Quantum Metropolis Sampling (0911.3635v2)

Abstract: The original motivation to build a quantum computer came from Feynman who envisaged a machine capable of simulating generic quantum mechanical systems, a task that is believed to be intractable for classical computers. Such a machine would have a wide range of applications in the simulation of many-body quantum physics, including condensed matter physics, chemistry, and high energy physics. Part of Feynman's challenge was met by Lloyd who showed how to approximately decompose the time-evolution operator of interacting quantum particles into a short sequence of elementary gates, suitable for operation on a quantum computer. However, this left open the problem of how to simulate the equilibrium and static properties of quantum systems. This requires the preparation of ground and Gibbs states on a quantum computer. For classical systems, this problem is solved by the ubiquitous Metropolis algorithm, a method that basically acquired a monopoly for the simulation of interacting particles. Here, we demonstrate how to implement a quantum version of the Metropolis algorithm on a quantum computer. This algorithm permits to sample directly from the eigenstates of the Hamiltonian and thus evades the sign problem present in classical simulations. A small scale implementation of this algorithm can already be achieved with today's technology

• The paper introduces a quantum Metropolis algorithm to sample Gibbs and ground states of many-body systems.
• The paper overcomes quantum measurement challenges by enabling state reversals without collapsing wave functions.
• The paper details a polynomial-time implementation using elementary quantum gates to resolve fermionic sign problems.

Quantum Metropolis Sampling

The paper introduces a quantum adaptation of the Metropolis algorithm that is designed for implementation on quantum computers. The original Metropolis algorithm, a pivotal development for simulating classical systems, has faced limitations when applied to quantum many-body systems due to the sign problem, especially in fermionic systems. This novel quantum algorithm aims to resolve such challenges, promising more direct simulations of quantum systems by avoiding the sign problem altogether.

Quantum computers, inspired by Feynman's vision, have the potential to simulate quantum mechanical systems far beyond the capabilities of classical computers. The ability to precisely simulate both the static and dynamic properties of these systems involves preparing quantum computers in significant states such as the Gibbs and ground states. Classical methods such as quantum Monte Carlo face scalability issues, particularly in systems without positive statistical weight mappings, such as those with fermionic particles.

This paper develops a direct quantum analog of the Metropolis algorithm. The primary contribution is an algorithm that allows sampling from the eigenstates of a given quantum Hamiltonian. It implements these steps using elementary quantum gates in polynomial time for certain classes of Hamiltonians. Importantly, this approach does not introduce additional errors from simulating thermodynamic interactions with a heat bath, like some previous suggestions.

Key Contributions:

1. Algorithm Design: The paper presents the algorithm’s design for simulating the Gibbs and ground states of quantum many-body systems. It includes procedures for performing random local unitary transformations and a method for deciding acceptance or rejection based on the quantum Metropolis criterion.
2. Overcoming Challenges: The algorithm addresses challenges associated with quantum measurements, such as how to reverse rejected moves without collapsing wave functions, which was historically a challenge due to the no-cloning theorem.
3. Implementation and Feasibility: The paper outlines how current quantum technology can implement a small-scale version of this algorithm. It revolves around coherent implementations of energy measurements followed by acceptance or rejection of state transitions.
4. Example Cases: The algorithm applies to a range of systems, including those relevant in condensed matter and high-energy physics, where previously the simulation was limited by the complexity of numerical approaches.

Future Implications and Applications

This quantum Metropolis framework has potential applications in quantum chemistry for calculating electronic binding energies, in condensed matter physics for exploring interacting fermionic systems, and in high-energy physics for problems like quark confinement.

Practically, the real power of this algorithm may fully manifest with more mature quantum computing technology that allows the simulation of larger and more complex many-body systems. The given examples provide a basis for future theoretical and computational studies to explore Hamiltonians that identify better "moves" to prompt faster convergence, perhaps finding quantum optimizations superior to classical techniques.

Moreover, the development of universality within the choice of unitary operations—that ensure ergodic evolution amongst system states—is poised to influence future algorithmic advancements. The implication of eradicating the fermionic sign problem is profound, opening possibilities for more efficient quantum simulations without exhaustive classical approximations.

This elegant merger of quantum computing and intricate thermodynamic simulations can inspire researchers broadly invested in quantum systems' statics and dynamics, prompting future studies that balance practical feasibility against algorithmic complexity.