Quantum search/amplitude amplification algorithms are designed to be able to amplify the amplitude in the target state linearly with the number of operations. Since the probability is the square of the amplitude, this results in the success probability rising quadratically with the number of operations. This paper presents a new kind of quantum search algorithm in which the amplitude of the target state, itself increases quadratically with the number of operations. However, the domain of applications of this is much more limited than standard amplitude amplification.
In this article, I show that a recent family of quantum algorithms, based on the quantum amplitude amplification algorithm, can be used to describe a cognitive heuristic called availability bias. The amplitude amplification algorithm is used to define quantitatively the ease of a memory task, while the quantum amplitude estimation and the quantum counting algorithms to describe cognitive tasks such as estimating probability or approximate counting.
The evolution of two-mode Gaussian state under symmetric amplification, non-symmetric damping and thermal noise is studied. The time dependent solution of the state characteristic function is obtained. The separability criterions are given for the final state of weak amplification as well as strong amplification.
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…
We consider the use of arbitrary phases in quantum amplitude amplification which is a generalization of quantum searching. We prove that the phase condition in amplitude amplification is given by $\tan(φ/2) = \tan(ϕ/2)(1-2a)$, where $ϕ$ and $ϕ$ are the phases used and where $a$ is the success probability of the given algorithm. Thus the choice of phases depends nontrivially and nonlinearly on the success probability. Utilizing this condition, we give methods for constructing quantum algorithms that succeed with certainty and for implementing arbitrary rotations. We also conclude that phase err…
There is a serious flaw in the proposal [arXiv:1603.06857] for the achievement of unity efficiency in SPDC. This is a replacement due to mistakes in the table of probabilities. Numbers have been corrected.
Tripartite three mode Gaussian state undergoes parametric amplification and amplitude damping as well as thermal noise is studied. In the case of a state totally symmetrically interacting with the environment, the time dependent correlation matrix of the state in evolution is given. The conditions for fully separability and fully entanglement of the final tripartite three mode Gaussian state are worked out.
Amplitude amplification is one of primary tools in building algorithms for quantum computers. This technique generalizes key ideas of the Grover search algorithm. Potentially useful modifications are connected with changing phases in the rotation operations and replacing the intermediate Hadamard transform with arbitrary unitary one. In addition, arbitrary initial distribution of the amplitudes may be prepared. We examine trade-off relations between measures of quantum coherence and the success probability in amplitude amplification processes. As measures of coherence, the geometric coherence …
Watrous had presented the first proof of zero-knowledge property of a proof system against a quantum verifier. The key of the proof is the construction of a quantum simulator. In the construction, the 'failure state' is rotated to the 'success' state by a tricky operation which is initially developped for the amplification of QMA proof systems. This manuscript presents a new and simpler construction of a simulator. In the construction, we simply amplify the success probability of a classical simulator using Grover's amplification.
We study an oracle operation, along with its circuit design, which combined with the Grover diffusion operator boosts the probability of finding minimum or maximum solutions on a weighted directed graph. We focus on a geometry of sequentially connected bipartite graphs, which naturally gives rise to solution spaces describable by gaussian distributions. We then demonstrate how an oracle which encodes these distributions can be used to solve for the optimal path via amplitude amplification. And finally, we explore the degree to which this algorithm is capable of solving cases which are generate…
We investigate the use of amplitude amplification on the gate-based model of quantum computing as a means for solving combinatorial optimization problems. This study focuses primarily on QUBO (quadratic unconstrained binary optimization) problems, which are well-suited for qubit superposition states. Specifically, we demonstrate circuit designs which encode QUBOs as `cost oracle' operations $U_{\textrm{C}}$, which when combined with the standard Grover diffusion operator $U_{\textrm{s}}$ lead to high probabilities of measurement for states corresponding to the optimal and near optimal solution…
Beyond Worst-Case Branching: Quantum Tree Search via Amplitude Amplification
Department of Computer Science and Engineering, INESC-ID & Instituto Superior Técnico, University of Lisbon（里斯本大学高等理工学院计算机科学与工程系，INESC-ID）；University of Lisbon（里斯本大学）
This paper generalizes the quantum amplitude amplification and amplitude estimation algorithms to work with non-boolean oracles. The action of a non-boolean oracle $U_φ$ on an eigenstate $|x\rangle$ is to apply a state-dependent phase-shift $φ(x)$. Unlike boolean oracles, the eigenvalues $\exp(iφ(x))$ of a non-boolean oracle are not restricted to be $\pm 1$. Two new oracular algorithms based on such non-boolean oracles are introduced. The first is the non-boolean amplitude amplification algorithm, which preferentially amplifies the amplitudes of the eigenstates based on the value of $φ(x)$. St…
We propose an optical scheme, employing optical parametric down-converters interlaced with nonlinear sign gates (NSGs), that completely converts an $n$-photon Fock-state pump to $n$ signal-idler photon pairs when the down-converters' crystal lengths are chosen appropriately. The proof of this assertion relies on amplitude amplification, analogous to that employed in Grover search, applied to the full quantum dynamics of single-mode parametric down-conversion. When we require that all Grover iterations use the same crystal, and account for potential experimental limitations on crystal-length pr…
Certain quantum operations can be built more efficiently through a procedure known as Repeat-Until-Success. Differently from other non-deterministic quantum operations, this procedure provides a classical flag which certifies the success or failure of the procedure and, in the latter case, a recovery step allows the restoration of the quantum state to its original condition. The procedure can then be repeated until success is achieved. After success is certified, the RUS procedure can be equated to a coherent gate. However, this is not the case when the operation needs to be conditioned on the…
Most of the schemes for "noiseless" amplification of coherent states, which have recently been attracting theoretical and experimental interest, share a common trait: the amplification is not truly noiseless, or perfect, for non-zero success probability. While this must hold true for all phase-independent amplification schemes, in this work we point out that truly noiseless amplification is indeed possible, provided that the states which we wish to amplify come from a finite set. Perfect amplification with unlimited average gain is then possible with finite success probability, for example usi…
The advantages of quantum information processing are in many cases obtained as consequences of quantum interactions, especially for computational tasks where two-qubit interactions are essential. In this work, we establish the framework of analyzing and quantifying loss or gain of information on a quantum system when the system interacts with its environment. We show that the information flow, the theoretical method of characterizing (non-)Markovianity of quantum dynamics, corresponds to the rate of the minimum uncertainty about the system given quantum side information. Thereafter, we analyze…
A probabilistic imaginary-time evolution (PITE) method was proposed as a nonvariational method to obtain a ground state on a quantum computer. In this formalism, the success probability of obtaining all imaginary-time evolution operators acting on the initial state decreases as the imaginary time proceeds. To alleviate the undesirable nature, we propose quantum circuits for PITE combined with the quantum amplitude amplification (QAA) method. We reduce the circuit depth in the combined circuit with QAA by introducing a pre-amplification operator. We successfully demonstrated that the combinatio…
Fock-state preparation based on amplitude amplification in cavity QED
Joint Quantum Institute, NIST/University of Maryland（联合量子研究所，美国国家标准与技术研究院/马里兰大学）；Joint Center for Quantum Information and Computer Science, NIST/University of Maryland（联合量子信息与计算科学中心，美国国家标准与技术研究院/马里兰大学）；Microsystems and Nanotechnology Division, Physical Measurement Laboratory, National Institute of Standards and Technology（微系统与纳米技术部，物理测量实验室，美国国家标准与技术研究院）
Amplitude amplification is a central tool used in Grover's quantum search algorithm and has been used in various forms in numerous quantum algorithms since then. It has been shown to completely eliminate one-sided error of quantum search algorithms where input is accessed in the form of black-box queries. We generalize amplitude amplification for two-sided error quantum algorithm for decision problems in the familiar form where input is accessed in the form of initial states of quantum circuits and where arbitrary projective measurements may be used to ascertain success or failure. This genera…
Distributed Exact Quantum Amplitude Amplification Algorithm for Arbitrary Quantum States
Yangtze Delta Industrial Innovation Center of Quantum Science and Technology（长三角量子科技产业创新中心）；Shanghai MiQro Era Digital Technology Co., Ltd.（上海微元时代数字科技有限公司）；Shenzhen University（深圳大学）；Quantum Science Center of Guangdong-Hong Kong-Macao Greater Bay Area (Guangdong)（粤港澳大湾区量子科学中心（广东））
Amplitude amplification provides a quadratic speed-up for an array of quantum algorithms when run on a quantum machine perfectly isolated from its environment. However, the advantage is substantially diminished as the NISQ-era quantum machines lack the large number of qubits necessary to provide error correction. Noise in the computation grows with the number of gate counts in the circuit with each iteration of amplitude amplification. After a certain number of amplifications, the loss in accuracy from the gate noise starts to overshadow the gain in accuracy due to amplification, forming an in…
We suggest and investigate a scheme for non-deterministic noiseless linear amplification of coherent states using successive photon addition, $(\hat a^{\dagger})^2$, where $\hat a^\dagger$ is the photon creation operator. We compare it with a previous proposal using the photon addition-then-subtraction, $\hat a \hat a^\dagger$, where $\hat a$ is the photon annihilation operator, that works as an appropriate amplifier only for weak light fields. We show that when the amplitude of a coherent state is $|α| \gtrsim 0.91$, the $(\hat a^{\dagger})^2$ operation serves as a more efficient amplifier co…
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 discuss how to build some partially entangled states of $n$ two-state quantum systems (qubits). The optimal partially entangled state with a high degree of symmetry is considered to be useful for overcoming a shot noise limit of Ramsey spectroscopy under some decoherence. This state is invariant under permutation of any two qubits and inversion between the ground state $|0\ket$ and an excited state $|1\ket$ for each qubit. We show that using selective phase shifts in certain basis vectors and Grover's inversion about average operations, we can construct this high symmetric entangled state b…
We present two new quantum algorithms. Our first algorithm is a generalization of amplitude amplification to the case when parts of the quantum algorithm that is being amplified stop at different times. Our second algorithm uses the first algorithm to improve the running time of Harrow et al. algorithm for solving systems of linear equations from O(kappa^2 log N) to O(kappa log^3 kappa log N) where κis the condition number of the system of equations.
We study search by quantum walk on a finite two dimensional grid. The algorithm of Ambainis, Kempe, Rivosh (quant-ph/0402107) takes O(\sqrt{N log N}) steps and finds a marked location with probability O(1/log N) for grid of size \sqrt{N} * \sqrt{N}. This probability is small, thus amplitude amplification is needed to achieve Θ(1) success probability. The amplitude amplification adds an additional O(\sqrt{log N}) factor to the number of steps, making it O(\sqrt{N} log N). In this paper, we show that despite a small probability to find a marked location, the probability to be within an O(\sqrt{N…
It is demonstrated that the nature of optical parametric amplification is a quantum phenomenon. The system Lagrangian can be constructed by the path integral of coherent state. The equations of motion for photon operators are indeed the Euler-Lagrange equations of a Lagrangian. The quantum state evolution equation can also be obtained without resorting to quantum Hamiltonian or Lagrangian. Starting with classical Newton equation, quantum transition amplitude of the system can be educed by surface integral.
The success probability in an ancilla based circuit generally decreases exponentially in the number of qubits consisted in the ancilla. Although the probability can be amplified through the amplitude amplification process, the input dependence of the amplitude amplification makes difficult to sequentially combine two or more ancilla based circuits. A new version of the amplitude amplification known as the oblivious amplitude amplification runs independently of the input to the system register. This allow us to to sequentially combine two or more ancilla based circuits. However, this type of th…
This paper focuses on the quantum amplitude estimation algorithm, which is a core subroutine in quantum computation for various applications. The conventional approach for amplitude estimation is to use the phase estimation algorithm, which consists of many controlled amplification operations followed by a quantum Fourier transform. However, the whole procedure is hard to implement with current and near-term quantum computers. In this paper, we propose a quantum amplitude estimation algorithm without the use of expensive controlled operations; the key idea is to utilize the maximum likelihood …
The landmark Grover algorithm for amplitude amplification serves as an essential subroutine in various type of quantum algorithms, with guaranteed quantum speedup in query complexity. However, there have been no proposal to realize the original motivating application of the algorithm, i.e., the database search or more broadly the pattern matching in a practical setting, mainly due to the technical difficulty in efficiently implementing the data loading and amplitude amplification processes. In this paper, we propose a quantum algorithm that approximately executes the entire Grover database sea…
We study the dynamics of a qubit-resonator system, when the resonator is driven by two signals. The interaction of the qubit with the high-amplitude driving we consider in terms of the qubit dressed states. Interaction of the dressed qubit with the second probing signal can essentially change the amplitude of this signal. We calculate the transmission amplitude of the probe signal through the resonator as a function of the qubit's energy and the driving frequency detuning. The regions of increase and attenuation of the transmitted signal are calculated and demonstrated graphically. We present …
Consider a Boolean function $χ: X \to \{0,1\}$ that partitions set $X$ between its good and bad elements, where $x$ is good if $χ(x)=1$ and bad otherwise. Consider also a quantum algorithm $\mathcal A$ such that $A |0\rangle= \sum_{x\in X} α_x |x\rangle$ is a quantum superposition of the elements of $X$, and let $a$ denote the probability that a good element is produced if $A |0\rangle$ is measured. If we repeat the process of running $A$, measuring the output, and using $χ$ to check the validity of the result, we shall expect to repeat $1/a$ times on the average before a solution is found. *A…
We explore an intriguing alternative for a fast and high-fidelity generation of steady-state entanglement. By exponentially enhancing the atom-cavity interaction, we obtain an exponentially-enhanced effective cooperativity of the system, which results in a high fidelity of the state generation. Meanwhile, we modulate the amplitudes of the driving fields to accelerate the population transfer to a target state, e.g., a Bell state. An exponentially-shortened stabilization time is thus predicted. Specifically, when the cooperativity of the system is $C=30$, the fidelity of the acceleration scheme …
The speedup of heavy numerical tasks by quantum computing is now actively investigated in various fields including data analysis in physics and astronomy. In this paper, we propose a new quantum algorithm for matched filtering in gravitational wave (GW) data analysis based on the previous work by Gao et al., Phys. Rev. Research 4, 023006 (2022) [arXiv:2109.01535]. Our approach uses the quantum algorithm for Monte Carlo integration for the signal-to-noise ratio (SNR) calculation instead of the fast Fourier transform used in Gao et al. and searches signal templates with high SNR by quantum ampli…
The master equation of quantum optical density operator is transformed to the equation of characteristic function. The parametric amplification and amplitude damping as well as the phase damping are considered. The solution for the most general initial quantum state is obtained for parametric amplification and amplitude damping. The purity of one mode Gaussian system and the entanglement of two mode Gaussian system are studied.
Experiments based on free space platform have demonstrated that the weak value amplification (WVA) technique can provide high sensitivity and precision for optical sensing and metrology. To promote this technique for real-world applications, it is more suitable to implement WVA based on optical fiber platform due to the lower cost, smaller scale and higher stability. In contrast to the free space platform, the birefringence in optical fiber is strong enough to cause polarization cross talk, and the amplitude-type noise must be taken into account. By theoretical analysis and experimental demons…
Coherent control of optomechanical entanglement and steering via dual parametric amplification
School of Physics and Astronomy, Beijing Normal University（北京师范大学物理学院）；Applied Optics Beijing Area Major Laboratory, Beijing Normal University（北京师范大学应用光学北京市重点实验室）；Key Laboratory of Multiscale Spin Physics (Ministry of Education), Beijing Normal University（北京师范大学多尺度自旋物理教育部重点实验室）
A randomly walking quantum particle searches in Grover's $Θ(\sqrt{N})$ iterations for a marked vertex on the complete graph of $N$ vertices by repeatedly querying an oracle that flips the amplitude at the marked vertex, scattering by a "coin" flip, and hopping. Physically, however, potential energy barriers can hinder the hop and cause the search to fail, even when the amplitude of not hopping decreases with $N$. We correct for these errors by interpreting the quantum walk search as an amplitude amplification algorithm and modifying the phases applied by the coin flip and oracle such that the …
Loss and decoherence are a major problem in the transmission of non-classical states of light over large distances. It was recently shown that the effects of decoherence can be reduced by applying a probabilistic noiseless attenuator before transmitting a quantum state through a lossy channel, followed by probabilistic noiseless amplification (M. Micuda et al, Phys. Rev. Lett. 109, 180503 (2012)). Here we show that similar results can be obtained for certain kinds of macroscopic quantum states by squeezing the signal before transmission, followed by deterministic amplification and anti-squeezi…
The idea of signal amplification is ubiquitous in the control of physical systems, and the ultimate performance limit of amplifiers is set by quantum physics. Increasing the amplitude of an unknown quantum optical field, or more generally any harmonic oscillator state, must introduce noise. This linear amplification noise prevents the perfect copying of the quantum state, enforces quantum limits on communications and metrology, and is the physical mechanism that prevents the increase of entanglement via local operations. It is known that non-deterministic versions of ideal cloning and local en…
Traditionally, quantum amplification limit refers to the property of inevitable noise addition on canonical variables when the field amplitude of an unknown state is linearly transformed through a quantum channel. Recent theoretical studies have determined amplification limits for cases of probabilistic quantum channels or general quantum operations by specifying a set of input states or a state ensemble. However, it remains open how much excess noise on canonical variables is unavoidable and whether there exists a fundamental trade-off relation between the canonical pair in a general amplific…
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.
