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Quantum Simulation and Dynamics

Rajat Kumar Vishwakarma

Published May 29, 2022• 0 citations
Publisher

Abstract

Today Quantum Computers have a wide range of applications, one of theprominent is quantum simulation. Simulations are used for many industrial andresearch purposes. Here we understand how can we simulate three atomsheisenberg model interaction on three qubits using real NISQ(NoisyIntermediate-Scale Quantum Computing) device ‘ibmq_jakarta’ first we performsimulation by classical method after that we will go on quantum approach,for 3qubit simulation its represented by 8×8 matrix. This is because there are 2³ = 8states in the N =3(N represents number of qubits used) system, if the simulation were50 particles(N=50) then it would be approx 10¹⁵×10¹⁵! Well beyond the capacity oftoday's classical computers, but for quantum computers this task will require only 15qubits.

DOI
10.31219/osf.io/6q37m

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Today Quantum Computers have a wide range of applications, one of theprominent is quantum simulation. Simulations are used for many industrial andresearch purposes. Here we understand how can we simulate three atomsheisenberg model interaction on three qubits using real NISQ(NoisyIntermediate-Scale Quantum Computing) device ‘ibmq_jakarta’ first we performsimulation by classical method after that we will go on quantum approach,for 3qubit simulation its represented by 8×8 matrix. This is because there are 2³ = 8states in the N =3(N represents number of qubits used) system, if the simulation were50 particles(N=50) then it would be approx 10¹⁵×10¹⁵! Well beyond the capacity oftoday's classical computers, but for quantum computers this task will require only 15qubits.
Today Quantum Computers have a wide range of applications, one of the prominent is quantum simulation. Simulations are used for many industrial and research purposes. Here we understand how can we simulate three atoms heisenberg model interaction on three qubits using real NISQ(Noisy Intermediate-Scale Quantum Computing) device ‘ibmq_jakarta’ first we perform simulation by classical method after that we will go on quantum approach,for 3 qubit simulation its represented by 8×8 matrix. This is because there are 2³ = 8 states in the N =3(N represents number of qubits used) system, if the simulation were 50 particles(N=50) then it would be approx 10¹⁵×10¹⁵! Well beyond the capacity of today's classical computers, but for quantum computers this task will require only 15 qubits.
Today Quantum Computers have a wide range of applications, one of the prominent is quantum simulation. Simulations are used for many industrial and research purposes. Here we understand how can we simulate three atoms heisenberg model interaction on three qubits using real NISQ(Noisy Intermediate-Scale Quantum Computing) device 'ibmq_jakarta' first we perform simulation by classical method after that we will go on quantum approach,for 3 qubit simulation its represented by 8×8 matrix. This is because there are 2³ = 8 states in the N =3(N represents number of qubits used) system, if the simulation were 50 particles(N=50) then it would be approx 10¹⁵×10¹⁵! Well beyond the capacity of today's classical computers, but for quantum computers this task will require only 15 qubits.
Quantum computing introduces a revolutionary approach to solving problems that are computationally beyond the reach of classical systems. Among its most promising applications is the simulation of many-body systems, where strong particle interactions lead to complex quantum behaviors. Traditional computational techniques often fail due to exponential resource requirements, but quantum simulators can efficiently model these interactions through entanglement and superposition. Recent advances in superconducting qubits, trapped ions, and photonic quantum devices have enabled small- scale experimental demonstrations of spin models, Hubbard systems, and correlated electron dynamics. These developments offer new insights into condensed matter physics and quantum phase transitions. As quantum technologies continue to evolve, they hold the potential to transform the study of materials, superconductivity, and other many-body phenomena with remarkable precision and scalability.
This chapter provides a basic overview of Quantum Computers (QCs) used in Quantum Computing (Quacomp) applications. Classical computers utilize silicon-based semiconductor devices. In recent years integrated circuit lateral feature sizes have been reduced to 10 nm, increasing chip density; however, as these sizes progress to be reduced to 5 nm and smaller, silicon-based circuits become impractical. QCs make use of quantum phenomena such as entanglement and superposition to increase and improve the computation speed as compared to classical computers, enabling them to solve some computationally complex problems in practical time windows. At this juncture QCs are not able to act as replacements for general- purpose computers, but they can outperform classical computers in a set of specific computational problems, such as optimization problems or for identifying the best solution from a set of feasible solutions. This chapter covers three QC approaches: the circuit model (or gate model), the adiabatic quantum computation model, and the categorical quantum model. Quacomp has experienced relatively rapid development in recent years, starting with the basic ideation in the early 1980s, then continuing with early proof of concepts for hardware in the next two decades, and now, in the 2020s, with QC hardware that can implement over one-thousand qubits (e.g., [ TIL22 ], [ FEY82 ], [ LEU00 ], [ NEG06 ], [ PLA07 ], [ MON11 ], [ MON16 ], [ CAS23 ]). Figure 5.1 (courtesy of IBM) depicts recent illustrative achievements in practical QC development at IBM, a key player in this arena (in the fall of 2023, IBM® announced the Condor prototype, the world’s first universal QC with 1,121 qubits 1 ). The generation of QCs at the time of this writing is known as Noisy-Intermediate Scale Quantum (NISQ) devices (some proponents cite this as being the third generation). These NISQ QCs are typically cooled to cryogenic temperatures (around −273° C) in order to operate efficiently; these environmental conditions are challenging to maintain in other than laboratory settings. In practical terms, QCs remain of somewhat limited and specialized functionality and are prone to computational errors. Considering these limitations, Hybrid QC (HQC) 170 171 172 arrangements and Hybrid Quantum Applications (HQApps) endeavor to combine classical and QC systems in order to undertake calculations directed at various (but specific) optimization or related use cases; tasks such as preparing inputs for quantum algorithms, storing data, or interacting with the user are typically still performed using classical computers. The number of operations in a QC typically equates with the number of quantum gates and the number of qubits supported; however, the larger the number of gates, the more likely it is that errors, including decoherence, are introduced into the quantum state. NISQ entails a somewhat limited number of logical qubits and associated physical qubits to support error correction schemes. Fortunately, a category of algorithms running on NISQ devices can proceed with only a small number of qubits and exhibit some degree of noise resilience; typically, they are hybrid algorithms, with a portion of the computational steps performed on a quantum device and with other computational steps running on a conventional computer. The bottom line is that with NISQ devices, the number of operations is typically moderate at this juncture, thus the scope and type of algorithms that can be implemented is relatively limited. However, there is research that shows that even these QCs can outperform conventional computers on a set of mathematical problems (e.g., [ PRE18 ], [ BRO19 ], [ MAD22 ]). The next step in the evolution is the interconnection of QCs over a distance network, as discussed in Chapter 6 . A classification of fundamental QC technologies has been proposed the European standards body CEN-CENELEC, specifically the Focus Group on Quantum Technologies (FGQT), which was established in June 2020 to coordinate and support the development of standards relevant for European industry and research [ OVA22 ]. See Table 5.1 .
An ultimate goal of quantum computing is to perform calculations beyond the reach of any classical computer.It is therefore imperative that useful quantum computers be very difficult to simulate classically, otherwise classical computers could be used for the applications envisioned for the quantum ones.Perfect quantum computers are unarguably exponentially difficult to simulate: the classical resources required grow exponentially with the number of qubits N or the depth D of the circuit.This difficulty has triggered recent experiments on deep, random circuits that aim to demonstrate that quantum devices may already perform tasks beyond the reach of classical computing.These real quantum computing devices, however, suffer from many sources of decoherence and imprecision which limit the degree of entanglement that can actually be reached to a fraction of its theoretical maximum.They are characterized by an exponentially decaying fidelity F ∼ ð1 -ϵÞ ND with an error rate ϵ per operation as small as ≈1% for current devices with several dozen qubits or even smaller for smaller devices.In this work, we provide new insight on the computing capabilities of real quantum computers by demonstrating that they can be simulated at a tiny fraction of the cost that would be needed for a perfect quantum computer.Our algorithms compress the representations of quantum wave functions using matrix product states, which are able to capture states with low to moderate entanglement very accurately.This compression introduces a finite error rate ϵ so that the algorithms closely mimic the behavior of real quantum computing devices.The computing time of our algorithm increases only linearly with N and D in sharp contrast with exact simulation algorithms.We illustrate our algorithms with simulations of random circuits for qubits connected in both one-and twodimensional lattices.We find that ϵ can be decreased at a polynomial cost in computing power down to a minimum error ϵ ∞ .Getting below ϵ ∞ requires computing resources that increase exponentially with ϵ ∞ =ϵ.For a two-dimensional array of N ¼ 54 qubits and a circuit with control-Z gates, error rates better than state-of-the-art devices can be obtained on a laptop in a few hours.For more complex gates such as a SWAP gate followed by a controlled rotation, the error rate increases by a factor 3 for similar computing time.Our results suggest that, despite the high fidelity reached by quantum devices, only a tiny fraction ð∼10 -8 Þ of the system Hilbert space is actually being exploited.
Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the near future. Quantum computers with 50-100 qubits may be able to perform tasks which surpass the capabilities of today's classical digital computers, but noise in quantum gates will limit the size of quantum circuits that can be executed reliably. NISQ devices will be useful tools for exploring many-body quantum physics, and may have other useful applications, but the 100-qubit quantum computer will not change the world right away - we should regard it as a significant step toward the more powerful quantum technologies of the future. Quantum technologists should continue to strive for more accurate quantum gates and, eventually, fully fault-tolerant quantum computing.