Sampling from the thermal quantum Gibbs state and evaluating partition functions with a quantum computer (0905.2199v2)

Abstract: We present a quantum algorithm to prepare the thermal Gibbs state of interacting quantum systems. This algorithm sets a universal upper bound D^alpha on the thermalization time of a quantum system, where D is the system's Hilbert space dimension and alpha < 1/2 is proportional to the Helmholtz free energy density of the system. We also derive an algorithm to evaluate the partition function of a quantum system in a time proportional to the system's thermalization time and inversely proportional to the targeted accuracy squared.

• The paper introduces quantum algorithms that prepare thermal Gibbs states with a universal thermalization time bound scaling as D^(α).
• It employs quantum phase estimation and Grover's technique to amplify the desired state's amplitude and manage energy estimate fluctuations.
• The algorithms accurately evaluate partition functions with relative precision, overcoming classical challenges such as the sign problem in complex systems.

Quantum Algorithms for Thermal Gibbs State Preparation and Partition Function Evaluation

The paper presents a significant advancement in quantum computing concerning the thermalization of quantum systems and the precise evaluation of partition functions. The authors, David Poulin and Pawel Wocjan, introduce two quantum algorithms, each targeting critical aspects of statistical physics in quantum systems.

Quantum Algorithm for Preparing the Thermal Gibbs State

The algorithm prepares the thermal Gibbs state of interacting quantum systems, offering a universal upper bound on the thermalization time, expressed as $D^\alpha$, where $D$ is the Hilbert space dimension and $\alpha \leq \frac{1}{2}$, proportional to the Helmholtz free energy density. The significance of this algorithm lies in its ability to efficiently thermalize quantum systems, which traditionally pose substantial computational challenges due to the exponential scaling of the Hilbert space.

The methodology utilizes quantum phase estimation (QPE) and Grover's technique to amplify the desired Gibbs state's amplitude, marking a purified Gibbs state that can be efficiently manipulated within quantum circuits. The approach accounts for fluctuations in energy estimates by calculating medians from multiple QPEs, ensuring robustness in the preparation process. The theoretical underpinning establishes an efficient route to simulate local Hamiltonians, emphasizing scalability proportional to the system's size and energy bounds.

Quantum Algorithm for Partition Function Evaluation

The paper introduces an algorithm for computing the partition function of quantum systems. This algorithm estimates the partition function with relative accuracy $\epsilon$, with complexity scaling inversely with $\epsilon^2$ and directly proportional to the thermalization time. Unlike classical methods, it is not affected by the sign problem, which impairs Monte Carlo methods for frustrated and fermionic systems.

Quantum counting techniques are used in conjunction with the Gibbs state preparation to evaluate partition function fractions sequentially, leveraging the purity and marking of the Gibbs states. The algorithm ensures that statistical fluctuations remain suppressed, providing a reliable evaluation for systems where traditional methods fail.

Implications and Speculations on Future Developments

The implications of these algorithms are profound for the future of quantum simulations, particularly in statistical physics and condensed matter theory. Practically, these algorithms could vastly enhance the simulation of quantum materials and interactions at finite temperatures, leading to more accurate modeling of complex quantum phenomena.

Theoretically, the universal establishment of bounds on thermalization time opens avenues for exploring quantum equilibration and entropy production with precision surpassed by classical systems. Future research may focus on refining these algorithms to consider special properties like energy gaps or correlation lengths, potentially optimizing the efficiency beyond what is currently achievable.

In conclusion, while these algorithms represent a significant stride in quantum computation, ongoing research needs to address the aspects where quantum systems exhibit unique characteristics that might simplify certain complex interactions. This exploration could lead to more generalized models capable of robustly simulating a broader array of quantum systems.