of the magnetic field at two different locations, performed with two independent groups of particles. This approach, however, is sensitive to fluctuations of the off-set field determining the level-splitting of the particles and results in collective dephasing. In this work we use the framework of quantum metrology to assess…
Given a blackbox for f, a smooth real scalar function of d real variables, one wants to estimate the gradient of f at a given point with n bits of precision. On a classical computer this requires a minimum of d+1 blackbox queries, whereas on a quantum computer it requires only one query regardless of d. The number of bits of precision to which f must be evaluated matches the classical requirement in the limit of large n.
QuGStep: Refining Step Size Selection in Gradient Estimation for Variational Quantum Algorithms
针对变分量子算法有限差分梯度估计受测量噪声和步长影响的问题，本文提出QuGStep，依据最优步长定理在给定shots预算下最小化梯度误差界，并在分子基态能量实验中减少逾94%的shots。
We study the precision limits of detecting a linear magnetic-field gradient by using W-states in the presence of different types of noises. We consider to use an atomic spin chain for probing the magnetic-field gradient, where a W-state is prepared. We compare this method with the measurement of using two uncorrelated atoms. For pure states, W-states can provide an improvement over uncorrelated states in determining the magnetic-field gradient up to four particles. We examine the effects of local dephasing and dissipations on the performances of detections. In presence of dephasing, the uncorr…
We present a novel method for determining gradients of parameterised quantum circuits (PQCs) in hybrid quantum-classical machine learning models by applying the multivariate version of the simultaneous perturbation stochastic approximation (SPSA) algorithm. The gradients of PQC layers can be calculated with an overhead of two evaluations per circuit per forward-pass independent of the number of circuit parameters, compared to the linear scaling of the parameter shift rule. These are then used in the backpropagation algorithm by applying the chain rule. We compare our method to the parameter sh…
In this paper, we explore an efficient online algorithm for quantum state estimation based on a matrix-exponentiated gradient method previously used in the context of machine learning. The state update is governed by a learning rate that determines how much weight is given to the new measurement results obtained in each step. We show convergence of the running state estimate in probability to the true state for both noiseless and noisy measurements. We find that in the latter case the learning rate has to be chosen adaptively and decreasing to guarantee convergence beyond the noise threshold. …
A precise estimation of the computational complexity in Shor's factoring algorithm under the condition that the large integer we want to factorize is composed by the product of two prime numbers, is derived by the results related to number theory. Compared with Shor's original estimation, our estimation shows that one can obtain the solution under such a condition, by less computational complexity.
Efficient adaptive Bayesian estimation of a slowly fluctuating Overhauser field gradient
针对III-V族半导体量子点中Overhauser场梯度引起的退相干问题，提出两种自适应贝叶斯估计方案，通过自由感应衰减实验实时估计梯度大小，显著提高精度并分析退相干与漂移影响。
Efficient Quantum Gradient and Higher-order Derivative Estimation via Generalized Hadamard Test
本文提出灵活 Hadamard 测试、k 重 Hadamard 测试和量子自动微分方法，通过反转生成元与可观测量角色、单电路计算高阶导数及参数级自适应选择梯度技术，显著降低 PQC 梯度估计的电路执行成本。
Efficient Gradient Estimation for Parameterized Quantum Systems with Lie Algebraic Symmetries
提出一种利用李代数结构和Hadamard测试的梯度估计框架，通过阴影层析技术实现测量次数随参数对数增长，指数级降低测量成本并多项式加速时间。
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