A. Carlini, A. Hosoya
We present a quantum probabilistic algorithm which tests with a polynomial computational complexity whether a given composite number is of the Carmichael type. We also suggest a quantum algorithm which could verify a conjecture by Pomerance, Selfridge and Wagstaff concerning the asymptotic distribution of Carmichael numbers smaller than a given integer.
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Quantum computers can solve many number theory problems efficiently. Using the efficient quantum algorithm for order finding as an oracle, this paper presents an algorithm that computes the Carmichael function for any integer $N$ with a probability as close to 1 as desired. The algorithm requires $O((\log n )^3n^3)$ quantum operations, or $O(\log\log n (\log n)^4 n^2)$ operations using fast multiplication. Verification, quantum optimizations and applications to RSA and primality tests are also discussed.
We present an algorithm to compute the number of solutions of the (constrained) number partitioning problem. A concrete implementation of the algorithm on an Ising-type quantum computer is given.
This work uses a simple quantum computer model to discuss the randomness of bit strings originated from integer sequences. The considered quantum computer model has three elements: a processing unit responsible for a mathematical operation, an initial equally weighted superposition and a quantum state used as resource. The randomness depends on the last.
We present a quantum algorithm for computing the Ramsey numbers whose computational complexity grows super-exponentially with the number of vertices of a graph on a classical computer. The problem is mapped to a decision problem on a quantum computer, a probe qubit is coupled to a register that represents the problem and detects the energy levels of the problem Hamiltonian. The decision problem is solved by determining whether the probe qubit exhibits resonance dynamics. The algorithm shows a quadratic speedup over its classical counterparts, and the degenerate ground state problem in the adia…
Realistic lower bounds for the factorization time of large numbers on a quantum computer
我们研究了量子计算机分解一个给定数所需的时间T，它取决于表示这个数所需的比特数L。我们强调这样一个事实，即在大多数情况下，必须考虑到单个量子门的执行时间与计算中涉及的量子位的退相干时间有关。虽然这里只展示了特殊系统，但退相干和计算时间的这种相互依赖性似乎是当前许多量子计算机模型的限制，并导致计算时间T与L的比例比以前预期的要大得多。
We discuss how to simulate simple quantum logic operations with a large number of qubits. These simulations are needed for experimental testing of scalable solid-state quantum computers. Quantum logic for remote qubits is simulated in a spin chain. Analytical estimates are presented for possible correlated errors caused by non-resonant transitions. A range of parameters is given in which non-resonant effects can be minimized.
In the framework of a model for quantum computer media, a nondigital implementation of the arithmetic of the real numbers is described. For this model, an elementary storage "cell" is an ensemble of qubits (quantum bits). It is found that to store an arbitrary real number it is sufficient to use four of these ensembles and the arithmetic operations can be implemented by fixed quantum circuits.
A new method for computing sums on a quantum computer is introduced. This technique uses the quantum Fourier transform and reduces the number of qubits necessary for addition by removing the need for temporary carry bits. This approach also allows the addition of a classical number to a quantum superposition without encoding the classical number in the quantum register. This method also allows for massive parallelization in its execution.
Shor's algorithm for factoring in polynomial time on a quantum computer\cite{Shor} gives an enormous advantage over all known classical factoring algorithm. We demonstrate how to factor products of large prime numbers using a compiled version of Shor's quantum factoring algorithm. Our technique can factor all products of $p,q$ such that $p,q$ are unequal primes greater than two, runs in constant time, and requires only two coherent qubits. This illustrates that the correct measure of difficulty when implementing Shor's algorithm is not the size of number factored, but the length of the period …
Using quantum computers to identify prime numbers via entanglement dynamics
针对谐振子纠缠动力学识别素数的理论构想，本文提出推广方法与确定性算法，使其可在容错量子比特量子计算机上实现，并证明所用对角酉运算仅具二次时间复杂度。
Certified Random Number Generation using Quantum Computers
针对单台量子计算机难以实施设备无关随机数认证的问题，本文利用违反 Leggett-Garg 不等式的时间关联协议，在现有设备上以单量子比特低深度电路实现安全的半设备无关随机数生成。
Partial Loopholes Free Device Independent Quantum Random Number Generator Using IBM's Quantum Computers
本文利用 IBM 云端量子计算机违背 CHSH 不等式，提出并实验验证一种尽可能封堵漏洞的设备无关量子随机数生成方案。
Does True Randomness Exist? Efficacy Testing IBM Quantum Computers via Statistical Randomness
本文提出利用量子随机数生成与十八项统计测试评估量子计算机效能，并对九台 IBM 系统测试，发现仅一个计算机—算法组合通过统计随机性检验。
We report the first simulations of the dynamics of quantum logic operations with a large number of qubits (up to 1000). A nuclear spin chain in which selective excitations of spins is provided by the gradient of the external magnetic field is considered. The spins interact with their nearest neighbors. We simulate the quantum control-not (CN) gate implementation for remote qubits which provides the long-distance entanglement. Our approach can be applied to any implementation of quantum logic gates involving a large number of qubits.
We study path integration on a quantum computer that performs quantum summation. We assume that the measure of path integration is Gaussian, with the eigenvalues of its covariance operator of order j^{-k} with k>1. For the Wiener measure occurring in many applications we have k=2. We want to compute an $\e$-approximation to path integrals whose integrands are at least Lipschitz. We prove: 1. Path integration on a quantum computer is tractable. 2. Path integration on a quantum computer can be solved roughly $\e^{-1}$ times faster than on a classical computer using randomization, and exponential…
As an application of fault-tolerant quantum computers, we consider radiation transport calculations in this study. Radiation transport calculation using Monte Carlo calculation can obtain a solution to even a problem difficult to solve analytically. However, it is time-consuming depending on the scale and precision of the problem. Because it is known that the computational complexity of Monte Carlo calculation can be square rooted by quantum amplitude estimation, acceleration can be expected if radiation transport calculation is run on a quantum computer. In this study, we designed and investi…
Quantum volume is a single-number metric which, loosely speaking, reports the number of usable qubits on a quantum computer. While improvements to the underlying hardware are a direct means of increasing quantum volume, the metric is "full-stack" and has also been increased by improvements to software, notably compilers. We extend this latter direction by demonstrating that error mitigation, a type of indirect compilation, increases the effective quantum volume of several quantum computers. Importantly, this increase occurs while taking the same number of overall samples. We encourage the adop…
The primary resource for quantum computation is Hilbert-space dimension. Whereas Hilbert space itself is an abstract construction, the number of dimensions available to a system is a physical quantity that requires physical resources. Avoiding a demand for an exponential amount of these resources places a fundamental constraint on the systems that are suitable for scalable quantum computation. To be scalable, the effective number of degrees of freedom in the computer must grow nearly linearly with the number of qubits in an equivalent qubit-based quantum computer.
At a given moment, the state of the hypothetical quantum computer with N qubits is characterized by 2^N quantum amplitudes, which are complex continuous variables restricted by the normalization condition only. Their values cannot be arbitrary, they must be under our control. For moderate N = 1000, the number of quantum amplitudes greatly exceeds the number of particles in the Universe. Thus the answer to the question in title is: When physicists and engineers will learn to keep under control this number of continuous parameters.
Estimating the ground state energy of a multiparticle system with relative error $\e$ using deterministic classical algorithms has cost that grows exponentially with the number of particles. The problem depends on a number of state variables $d$ that is proportional to the number of particles and suffers from the curse of dimensionality. Quantum computers can vanquish this curse. In particular, we study a ground state eigenvalue problem and exhibit a quantum algorithm that achieves relative error $\e$ using a number of qubits $C^\prime d\log \e^{-1}$ with total cost (number of queries plus oth…
Recently Quantum Computation has generated a lot of interest due to the discovery of a quantum algorithm which can factor large numbers in polynomial time. The usefulness of a quantum com puter is limited by the effect of errors. Simulation is a useful tool for determining the feasibility of quantum computers in the presence of errors. The size of a quantum computer that can be simulat ed is small because faithfully modeling a quantum computer requires an exponential amount of storage and number of operations. In this paper we define simulation models to study the feasibility of quantum comput…
We examine the problem of simulating lattice gauge theories on a universal quantum computer. The basic strategy of our approach is to transcribe lattice gauge theories in the Hamiltonian formulation into a Hamiltonian involving only Pauli spin operators such that the simulation can be performed on a quantum computer using only one and two qubit manipulations. We examine three models, the U(1), SU(2), and SU(3) lattice gauge theories which are transcribed into a spin Hamiltonian up to a cutoff in the Hilbert space of the gauge fields on the lattice. The number of qubits required for storing a p…
Non-Hermitian Computers Need No Complex Numbers
本文研究非厄米量子计算中实门集${H, \text{CCNOT}, G}$的威力，证明其能在多项式时间内解决$\text{P}^{\sharp\text{P}}$问题，表明非酉性而非通用性是关键资源，且复数并非必要。
We discuss the scalability of superconducting quantum computers, especially in a wiring problem. The number of wiring inside a cryostat is almost proportional to the number of qubits in current wiring architectures. We introduce regularity, modularity, and hierarchy to an architecture design of superconducting quantum computers. The key to the wiring elimination is found in the quantum error correction codes having thresholds and spatial translational symmetry, i.e., the surface code. We show a superconducting-digital-logic-based architecture and introduce a stacked heterogeneous structure of …
We study the complexity of approximating the smallest eigenvalue of a univariate Sturm-Liouville problem on a quantum computer. This general problem includes the special case of solving a one-dimensional Schroedinger equation with a given potential for the ground state energy. The Sturm-Liouville problem depends on a function q, which, in the case of the Schroedinger equation, can be identified with the potential function V. Recently Papageorgiou and Wozniakowski proved that quantum computers achieve an exponential reduction in the number of queries over the number needed in the classical wors…
Testing the necessity of complex numbers in traditional quantum theory with quantum computers
本研究使用IBM量子计算机复现了测试量子理论中复数必要性的实验，证明实值量子理论在修改的CHSH贝尔测试中预测不同，且某些低误差设备能证伪实值表述，展示了用免费量子设备探索量子力学基础特征的可行性。
In existing general-purpose architectures for surface-code-based fault-tolerant quantum computers, the cost of a quantum computation is determined by the circuit volume, i.e., the number of qubits multiplied by the number of non-Clifford gates. We introduce an architecture using non-2D-local connections in which the cost does not scale with the number of qubits, and instead only with the number of logical operations. Each logical operation has an associated active volume, such that the cost of a quantum computation can be quantified as a sum of active volumes of all operations. For quantum com…
It is well known, and appreciated, that quantum computers have the potential to be the most powerful computational devices ever created. This newfound power comes from a quantum parallelism effect that allows the computer to be in multiple states at the same time. This property of quantum parallelism, while suited to handle common problems such as factoring and searching an unorganized database, is extremely well-suited to handle the task of solving a binary maze. I propose an algorithm that can be used to solve a binary maze on a quantum computer, with guaranteed accuracy. While it does work,…
Adaptive Quantum Computers: decoding and state preparation
本文提出自适应量子计算机模型，证明其在损坏数据解码上优于经典计算机，并给出高效制备均匀叠加态、GHZ态、W态和Dicke态的自适应量子算法，理论与硬件实验验证其优势。
Investigating methods to solve large windfarm optimization problems with a minimum number of qubits using circuit-based quantum computers
本文提出两种编码方法，利用少量量子比特在量子计算机上解决大型风电场布局优化问题，并展示其良好的扩展性能。
Considering its relevance in the field of cryptography, integer factorization is a prominent application where Quantum computers are expected to have a substantial impact. Thanks to Shor's algorithm this peculiar problem can be solved in polynomial time. However, both the number of qubits and applied gates detrimentally affect the ability to run a particular quantum circuit on the near term Quantum hardware. In this work, we help addressing both these problems by introducing a reduced version of Shor's algorithm that proposes a step forward in increasing the range of numbers that can be factor…
Simulating quantum chaos on a quantum computer
本文提出一种经典-量子混合方法，利用NISQ量子计算机模拟量子 kicked top 的混沌行为，展示了其在不同混沌参数范围内的实验探索能力，并验证了纠缠与去局部化的理论预测。
Efficient Floating-Point Arithmetic on Fault-Tolerant Quantum Computers
本文提出一种基于二进制补码的浮点编码方案，用于容错量子计算机上的高效算术运算，并通过模拟验证了其快速收敛和减少辅助量子比特的优势。
Practical randomness amplification and privatisation with implementations on quantum computers
提出基于Bell测试的端到端随机性放大与私有化协议，构建设备无关随机数生成器，在多台量子计算机上以半设备无关方式生成近乎无偏且私有的随机数。
First solid state quantum computer was built using transmons (cooper pair boxes). The operation of the computer is limited because of using a number of the rigit cooper boxes working with fixed frequency at temperatures of superconducting material. Here, we propose a novel architecture of quantum computer based on a flexible wire circuit of many coupled quantum nodes containing controlled atomic (molecular) ensembles. We demonstrate wide opportunities of the proposed computer. Firstly, we reveal a perfect storage of external photon qubits to multi-mode quantum memory node and demonstrate a rev…
In recent years a number of quantum computing devices with small numbers of qubits became available. We present a hybrid quantum local search (QLS) approach that combines a classical machine and a small quantum device to solve problems of practical size. The proposed approach is applied to the network community detection problem. QLS is hardware-agnostic and easily extendable to new quantum computing devices as they become available. We demonstrate it to solve the 2-community detection problem on graphs of size up to 410 vertices using the 16-qubit IBM quantum computer and D-Wave 2000Q, and co…
We demonstrate the implementation of a quantum algorithm for estimating the number of matching items in a search operation using a two qubit nuclear magnetic resonance (NMR) quantum computer.
We report experimental implementation of discrete Fourier transformation(DFT) on a nuclear magnetic resonance(NMR) quantum computer. Experimental results agree with theoretical results. Using the pulse sequences we introduced, DFT can be realized on any L-bit quantum number in principle.
The security of messages encoded via the widely used RSA public key encryption system rests on the enormous computational effort required to find the prime factors of a large number N using classical (i.e., conventional) computers. In 1994, however, Peter Shor showed that for sufficiently large N a quantum computer would be expected to perform the factoring with much less computational effort. This paper endeavors to explain, in a fashion comprehensible to the non-expert readers of this journal: (i) the RSA encryption protocol; (ii) the various quantum computer manipulations constituting the S…
Quantum chemistry simulations on a quantum computer suffer from the overhead needed for encoding the fermionic problem in a bosonic system of qubits. By exploiting the block diagonality of a fermionic Hamiltonian, we show that the number of required qubits can be reduced by a factor of two or more. There is no need to go into the basis of the Hilbert space for this reduction because all operations can be performed in the operator space. The scheme is conceived as a pre-computational step that would be performed on a classical computer prior to the actual quantum simulation. We apply this schem…
We propose a quantum computer structure based on coupled asymmetric single-electron quantum dots. Adjacent dots are strongly coupled by means of electric dipole-dipole interactions enabling rapid computation rates. Further, the asymmetric structures can be tailored for a long coherence time. The result maximizes the number of computation cycles prior to loss of coherence.
Most quantum gate errors can be characterized by two error models, namely the probabilistic error model and the Kraus error model. We proved that for a quantum circuit with either of those two models or a mix of both, the propagation error in terms of Frobenius norm is upper bounded by $2(1 - (1 - r)^m)$, where $0 \le r < 1$ is a constant independent of the qubit number and circuit depth, and $m$ is the number of gates in the circuit. Numerical experiments of synthetic quantum circuits and quantum Fourier transform circuits are performed on the simulator of the IBM Vigo quantum computer to ver…
The quantum adder is an essential attribute of a quantum computer, just as classical adder is needed for operation of a digital computer. We model the quantum full adder as a realistic complex algorithm on a large number of qubits in an Ising-spin quantum computer. Our results are an important step toward effective modeling of the quantum modular adder which is needed for Shor's and other quantum algorithms. Our full adder has the following features: (i) The near-resonant transitions with small detunings are completely suppressed, which allows us to decrease errors by several orders of magnitu…
A Programmable True Random Number Generator Using Commercial Quantum Computers
提出一种在商用门模型量子计算机上实现可编程真随机数生成器（TRNG）的自动化方法，通过编译工具根据用户指定的概率质量函数生成优化的OpenQASM量子电路，实验验证了其多比特随机性生成能力。
A quantum algorithm is a set of instructions for a quantum computer, however, unlike algorithms in classical computer science their results cannot be guaranteed. Quantum search algorithm can be described as the rotation of state vectors in a Hilbert space. The state vectors uniformly rotate by iterative sequences until they hit the target position. To optimize the algorithm, it is necessary to have the precise knowledge about some parameters like the number of target positions and total number of states. Here, we demonstrate the implementation of optimal fixed-point quantum search algorithm in…
Construction of virtual quantum states became possible due to the hypothesis on the nature of quantum states quant-ph/0212139. This study considers a stochastic geometrical background (stochastic gravitational background) generating correlation (or, coherency) of various quantum non-interacting objects. In the quantum state virtual model, a simple method of generating of two (or more) dichotomic signals with controlled mutual correlation factor from a single continuous stochastic process is implemented. Basing on the system random number generator of the computer, a model of the stationary ran…
In this paper we discuss the computation of Casimir energy on a quantum computer. The Casimir energy is an ideal quantity to calculate on a quantum computer as near term hybrid classical quantum algorithms exist to calculate the ground state energy and the Casimir energy gives physical implications for this quantity in a variety of settings. Depending on boundary conditions and whether the field is bosonic or fermionic we illustrate how the Casimir energy calculation can be set up on a quantum computer and calculated using the Variational Quantum Eigensolver algorithm with IBM QISKit. We compa…
Lasers and Bose-Einstein condensates (BECs) exhibit macroscopic quantum coherence in seemingly unrelated ways. Lasers possess a well-defined global phase and are characterized by large fluctuations in the number of photons. In BECs of atoms, instead, the number of particles is conserved and the global phase is undefined. Here, we present a unified framework to simulate lasers and BECs states in gate-based quantum computers, by mapping bosonic particles to qubit excitations. Our approach relies on a scalable circuit that measures the total number of particles without destroying long-range coher…
Analogue computers use continuous properties of physical system for modeling. In the paper is described possibility of modeling by analogue quantum computers for some model of data analysis. It is analogue associative memory and a formal neural network. A particularity of the models is combination of continuous internal processes with discrete set of output states. The modeling of the system by classical analogue computers was offered long times ago, but now it is not very effectively in comparison with modern digital computers. The application of quantum analogue modelling looks quite possibl…
The energy gap is calculated for the ground state quantum computer circuit, which was recently proposed by Mizel et.al. When implementing a quantum algorithm by Hamiltonians containing only pairwise interaction, the inverse of energy gap $1/Δ$ is proportional to $N^{4k}$, where $N$ is the number of bits involved in the problem, and $N^k$ is the number of control operations performed in a standard quantum paradigm. Besides suppressing decoherence due to the energy gap, in polynomial time ground state quantum computer can finish the quantum algorithms that are supposed to be implemented by stand…
Inspired by non-abelian vortex anyons in spinor Bose-Einstein condensates, we consider the quantum double $\mathcal{D}(Q_8)$ anyon model as a platform to carry out a particular instance of Shor's factorization algorithm. We provide the excitation spectrum, the fusion rules, and the braid group representation for this model, and design a circuit architecture that facilitates the computation. All necessary quantum gates, less one, can be compiled exactly for this hybrid topological quantum computer, and to achieve universality the last operation can be implemented in a non-topological fashion. T…
As quantum computing technology improves and quantum computers with a small but non-trivial number of N > 100 qubits appear feasible in the near future the question of possible applications of small quantum computers gains importance. One frequently mentioned application is Feynman's original proposal of simulating quantum systems, and in particular the electronic structure of molecules and materials. In this paper, we analyze the computational requirements for one of the standard algorithms to perform quantum chemistry on a quantum computer. We focus on the quantum resources required to find …
Almost all public secure communication relies on the inability to factor large numbers. There is no known analytic or classical numeric method to rapidly factor large numbers. Shor[1] has shown that a quantum computer can factor numbers in polynomial time but there is no practical quantum computer that can yet do such computations. We show that a simulated annealing[2] approach can be adapted to find factors of large numbers.
It is proposed to modify the Cirac-Zoller proposal of quantum computer with cold ions in a global oscillator trap potential by adding a periodic potential with an incommensurate average ratio of number of ions to number of periods being order of unity. With the increase of the periodic potential amplitude the system enters in the Aubry pinned phase characterized by quasi-frozen positions of ions and a gap of their first phonon excitations becomes independent of number of ions. This gives hopes that this quantum computer will be really scalable. It is argued that the usual single- and two-qubit…
Classical simulation is important because it sets a benchmark for quantum computer performance. Classical simulation is currently the only way to exercise larger numbers of qubits. To achieve larger simulations, sparse matrix processing is emphasized below while trading memory for processing. It performed well within NCSA supercomputers, giving a state vector in convenient continuous portions ready for post processing.
Programmable quantum processors are suitable platforms for simulating quantum systems, of which topological phases are of particular interest. We simulate the quench dynamics of a one-dimensional system on IBM Q devices. The topological properties of the dynamics are described by the dynamical topological invariants, the dynamical winding number and the time-dependent Berry phase, which are simulated with the quantum circuit model. The results show that despite the noise present in the current quantum computers, the dynamical topological invariants are robust. Moreover, to investigate the infl…
Quantum computing opens up new possibilities for the simulation of many-body nuclear systems. As the number of particles in a many-body system increases, the size of the space if the associated Hamiltonian increases exponentially. This presents a challenge when performing calculations on large systems when using classical computing methods. By using a quantum computer, one may be able to overcome this difficulty thanks to the exponential way the Hilbert space of a quantum computer grows with the number of quantum bits (qubits). Our aim is to develop quantum computing algorithms which can repro…
We describe the construction of quantum gates (unitary operators) from boolean functions and give a number of applications. Both non-reversible and reversible boolean functions are considered. The construction of the Hamilton operator for a quantum gate is also described with the Hamilton operator expressed as spin system. Computer algebra implementations are provided.
Optimal Qubit Reuse for Near-Term Quantum Computers
针对近期量子计算机的高错误率与资源限制，本文提出首个量子比特重用优化的形式化模型，在深度、比特数和swap门数量上提供可证明的最优解，并引入包含重置错误的成本函数以优化比特分配，有效提升了线路性能。
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…
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