Charlee Stefanski, Vanio Markov, Constantin Gonciulea
In this paper we present a method for representing continuous signals with high precision by interpolating quantum state amplitudes. The method is inspired by the Nyquist-Shannon sampling theorem, which links continuous and discrete time signals. This method extends our previous method of computing generalized inner products from integer-valued functions to real-valued functions.
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Pricing of financial derivatives, in particular early exercisable options such as Bermudan options, is an important but heavy numerical task in financial institutions, and its speed-up will provide a large business impact. Recently, applications of quantum computing to financial problems have been started to be investigated. In this paper, we first propose a quantum algorithm for Bermudan option pricing. This method performs the approximation of the continuation value, which is a crucial part of Bermudan option pricing, by Chebyshev interpolation, using the values at interpolation nodes estima…
Energy Optimal Interpolation in Quantum Evolution
我们在量子演化中引入了插值的概念，并提出了一个通用的框架，以找到能量最优的哈密顿量的量子系统之间的一组给定的中间状态，使用变分和几何方法。一些特殊的情况进行了仔细研究。量子最速线问题被证明是一个特例。
We present efficient methods to interpolate data with a quantum computer that complement uploading techniques and quantum post-processing. The quantum algorithms are supported by the efficient Quantum Fourier Transform (QFT) and classical signal and imaging processing techniques, and open the door of quantum advantage to relevant families of data. We showcase a QFT interpolation method, a Quantum Cosine Transform (QCT) interpolation geared towards natural data, and we improve upon them by utilizing a quantum circuit's capabilities of processing data in superposition. A novel circuit for the QC…
In this paper, we present an interpolation scheme for FRQI images based on bilinear interpolation. To accomplish this, we formulated several quantum modules, i.e., assignment module, increment module, and quarter module, and suffused them into our proposed quantum image interpolation circuit. The concrete quantum circuits to accomplish up-scaling and down-scaling based on bilinear for FRQI images are designed and the network complexities of them are analyzed. Finally, to validate the proposed method, simulation experiments to enlarge and reduce the test images are executed, whose results are c…
The paper is devoted to the problem of multivariate polynomial interpolation and its application to quantum secret sharing. We show that using quantum Fourier transform one can produce the protocol for quantum secret sharing distribution.
Extracting the outcome of a quantum computation is a difficult task. In many cases, the quantum phase estimation algorithm is used to digitally encode a value in a quantum register whose amplitudes' magnitudes reflect the discrete sinc function. In the standard implementation the value is approximated by the most frequent outcome, however, using the frequencies of other outcomes allows for increased precision without using additional qubits. One existing approach is to use Maximum Likelihood Estimation, which uses the frequencies of all measurement outcomes. We provide and analyze several alte…
In this work we study the encoding of smooth, differentiable multivariate functions in quantum registers, using quantum computers or tensor-network representations. We show that a large family of distributions can be encoded as low-entanglement states of the quantum register. These states can be efficiently created in a quantum computer, but they are also efficiently stored, manipulated and probed using Matrix-Product States techniques. Inspired by this idea, we present eight quantum-inspired numerical analysis algorithms, that include Fourier sampling, interpolation, differentiation and integ…
A method for synthesizing quantum gates is presented based on interpolation methods applied to operators in Hilbert space. Starting from the diagonal forms of specific generating seed operators with non-degenerate eigenvalue spectrum one obtains for arity-one a complete family of logical operators corresponding to all the one-argument logical connectives. Scaling-up to n-arity gates is obtained by using the Kronecker product and unitary transformations. The quantum version of the Fourier transform of Boolean functions is presented and a Reed-Muller decomposition for quantum logical gates is de…
In this paper we study the functions that can be learned through the polynomial interpolation quantum algorithm designed by Childs et al. This algorithm was initially intended to find the coefficients of a multivariate polynomial function defined on finite fields $\mathbb{F}_q$. We extend its scope to vector inner product functions of the form $\mathcal{O}_{\mathbf{s}}(\mathbf{v}) = \mathbf{s}\cdot\mathbf{v}$ where the goal is to find the vector $\mathbf{s} \in \mathbb{F}_q^n$. We examine the necessary conditions on the domain $\mathcal{V}$ of $\mathcal{O}_{\mathbf{s}}$ and prove that the algo…
HHL algorithm \cite{harrow} to solve linear system is a powerful and efficient quantum technique to deal with many matrix operations (such as matrix multiplication, powers and inversion). It inspires many applications in quantum machine learning \cite{biamonte, dunjko}. However, due to the restrictions of HHL algorithm itself, many quantum machine learning algorithms also share one or two restrictions. The most common restrictions include quantum state preparation, condition number and Hamiltonian simulation. In this work, we first give an efficient quantum algorithm to achieve quantum state p…
Canonical Partition Function on a Quantum Computer through Trotter Interpolation
本文提出基于 Trotter 插值的 Gibbs 态可观测量估计算法，将计算成本降至 $\tilde{O}(β\log{1/ε})$，并在 SYK 模型上验证其优势。
Simplifying a classical-quantum algorithm interpolation with quantum singular value transformations
本文利用量子奇异值变换框架重新推导并简化了α-QPE算法的标度关系，揭示了相干预言机调用次数与符号函数多项式逼近质量之间的内在联系，为经典-量子插值提供了新的解释和统一视角。
We consider quantum interpolation of polynomials. We imagine a quantum computer with black-box access to input/output pairs (x_i, f(x_i)), where f is a degree-d polynomial, and we wish to compute f(0). We give asymptotically tight quantum lower bounds for this problem, even in the case where 0 is among the possible values of x_i.
Recent advances in engineering and control of nanoscale quantum sensors have opened new paradigms in precision metrology. Unfortunately, hardware restrictions often limit the sensor performance. In nanoscale magnetic resonance probes, for instance, finite sampling times greatly limit the achievable sensitivity and spectral resolution. We develop a technique for coherent quantum interpolation that can overcome these problems. Using a quantum sensor associated with the Nitrogen Vacancy center in diamond, we experimentally demonstrate that quantum interpolation can achieve spectroscopy of classic…
Improved Accuracy for Trotter Simulations Using Chebyshev Interpolation
本文提出利用切比雪夫多项式插值将 Trotter 化模拟结果外推至零步长，以抑制算法误差，并证明该方法在本征值和期望值估计中可在多对数因子内达到海森堡极限。
Interpolation by Different Types of Quantum Channels Using Conic Programs
本文利用锥规划刻画量子信道插值问题，并推广到凸集内信道，证明了正交输入输出矩阵集合下纠缠破坏信道的存在性。
Closest Accessible Symmetry reduction: a tool for Hamiltonian interpolation analysis
提出基于可及对称性概念的方法，通过投影到最近对称性扇区分析哈密顿量插值谱，捕捉量子相变特征并估计其位置。
Quantum circuits for basic image processing functions such as bilinear interpolation are required to implement image processing algorithms on quantum computers. In this work, we propose quantum circuits for the bilinear interpolation of NEQR encoded images based on Clifford+T gates. Quantum circuits for the scale up operation and scale down operation are illustrated. The proposed quantum circuits are based on quantum Clifford+T gates and are optimized for T-count. Quantum circuits based on Clifford+T gates can be made fault tolerant but the T gate is very costly to implement. As a result, redu…
Classical reversible circuits, acting on $w$~bits, are represented by permutation matrices of size $2^w \times 2^w$. Those matrices form the group P($2^w$), isomorphic to the symmetric group {\bf S}$_{2^w}$. The permutation group P($n$), isomorphic to {\bf S}$_n$, contains cycles with length~$p$, ranging from~1 to $L(n)$, where $L(n)$ is the so-called Landau function. By Lagrange interpolation between the $p$~matrices of the cycle, we step from a finite cyclic group of order~$p$ to a 1-dimensional Lie group, subgroup of the unitary group U($n$). As U($2^w$) is the group of all possible quantum…
Quantum-classical algorithm interpolation for spatial search
该研究针对空间搜索问题，提出可在经典与量子 regime 间插值的混合量子-经典算法，实现了电路深度与查询复杂度的权衡，并推导了定点振幅放大的最优深度-查询权衡。
A Number Theoretic Interpolation Between Quantum and Classical Complexity Classes
我们揭示了一个自然的代数问题，其复杂性似乎介于著名的复杂性类BQP和NP之间：（*）决定一个具有正好m个单项项的一元多项式是否有一个p-adic有理根。特别地，我们证明了当m=2时，while（*）在量子随机多项式时间内是可行的（并且没有已知的经典随机多项式时间算法），（*）对于一般m几乎是NP困难的：在一个合理的假设下，（由某些分圆域的广义黎曼假设所暗示），（*）的随机多项式时间算法将意味着广泛不被相信的包含NP \subseteq BPP。这种类型的量子/经典插值现象似乎是新的。
The variational quantum eigensolver (VQE) is an attracting possible application of near-term quantum computers. Originally, the aim of the VQE is to find a ground state for a given specific Hamiltonian. It is achieved by minimizing the expectation value of the Hamiltonian with respect to an ansatz state by tuning parameters \(\bmθ\) on a quantum circuit which constructs the ansatz. Here we consider an extended problem of the VQE, namely, our objective in this work is to "generalize" the optimized output of the VQE just like machine learning. We aim to find ground states for a given set of Hami…
Improving the trainability of VQE on NISQ computers for solving portfolio optimization using convex interpolation
本文利用凸插值改进变分量子本征求解器（VQE）的可训练性，通过近解初始化、正则代价函数景观和递归拟设平衡分区策略，在仅用10个超导量子比特实现40量子比特实验，并融合贪心算法实现全局与局部优化的互补。
Interpolation-based coordinate descent method for parameterized quantum circuits
本文提出基于插值的坐标下降法用于优化参数化量子电路，通过插值恢复三角结构并减少统计误差，提升求解效率。
Computing Quantum Resources using Tensor Cross Interpolation
提出基于张量交叉插值（TCI）算法的通用框架，用于高效计算强关联系统中的量子信息量化指标，并在1D和2D铁磁Ising模型中成功验证了计算SRE和REC的有效性。
Lie Algebra-Based Quantum Optimal Controls Interpolation
提出结合李群理论与前馈神经网络的方法，无需推理时优化即可高效生成超导量子比特系统中任意酉操作的量子最优控制脉冲，解决了Trotterized量子模拟中大量传播子处理的瓶颈。
A Random Matrix Model of Adiabatic Quantum Computing
我们提出了一个分析的量子绝热算法解决困难的情况下，3-SAT（NP完全问题）的随机矩阵理论（RMT）。我们确定的全局规律性的瞬时哈密顿量的频谱波动的初始哈密顿量和那些的基态编码的解决方案的计算问题之间的插值过程中遇到的。在每个插值点，我们通过其Brody参数量化平均光谱分布的规则性程度，Brody参数是区分规则（即，Poissonian）从混沌（即，Wigner型）分布的归一化最近邻间距。我们发现，对于困难的问题实例，即，那些具有子句与变量的临界比率的情况下，谱波动通常在插值参数的连续区域上变得不规则，而谱对于简单的实例是规则的。在硬区域内，RMT可以被应用以获得由于非绝热朗道-齐纳型转变而导致的绝热算法的避免的能级交叉和伴随的故障率的概率的数学模型。我们的模型预测，如果插值是在一个统一的速度，量子绝热算法的平均故障率，当平均超过硬问题的情况下，随着问题的大小呈指数级增长。
Using optimal control to guide neural-network interpolation of continuously-parameterized gates
本文提出将量子最优控制与物理信息神经网络结合，通过主动学习预训练和迁移学习快速综合连续参数化量子门控制曲面，超越线性插值并提升变分量子算法的噪声鲁棒性与运行速度。
Interpolation of unitaries with time-dependent Hamiltonians via Deep Learning
通过深度学习方法，有效插值时间依赖哈密顿量的单位元，提升量子系统模拟的准确性与效率。
Vertex coupling interpolation in quantum chain graphs
本文研究量子链图中破坏时间反演不变性的顶点耦合插值，发现平坦能带通常缺失且负谱非空，并确定了能带的高能渐近行为。
Exponentiation of Parametric Hamiltonians via Unitary interpolation
针对量子系统时间演化所需的矩阵指数生成限制相关问题，提出将Suzuki-Trotter乘积公式改为插值方案、在参数网格上定义插值的方法，提升精度与计算效率，且插值非保真度呈4阶收敛。
Quantum adiabatic evolutions that can't be used to design efficient algorithms
量子绝热计算是一种新的量子算法设计范例，通常用于求经典函数的极小值。本文证明了，如果具有内插路径的量子绝热演化的初始哈密顿量太简单，则这种量子绝热演化的基态与第一激发态之间的最小能隙是一个反指数距离。因此，这种量子绝热演化不能用来设计有效的量子算法。类似地，我们证明了具有简单最终哈密顿量的量子绝热演化也有很长的运行时间，这表明某些函数不能被任何具有插值路径的量子绝热演化有效地最小化。
Continuous-time evolution via probabilistic angle interpolation and its applications
本文提出基于概率角度插值的随机时间演进算法，通过连续时间极限消除Trotter误差，应用于氢分子基态能估计和稀疏Sachdev-Ye-Kitaev模型的非时间有序关联子计算。
Probabilistic Interpolation of Quantum Rotation Angles
针对量子设备旋转角度分辨率受限的问题，提出概率角度插值（PAI）方法，通过随机选择离散门设置并后处理输出实现连续旋转，在最小开销下显著放宽了硬件要求。
Interpolation of Quantum Polar Codes and Quantum Reed-Muller Codes
本文提出在量子极化码与量子Reed-Muller码之间进行插值的策略，以解决无需纠缠辅助构造有效量子极化码并提升其有限长度性能的问题。
Tensor Cross Interpolation of Purities in Quantum Many-Body Systems
提出利用张量交叉插值算法，以多项式样本量学习量子多体系统的纠缠特征（即所有子区域纯度的编码），并验证其在Haar随机态和MPS态上的有效性，进而应用于一维量子系统本征态，展示其在量化纠缠模式距离和优化物理指标排序中的潜力。
We propose a method to interpolate dynamics of von Neumann and classical master equations with an arbitrary mixing parameter to investigate the thermal effects in quantum dynamics. The two dynamics are mixed by intervening to continuously modify their solutions, thus coupling them indirectly instead of directly introducing a coupling term. This maintains the quantum system in a pure state even after the introduction of thermal effects and obtains not only a density matrix but also a state vector representation. Further, we demonstrate that the dynamics of a two-level system can be rewritten as…
Black hole/quantum machine learning correspondence
本文将黑洞信息恢复表述为黑洞微观态上的量子线性回归，用 Marchenko–Pastur 定律说明 Page 时间对应双下降插值阈值，并将该转变联系于子系统秩结构变化。
Characterizing a system often demands learning its response function to an applied field. Such knowledge is rooted on the experimental evaluation of punctual fiducial response and interpolation to access prediction at arbitrary values. Quantum metrological resources are known to provide enhancement in assessing these fiducial points, but the implications for improved function estimation have only recently been explored, and have not been yet demonstrated. Here we show an experimental realization of function estimation based on a photonic achitecture. The phase response of a liquid-crystal to a…
Shaping double descent in quantum extreme learning through model design
本研究通过解析刻画经典线性回归的双下降曲线，并将其推广至量子极端学习机，揭示编码强度、输入重上传和测量次数三个设计参数可正则化输出协方差并稳定插值阈值，为QELMs提供超越插值的泛化设计原则。
Performing efficient quantum computer tuneup and calibration is essential for growth in system complexity. In this work we explore the link between facilitating such capabilities and the underlying architecture of the physical hardware. We focus on the specific challenge of measuring (``mapping'') spatially inhomogeneous quasi-static calibration errors using spectator qubits dedicated to the task of sensing and calibration. We introduce a novel architectural concept for such spectator qubits: arranging them spatially according to prescriptions from optimal 2D approximation theory. We show that…
We consider the number of quantum queries required to determine the coefficients of a degree-d polynomial over GF(q). A lower bound shown independently by Kane and Kutin and by Meyer and Pommersheim shows that d/2+1/2 quantum queries are needed to solve this problem with bounded error, whereas an algorithm of Boneh and Zhandry shows that d quantum queries are sufficient. We show that the lower bound is achievable: d/2+1/2 quantum queries suffice to determine the polynomial with bounded error. Furthermore, we show that d/2+1 queries suffice to achieve probability approaching 1 for large q. Thes…
Quantum amplitude amplification algorithm: an explanation of availability bias
在这篇文章中，我表明，最近的家庭的量子算法，量子振幅放大算法的基础上，可以用来描述一个认知启发式称为可用性偏差。振幅放大算法用于定量地定义记忆任务的难易程度，量子振幅估计和量子计数算法用于描述概率估计或近似计数等认知任务。
We show that two qubits initially in completely classical state can create quantum discord through a local generalized amplitude damping channel, but high temperature will impede the creating of quantum discord.
Quantum control could be implemented by varying the system Hamiltonian. According to adiabatic theorem, a slowly changing Hamiltonian can approximately keep the system at the ground state during the evolution if the initial state is a ground state. In this paper we consider this process as an interpolation between the initial and final Hamiltonians. We use the mean value of a single operator to measure the distance between the final state and the ideal ground state. This measure could be taken as the error of adiabatic approximation. We prove under certain conditions, this error can be precise…
Are classical probabilities instances of quantum amplitudes?
提出并讨论了关于经典概率的量子性质的一个猜想。
Quantum algorithm involves the manipulation of amplitudes and computational basis, of which manipulating basis is largely a quantum analogue of classical computing that is always a major contributor to the complexity. In order to make full use of quantum mechanical speedup, more transformation should be implemented on amplitudes. Here we propose the notion of quantum amplitude arithmetic (QAA) that intent to evolve the quantum state by performing arithmetic operations on amplitude. Based on the basic design of multiplication and addition operations, QAA can be applied to solve the black-box qu…
In this work, we show the characterization of quantum iterations that would generally construct quantum amplitude amplification algorithms with a quadratic speedup, namely, quantum amplitude amplification operators (QAAOs). Exact quantum search algorithms that find a target with certainty and with a quadratic speedup can be composed of sequential applications of QAAO: existing quantum amplitude amplification algorithms thus turn out to be sequences of QAAOs. We show that an optimal and exact quantum amplitude amplification algorithm corresponds to the Grover algorithm together with a single it…
Iterative Interpolation Schedules for Quantum Approximate Optimization Algorithm
针对QAOA算法在深层电路下参数优化困难的问题，提出迭代插值方法，利用正交函数基表示最优参数调度，以更少的优化步骤为大规模电路构建高质量调度，在多个问题上超越现有方法并实现超千层深度。
Quantum Mechanics without amplitudes
振幅是量子理论的主要逻辑对象。尽管如此，它们并没有呈现出物理实在性，因此只有可观测的东西才能被实验检验。我们讨论了一种能够充分解释量子现象的量子概率理论的可能性。这个公式的优点是用振幅证明了orotodox公式没有描述的物理过程，以及量子力学完全算法化的可能性。
We present a quantum algorithm for the calculation of scattering amplitudes of massive charged scalar particles in scalar quantum electrodynamics. Our algorithm is based on continuous-variable quantum computing architecture resulting in expo- nential speedup over classical methods. We derive a simple form of the Hamiltonian including interactions, and a straightforward implementation of the constraint due to gauge invariance.
We propose to perform amplitude estimation with the help of constant-depth quantum circuits that variationally approximate states during amplitude amplification. In the context of Monte Carlo (MC) integration, we numerically show that shallow circuits can accurately approximate many amplitude amplification steps. We combine the variational approach with maximum likelihood amplitude estimation [Y. Suzuki et al., Quantum Inf. Process. 19, 75 (2020)] in variational quantum amplitude estimation (VQAE). VQAE typically has larger computational requirements than classical MC sampling. To reduce the v…
We show how phase and amplitude estimation algorithms can be parallelized. This can reduce the gate depth of the quantum circuits to that of a single Grover operator with a small overhead. Further, we show that for quantum amplitude estimation, the parallelization can lead to vast improvements in resilience against quantum errors. The resilience is not caused by the lower gate depth, but by the structure of the algorithm. Even in cases with errors that make it impossible to read out the exact or approximate solutions from conventional amplitude estimation, our parallel approach provided the co…
We introduce the Real Quantum Amplitude Estimation (RQAE) algorithm, an extension of Quantum Amplitude Estimation (QAE) which is sensitive to the sign of the amplitude. RQAE is an iterative algorithm which offers explicit control over the amplification policy through an adjustable parameter. We provide a rigorous analysis of the RQAE performance and prove that it achieves a quadratic speedup, modulo logarithmic corrections, with respect to unamplified sampling. Besides, we corroborate the theoretical analysis with a set of numerical experiments.
Application of a Quantum Amplitude Redistribution Algorithm to the Data Filtering Problem
本文探讨了量子振幅重新分配算法在数据滤波中的应用，并对比了该算法与中值滤波器的性能。
How many quantum queries are required to determine the coefficients of a degree-$d$ polynomial in $n$ variables? We present and analyze quantum algorithms for this multivariate polynomial interpolation problem over the fields $\mathbb{F}_q$, $\mathbb{R}$, and $\mathbb{C}$. We show that $k_{\mathbb{C}}$ and $2k_{\mathbb{C}}$ queries suffice to achieve probability $1$ for $\mathbb{C}$ and $\mathbb{R}$, respectively, where $k_{\mathbb{C}}=\smash{\lceil\frac{1}{n+1}{n+d\choose d}\rceil}$ except for $d=2$ and four other special cases. For $\mathbb{F}_q$, we show that $\smash{\lceil\frac{d}{n+d}{n+d…
The set of Multi-level Amplitude Damping (MAD) quantum channels is introduced as a generalization of the standard qubit Amplitude Damping Channel to quantum systems of finite dimension $d$. In the special case of $d=3$, by exploiting degradability, data-processing inequalities, and channel isomorphism, we compute the associated quantum and private classical capacities for a rather wide class of maps, extending the set of solvable models known so far. We proceed then to the evaluation of the entanglement assisted, quantum and classical, capacities.
How to achieve an arbitrary real-valued probability amplitude in the general single-partite or multipartite quantum system without measuring any other quantum state's probability amplitude? How to achieve an arbitrary real-valued probability amplitude with the deterministic polynomial time's complexity under a small given error? In this paper, one novel quantum measurement scheme is proposed to solve these questions based on the idea of binary searching. First, the measurement algorithm with the exponential speed-up on the quantum state with one single qubit is well-designed. Then, the measure…
Amplitude estimation algorithms are based on Grover's algorithm: alternating reflections about the input state and the desired outcome. But what if we are given the ability to perform arbitrary rotations, instead of just reflections? In this situation, we find that quantum signal processing lets us estimate the amplitude in a more flexible way. We leverage this technique to give improved and simplified algorithms for many amplitude estimation tasks: we perform non-destructive estimation without any assumptions on the amplitude, develop an algorithm with improved performance in practice, presen…
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。
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