The Promise of Digital Quantum Simulation
Simulating quantum many-body systems is a key application for emerging quantum processors. While analog quantum simulation has already demonstrated quantum advantage, its digital counterpart has recently become the focus of intense research interest due to the availability of devices that aim to realize general-purpose quantum computers.
In this article, we’ll take a closer look at the current state of digital quantum simulation (DQS) and explore the potential it holds for solving complex problems in physics, chemistry, and materials science. We’ll discuss the various hardware platforms being explored, the algorithmic advancements that are enabling new breakthroughs, and the challenges that need to be overcome to unlock the full power of DQS.
The Computational Challenge of Quantum Many-Body Systems
The reason that quantum simulation is a computationally hard problem is due to the tensor product structure of the Hilbert space for combined quantum systems. As the size of a quantum system increases, the dimension of the Hilbert space grows exponentially with the number of degrees of freedom.
Modern classical algorithms, such as tensor networks and neural network quantum states, attempt to limit the impact of this exponential scaling by using a low-rank representation of quantum states. However, this approach is not always applicable, as many physical states do not possess such a representation, especially for highly entangled quantum matter in two and higher dimensions and in systems out of equilibrium.
One way to tackle this issue is to find a quantum platform that acts as a surrogate for the system one wants to investigate, i.e., analog quantum simulation. Control parameters are used to tune the behavior of the physical quantum platform to match that of the system, and by measuring the state on the platform, it is possible to learn about the dynamics of the system being studied.
The Rise of Digital Quantum Simulation
In contrast to analog quantum simulation, digital quantum simulation (DQS) uses a gate-based quantum computer to simulate the quantum system. In its original formulation, it involves discretizing the time evolution of the system and breaking it up into a series of small time steps, which can be implemented using quantum gates.
DQS has the potential advantage over analog quantum simulation that it allows for universal simulation of many-body dynamics, particularly for systems that do not ‘fit’ onto analog quantum simulators. For example, it might be impossible to simulate certain many-body interactions as they are physically not realizable in hardware, but their digital decomposition can be implemented on a gate-based platform.
Hardware Platforms for Digital Quantum Simulation
Various technologies have emerged as potential platforms for gate-based quantum computing and DQS. Here’s a brief overview of some of the most promising approaches:
Laser-Cooled Neutral Atoms: This technology has recently reached a level that allows for digital quantum computation, with qubits realized as electronic states of the atoms, which are entangled via long-range Rydberg states.
Trapped Ions: Trapped ions were one of the first platforms proposed as practical quantum computers, with the first quantum gate implemented in 1995. Trapped ions have a long coherence time, as the qubits are formed by internal electronic states which are well isolated.
Superconducting Qubits: Over the past two decades, the coherence time of superconducting qubits has seen significant improvement, and the artificial nature of these qubits has positioned superconducting quantum processors at the forefront in terms of sheer qubit count.
Linear Optics: Recent developments have focused on using single photon degrees of freedom for quantum computing in integrated photonic circuits, where beam splitters, mirrors, phase shifters, and non-linear interactions in matter can be used to manipulate the quantum states of the photons.
Other Solid-State Approaches: Platforms such as nitrogen-vacancy centers in diamond and quantum dots are also being explored as possible quantum computing platforms, with some already used for analog quantum simulation.
While each of these platforms has its own strengths and limitations, the field of digital quantum simulation is rapidly evolving, and researchers are working to address the key challenges facing these technologies.
Algorithmic Advances in Digital Quantum Simulation
The development of digital quantum simulation has been driven by both non-variational and variational approaches. Let’s explore some of the key results in each of these areas.
Non-Variational Approaches
The Trotterized approach to simulate the time-evolution of quantum many-body systems was one of the first proposed algorithms for quantum simulation with a provable quantum advantage. This approach relies on the assumption that the Hamiltonian of the system can be decomposed into polynomially many, local operators.
Experiments using superconducting transmon qubits and trapped ions have demonstrated the feasibility of this approach, simulating Ising, Heisenberg, and other spin-1/2 models. Fermionic models have also been successfully implemented, encoding them via the Jordan-Wigner transformation or tree-based encodings.
Another interesting application of non-variational DQS is the simulation of topological systems, which are of great interest for understanding quantum error correction codes. Experiments have shown the ability to prepare the topologically ordered ground state of the toric code and measure the entanglement entropy, as well as simulate topological Floquet phases.
Variational Approaches
To overcome the limitations of noise and gate errors in NISQ devices, a novel paradigm emerged: the concept of variational quantum algorithms. These hybrid quantum-classical approaches combine the ability of a classical computer to efficiently optimize scalar functions of multiple, real variables and of a quantum computer to represent states in high-dimensional Hilbert spaces and measure corresponding expectation values.
The variational quantum eigensolver (VQE) was one of the first variational algorithms to be employed on a photonic quantum computer. The VQE approach approximates the ground state of a quantum system via a parameterized quantum circuit, with the target function being the energy of the variational state.
Variational approaches have also been explored for computing excited states, as well as for implementing time evolution. Techniques like variational fast forwarding and hybrid algorithms that utilize a qubit representation of the density matrix have shown promising results in simulating the dynamics of quantum many-body systems on NISQ devices.
Overcoming the Challenges of NISQ Devices
While the advancements in digital quantum simulation are exciting, the current limitations of Noisy Intermediate-Scale Quantum (NISQ) devices pose significant challenges. The primary sources of errors in these devices are decoherence and gate imperfections, which can quickly lead to the breakdown of the potential quantum advantage.
To address these issues, researchers have developed various error mitigation techniques, such as readout error mitigation, zero noise extrapolation, and optimized circuit compilation. By combining these methods, it is possible to significantly enhance the performance of NISQ devices and push the boundaries of what can be achieved with digital quantum simulation.
The Promise of Quantum Simulation for Practical Applications
As we look to the future, the potential of digital quantum simulation to unlock new discoveries in physics, chemistry, and materials science is particularly promising. While the pursuit of improved quantitative accuracy is essential, I believe that problems of a qualitative nature are much more suitable for near-term devices.
For example, gaining a better understanding of the complex phase diagram of high-temperature superconductors, unraveling the mechanisms behind many-body localization, and exploring the behavior of quantum systems out of equilibrium are all areas where digital quantum simulation could provide valuable insights, even if the initial energy measurements have large error bars.
By shifting our focus towards fundamental problems that involve the nature of quantum states, their stability, phase transitions, and symmetries, we can leverage the unique capabilities of digital quantum simulation to make progress in areas that have remained challenging for classical computational methods.
Moreover, the advancements in digital quantum simulation could potentially enable a full quantum technology circle, where quantum computing is used to enhance the simulation capabilities, which in turn feed back into the development of novel quantum hardware and algorithms. Fostering this synergistic relationship between quantum simulation, quantum matter, and quantum computing could unlock a new era of scientific discovery and technological innovation.
