We investigate the problem of simulating classical stochastic processes through quantum dynamics, and present three scenarios where memory or time quantum advantages arise. First, by introducing and analysing a quantum version of the embeddability problem for stochastic matrices, we show that quantum memoryless dynamics can simulate classical processes that necessarily require memory. Second, by extending the notion of space-time cost of a stochastic process $P$ to the quantum domain, we prove an advantage of the quantum cost of simulating $P$ over the classical cost. Third, we demonstrate that the set of classical states accessible via Markovian master equations with quantum controls is larger than the set of those accessible with classical controls, leading, e.g., to a potential advantage in cooling protocols.
We demonstrate how to simulate both discrete and continuous stochastic evolution of a quantum many body system subject to measurements using matrix product states. A particular, but generally applicable, measurement model is analyzed and a simple representation in terms of matrix product operators is found. The technique is exemplified by numerical simulations of the anti-ferromagnetic Heisenberg spin-chain model subject to various instances of the measurement model. In particular we focus on local measurements with small support and non-local measurements which induces long range correlations…
Stochastic processes play a fundamental role in physics, mathematics, engineering and finance. One potential application of quantum computation is to better approximate properties of stochastic processes. For example, quantum algorithms for Monte Carlo estimation combine a quantum simulation of a stochastic process with amplitude estimation to improve mean estimation. In this work we study quantum algorithms for simulating stochastic processes which are compatible with Monte Carlo methods. We introduce a new ``analog'' quantum representation of stochastic processes, in which the value of the p…
Noisy quantum simulation is challenging since one has to take into account the stochastic nature of the process. The dominating method for it is the density matrix approach. In this paper, we evaluate conditions for which this method is inferior to a substantially simpler way of simulation. Our approach uses stochastic ensembles of quantum circuits, where random Kraus operators are applied to original quantum gates to represent random errors for modeling quantum channels. We show that our stochastic simulation error is relatively low, even for large numbers of qubits. We implemented this appro…
Simulating complex molecules and materials is an anticipated application of quantum devices. With strong quantum advantage demonstrated in artificial tasks, we examine how such advantage translates into modeling physical problems of correlated electronic structure. We simulate static and dynamical electronic structure on a superconducting quantum processor derived from Google's Sycamore architecture for two representative correlated electron problems: the nitrogenase iron-sulfur molecular clusters, and $α$-ruthenium trichloride, a proximate spin-liquid material. To do so, we simplify the elect…
Classical stochastic processes can be generated by quantum simulators instead of the more standard classical ones, such as hidden Markov models. One reason for using quantum simulators is that they generally require less memory than their classical counterparts. Here, we examine this quantum advantage for strongly coupled spin systems---the Dyson-like one-dimensional Ising spin chain with variable interaction length. We find that the advantage scales with both interaction range and temperature, growing without bound as interaction increases. Thus, quantum systems can very efficiently simulate …
No-go theorems on simulating uncertainty principle's signatures
H. H. Wills Physics Laboratory, University of Bristol（布里斯托大学）；Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）；Department of Physics, National Chung Hsing University（国立中兴大学物理系）；Physics Division, National Center for Theoretical Sciences（国家理论科学中心物理组）
As research on building scalable quantum computers advances, it is important to be able to certify their correctness. Due to the exponential hardness of classically simulating quantum computation, straight-forward verification through classical simulation fails. However, we can classically simulate small scale quantum computations and hence we are able to test that devices behave as expected in this domain. This constitutes the first step towards obtaining confidence in the anticipated quantum-advantage when we extend to scales which can no longer be simulated. Realistic devices have restricti…
Error mitigation is likely to be key in obtaining near term quantum advantage. In this work we present one of the first implementations of several Clifford data regression based methods which are used to mitigate the effect of noise in real quantum data. We explore the dynamics of the 1-D Ising model with transverse and longitudinal magnetic fields, highlighting signatures of confinement. We find in general Clifford data regression based techniques are advantageous in comparison with zero-noise extrapolation and obtain quantitative agreement with exact results for systems of 9 qubits with circ…
Recent years have seen unprecedented advance in the design and control of quantum computers. Nonetheless, their applicability is still restricted and access remains expensive. Therefore, a substantial amount of quantum algorithms research still relies on simulating quantum circuits on classical hardware. However, due to the sheer complexity of simulating real quantum computers, many simulators unrealistically simplify the problem and instead simulate perfect quantum hardware, i.e., they do not consider errors caused by the fragile nature of quantum systems. Stochastic quantum simulation provid…
Computer simulation of observable phenomena is an indispensable tool for engineering new technology, understanding the natural world, and studying human society. Yet the most interesting systems are often complex, such that simulating their future behaviour demands storing immense amounts of information regarding how they have behaved in the past. For increasingly complex systems, simulation becomes increasingly difficult and is ultimately constrained by resources such as computer memory. Recent theoretical work shows quantum theory can reduce this memory requirement beyond ultimate classical …
With the increasing crossover between quantum information and machine learning, quantum simulation of neural networks has drawn unprecedentedly strong attention, especially for the simulation of associative memory in Hopfield neural networks due to their wide applications and relatively simple structures that allow for easier mapping to the quantum regime. Quantum stochastic walk, a strikingly powerful tool to analyze quantum dynamics, has been recently proposed to simulate the firing pattern and associative memory with a dependence on Hamming Distance. We successfully map the theoretical sche…
The difficulty of simulating quantum systems, well-known to quantum chemists, prompted the idea of quantum computation. One can avoid the steep scaling associated with the exact simulation of increasingly large quantum systems on conventional computers, by mapping the quantum system to another, more controllable one. In this review, we discuss to what extent the ideas in quantum computation, now a well-established field, have been applied to chemical problems. We describe algorithms that achieve significant advantages for the electronic-structure problem, the simulation of chemical dynamics, p…
Quantum information processing provides remarkable advantages over its classical counterpart. Quantum optical systems are proved to be sufficient for realizing general quantum tasks, which however often rely on single photon sources. In practice, imperfect single photon sources, such as weak coherent state source, are used instead, which will inevitably limit the power in demonstrating quantum effects. For instance, with imperfect photon sources, the key rate of the BB84 quantum key distribution protocol will be very low, which fortunately can be resolved by utilizing the decoy state method. A…
On the quantum computational complexity of classical linear dynamics with geometrically local interactions: Dequantization and universality
Graduate School of Engineering Science, The University of Osaka（大阪大学工学研究科）；Center for Quantum Information and Quantum Biology, The University of Osaka（大阪大学量子信息与量子生物学中心）；Center for Quantum Computing, RIKEN（理化学研究所量子计算中心）
Quantum computing is an emerging technology, promising a paradigm shift in computing, and allowing for speedups in many different problems. However, quantum devices are still in their early stages, most with only a small number qubits. This places a reliance on simulation to develop quantum algorithms and to verify these devices. While there exists many algorithms for the simulation of quantum circuits, there is (at the time of writing) no tools which use OpenCL to parallelize this simulation, thereby taking advantage of devices such as GPUs while still remaining portable. In this paper, such …
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…
Many developing quantum technologies make use of quantum networks of different types. Even linear quantum networks are nontrivial, as the output photon distributions can be exponentially complex. Despite this, they can still be computationally simulated. The methods used are transformations into equivalent phase-space representations, which can then be treated probabilistically. This provides an exceptionally useful tool for the prediction and validation of experimental results, including decoherence. As well as experiments in Gaussian boson sampling, which are intended to demonstrate quantum …
Entanglement appears in two different ways in quantum mechanics, namely as a property of states and as a property of measurement outcomes in joint measurements. By combining these two aspects of entanglement, it is possible to generate nonlocality between particles that never interacted, using the protocol of entanglement swapping. We investigate the communication cost of classically simulating this process. While the communication cost of simulating nonlocal correlations of entangled states appears to be generally quite low, we show here that infinite communication is required to simulate ent…
The coherent Ising machine (CIM) enables efficient sampling of low-lying energy states of the Ising Hamiltonian with all-to-all connectivity by encoding the spins in the amplitudes of pulsed modes in an optical parametric oscillator (OPO). The interaction between the pulses is realized by means of measurement-based optoelectronic feedforward which enhances the gain for lower-energy spin configurations. We present an efficient method of simulating the CIM on a classical computer that outperforms the CIM itself as well as the noisy mean-field annealer in terms of both the quality of the samples …
Presented here is an algorithm for a type-II quantum computer which simulates the Ising model in one and two dimensions. It is equivalent to the Metropolis Monte-Carlo method and takes advantage of quantum superposition for random number generation. This algorithm does not require the ensemble of states to be measured at the end of each iteration, as is required for other type-II algorithms. Only the binary result is measured at each node which means this algorithm could be implemented using a range of different quantum computing architectures. The Ising model provides an example of how cellul…
The presented paper describes QSWalk.jl package for Julia programming language, developed for the purpose of simulating the evolution of open quantum systems. The package enables the study of quantum procedures developed using stochastic quantum walks on arbitrary directed graphs. We provide a detailed description of the implemented functions, along with a number of usage examples. The package is compared with the existing software offering a similar functionality.
The search for new, application-specific quantum computers designed to outperform any classical computer is driven by the ending of Moore's law and the quantum advantages potentially obtainable. Photonic networks are promising examples, with experimental demonstrations and potential for obtaining a quantum computer to solve problems believed classically impossible. This introduces a challenge: how does one design or understand such photonic networks? One must be able to calculate observables using general methods capable of treating arbitrary inputs, dissipation and noise. We develop novel com…
Stochastic processes underlie a vast range of natural and social phenomena. Some processes such as atomic decay feature intrinsic randomness, whereas other complex processes, e.g. traffic congestion, are effectively probabilistic because we cannot track all relevant variables. To simulate a stochastic system's future behaviour, information about its past must be stored and thus memory is a key resource. Quantum information processing promises a memory advantage for stochastic simulation that has been validated in recent proof-of-concept experiments. Yet, in all past works, the memory saving wo…
Stochastic differential equations (SDEs), which models uncertain phenomena as the time evolution of random variables, are exploited in various fields of natural and social sciences such as finance. Since SDEs rarely admit analytical solutions and must usually be solved numerically with huge classical-computational resources in practical applications, there is strong motivation to use quantum computation to accelerate the calculation. Here, we propose a quantum-classical hybrid algorithm that solves SDEs based on variational quantum simulation (VQS). We first approximate the target SDE by a tri…
While quantum algorithms for simulation exhibit better asymptotic scaling than their classical counterparts, they currently cannot be implemented on real-world devices. Instead, chemists and computer scientists rely on costly classical simulations of these quantum algorithms. In particular, the quantum phase estimation (QPE) algorithm is among several approaches that have attracted much attention in recent years for its genuine quantum character. However, it is memory-intensive to simulate and intractable for moderate system sizes. This paper discusses the performance and applicability of QPES…
Non-signalling quantum channels -- relevant in, e.g., the study of Bell and Einstein-Podolsky-Rosen scenarios -- may be simulated via affine combinations of local operations in bipartite scenarios. Moreover, when these channels correspond to stochastic maps between classical variables, such simulation is possible even in multipartite scenarios. These two results have proven useful when studying the properties of these channels, such as their communication and information processing power, and even when defining measures of the non-classicality of physical phenomena (such as Bell non-classicali…
Variational quantum eigensolver (VQE) is promising to show quantum advantage on near-term noisy-intermediate-scale quantum (NISQ) computers. One central problem of VQE is the effect of noise, especially the physical noise on realistic quantum computers. We study systematically the effect of noise for the VQE algorithm, by performing numerical simulations with various local noise models, including the amplitude damping, dephasing, and depolarizing noise. We show that the ground state energy will deviate from the exact value as the noise probability increase and normally noise will accumulate as…
Digital quantum computers are potentially an ideal platform for simulating open quantum many-body systems beyond the digital classical computers. Many studies have focused on obtaining the ground state by simulating time dynamics or variational approaches of closed quantum systems. However, dynamics of open quantum systems has not been given much attention with a reason being their non-unitary dynamics not natural to simulate on a set of unitary gate operations in quantum computing. Here we study prototypical models in open quantum systems with Trotterisations for the modified stochastic Schr{…
Improved precision scaling for simulating coupled quantum-classical dynamics
University of Toronto（多伦多大学）；Pacific Northwest National Laboratory（太平洋西北国家实验室）；Canadian Institute for Advanced Studies（加拿大高级研究所）；Boehringer Ingelheim（勃林格殷格翰）；University of Innsbruck（因斯布鲁克大学）
Simulating the stochastic evolution of real quantities on a digital computer requires a trade-off between the precision to which these quantities are approximated, and the memory required to store them. The statistical accuracy of the simulation is thus generally limited by the internal memory available to the simulator. Here, using tools from computational mechanics, we show that quantum processors with a fixed finite memory can simulate stochastic processes of real variables to arbitrarily high precision. This demonstrates a provable, unbounded memory advantage that a quantum simulator can e…
A Quantum Walk Inspired Qubit Lattice Algorithm for Simulating Electromagnetic Wave Propagation and Scattering in Conservative and Dissipative Magnetized Plasmas
Large-scale stochastic simulation of open quantum systems
Technical University of Munich（慕尼黑工业大学）；Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机性研究所）；Freie Universität Berlin（柏林自由大学）；Helmholtz-Zentrum Berlin für Materialien und Energie（亥姆霍兹柏林材料与能源中心）；Zuse Institute Berlin（祖斯研究所柏林）；Munich Quantum Software Company GmbH（慕尼黑量子软件有限公司）；Software Competence Center Hagenberg GmbH (SCCH)（哈根贝格软件能力中心有限公司）
We develop a workflow to use current quantum computing hardware for solving quantum many-body problems, using the example of the fermionic Hubbard model. Concretely, we study a four-site Hubbard ring that exhibits a transition from a product state to an intrinsically interacting ground state as hopping amplitudes are changed. We locate this transition and solve for the ground state energy with high quantitative accuracy using a variational quantum algorithm executed on an IBM quantum computer. Our results are enabled by a variational ansatz that takes full advantage of the maximal set of commu…
Dynamically Optimal Unraveling Schemes for Simulating Lindblad Equations
Institute of Natural Sciences, Shanghai Jiao Tong University（上海交大自然科学研究院）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）；Ministry of Education Key Laboratory in Scientific and Engineering Computing, Shanghai Jiao Tong University（上海交通大学教育部计算科学与工程技术重点实验室）；Zhiyuan College, Shanghai Jiao Tong University（上海交通大学致远学院）；Department of Mathematics, Pennsylvania State University（宾夕法尼亚州立大学数学系）
The possibility of simulating a stochastic process by the intrinsic randomness of quantum system is investigated. Two simulations of Markov Chains by the measurements of quantum systems are proposed.
Classical simulation of quantum computation is necessary for studying the numerical behavior of quantum algorithms, as there does not yet exist a large viable quantum computer on which to perform numerical tests. Tensor network (TN) contraction is an algorithmic method that can efficiently simulate some quantum circuits, often greatly reducing the computational cost over methods that simulate the full Hilbert space. In this study we implement a tensor network contraction program for simulating quantum circuits using multi-core compute nodes. We show simulation results for the Max-Cut problem o…
