In this paper we present the computational model underlying the one-way quantum computer which we introduced recently [Phys. Rev. Lett. 86, 5188 (2001)]. The one-way quantum computer has the property that any quantum logic network can be simulated on it. Conversely, not all ways of quantum information processing that are possible with the one-way quantum computer can be understood properly in network model terms. We show that the logical depth is, for certain algorithms, lower than has so far been known for networks. For example, every quantum circuit in the Clifford group can be performed on the one-way quantum computer in a single step.
A computational problem fed into a gate-model quantum computer identifies an objective function with a particular computational pathway (objective function connectivity). The solution of the computational problem involves identifying a target objective function value that is the subject to be reached. A bottleneck in a gate-model quantum computer is the requirement of several rounds of quantum state preparations, high-cost run sequences, and multiple rounds of measurements to determine a target (optimal) state of the quantum computer that achieves the target objective function value. Here, we …
The one-way quantum computer (QCc) is a universal scheme of quantum computation consisting only of one-qubit measurements on a particular entangled multi-qubit state, the cluster state. The computational model underlying the QCc is different from the quantum logic network model and it is based on different constituents. It has no quantum register and does not consist of quantum gates. The QCc is nevertheless quantum mechanical since it uses a highly entangled cluster state as the central physical resource. The scheme works by measuring quantum correlations of the universal cluster state.
The field of quantum algorithms aims to find ways to speed up the solution of computational problems by using a quantum computer. A key milestone in this field will be when a universal quantum computer performs a computational task that is beyond the capability of any classical computer, an event known as quantum supremacy. This would be easier to achieve experimentally than full-scale quantum computing, but involves new theoretical challenges. Here we present the leading proposals to achieve quantum supremacy, and discuss how we can reliably compare the power of a classical computer to the po…
We investigate if physical laws can impose limit on computational time and speed of a quantum computer built from elementary particles. We show that the product of the speed and the running time of a quantum computer is limited by the type of fundamental interactions present inside the system. This will help us to decide as to what type of interaction should be allowed in building quantum computers in achieving the desired speed.
We found that the actual computational time-cost of the QFT is O(n 2^n) for large n in a quantum computer using nuclear spins. The computational cost of a quantum algorithm has usually been estimated as the sum of the universal gates required in such ideal mathematical models as the Quantum Turing Machine(QTM) and the quantum circuit. This cost is proportional to an actual time-cost in the physical implementation where all quantum operations can be achieved in the same time. However, if the implementation takes a different time for each quantum gate, there is a possibility that the actual time…
Quantum computers promise to efficiently solve important problems that are intractable on a conventional computer. Quantum computational algorithms have the potential to be an exciting new way of studying quantum cosmology. In quantum cosmology, we learn about the dynamics of the universe without constructing a complete theory of quantum gravity. Since the universal wavefunction exists in an infinite-dimensional superspace over all possible 3D metrics and modes of matter configurations, we take minisuperspaces for our work by constraining the degrees of freedom to particular 3D metrics and uni…
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…
We present computational chemistry data for small molecules ($CO$, $HCl$, $F_2$, $NH_4^+$, $CH_4$, $NH_{3}$, $H_3O^+$, $H{_2}O$, $BeH_{2}$, $LiH$, $OH^-$, $HF$, $HeH^+$, $H_2$), obtained by implementing the Unitary Coupled Cluster method with Single and Double excitations (UCCSD) on a quantum computer simulator. We have used the Variational Quantum Eigensolver (VQE) algorithm to extract the ground state energies of these molecules. This energy data represents the expected ground state energy that a quantum computer will produce for the given molecules, on the STO-3G basis. Since there is a lot…
Set theory reduces all processes to assembly and disassembly. A similar architecture is proposed for nature as quantum computer. It resolves the classical space-time underlying Feynman diagrams into a quantum network of creation and annihilation processes, reducing kinematics to quantum statistics, and regularizing the Lie algebra of the Einstein diffeomorphism group. The usually separate and singular Lie algebras of kinematics, statistics, and conserved currents merge into one regular statistics Lie algebra.
Simplification of tensor updates toward performance-complexity balanced quantum computer simulation
Information Technology R&D Center, Mitsubishi Electric Corporation（三菱电机株式会社信息技术研发中心）；Institute for Physics of Intelligence, The University of Tokyo（东京大学物理智能研究所）；Department of Physics, University of Tokyo（东京大学物理学系）；Institute for Solid State Physics, The University of Tokyo（东京大学固体物理研究所）
Anyons obtained from a finite gauge theory have a computational power that depends on the symmetry group. The relationship between group structure and computational power is discussed in this paper. In particular, it is shown that anyons based on finite groups that are solvable but not nilpotent are capable of universal quantum computation. This extends previously published results to groups that are smaller, and therefore more practical. Additionally, a new universal gate-set is built out of an operation called a probabilistic projection, and a quasi-universal leakage correction scheme is dis…
Shor's and Grover's famous quantum algorithms for factoring and searching show that quantum computers can solve certain computational problems significantly faster than any classical computer. We discuss here what quantum computers_cannot_ do, and specifically how to prove limits on their computational power. We cover the main known techniques for proving lower bounds, and exemplify and compare the methods.
This study examines the simulation of quantum algorithms on a classical computer. The program code implemented on a classical computer will be a straight connection between the mathematical formulation of quantum mechanics and computational methods. The computational language will include formulations such as quantum state, superposition and quantum operator.
Equivalence of continuous- and discrete-variable gate-based quantum computers with finite energy
Chalmers University of Technology（查尔姆斯理工大学）；Technical University of Denmark（丹麦技术大学）；Johannes-Gutenberg University of Mainz（约翰内斯·古腾堡美因茨大学）；University of Oxford（牛津大学）
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…
Quantum computers have an exponential speed-up advantage over classical computers. One of the most prominent utilities of quantum computers is their ability to study complex quantum systems in various fields using quantum computational algorithms. Quantum computational algorithms can be used to study cosmological systems and how they behave with variations in the different parameters of the system. Here, we use the variational quantum eigensolver (VQE) algorithm to simulate the Hawking radiation phenomenon. VQE algorithm is a combination of quantum and classical computation methods used to obt…
Accounting for resources is the central issue in computational efficiency. We point out physical constraints implicit in information readout that have been overlooked in classical computing. The basic particle-counting mode of read-out sets a lower bound on the resources needed to implement a quantum computer. As a consequence, computers based on classical waves are as efficient as those based on single quantum particles.
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.
Quantum computational chemistry is a potential application of quantum computers that is expected to effectively solve several quantum-chemistry problems, particularly the electronic structure problem. Quantum computational chemistry can be compared to the conventional computational devices. This review comprehensively investigates the applications and overview of quantum computational chemistry, including a review of the Hartree-Fock method for quantum information scientists. Quantum algorithms, quantum phase estimation, and variational quantum eigensolver, have been applied to the post-Hartre…
Hypothesis elimination is a special case of Bayesian updating, where each piece of new data rules out a set of prior hypotheses. We describe how to use Grover's algorithm to perform hypothesis elimination for a class of probability distributions encoded on a register of qubits, and establish a lower bound on the required computational resources.
In the past few years it has been shown that universal quantum computation can be obtained by projective measurements alone, with no need for unitary gates. This suggests that the underlying logic of quantum computing may be an algebra of sequences of quantum measurements rather than an algebra of products of unitary operators. Such a Sequential Quantum Logic (SQL) was developed in the late 70's and has more recently been applied to the consistent histories framework of quantum mechanics as a possible route to the theory of quantum gravity. In this letter, I give a method for deciding the trut…
In this work we advance a generalization of quantum computational logics capable of dealing with some important examples of quantum algorithms. We outline an algebraic axiomatization of these structures.
There is no unique way to encode a quantum algorithm into a quantum circuit. With limited qubit counts, connectivity, and coherence times, a quantum circuit optimization is essential to make the best use of near-term quantum devices. We introduce a new circuit optimizer called AQCEL, which aims to remove redundant controlled operations from controlled gates, depending on initial states of the circuit. Especially, the AQCEL can remove unnecessary qubit controls from multi-controlled gates in polynomial computational resources, even when all the relevant qubits are entangled, by identifying zero…
Quantum computers are a promising candidate to radically expand computational science through increased computing power and more effective algorithms. In particular quantum computing could have a tremendous impact in the field of quantum cosmology. The goal of quantum cosmology is to describe the evolution of the Universe through the Wheeler-DeWitt equation or path integral methods without having to first formulate a full theory of quantum gravity. The quantum computer provides an advantage in this endeavor because it can perform path integrals in Lorentzian space and does not require construc…
Computational devices may be supplied with external sources of information (oracles). Quantum oracles may transmit phase information which is available to a quantum computer but not a classical computer. One consequence of this observation is that there is an oracle which is of no assistance to a classical computer but which allows a quantum computer to solve undecidable problems. Thus useful relativized separations between quantum and classical complexity classes must exclude the transmission of phase information from oracle to computer.
Numerous reports claim that quantum advantage, which should emerge as a direct consequence of the advent of quantum computers, will herald a new era of chemical research because it will enable scientists to perform the kinds of quantum chemical simulations that have not been possible before. Such simulations on quantum computers, promising a significantly greater accuracy and speed, are projected to exert a great impact on the way we can probe reality, predict the outcomes of chemical experiments, and even drive design of drugs, catalysts, and materials. In this work we review the current stat…
The quest for quantum computers is motivated by their potential for solving problems that defy existing, classical, computers. The theory of computational complexity, one of the crown jewels of computer science, provides a rigorous framework for classifying the hardness of problems according to the computational resources, most notably time, needed to solve them. Its extension to quantum computers allows the relative power of quantum computers to be analyzed. This framework identifies families of problems which are likely hard for classical computers (``NP-complete'') and those which are likel…
This article reviews the history of digital computation, and investigates just how far the concept of computation can be taken. In particular, I address the question of whether the universe itself is in fact a giant computer, and if so, just what kind of computer it is. I will show that the universe can be regarded as a giant quantum computer. The quantum computational model of the universe explains a variety of observed phenomena not encompassed by the ordinary laws of physics. In particular, the model shows that the the quantum computational universe automatically gives rise to a mix of rand…
A global race towards developing a gate-based, universal quantum computer that one day promises to unlock the never before seen computational power has begun and the biggest challenge in achieving this goal arguably is the quality implementation of a two-qubit gate. In a trapped-ion quantum computer, one of the leading quantum computational platforms, a two-qubit gate is typically implemented by modulating the individual addressing beams that illuminate the two target ions, which, together with others, form a linear chain. The required modulation, expectedly so, becomes increasingly more compl…
Gate-model quantum computers provide an experimentally implementable architecture for near term quantum computations. To design a reduced quantum circuit that can simulate a high complexity reference quantum circuit, an optimization should be taken on the number of input quantum states, on the unitary operations of the quantum circuit, and on the number of output measurement rounds. Besides the optimization of the physical layout of the hardware layer, the quantum computer should also solve difficult computational problems very efficiently. To yield a desired output system, a particular object…
We give a set of sufficient conditions on the experimental Boson-Sampling computer to satisfy Theorem 1.3 of Aaronson & Arkhipov (Theory of Computing \textbf{9}, 143 (2013)) stating a computational problem whose simulation on a classical computer would collapse the polynomial hierarchy of the computational complexity to the third level. This implies that such an experimental device is in conflict with the Extended Church-Turing thesis. In practical terms, we give a set of sufficient conditions for the scalability of the experimental Boson-Sampling computer beyond the power of the classical com…
Spectroscopy is an indispensable tool in understanding the structures and dynamics of molecular systems. However computational modelling of spectroscopy is challenging due to the exponential scaling of computational complexity with system sizes unless drastic approximations are made. Quantum computer could potentially overcome these classically intractable computational tasks, but existing approaches using quantum computers to simulate spectroscopy can only handle isolated and static molecules. In this work we develop a workflow that combines multi-scale modeling and time-dependent variational…
Nonlinear stochastic modeling is useful for describing complex engineering systems. Meanwhile, neuromorphic (brain-inspired) computing paradigms are developing to tackle tasks that are challenging and resource intensive on digital computers. An emerging scheme is reservoir computing which exploits nonlinear dynamical systems for temporal information processing. This paper introduces reservoir computers with output feedback as stationary and ergodic infinite-order nonlinear autoregressive models. We highlight the versatility of this approach by employing classical and quantum reservoir computer…
Following the recent great advance of quantum computing technology, there are growing interests in its applications to industries, including finance. In this paper, we focus on derivative pricing based on solving the Black-Scholes partial differential equation by finite difference method (FDM), which is a suitable approach for some types of derivatives but suffers from the {\it curse of dimensionality}, that is, exponential growth of complexity in the case of multiple underlying assets. We propose a quantum algorithm for FDM-based pricing of multi-asset derivative with exponential speedup with…
In [Sch05a], it is argued that Boltzmann's intuition, that the psychological arrow of time is necessarily aligned with the thermodynamic arrow, is correct. Schulman gives an explicit physical mechanism for this connection, based on the brain being representable as a computer, together with certain thermodynamic properties of computational processes. [Haw94] presents similar, if briefer, arguments. The purpose of this paper is to critically examine the support for the link between thermodynamics and an arrow of time for computers. The principal arguments put forward by Schulman and Hawking will…
Tensor networks permit computational and entanglement resources to be concentrated in interesting regions of Hilbert space. Implemented on NISQ machines they allow simulation of quantum systems that are much larger than the computational machine itself. This is achieved by parallelising the quantum simulation. Here, we demonstrate this in the simplest case; an infinite, translationally invariant quantum spin chain. We provide Cirq and Qiskit code that translate infinite, translationally invariant matrix product state (iMPS) algorithms to finite-depth quantum circuit machines, allowing the repr…
My 2018 lecture at the ICA workshop in Singapore dealt with quantum computation as a meeting point of the laws of computation and the laws of quantum mechanics. We described a computational complexity argument against the feasibility of quantum computers: we identified a very low-level complexity class of probability distributions described by noisy intermediate-scale quantum computers, and explained why it would allow neither good-quality quantum error-correction nor a demonstration of "quantum supremacy," namely, the ability of quantum computers to make computations that are impossible or ex…
Over the last century, a large number of physical and mathematical developments paired with rapidly advancing technology have allowed the field of quantum chemistry to advance dramatically. However, the lack of computationally efficient methods for the exact simulation of quantum systems on classical computers presents a limitation of current computational approaches. We report, in detail, how a set of pre-computed molecular integrals can be used to explicitly create a quantum circuit, i.e. a sequence of elementary quantum operations, that, when run on a quantum computer, to obtain the energy …
Current and near term quantum computers (i.e. NISQ devices) are limited in their computational power in part due to qubit decoherence. Here we seek to take advantage of qubit decoherence as a resource in simulating the behavior of real world quantum systems, which are always subject to decoherence, with no additional computational overhead. As a first step toward this goal we simulate the thermal relaxation of quantum beats in radical ion pairs (RPs) on a quantum computer as a proof of concept of the method. We present three methods for implementing the thermal relaxation, one which explicitly…
The computational power of a quantum computer is limited by the number of qubits available for information processing. Increasing this number within a single device is difficult; it is widely accepted that distributed modular architectures are the solution to large scale quantum computing. The major challenge in implementing such architectures is the need to exchange quantum information between modules. In this work, we show that a distributed quantum computing architecture with {\it limited} capacity to exchange information between modules can accurately solve quantum computational problems. …
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.
Max Planck Institute for Informatics（马克斯·普朗克信息学研究所）；Saarland Informatics Campus（萨尔州信息学园区）
Modern adiabatic quantum computers (AQC) are already used to solve difficult combinatorial optimisation problems in various domains of science. Currently, only a few applications of AQC in computer vision have been demonstrated. We review AQC and derive a new algorithm for correspondence problems on point sets suitable for execution on AQC. Our algorithm has a subquadratic computational complexity of the state preparation. Examples of successful transformation estimation and point set alignment by simulated sampling are shown in the systematic experimental evaluation. Finally, we analyse the d…
In this article we develop an algorithm which computes a divisor of an integer $N$, which is assumed to be neither prime nor the power of a prime. The algorithm uses discrete time heat diffusion on a finite graph. If $N$ has $m$ distinct prime factors, then the probability that our algorithm runs successfully is at least $p(m) = 1-(m+1)/2^{m}$. We compute the computational complexity of the algorithm in terms of classical, or digital, steps and in terms of diffusion steps, which is a concept that we define here. As we will discuss below, we assert that a diffusion step can and should be consid…
In this work we present a quantum algorithm for multiobjective combinatorial optimization. We show how to map a convex combination of objective functions onto a Hamiltonian and then use that Hamiltonian to prove that the quantum adiabatic algorithm of Farhi \emph{et al.} [arXiv:quant-ph/0001106] can find Pareto-optimal solutions in finite time provided certain convex combinations of objectives are used and the underlying multiobjective problem meets certain restrictions.
Spacetime foam, also known as quantum foam, has its origin in quantum fluctuations of spacetime. Arguably it is the source of the holographic principle, which severely limits how densely information can be packed in space. Its physics is also intimately linked to that of black holes and computation. In particular, the same underlying physics is shown to govern the computational power of black hole quantum computers.
