Abstract We argue that a complete description of quantum annealing implemented with continuous variables must take into account the non-adiabatic Aharonov-Anandan geometric phase that arises when the system Hamiltonian changes during the anneal. We show that this geometric effect leads to the appearance of non-stoquasticity in the effective quantum Ising Hamiltonians that are typically used to describe quantum annealing with flux qubits. We explicitly demonstrate the effect of this geometric non-stoquasticity when quantum annealing is performed with a system of one and two coupled flux qubits. The realization of non-stoquastic Hamiltonians has important implications from a computational complexity perspective, since it is believed that in many cases quantum annealing with stoquastic Hamiltonians can be efficiently simulated via classical algorithms such as Quantum Monte Carlo. It is well known that the direct implementation of non-stoquastic Hamiltonians with flux qubits is particularly challenging. Our results suggest an alternative path for the implementation of non-stoquasticity via geometric phases that can be exploited for computational purposes.
Abstract We argue that a complete description of quantum annealing implemented with continuous variables must take into account the non-adiabatic Aharonov-Anandan geometric phase that arises when the system Hamiltonian changes during the anneal. We show that this geometric effect leads to the appearance of non-stoquasticity in the effective quantum Ising Hamiltonians that are typically used to describe quantum annealing with flux qubits. We explicitly demonstrate the effect of this geometric non-stoquasticity when quantum annealing is performed with a system of one and two coupled flux qubits. The realization of non-stoquastic Hamiltonians has important implications from a computational complexity perspective, since it is believed that in many cases quantum annealing with stoquastic Hamiltonians can be efficiently simulated via classical algorithms such as Quantum Monte Carlo. It is well known that the direct implementation of non-stoquastic Hamiltonians with flux qubits is particularly challenging. Our results suggest an alternative path for the implementation of non-stoquasticity via geometric phases that can be exploited for computational purposes.
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 (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 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 challenge the notion that a stoquastic binary governs fundamental computational boundaries in quantum computing and classical simulation of quantum systems. We argue that vanishing geometric phase (VGP), a geometric condition on the Hamiltonian's transition graph, more adequately captures these boundaries. To distinguish VGP from stoquasticity, we construct VGP 3-local Hamiltonians that are formally hard to stoquastize, yet belong to a family admitting polynomial-time recognition of the VGP property. Without constructing a stoquastizing unitary, we prove that the local Hamiltonian problem is StoqMA-complete under the promise that the input Hamiltonian has VGP, and that a frustration-free variant is in MA under the same promise. We use this result to argue that non-VGP is necessary for any claimed adiabatic advantage justified by escaping the StoqMA regime. Further, we identify natural settings where the VGP property can be recognized in polynomial time. In contrast, we show that recognition of VGP is PSPACE-complete in general for geometrically local Hamiltonians. Our results show that the computational boundaries MA⊆StoqMA⊆QMA traditionally attributed to stoquasticity are better understood as boundaries between vanishing and non-vanishing geometric phase structure.
Quantum annealing (QA) is a promising method for solving combinatorial optimization problems whose solutions are embedded into a ground state of the Ising Hamiltonian. This method employs two types of Hamiltonians: a driver Hamiltonian and a problem Hamiltonian. After a sufficiently slow change from the driver Hamiltonian to the problem Hamiltonian, we can obtain the target ground state that corresponds to the solution. The inclusion of non-stoquastic terms in the driver Hamiltonian is believed to enhance the efficiency of the QA. Meanwhile, decoherence is regarded as of the main obstacles for QA. Here, we present examples showing that non-stoaquastic Hamiltonians can lead to catastrophic failure of QA, whereas a certain decoherence process can be used to avoid such failure. More specifically, when we include anti-ferromagnetic interactions (i.e., typical non-stoquastic terms) in the Hamiltonian, we are unable to prepare the target ground state even with an infinitely long annealing time for some specific cases. In our example, owing to a symmetry, the Hamiltonian is block-diagonalized, and a crossing occurs during the QA, which leads to a complete failure of the ground-state search. Moreover, we show that, when we add a certain type of decoherence, we can obtain the ground state after QA for these cases. This is because, even when symmetry exists in isolated quantum systems, the environment breaks the symmetry. Our counter intuitive results provide a deep insight into the fundamental mechanism of QA.
Quantum annealers are computing machines that utilize quantum effects to solve hard optimization problems. Understanding the circumstances under which quantum annealers are more efficient than other optimization methods is a challenging open problem. A clue to this conundrum comes from studying the complexity of quantum annealers. In general, the complexity of a physical system relates to how efficiently the system can be simulated by classical algorithms. One then asks: if some class of quantum annealers are hard to simulate classically, are they also more powerful optimization machines? Here, the authors study quantum annealers belonging to two different complexity classes: the so-called ``stoquastic'' systems, which can be efficiently simulated with classical algorithms, and ``nonstoquastic'' systems, currently with no efficient classical treatment. Applied to a prototypical optimization problem, the authors observe that, on average, the stoquastic systems perform better on easier instances of the problem, while the more complex nonstoquastic annealers show significant advantage when applied to the hardest instances. The authors conjecture that the performance of the nonstoquastic Hamiltonians is closely related to their internal structure during the annealing process and elucidate how properties such as magnetic frustration can play a key role in devising more powerful quantum annealers.