In this work, we present a quantum Markov chain algorithm for many-body systems that utilizes a special phase of matter known as the Many-Body Localized (MBL) phase. We show how the properties of the MBL phase enable one to address the conditions for ergodicity and sampling from distributions of quantum states. We demonstrate how to exploit the thermalized-to-localized transition to tune the acceptance rate of the Markov chain, and apply the algorithm to solve a range of combinatorial optimization problems of quadratic order and higher. The algorithm can be implemented on any quantum processing unit capable of simulating the Floquet dynamics of a one-dimensional Ising chain with nearest-neighbor interactions, providing a practical way of sampling from thermal distributions of Hamiltonians that cannot be natively implemented on the quantum hardware.
In this work, we present a quantum Markov chain algorithm for many-body systems that utilizes a special phase of matter known as the Many-Body Localized (MBL) phase. We show how the properties of the MBL phase enable one to address the conditions for ergodicity and sampling from distributions of quantum states. We demonstrate how to exploit the thermalized-to-localized transition to tune the acceptance rate of the Markov chain, and apply the algorithm to solve a range of combinatorial optimization problems of quadratic order and higher. The algorithm can be implemented on any quantum processing unit capable of simulating the Floquet dynamics of a one-dimensional Ising chain with nearest-neighbor interactions, providing a practical way of sampling from thermal distributions of Hamiltonians that cannot be natively implemented on the quantum hardware.
Quantum algorithms present a quadratically improved complexity over classical ones for certain sampling tasks. For instance, the Quantum Amplitude Estimation (QAE) algorithm promises to speed up the estimation of the mean of certain functions, given access to the quantum state corresponding to the probability distribution to be sampled from. Classically, samples are often obtained by simulating a Markov chain. In this work, we experimentally use encodings of Markov chains to prepare quantum states and run a quantum Markov Chain Monte Carlo (qMCMC) algorithm on Quantinuum System Model H2 and Quantinuum Helios quantum computers. We first demonstrate that it is possible to obtain accurate results by operating directly on the physical qubits. Crucially, to pave the way toward fault-tolerant execution, we also implement these algorithms using logical qubits protected by the Iceberg Quantum Error Detection (QED) code. By applying multiple rounds of mid-circuit QED during a random walk, we successfully catch and filter out hardware errors. These experiments demonstrate the feasibility of qMCMC on quantum hardware, bringing us closer to realizing quadratic quantum speedups with broad applications ranging from molecular modeling to machine learning and finance.
Quantum computing offers an alternative paradigm for addressing combinatorial optimization problems compared to classical computing. Despite recent hardware improvements, the execution of empirical quantum optimization experiments at scales known to be hard for state-of-the-art classical solvers is not yet in reach. In this work, we offer a different way to approach combinatorial optimization with near-term quantum computing. Motivated by the promising results observed in using quantum-enhanced Markov chain Monte Carlo (QeMCMC) for approximating complicated probability distributions, we combine ideas of sampling from the device with QeMCMC together with warm-starting and parallel tempering, in the context of combinatorial optimization. We demonstrate empirically that our algorithm recovers the global optima for instances of the Maximum Independent Set problem (MIS) up to 117 decision variables using 117 qubits on IBM quantum hardware. We show early evidence of a scaling advantage of our algorithm compared to similar classical methods for the chosen instances of MIS. MIS is practically relevant across domains like financial services and molecular biology, and, in some cases, already difficult to solve to optimality classically with only a few hundred decision variables.
Our discussion of Markov chains, with the exception of mentioning the Metropolis and heat-bath algorithms, has so far been very general with little contact with issues and opportunities related to specific applications. In this chapter, we recall that our target is many-body problems defined on a lattice and introduce several frameworks exploiting what is special about Markov processes for these types of problems. We consider here classical many-body problems, using the Ising model as the representative. Our discussion will be extended to various quantum many-body problems and algorithms in subsequent chapters. Many-body phase space The numerical difficulty of studying many-body problems on a lattice arises from the fact that their phase space Ω is a direct product of many phase spaces of local degrees of freedom. Generally, a local phase space is associated with each lattice site. If n is the size of this phase space and N is the number of lattice sites, then the number of states |Ω| available to the whole system is n N . In other words, the number of states in the phase space grows exponentially fast with the physical size of the system. For example, in the Ising model, the Ising spin s i on each site can take one of the two values ±1, and hence the number of states in the total phase space is 2 N . In Chapter 1, we noted that this exponential scaling thwarts deterministic solutions and is a reason why we use the Monte Carlo method. The exponentially large number of states implies that the enumeration of all states, which requires a computational effort proportional to the size of the phase space, is not an option for a problem with a large number of sites. As discussed in Section 2.7, the Monte Carlo method generally samples from a compressed phase space, avoiding the need to solve the entire problem. However, even with this advantage, can we reduce the computational effort to a manageable level? An obvious requirement is that we can equilibrate the simulation in a reasonable amount of computer time. We know that the Markov process converges to a stationary state sampling some distribution (Section 2.4), and for detailed balance algorithms, Rosenbluth's theorem (Section 2.6) guarantees monotonic convergence.
Sampling from complicated probability distributions is a hard computational problem arising in many fields, including statistical physics, optimization, and machine learning. Quantum computers have recently been used to sample from complicated distributions that are hard to sample from classically, but which seldom arise in applications. Here we introduce a quantum algorithm to sample from distributions that pose a bottleneck in several applications, which we implement on a superconducting quantum processor. The algorithm performs Markov chain Monte Carlo (MCMC), a popular iterative sampling technique, to sample from the Boltzmann distribution of classical Ising models. In each step, the quantum processor explores the model in superposition to propose a random move, which is then accepted or rejected by a classical computer and returned to the quantum processor, ensuring convergence to the desired Boltzmann distribution. We find that this quantum algorithm converges in fewer iterations than common classical MCMC alternatives on relevant problem instances, both in simulations and experiments. It therefore opens a new path for quantum computers to solve useful--not merely difficult--problems in the near term.
Monte Carlo methods are widely used to estimate observables in many-body quantum systems. However, conventional sampling schemes often require a large number of samples to achieve sufficient accuracy. In this work we propose the concentrated Monte Carlo sampling approach, which builds on the idea that in systems with only short range correlations, to obtain accurate expectation values for local observables, one would favor detailed information in the surroundings of this observable compared to far away from it. In this approach we consider all possible configurations in the surroundings of a local observable, and unique samples from the remaining of the setup drawn using Markov chain Monte Carlo. We have tested the performance of this approach for ground states of the spin-1/2 tilted Ising model in different phases, and also for thermal states in the a spin-1 bilinear-biquadratic model. Our results demonstrate that CMCS yields higher accuracy for local observables in short-range correlated states while requiring substantially fewer samples, showcasing in which regimes one can obtain acceleration for the evaluation of expectation values.
The Markov-chain Monte Carlo method is at the heart of efficient approximation schemes for a wide range of problems in combinatorial enumeration and statistical physics. It is therefore very natural and important to determine whether quantum computers can speed up classical mixing processes based on Markov chains. To this end, we present a quantum algorithm, making it possible to prepare a quantum sample---i.e., a coherent version of the stationary distribution of a reversible Markov chain. Our algorithm has a significantly better running time than that of a previous algorithm based on adiabatic-state generation. We also show that our methods provide a greater speedup over a recently proposed method for obtaining the ground states of (classical) Hamiltonians.