It is shown that a class of separately frustration-free (SFF) Hamiltonians can be Monte Carlo simulated efficiently on a classical computing machine, because such an SFF Hamiltonian corresponds to a Gibbs wavefunction whose nodal structure is efficiently computable by solving a small subsystem associated with a low-dimensional configuration subspace. It is further demonstrated that SFF Hamiltonians can be designed to implement universal ground state quantum computation. The two results combined have effectively solved the notorious sign problem in Monte Carlo simulations, and proved that all bounded-error quantum polynomial time algorithms admit bounded-error probabilistic polynomial time simulations.
It is shown that a class of separately frustration-free (SFF) Hamiltonians can be Monte Carlo simulated efficiently on a classical computing machine, because such an SFF Hamiltonian corresponds to a Gibbs wavefunction whose nodal structure is efficiently computable by solving a small subsystem associated with a low-dimensional configuration subspace. It is further demonstrated that SFF Hamiltonians can be designed to implement universal ground state quantum computation. The two results combined have effectively solved the notorious sign problem in Monte Carlo simulations, and proved that all bounded-error quantum polynomial time algorithms admit bounded-error probabilistic polynomial time simulations.
A restricted path integral method is proposed to simulate a type of quantum system or Hamiltonian called a sum of controlled few-fermions on a classical computer using Monte Carlo without a numerical sign problem. Then a universality is proven to assert that any bounded-error quantum polynomial time (BQP) algorithm can be encoded into a sum of controlled few-fermions and simulated efficiently using classical Monte Carlo. Therefore, BQP is precisely the same as the class of bounded-error probabilistic polynomial time (BPP), namely, BPP = BQP.
Quantum Monte Carlo simulations, while being efficient for bosons, suffer from the "negative sign problem" when applied to fermions--causing an exponential increase of the computing time with the number of particles. A polynomial time solution to the sign problem is highly desired since it would provide an unbiased and numerically exact method to simulate correlated quantum systems. Here we show that such a solution is almost certainly unattainable by proving that the sign problem is nondeterministic polynomial (NP) hard, implying that a generic solution of the sign problem would also solve all problems in the complexity class NP in polynomial time.
The simulation of quantum systems is the most promising candidate for computational advantagesof large scale, fault-tolerant quantum computers over classical computers. Two fundamental prob- lems that appear when simulating quantum systems on quantum computers are the preparation of good input states to the simulation and the problem of decomposing the time evolution operator ???? into primitive gates that are available on the quantum computer. This thesis presents two new algorithms for solving these problems. The first is a time evolution algorithm that approximates the time evolution operator by combining two previously developed techniques. We bound the error of this composite approach and determine analytic conditions on when this can lead to cost savings of the overall simulation. These savings are then verified through extensive numerics. This simulation algorithm can then be used as a subroutine to prepare thermal states, which are typical input states that have a high overlap with the ground state. Our algorithm is inspired by Hamiltonian Monte Carlo, which is a classical sampling algorithm that utilizes Hamiltonian dynamics to sample from arbitrary probability distributions. We bound the cost of our quantum thermal state preparation algorithm in terms of the total amount of time needed to simulate the Hamiltonian of the system in four scenarios: single qubit systems, truncated harmonic oscillators, non-degenerate systems where one knows the eigenvalues, and non-degenerate systems where no eigenvalue knowledge is present. We numerically demonstrate that this algorithm works as guaranteed and can significantly outperform analytic worst-case bounds.
Quantum Monte Carlo and quantum simulation are both important tools for
understanding quantum many-body systems. As a classical algorithm, quantum
Monte Carlo suffers from the sign problem, preventing its applications to most
fermion systems and real-time dynamics. In contrast, on a quantum computer, we
can simulate time evolution of general systems with local interactions, yet
such algorithms usually require the development of fault-tolerant quantum
technologies. In this paper, we introduce a novel non-variational algorithm
using quantum computing as a subroutine to accelerate quantum Monte Carlo. On a
fault-tolerant quantum computer, the complexity of our algorithm is polynomial
as the same as conventional quantum simulation algorithms. On a noisy quantum
computer, our algorithm incorporating error mitigation eases the sign problem.
Even when the circuit noise is significant, our algorithm can reduce the Monte
Carlo variance by orders of magnitude.
Reliable simulations of correlated quantum systems, including high-temperature superconductors and frustrated magnets, are increasingly desired nowadays to further our understanding of essential features in such systems. Quantum Monte Carlo (QMC) is a unique numerically exact and intrinsically unbiased method to simulate interacting quantum many-body systems. More importantly, when QMC simulations are free from the notorious fermion sign problem, they can reliably simulate interacting quantum models with large system size and low temperature to reveal low-energy physics such as spontaneously broken symmetries and universal quantum critical behaviors. Here, we concisely review recent progress made in developing new sign-problem-free QMC algorithms, including those employing Majorana representation and those utilizing hot-spot physics. We also discuss applications of these novel sign-problem-free QMC algorithms in simulations of various interesting quantum many-body models. Finally, we discuss possible future directions of designing sign-problem-free QMC methods.
Understanding strongly correlated electrons is an important long-term goal, not only for uncovering fundamental physics behind, but also for their emergence of lots of novel states which have potential applications in quantum control and quantum computations. Meanwhile, the strongly correlated electrons are usually extremely hard problems, and it is generally impossible to understand them unbiasedly. Quantum Monte Carlo is a typical unbiased numeric method, which does not depend on any perturbation, and it can help us to exactly understand the strongly correlated electrons, so that it is widely used in high energy and condensed matter physics. However, quantum Monte Carlo usually suffers from the notorious sign problem. In this paper, we introduce general ideas to design sign problem free models and discuss the sign bound theory we proposed recently. In the sign bound theory, we build a direct connection between the average sign and the ground state properties of the system. We find usually the average sign has the conventional exponential decay with system size increasing, leading to exponential complexity; but for some cases it can have algebraic decay, so that quantum Monte Carlo simulation still has polynomial complexity. By designing sign problem free or algebraic sign behaved strongly correlated electron models, we can approach to several long outstanding problems, such as the itinerant quantum criticality, the competition between unconventional superconductivity and magnetism, as well as the recently found correlated phases and phase transitions in moiré quantum matter.