Simulation Of Quantum Systems Research Articles

Average-Case Speedup for Product Formulas

Quantum simulation is a promising application of future quantum computers. Product formulas, or Trotterization, are the oldest and still remain an appealing method to simulate quantum systems. For an accurate product formula approximation, the state-of-the-art gate complexity depends on the number of terms in the Hamiltonian and a local energy estimate. In this work, we give evidence that product formulas, in practice, may work much better than expected. We prove that the Trotter error exhibits a qualitatively better scaling for the vast majority of input states, while the existing estimate is for the worst states. For general k-local Hamiltonians and higher-order product formulas, we obtain gate count estimates for input states drawn from any orthogonal basis. The gate complexity significantly improves over the worst case for systems with large connectivity. Our typical-case results generalize to Hamiltonians with Fermionic terms, with input states drawn from a fixed-particle number subspace, and with Gaussian coefficients (e.g., the SYK models). Technically, we employ a family of simple but versatile inequalities from non-commutative martingales called uniform smoothness, which leads to Hypercontractivity, namely p-norm estimates for k-local operators. This delivers concentration bounds via Markov’s inequality. For optimality, we give analytic and numerical examples that simultaneously match our typical-case estimates and the existing worst-case estimates. Therefore, our improvement is due to asking a qualitatively different question, and our results open doors to the study of quantum algorithms in the average case.

Bounding entanglement dimensionality from the covariance matrix

High-dimensional entanglement has been identified as an important resource in quantum information processing, and also as a main obstacle for simulating quantum systems. Its certification is often difficult, and most widely used methods for experiments are based on fidelity measurements with respect to highly entangled states. Here, instead, we consider covariances of collective observables, as in the well-known Covariance Matrix Criterion (CMC) \cite{guhnecova} and present a generalization of the CMC for determining the Schmidt number of a bipartite system. This is potentially particularly advantageous in many-body systems, such as cold atoms, where the set of practical measurements is very limited and only variances of collective operators can typically be estimated. To show the practical relevance of our results, we derive simpler Schmidt-number criteria that require similar information as the fidelity-based witnesses, yet can detect a wider set of states. We also consider paradigmatic criteria based on spin covariances, which would be very helpful for experimental detection of high-dimensional entanglement in cold atom systems. We conclude by discussing the applicability of our results to a multiparticle ensemble and some open questions for future work.

Unveiling the potential of phonons and photons in quantum computing and communication

This article explores the cutting-edge advancements in quantum technologies, focusing on the innovative use of atomic interactions and the role of phonons and photons in quantum systems. We delve into the recent discovery by researchers at the University of Washington, who have demonstrated the potential of atomic vibrations for tracking and transmitting quantum information, laying the groundwork for new communication systems, high-precision sensors, and powerful computational architectures. The discussion extends to the domain of optomechanics, where light and mechanical vibrations are intricately linked, offering new quantum phenomena for controlling single photons through integrated optical circuits. The study of excitons and their interaction with photons highlights the crucial role these quasi-particles play in the absorption and emission of light in semiconductors, and their potential for encoding and transmitting quantum information. Furthermore, the article addresses the challenges and solutions in leveraging phonons within quantum technologies, emphasizing their application in quantum computing, communication, and sensing. The inherent challenges of simulating quantum systems on classical computers are also discussed, alongside the pivotal role of software simulations in quantum algorithm development, education, and research. In conclusion, the article underscores the ongoing journey of quantum technology development, marked by significant challenges but driven by the immense potential to revolutionize computing, communication, and sensing. The integration of quantum principles into practical applications continues to push the boundaries of what is technologically feasible, heralding a new era of innovation and discovery in the quantum realm.

Digital Simulation of Single Qubit Markovian Open Quantum Systems: A Tutorial

One of the first proposals for the use of quantum computers was the simulation of quantum systems. Over the past three decades, great strides have been made in the development of algorithms for simulating closed quantum systems and the more complex open quantum systems. In this tutorial, we introduce the methods used in the simulation of single qubit Markovian open quantum systems. It combines various existing notations into a common framework that can be extended to more complex open system simulation problems. The only currently available algorithm for the digital simulation of single qubit open quantum systems is discussed in detail. A modification to the implementation of the simpler channels is made that removes the need for classical random sampling, thus making the modified algorithm a strictly quantum algorithm. The modified algorithm makes use of quantum forking to implement the simpler channels that approximate the total channel. This circumvents the need for quantum circuits with a large number of CNOT gates.Quanta 2023; 12: 131–163.

Complexity Phase Transitions Generated by Entanglement.

Entanglement is one of the physical properties of quantum systems responsible for the computational hardness of simulating quantum systems. But while the runtime of specific algorithms, notably tensor network algorithms, explicitly depends on the amount of entanglement in the system, it is unknown whether this connection runs deeper and entanglement can also cause inherent, algorithm-independent complexity. In this Letter, we quantitatively connect the entanglement present in certain quantum systems to the computational complexity of simulating those systems. Moreover, we completely characterize the entanglement and complexity as a function of a system parameter. Specifically, we consider the task of simulating single-qubit measurements of k-regular graph states on n qubits. We show that, as the regularity parameter is increased from 1 to n-1, there is a sharp transition from an easy regime with low entanglement to a hard regime with high entanglement at k=3, and a transition back to easy and low entanglement at k=n-3. As a key technical result, we prove a duality for the simulation complexity of regular graph states between low and high regularity.

Digital quantum simulation of the BCS model with a central-spin-like quantum processor

The simulation of quantum systems is one of the most promising applications of quantum computers. In this paper we present a quantum algorithm to perform digital quantum simulations of the BCS model on a quantum register with a star shaped connectivity map, as it is, e.g., featured by color centers in diamond. We show how to effectively translate the problem onto the quantum hardware and implement the algorithm using only the native interactions between the qubits. Furthermore we discuss the complexity of the circuit. We use the algorithm to simulate the dynamics of the BCS model by subjecting its mean-field ground state to a time-dependent perturbation. The quantum simulation algorithm is studied using a classical simulation.

Perturbation theory with quantum signal processing

Perturbation theory is an important technique for reducing computational cost and providing physical insights in simulating quantum systems with classical computers. Here, we provide a quantum algorithm to obtain perturbative energies on quantum computers. The benefit of using quantum computers is that we can start the perturbation from a Hamiltonian that is classically hard to solve. The proposed algorithm uses quantum signal processing (QSP) to achieve this goal. Along with the perturbation theory, we construct a technique for ground state preparation with detailed computational cost analysis, which can be of independent interest. We also estimate a rough computational cost of the algorithm for simple chemical systems such as water clusters and polyacene molecules. To the best of our knowledge, this is the first of such estimates for practical applications of QSP. Unfortunately, we find that the proposed algorithm, at least in its current form, does not exhibit practical numbers despite of the efficiency of QSP compared to conventional quantum algorithms. However, perturbation theory itself is an attractive direction to explore because of its physical interpretability; it provides us insights about what interaction gives an important contribution to the properties of systems. This is in sharp contrast to the conventional approaches based on the quantum phase estimation algorithm, where we can only obtain values of energy. From this aspect, this work is a first step towards “explainable'' quantum simulation on fault-tolerant quantum computers.

Grover’s Algorithm – Implementations and Implications

Ever since the realisation of the potential computational power of quantum-natured circuits, multiple quantum algorithms have been proposed, exploiting quantum superposition or quantum entanglement features to simulate quantum systems that classical computers cannot efficiently probe. The most notable among these strategies is Shor’s algorithm, which takes advantage of the quantum Fourier transform and has been shown capable of solving integer factorisation problems within polynomial time. This article’s focus is on one type of quantum algorithm based on amplitude amplification, namely Grover’s algorithm. This algorithm’s working principle is explained, and a discussion of its recent developments and possible direction of future reach is provided. In addition, implementations of Grover’s algorithm have been performed using the IBM Quantum Lab, followed by a reference to its connection to life sciences.

Efficient few-body calculations in finite volume

Simulating quantum systems in a finite volume is a powerful theoretical tool to extract information about them. Real-world properties of the system are encoded in how its discrete energy levels change with the size of the volume. This approach is relevant not only for nuclear physics, where lattice methods for few- and many-nucleon states complement phenomenological shell-model descriptions and ab initio calculations of atomic nuclei based on harmonic oscillator expansions, but also for other fields such as simulations of cold atomic systems. This contribution presents recent progress concerning finite-volume simulations of few-body systems. In particular, it discusses details regarding the efficient numerical implementation of separable interactions and it presents eigenvector continuation as a method for performing robust and efficient volume extrapolations.

MuSpinSim: spin dynamics calculations for muon science

MuSpinSim is a Python software to simulate muon (µSR) experiments. In particular, it simulates the spin dynamics of a system of a muon plus other spins such as electrons and atomic nuclei. MuSpinSim can simulate various common experimental setups used in µSR, such as zero, transverse and longitudinal field experiments; and it can simulate µSR experiments that are resolved in time, field, or temperature. Furthermore, MuSpinSim can account for the effects of hyperfine, dipolar, quadrupolar and Zeeman couplings, as well as simulate quantum systems exchanging energy with the environment with the Lindblad master equation. Finally, MuSpinSim can be used to fit experimental µSR data with simulations that use all of the capabilities described above. In this work, we present the Python package MuSpinSim with all the utilities it provides to facilitate simulations of µSR experiments, and we demonstrate the effectiveness of the method with some chosen example systems.

Current Challenge and Limitations in Quantum Computation

This paper reviews various engineering hurdles facing the field of quantum computing. Specifically, problems related to decoherence, state preparation, error correction, and implement ability of gates are considered. Computer computation is very advanced computers used in recent days for various applications, including cryptography, optimization problems, financial modelling, route and traffic optimization, manufacturing drug, chemical research, batteries, simulation of quantum systems, and even machine learning. In a quantum computer, two qubits can also represent the exact same four states (00,01,10 or 11). It’s an exciting field with lots of potential. But it faces lots of challenges and problems like maintaining qubits, interaction with the environment can cause decoherence. However, the adventure to harness the enormous power of quantum computer systems is packed with challenges and boundaries. In this discussion, we can embark on an exploration of the current boundaries that researchers and engineers face within the world of quantum computing, losing mild on the complexities and constraints that ought to be overcome to fully free up its ability. Reacher’s are working on error correction techniques to address this issue and make quantum computations more reliable. It’s fascinating area of ongoing research. In recent development in quantum computing on 4 October 2022 The Royal Swedish Academy of Sciences has decided to award the Nobel Prize in Physics 2022 to Alain Aspect (Institute d’Optique Graduate School – Université Paris-Saclay and École Polytechnique, Palaiseau, France ), John F. Clauser ( J.F. Clauser & Assoc., Walnut Creek, CA, USA ) Anton Zeilinger University of Vienna, Austria“for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum

Error mitigation of quantum teleportation on IBM quantum computers

Quantum computers can perform computational tasks beyond the capability of classical computers, such as simulating quantum systems in materials science and chemistry. Quantum teleportation is the transfer of quantum information across distances, relying on entangled states generated by quantum computing. It is becoming a more secure way of sending information, but there is noise in the results. We sought to mitigate the error of quantum teleportation which was simulated on IBM cloud quantum computers. We hypothesized that the noise on all IBM quantum computers could be mitigated with noise-mitigation matrices. We created a quantum teleportation circuit which ran with shots of 500, 1000, 5000, and 8192 for four different qubit states. We studied general error trends in each machine and created two types of noise-mitigation matrices: universal and machine-specific. We then compared the mitigated results of both types of matrices. We found that there was noise for every IBM quantum computer during the trials. The universal noise-mitigation matrix for quantum teleportation for the three tested machines decreased the error for most trials and varied between 1%-5% for most cases after mitigation. The machine-specific noise-mitigation matrix mitigated the error of most machines to just 1-2%, which was a dramatic decrease from unmitigated results (varied between 1%-16% across three machines). The error rates for the machine-specific matrices have less variability than the universal mitigation matrix. We concluded that a universal mitigation matrix could be found for the three machines, but the machine-specific noise mitigation matrices were able to achieve more accurate results.

Insight Into Quantum Computing

Newtonian physics has provided an explanation for the interactions occurring in the everyday world while encouraging the development of technology, such as computers, that utilize our understanding of physics. However, classical computers based on Newtonian physics fail to operate on an advanced level necessary in the growing fields of dark matter or machine learning. As an alternative, quantum mechanics provide the potential for an improved computing system. The nonlinearity and superposition properties of quantum computing ensure that quantum computers will be better suited in certain matters, such as simulating quantum systems or factoring insurmountable numbers, when compared to classical computers. Advances in machine learning and dark matter detection with the use of quantum technology exemplify the true potential of quantum computing. While quantum computing has the capability to advance the world, the realistic means of creating a quantum computer is hindered by the difficulties of quantum mechanics. For example, measuring qubits through entanglement and isolating a quantum computer are both tasks posing as barriers in the path of quantum computing. Nevertheless, experiments overcoming these obstacles demonstrate a means to achieve the implementation of quantum computers to solve scientific mysteries. This paper will review the potential applications of quantum computing while also considering the problems it faces. It will stress the importance of research in quantum mechanics and a future focused on improving quantum technology. 

Verification of colorable hypergraph states with stabilizer test

Many-body quantum states, as a matter of fact, are extremely essential to solve certain mathematical problems or simulate quantum systems in measurement-based quantum computation. However, how to verify large-scale quantum states, such as hypergraph states, is an exceedingly hard task for many-body quantum systems. Here, we propose a novel fault-tolerant solution for the verification of colorable hypergraph states by using the stabilizer test. Furthermore, our protocol is dramatically facilitated by making only Pauli-X and Pauli-Z measurements. For geometric structure hypergraph states, the computational complexity of our protocol is polynomial. As to appliance, it will be also applied to blind quantum computing based on the no-signaling principle.

Preface to the Special Issue on “Quantum Computing Algorithms and Computational Complexity”

In 1982, Richard Feynman stated that in order to simulate quantum systems, we would rather go for a sort of brand-new powered quantum processor instead of a classical one [...]

Zoo of silicon-based quantum bits

Zoo of silicon-based quantum bits

Recompilation-enhanced simulation of electron–phonon dynamics on IBM quantum computers

Simulating quantum systems is believed to be one of the first applications for which quantum computers may demonstrate a useful advantage. For many problems in physics, we are interested in studying the evolution of the electron–phonon Hamiltonian, for which efficient digital quantum computing schemes exist. Yet to date, no accurate simulation of this system has been produced on real quantum hardware. In this work, we consider the absolute resource cost for gate-based quantum simulation of small electron–phonon systems as dictated by the number of Trotter steps and bosonic energy levels necessary for the convergence of dynamics. We then apply these findings to perform experiments on IBM quantum hardware for both weak and strong electron–phonon coupling. Despite significant device noise, through the use of approximate circuit recompilation we obtain electron–phonon dynamics on current quantum computers comparable to exact diagonalisation. Our results represent a significant step in utilising near term quantum computers for simulation of quantum dynamics and highlight the novelty of approximate circuit recompilation as a tool for reducing noise.

Greedy algorithm based circuit optimization for near-term quantum simulation

Simulating quantum systems is believed to be one of the most important applications of quantum computers. On noisy intermediate-scale quantum (NISQ) devices, the high-level circuit designed by quantum algorithms for Hamiltonian simulation needs to consider hardware limitations such as gate errors and circuit depth before it can be efficiently executed. In this work, we develop a hardware-agnostic circuit optimization algorithm to reduce the overall circuit cost for Hamiltonian simulation problems. Our method employ a novel sub-circuit synthesis in intermediate representation and propose a greedy ordering scheme for gate cancellation to minimize the gate count and circuit depth. To quantify the benefits of this approach, we benchmark proposed algorithm on different Hamiltonian models. Compared with state-of-the-art generic quantum compilers and specific quantum simulation compiler, the benchmarking results of our algorithm show an average reduction in circuit depth by 16.5× (up to 64.1×) and in gate count by 7.8× (up to 23.7×). This significant improvement helps enhance the performance of Hamiltonian simulation in the NISQ era.

Entanglement from Tensor Networks on a Trapped-Ion Quantum Computer.

The ability to selectively measure, initialize, and reuse qubits during a quantum circuit enables a mapping of the spatial structure of certain tensor-network states onto the dynamics of quantum circuits, thereby achieving dramatic resource savings when simulating quantum systems with limited entanglement. We experimentally demonstrate a significant benefit of this approach to quantum simulation: the entanglement structure of an infinite system-specifically the half-chain entanglement spectrum-is conveniently encoded within a small register of "bond qubits" and can be extracted with relative ease. Using Honeywell's model H0 quantum computer equipped with selective midcircuit measurement and reset, we quantitatively determine the near-critical entanglement entropy of a correlated spin chain directly in the thermodynamic limit and show that its phase transition becomes quickly resolved upon expanding the bond-qubit register.

Simplification of the Gram Matrix Eigenvalue Problem for Quadrature Amplitude Modulation Signals.

In quantum information science, it is very important to solve the eigenvalue problem of the Gram matrix for quantum signals. This allows various quantities to be calculated, such as the error probability, mutual information, channel capacity, and the upper and lower bounds of the reliability function. Solving the eigenvalue problem also provides a matrix representation of quantum signals, which is useful for simulating quantum systems. In the case of symmetric signals, analytic solutions to the eigenvalue problem of the Gram matrix have been obtained, and efficient computations are possible. However, for asymmetric signals, there is no analytic solution and universal numerical algorithms that must be used, rendering the computations inefficient. Recently, we have shown that, for asymmetric signals such as amplitude-shift keying coherent-state signals, the Gram matrix eigenvalue problem can be simplified by exploiting its partial symmetry. In this paper, we clarify a method for simplifying the eigenvalue problem of the Gram matrix for quadrature amplitude modulation (QAM) signals, which are extremely important for applications in quantum communication and quantum ciphers. The results presented in this paper are applicable to ordinary QAM signals as well as modified QAM signals, which enhance the security of quantum cryptography.