Non-stoquastic Hamiltonians have both positive and negative signs in off-diagonal elements in their matrix representation in the standard computational basis and thus cannot be simulated efficiently by the standard quantum Monte Carlo method due to the sign problem. We describe our analytical studies of this type of Hamiltonians with infinite-range non-random as well as random interactions from the perspective of possible enhancement of the efficiency of quantum annealing or adiabatic quantum computing. It is shown that multi-body transverse interactions like XX and XXXXX with positive coefficients appended to a stoquastic transverse-field Ising model render the Hamiltonian non-stoquastic and reduce a first-order quantum phase transition in the simple transverse-field case to a second-order transition. This implies that the efficiency of quantum annealing is exponentially enhanced, because a first-order transition has an exponentially small energy gap (and therefore exponentially long computation time) whereas a second-order transition has a polynomially decaying gap (polynomial computation time). The examples presented here represent rare instances where strong quantum effects, in the sense that they cannot be efficiently simulated in the standard quantum Monte Carlo, have analytically been shown to exponentially enhance the efficiency of quantum annealing for combinatorial optimization problems.
Non-stoquastic Hamiltonians have both positive and negative signs in off-diagonal elements in their matrix representation in the standard computational basis and thus cannot be simulated efficiently by the standard quantum Monte Carlo method due to the sign problem. We describe our analytical studies of this type of Hamiltonians with infinite-range non-random as well as random interactions from the perspective of possible enhancement of the efficiency of quantum annealing or adiabatic quantum computing. It is shown that multi-body transverse interactions like XX and XXXXX with positive coefficients appended to a stoquastic transverse-field Ising model render the Hamiltonian non-stoquastic and reduce a first-order quantum phase transition in the simple transverse-field case to a second-order transition. This implies that the efficiency of quantum annealing is exponentially enhanced, because a first-order transition has an exponentially small energy gap (and therefore exponentially long computation time) whereas a second-order transition has a polynomially decaying gap (polynomial computation time). The examples presented here represent rare instances where strong quantum effects, in the sense that they cannot be efficiently simulated in the standard quantum Monte Carlo, have analytically been shown to exponentially enhance the efficiency of quantum annealing for combinatorial optimization problems.
Quantum annealing promises to solve combinatorial optimization problems by preparing the ground state of a target Hamiltonian. Standard annealing protocols are, however, stoquastic and can thus be simulated by sign-problem-free quantum Monte-Carlo methods. To obtain a true quantum advantage, it has been proposed to use non-stoquastic catalyst Hamiltonians. Active only at intermediate stages of the protocol, these can, for certain problems, convert first-order into second-order quantum phase transitions and thus permit an exponential speedup over the stoquastic protocol. At the same time, the non-stoquastic catalyst renders quantum Monte-Carlo methods inefficient. It remains, however, an open question how other classical computation methods are affected by the non-stoquastic terms. We address this question by computing quantum resources -- entanglement entropy and stabilizer Rényi entropy -- whose presence makes classical computations based on tensor networks and stabilizer-tableau methods exponentially hard. We compare these with the spectral gap along the annealing path for two paradigmatic benchmark models, the fully connected p-spin model and a geometrically local Ising model. While the exact behavior shows a subtle dependency on the underlying model and the annealing path, our numerics suggest consistently that the scaling of entanglement and non-stabilizerness is at least maintained in the deeply non-stoquastic regime and in some cases even significantly enhanced. Our results thus suggest that improvements of quantum performance in non-stoquastic annealing coincide with significant presence of quantum computational resources.
We show, for quantum annealing, that a certain type of inhomogeneous driving of the transverse field erases first-order quantum phase transitions in the p-body interacting mean-field-type model with and without longitudinal random field. Since a first-order phase transition poses a serious difficulty for quantum annealing (adiabatic quantum computing) due to the exponentially small energy gap, the removal of first-order transitions means an exponential speedup of the annealing process. The present method may serve as a simple protocol for the performance enhancement of quantum annealing, complementary to non-stoquastic Hamiltonians.
Quantum Annealing (QA) has already put into practical use and considered useful for solving many social issues, such as reduction of traffic congestion and delivery optimization. But in QA, when the energy difference between the ground state and the first excited state is small, the transition probability between them increases. Therefore, in order to decrease the transition, QA has to be performed at extremely low temperatures. To simulate the situation, it has the problem in that it takes much time. Many studies are underway to accelerate QA theoretically, one of which is to incorporate non-stoquastic Hamiltonian. On the other hand, I proposed Nested Simulated Annealing (NSA) inspired by Quantum Monte Carlo (QMC). I showed the computational speedup could be achieved dramatically, by considering the idea that the effect of flipping a spin preferentially influenced the spins directly interacting with it. Although NSA was based on such classical concept of causality, the hybrid computation both quantum and classical approach worked well. In this paper, in order to discuss the relationship between NSA and non-stoquastic Hamiltonian like XX-interaction, the spins are treated as continuous variables. I derive the formula to calculate the total energy in both with a problem Hamiltonian and the perturbation Hamiltonian induced by a transverse electromagnetic field. And I will show a clear relationship between local-maxima and the convergence speed when calculating it with the binary spin. More precisely, the annealing path with the smallest local-maxima converges fast, even though it consumes fewer computational resources. Therefore, it is possible to induce an optimal adiabatic phase transition by selecting NSA parameters appropriately. This paper shows an effective method to choose the parameters when simulating QMC on a classical computer.
Quantum annealing is a generic solver of the optimization problem that uses fictitious quantum fluctuation. Its simulation in classical computing is often performed using the quantum Monte Carlo simulation via the Suzuki-Trotter decomposition. However, the negative sign problem sometimes emerges in the simulation of quantum annealing with an elaborate driver Hamiltonian, since it belongs to a class of non-stoquastic Hamiltonians. In the present study, we propose an alternative way to avoid the negative sign problem involved in a particular class of the non-stoquastic Hamiltonians. To check the validity of the method, we demonstrate our method by applying it to a simple problem that includes the anti-ferromagnetic XX interaction, which is a typical instance of the non-stoquastic Hamiltonians.
One of the distinct features of quantum mechanics is that the probability amplitude can have both positive and negative signs, which has no classical counterpart as the classical probability must be positive. Consequently, one possible way to achieve quantum speedup is to explicitly harness this feature. Unlike a stoquastic Hamiltonian whose ground state has only positive amplitudes (with respect to the computational basis), a non-stoquastic Hamiltonian can be eventually stoquastic or properly non-stoquastic when its ground state has both positive and negative amplitudes. In this paper, we describe that, for some hard instances which are characterized by the presence of an anti-crossing (AC) in a transverse-field quantum annealing (QA) algorithm, how to design an appropriate XX-driver graph (without knowing the prior problem structure) with an appropriate XX-coupler strength such that the resulting non-stoquastic QA algorithm is proper-non-stoquastic with two bridged anti-crossings (a double-AC) where the spectral gap between the first and second level is large enough such that the system can be operated diabatically in polynomial time. The speedup is exponential in the original AC-distance, which can be sub-exponential or exponential in the system size, over the stoquastic QA algorithm, and possibly the same order of speedup over the state-of-the-art classical algorithms in optimization. This work is developed based on the novel characterizations of a modified and generalized parametrization definition of an anti-crossing in the context of quantum optimization annealing introduced in [4].
Establishing quantum speedup for computationally hard problems of practical relevance, particularly combinatorial optimization problems, remains a central challenge in quantum computation. In this work, we identify a structurally defined family of classically hard maximum independent set (MIS) instances, and design and analyze a non-stoquastic adiabatic quantum optimization algorithm that exploits this structure. The algorithm runs in polynomial time and achieves an exponential speedup over both transverse-field quantum annealing and state-of-the-art classical solvers on these instances, under assumptions supported by analytical and numerical evidence. We identify the essential quantum mechanism enabling the speedup as the use of a non-stoquastic XX-driver to access a larger sign-structured admissible subspace beyond the stoquastic regime, which allows sign-generating quantum interference to create smooth evolution paths that bypass tunneling. This identifies a distinctive quantum mechanism underlying the speedup and explains why no efficient classical analogue is likely to exist. In addition, our analysis produces scalable small-scale models, derived from our structural reduction, that capture the essential dynamics of the algorithm. These models provide a concrete opportunity for verification of the quantum advantage mechanism on currently available universal quantum computers.