Researchers Simulate Systems with Memory Using Quantum Algorithms

Until now, quantum algorithms have efficiently simulated Markovian dynamical systems, where a system’s future depends only on its current state. IBM Research has, for the first time, developed quantum algorithms to efficiently simulate non-Markovian systems, where future evolution depends on past history. These algorithms provide an exponential speedup in system size compared to existing classical methods when the strength of the memory term, denoted as M, is less than one.

Researchers have created new quantum algorithms that model systems influenced by their past states, a characteristic called non-Markovian dynamics. Previously, quantum algorithms could only efficiently simulate systems where only the present state mattered; this work expands those capabilities to a broader range of complex phenomena. These algorithms efficiently simulate linear Volterra integro-differential equations, which describe systems with ‘memory effects’ that are challenging for standard computers to handle.

The significance of this advancement lies in its potential to model a wider array of physical and chemical processes accurately, as many real-world systems exhibit non-Markovian behaviour. Classical simulations of such systems often require immense computational resources, scaling polynomially with system size, making them intractable for all but the simplest cases.

Researchers at IBM Research have developed new quantum algorithms capable of simulating systems where the future state depends not only on the present, but also on a ‘memory’ of the past. However, simulating these systems becomes computationally difficult when the memory effect is strong, prompting the researchers to explore techniques for converting complex problems into simpler forms, a process they term Markovianization.

The ability to accurately model non-Markovian dynamics is crucial in fields like quantum chemistry, where the interactions between electrons can exhibit memory effects, and in materials science, where the history of a material influences its current properties.

Quantum Markovianization overcomes limitations in simulating systems with substantial memory effects

An exponential speedup in system size occurred when simulating linear Volterra integro-differential equations, or VIDEs, moving from polynomial scaling with classical algorithms to exponential. Previously, classical computers struggled with the computational demands of VIDEs due to their inherent ‘memory’ of past states, but this breakthrough initially happened when the strength of the memory effect, quantified by the parameter M, was less than one. This initial success hinged on the ability to efficiently represent and manipulate the integral kernel within the VIDE on a quantum computer.

The integral kernel defines how past states contribute to the present state, and its complexity often dictates the computational cost of the simulation. The exponential speedup arises from the quantum algorithms’ ability to perform certain calculations on the kernel in a fundamentally more efficient manner than classical algorithms. Efficiently transforming strongly-memory-affected VIDEs into solvable ordinary differential equations is achieved through a technique called Markovianization, extending this capability.

Markovianization involves approximating the non-Markovian dynamics with an effective Markovian description, effectively ‘tracing out’ the memory effects under certain conditions. This allows the problem to be recast as a standard quantum simulation of an ordinary differential equation, which is well-understood and efficiently solvable on a quantum computer.

Researchers at Massachusetts Institute of Technology and IBM Research have demonstrated a quantum algorithm capable of simulating linear Volterra integro-differential equations, or VIDEs, even when the system possesses strong memory effects, quantified by a parameter M greater than or equal to one. The conversion allows for exponential speedup in system size compared to classical methods, and the team circumvented previously established lower bounds demonstrating intractability for general-kernel VIDEs when M ≥ 1.

This is a significant result, as it challenges the conventional wisdom that strongly non-Markovian systems are inherently difficult to simulate. The technique transforms the complex VIDEs into a set of ordinary differential equations, enabling efficient quantum simulation. Applying the algorithm to the Mori-Zwanzig formalism, commonly used in modelling open quantum systems and fluid dynamics, served as a practical application. The Mori-Zwanzig formalism provides a hierarchical approach to describing non-Markovian dynamics, and the quantum algorithm can efficiently simulate the lower levels of this hierarchy.

However, the algorithms become less efficient when simulating systems with very strong “memory”, where past events heavily influence the future, as computational demands increase sharply with memory strength unless the system has a particular, simplifying structure. The efficiency of Markovianization depends on the specific form of the integral kernel and the strength of the memory effect; in some cases, the approximation may introduce significant errors.

Quantum simulation extends to systems exhibiting historical dependence

The researchers and IBM Research have unlocked a new capability for quantum computers, simulating systems where a system’s past significantly influences its future behaviour. Quantum simulation, previously focused on systems where only the present state mattered, now addresses a long-standing limitation. This advancement opens up new avenues for exploring complex phenomena in various scientific disciplines, including chemistry, physics, and engineering.

The parameter M represents the magnitude of the memory kernel, and its value dictates the computational resources required for the simulation. When M is large, the quantum algorithm requires more qubits and longer circuit depths, making it more susceptible to errors.

Quantum computers can now model systems possessing ‘memory effects’, where past states influence future behaviour, as the team have extended their capabilities. This represents a shift from previous quantum algorithms focused on Markovian dynamics, which only considered the present state when predicting evolution. The ability to accurately capture these memory effects is crucial for understanding a wide range of physical and chemical processes, such as the dynamics of open quantum systems, the behaviour of viscoelastic materials, and the evolution of biological systems. Their new algorithms efficiently simulate linear Volterra integro-differential equations, achieving speed improvements over classical methods, and are applicable to a range of physical systems exhibiting historical dependence. The potential applications of this technology are vast, ranging from the design of new materials with tailored properties to the development of more accurate models for predicting climate change and understanding complex biological processes. Further research will focus on extending these algorithms to handle more complex non-Markovian systems and improving their robustness to noise.

Researchers developed quantum algorithms capable of simulating systems with memory effects, where a system’s past influences its future state. This extends the scope of quantum simulation beyond systems dependent only on present conditions, allowing for the modelling of more complex dynamics.

The algorithms efficiently solve linear Volterra integro-differential equations, offering a computational advantage over classical approaches when the strength of the memory, represented by the parameter M, is sufficiently low. When M is large, the algorithms require specific system structures to remain efficient, and the authors intend to extend these methods to more complex systems in future work.