令$f$表示关于字的长度保持函数。经典算法可以被认为是表示$f$的黑盒的$T$迭代应用，从长度为$n$的输入字$x$开始。证明了如果随机选择$T=O(2^{n/(7+e)}), e >0$和$f$，则每个量子计算机以概率1需要对$f$进行不少于$T$次的计算，以获得经典计算的结果.这意味着允许量子加速的经典算法集的概率测度为零。第二个结果是，对于以概率1随机选择的任意经典时间复杂度$T$和$f$，经典计算的每个量子模拟至少需要$f$的$Ω(\sqrt {T})$评估。
英文摘要：
Let $f$ denote length preserving function on words. A classical algorithm can be considered as $T$ iterated applications of black box representing $f$, beginning with input word $x$ of length $n$.
It is proved that if $T=O(2^{n/(7+e)}), e >0$, and $f$ is chosen randomly then with probability 1 every quantum computer requires not less than $T$ evaluations of $f$ to obtain the result of classical computation. It means that the set of classical algorithms admitting quantum speeding up has probability measure zero.
The second result is that for arbitrary classical time complexity $T$ and $f$ chosen randomly with probability 1 every quantum simulation of classical computation requires at least $Ω(\sqrt {T})$ evaluations of $f$.
Defining the right metrics to properly represent the performance of a quantum computer is critical to both users and developers of a computing system. In this white paper, we identify three key attributes for quantum computing performance: quality, speed, and scale. Quality and scale are measured by quantum volume and number of qubits, respectively. We propose a speed benchmark, using an update to the quantum volume experiments that allows the measurement of Circuit Layer Operations Per Second (CLOPS) and identify how both classical and quantum components play a role in improving performance. …
Quantum Computer Can Not Speed Up Iterated Applications of a Black Box
让一个经典的算法是由执行该算法的一个步骤的黑盒的顺序应用程序确定。如果我们把这个黑盒看作一个预言机，它为任何查询a给出一个值F（a），我们可以在一台经典计算机上计算T个F在时间T上相对于这个预言机的顺序应用。证明了当T=O（2^{n/7}），其中n为输入长度时，F的T个顺序应用的结果不能在具有Oracle for F的量子计算机上以比时间Ω（T）更快的速度计算.这意味着没有一种量子加速经典算法的通用方法，在这种通用方法中，经典算法被视为给定黑盒的迭代应用。对于任意的时间复杂度T，量子模拟的时间下限为Ω（T^{1/2}）.
A probabilistic algorithm for preparing Bethe eigenstates of the spin-1/2 Heisenberg spin chain on a quantum computer has recently been found. We derive an exact formula for the success probability of this algorithm in terms of the Gaudin determinant, and we study its large-length limit. We demonstrate the feasibility of computing antiferromagnetic ground-state spin-spin correlation functions for short chains. However, the success probability decreases exponentially with the chain length, which precludes the computation of these correlation functions for chains of moderate length. Some conject…
We propose protocols for determining the distances in Hilbert space between pure and mixed quantum states prepared on a quantum computer. In the case of pure quantum states, the protocol is based on measuring the square of modulus of scalar product between certain states. Determination of the distance between mixed quantum states is reduced to measuring the squares of modules of scalar products between all pure states included in the mixed states. In addition, we develop a protocol that allows one to determine the speed of evolution of the spin system simulated by a quantum computer. These pro…
Quantum computing is an emerging field of science which will eventually lead us to new and powerful logic devices with capabilities far beyond the limits of current transistor-based technology. There are certain types of problems which quantum computers can solve fundamentally faster than the tradition digital computers. There are quantum algorithms which require both superposition and entanglement (e.g. Shor algorithm). But neither the Grover algorithm nor the very first quantum algorithm due to Deutsch and Jozsa need entanglement. Is it possible to utilize classical wave superposition to spe…
With quantum computers of significant size now on the horizon, we should understand how to best exploit their initially limited abilities. To this end, we aim to identify a practical problem that is beyond the reach of current classical computers, but that requires the fewest resources for a quantum computer. We consider quantum simulation of spin systems, which could be applied to understand condensed matter phenomena. We synthesize explicit circuits for three leading quantum simulation algorithms, employing diverse techniques to tighten error bounds and optimize circuit implementations. Quan…
From the 1970s up to now, Mastermind, a classic two-player game, has attracted plenty of attention, not only from the public as a popular game, but also from the academic community as a scientific issue. Mastermind with n positions and k colors is formally described as: the codemaker privately chooses a secret $s\in [k]^n$, and the coderbreaker want to determine $s$ in as few queries like $f_s(x)$ as possible to the codemaker, where $f_s(x)$ indicates how x is close to s. The complexity of a strategy is measured by the number of queries used. In this work we study playing Mastermind on quantum…
Boson sampling is a sub-universal model used to show quantum speed-up. However, the methods of validation to prove quantum speedup are not robust and accurate. All verification methods involve additional or little studied assumptions. Here, we use the protocols given in the paper [arXiv:2006.03520] to construct a boson sampling experiment using discrete quantum states on IBM quantum computer and verify the fidelity of the output states using heterodyne detection. We demonstrate the protocols for single mode fidelity estimation, multi mode fidelity estimation and a verification protocol using I…
We introduce nebit, a classical bit with a signed probability distribution. We quantify its entropy using Szekely theorem on signed probability measures and show that classical stochastic dynamics supplemented with nebits can achieve or even exceed the speedup of Grover's quantum search algorithm. The proposed classical dynamics never reveals negativity of nebits and thus we do not need any operational interpretation of negative probabilities. We argue that nebits can be useful as a measure of non-classicality as well as a tool to find new quantum algorithms.
Quantum algorithms that can speed up certain tasks, such as factorisation and unstructured search, have driven a decades-long development of quantum computers and quantum technologies. Yet, outside specialized applications, quantum computers are believed to offer no advantage over classical computers. Here, we present a method which exploits quantum effects to speed up all possible classical computations. This method-which we call Coherent Parallelization (CP)-exploits quantum correlations generated by higher-order Hamiltonians to speed up any possible classical computation by a factor that de…
Recent works have shown that quantum computers can polynomially speed up certain SAT-solving algorithms even when the number of available qubits is significantly smaller than the number of variables. Here we generalise this approach. We present a framework for hybrid quantum-classical algorithms which utilise quantum computers significantly smaller than the problem size. Given an arbitrarily small ratio of the quantum computer to the instance size, we achieve polynomial speedups for classical divide-and-conquer algorithms, provided that certain criteria on the time- and space-efficiency are me…
The reverse quantum speed limit (RQSL) gives an upper limit to the time for evolution between initial and final quantum states. We show that, in conjunction with the existence of a minimum time scale, the RQSL implies a lower limit to the norm of the change in a quantum state, and confirm that this limit is satisfied in two-state and ideal-measurement models. Such a lower limit is of relevance for interpretational issues in probability and for understanding the meaning of probability in Everett quantum theory.
Noisy quantum computers suffer from readout or measurement error. It is a classical bit-flip error due to which state "1" is read out as "0" and vice-versa. The probability of readout error shows a state dependence i.e., flipping probability of state "1" may differ from flipping probability of state "0". Moreover, the probability shows correlation across qubits. These state-dependent and correlated error probability introduces a signature of victim outputs on adversary output when two programs are run simultaneously on the same quantum computer. This can be exploited to sense victim output whi…
Currently there are three major paradigms of quantum computation, the gate model, annealing, and walks on graphs. The gate model and quantum walks on graphs are universal computation models, while annealing plays within a specific subset of scientific and numerical computations. Quantum walks on graphs have, however, not received such widespread attention and thus the door is wide open for new applications and algorithms to emerge. In this paper we explore teaching a coined discrete time quantum walk on a regular graph a probability distribution. We go through this exercise in two ways. First …
It is shown that under reasonable assumptions a Drake-style equation can be obtained for the probability that our universe is the result of a deliberate simulation. Evaluating loose bounds for certain terms in the equation shows that the probability is unlikely to be as high as previously reported in the literature, especially in a scenario where the simulations are recursive. Furthermore, we investigate the possibility of eavesdropping from the outside of such a simulation and introduce a general attack that can circumvent attempts at quantum cryptography inside the simulation, even if the qu…
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…
Max Planck Institut für Quantenoptik（马克斯·普朗克量子光学研究所）；Leiden University（莱顿大学）
Suppose we have a small quantum computer with only M qubits. Can such a device genuinely speed up certain algorithms, even when the problem size is much larger than M? Here we answer this question to the affirmative. We present a hybrid quantum-classical algorithm to solve 3SAT problems involving n>>M variables that significantly speeds up its fully classical counterpart. This question may be relevant in view of the current quest to build small quantum computers.
Computational quantum technologies are entering a new phase in which noisy intermediate-scale quantum computers are available, but are still too small to benefit from active error correction. Even with a finite coherence budget to invest in quantum information processing, noisy devices with about 50 qubits are expected to experimentally demonstrate quantum supremacy in the next few years. Defined in terms of artificial tasks, current proposals for quantum supremacy, even if successful, will not help to provide solutions to practical problems. Instead, we believe that future users of quantum co…
Software under test can be analyzed dynamically, while it is being executed, to find defects. However, as the number and possible values of input parameters increase, the cost of dynamic testing rises. This paper examines whether quantum computers (QCs) can help speed up the dynamic testing of programs written for classical computers (CCs). To accomplish this, an approach is devised involving the following three steps: (1) converting a classical program to a quantum program; (2) computing the number of inputs causing errors, denoted by $K$, using a quantum counting algorithm; and (3) obtaining…
