# Quantum Simulation - Key Challenges and Effective Solutions

Quantum simulation is a major focus of quantum computing research due to the goal of solving classically-intractable problems in predicting the behavior of complex quantum systems* *across a broad spectrum of applications, including material science and drug discovery. To get there, key challenges need to be overcome:* *

- Error correction codes and improved hardware are needed to overcome limitations such as high error rates, limited qubit connectivity, and short coherence times
- Error suppression and advanced calibration are needed to provide accurate digital quantum simulation
- Error mitigation and noise-adaptive quantum simulation software are needed to counter the effects of noise
- Specialized algorithms may be needed so that quantum simulation technologies may solve specific problems
- Analog quantum simulation may allow larger problems to be addressed than a quantum computer simulator can handle due to exponential memory requirements
- Overall error-handling efficiency is needed to reduce the overhead of fault tolerance
- Quantum simulation software needs classical benchmarks to verify and validate results
- More quantum simulation technologies need to be accessible via the cloud to improve availability and to lower costs to the research community

* *It is important to note that none of these challenges are insurmountable. All of these solutions are believed to be achievable. The potential of quantum simulation is within reach.

**The Quantum Simulation Landscape**

Although there are different approaches to take, analog quantum simulation is the most promising. An engineered quantum system, typically consisting of an array of ultracold atoms, mimics a target quantum system. This system can then be precisely controlled to reproduce the desired behaviors and interactions. Although digital quantum simulation also uses quantum systems, it does so to solve equations, not mimic behaviors, and requires a level or fault tolerance that does not yet exist. Other methods require classical computation, and face the very challenges that are fueling research into quantum computers.

Recent research can shine a light on the limits of what is currently possible, as well as offer a glimpse of what might be coming next.

- Controlling quantum many-body dynamics in driven Rydberg atom arrays by a team of researchers from Harvard University, QuEra Computing, IST Austria, Stanford University, the University of California Berkeley, and the Massachusetts Institute of Technology discovered that periodically-excited Rydberg atoms, whether in 1D or 2D arrays, formed apparent time crystals. This could lead to new understandings of quantum thermalization and better control of complex entangled states in many-body systems. The potential applications include both quantum metrology and quantum information science.
- Quantum phases of matter on a 256-atom programmable quantum simulator by a team of researchers from Harvard University, QuEra Computing, the University of Innsbruck, the Austrian Academy of Sciences, Stanford University, the University of California Berkeley, and the Massachusetts Institute of Technology laid the groundwork for studying exotic quantum phases and non-equilibrium entanglement dynamics
*.* - Probing topological spin liquids on a programmable quantum simulator by a team of researchers from Harvard University, QuEra Computing, the University of Innsbruck, the Austrian Academy of Sciences, the Institute for Advanced Study, and the Massachusetts Institute of Technology engineered topologically-ordered toric code-type quantum spin liquids in 2D arrays of Rydberg atoms to study the ground state and excitations. This could be a step toward topological qubits and fault-tolerant quantum computation.
- Realizing quantum spin liquid phase on an analog Hamiltonian Rydberg simulator by representatives of QuEra Computing and Amazon Braket describes the use of Analog Hamiltonian Simulation (AHS) to emulate quantum mechanical systems, such as topological spin liquids, that are hard to simulate classically. Once the atoms are under control, they undergo time evolution intended to mimic the behavior of the target system. Being able to precisely control the parameters of the experiment allows for a deeper understanding of the observations. The analog approach allows larger scale problems as compared to digitally-controlled approaches, thus allowing spin liquids to be studied experimentally for the first time.

“Nature isn’t classical […] and if you want to make a simulation of Nature, you’d better make it quantum mechanical, and by golly it’s a wonderful problem because it doesn’t look so easy. […] I want to talk about the possibility that there is to be an exact simulation, that the computer will do exactly the same as nature.” (emphasis in original)

– Richard P. Feynman, International Journal of Theoretical Physics, Vol 21, Nos. 6/7, 1982

For a deeper overview than can be presented here, a video titled A review of Harvard and QuEra quantum simulation results with neutral atoms reviews several recently-published results that were obtained using neutral-atom machines. The 58:10 presentation, dated March 28, 2023, includes a review by three QuEra researchers, with a following Q&A period.

**Key Challenges in Quantum Simulations**

The most commonly cited challenges to quantum simulation, as well as quantum computation, are noise and decoherence, scalability to large numbers of qubits, high-error operations, developing efficient algorithms, mapping problems to available hardware, optimizing parameters with classical machine learning algorithms, prioritizing quantum systems to simulate, validating simulations in the absence of comparisons, allocating quantum and classical processing, reducing estimated resource requirements, and achieving computational advantages that can withstand classical challenges.

In the paper The inherent problem in quantum simulations — and how to tackle it, Jukka Knuutinen and Dr. Ljubomir Budinski of Quanscient address a more fundamental problem. Quantum mechanics is natively linear, but many of the problems that need to be solved involve non-linear equations. One approach they identify is to linearize (Carleman linearization) those equations and then solve those linear equations, but information loss can cause inaccuracy. Another approach they identify is the lattice Boltzmann method (LBM), a computational fluid dynamics (CFD) solver, which divides the problem into independent computational points. Parallelization allows significant computational speed-ups, but quantum-unsolvable non-linear terms remain. The elimination of non-linear terms and some hybrid quantum-classical approaches are hypothetical or not well tested.

**Effective Solutions for Quantum Simulations**

Attempts to overcome these challenges take many forms. The most notable ones include the classical optimization of parameterized quantum circuits, classical algorithms for approximating solutions, quantum error correction (QEC) codes for gate-based quantum algorithms, evolving quantum systems through adiabatic quantum computing, tensor network techniques such as Matrix Product States (MPS), quantum-inspired classical algorithms, hybrid classical-quantum algorithms, task-specific software frameworks and libraries, and task-specific algorithms.

In the article Scientists double the size of quantum simulations with entanglement forging, Robert Davis of IBM Research discusses a relatively new approach to effectually doubling the size of the systems that can be studied. An IBM team calculated the ground state energy of a 10-spin-orbital water molecule using only 5 qubits, halving the traditional qubit requirement. Whereas one qubit per feature is normally needed, the team was able to classically “cut” the problem into equal-sized groups and then classically “knit” the results together. The team identified weak entanglement, the limited entanglement between spin-up and spin-down orbitals, and divided the problem there. In principle, this approach can be applied to other problems and even to strong correlations, although the latter is considerably more challenging.