We show agreement In problems with first order phase transition, ... We demonstrate this next with some striking cases. instance. An initial Hamiltonian which insufficiently mixes computational basis states is analogous to a poorly mixing Markov transition rule. We explicitly show the high figures of merit of the QCD via the quantum gain and the signal-to-quantum noise ratio. Express 27, 10482 (2019)], functions by exploiting high sensitivity near the critical point of first-order quantum phase transitions. We define two new magnetic order parameters to quantitatively characterize the first-order QPT of the interacting spins in the detector. We provide a detailed comparison of experiments from a D-Wave device, simulations of the quantum adiabatic master equation and a classical analogue of quantum annealing, spin-vector Monte Carlo, and we observe qualitative agreement, showing that the characteristic increase in success probability when pausing is not a uniquely quantum phenomena. mechanisms into two sets: small gaps due to quantum phase transitions and small Thus, a dynamical detection event may have totally different sensitivity scaling. Adiabatic quantum computation (AQC) was first pro-, proved that AQC is polynomially equivalent to conven-, In AQC, the system’s Hamiltonian, usually written as, is assumed to have an easily accessible ground, which the system is initialized, while the ground state, state with high fidelity, the adiabatic theorem requires, most fundamental problem in AQC is therefore how to, unveil the quantum evolution blackbox by relating the. We shall discuss first-order transition in … lower-level wave function. Such problems, therefore, are not suitable for adiabatic quantum computation. The spin glass ground state is found to be replica symmetric, with replica symmetry breaking appearing only at finite temperatures. constraints that affect the performance of a realistic system. Authors: M. H. S. Amin, V. Choi. Adiabatic quantum optimization has attracted a lot of attention because small scale simulations gave hope that it would allow to solve NP-complete problems efficiently. Using the Maximum-weighted Independent Set (MIS) problem in which there are flexible parameters (energy penalties J between pairs of edges) in an Ising formulation as the model problem, we construct examples to show that by changing the value of J, we can change the quantum evolution from one that has an anti-crossing (that results in an exponential small min-gap) to one that does not have, or the other way around, and thus drastically change (increase or decrease) the min-gap. All rights reserved. We compare the spectral gaps of these adiabatic paths and find both theoretically and numerically that the paths based on non-stoquastic Hamiltonians have generically smaller spectral gaps between the ground and first excited states, suggesting they are less useful than stoquastic Hamiltonians for quantum adiabatic optimization. the minimum gap changes by 6 orders of ma, is exponentially sensitive to parameters that, of bit flip within them compared to the global minim, Our perturbative calculation indicates that the g. the two minima involved in the phase transition. This shows that if the adiabatic time scale were to Anti-crossing vs perturbative crossing Our definition of anti-crossing is more general than the perturbative crossing in, ... Anti-crossing and min-gap size The min-gap size is expected to be exponentially small in O(b k ) where k = ||FS − GS|| for some 0 < b < 1. The presence of this line of impurities makes the system anisotropic and the interactions highly nonlocal. What is meant by an adiabatic process? systems with up to 20 qubits independently coupled to this environment via two optimization problems. problems in finite dimensions. In spite of this, there was still hope that this would not happen for random instances of NP-complete problems. The exponent z results from the presence of anisotropy in the system. We establish a relation between our model and a quantum model in one less dimension with random pointlike impurities. entanglement, where in each step the, We present a perturbative method to estimate the spectral gap for adiabatic quantum optimization, based on the structure of the energy levels in the problem Hamiltonian. On the contrary, we also argue that, if the phase surrounding The localization transition in QA is a potential phase transition that is related to computational complexity. We define two new magnetic order parameters to quantitatively characterize the first-order QPT of the interacting spins in the detector. Unfortunately, some examples indicating that a quantum first order phase transition tends to occur during the adiabatic computation [14][15] [16] [17][18][19][20] have been found. For a suitable distribution of exchange constants, these models display spin glass and quantum paramagnet phases and a zero temperature quantum transition between them. Nonstoquastic Hamiltonians are hard to simulate due to the sign problem in quantum Monte Carlo simulation. We compute the coefficient in the exponential closure of the gap using, Finding a succinct representation to describe the ground state of a The first three orders are given in the figure. An earlier exact solution for the critical properties of a model with infinite-range interactions can be reproduced by minimization of a Landau effective-action functional for the model in finite $d$ with short-range interactions. architecture are about 4 and 6 orders of magnitude shorter than the two In this paper we show that the performance of the quantum adiabatic algorithm is determined by phase transitions in underlying problem in the presence of transverse magnetic field $\Gamma$. March Meeting in Denver, Colorado (2007). endobj No matter how slow the transition is the defect density remains more or less the same. We compare the median adiabatic times with number of qubits. We study a physical system -- the Ising quantum chain with alternating sector interaction defects, but constant transverse field -- which is equivalent to applying the quantum adiabatic algorithm to a particular SAT problem. General scaling relations that should be valid even at the strong coupling fixed point are proposed and compared with Monte Carlo simulations. adiabatic quantum computation may then be accessed only via the local adiabatic evolution, which requires phase coherence throughout the evolution and knowledge of the spectrum. In the system studied here, only for short pauses is there expected to be an improvement. Applying the adiabatic theorem therefore takes exponential time, even for this simple problem. 2), which can be understood in terms of a perturbative crossing. iteration an ansatz of multilayer wave functions, with different levels of Alternatively, the longitudinal fields can also be applied as an antibias which filters out unwanted contributions from the final state. specific instances of 3-satisfiability problems (Exact Cover) and make a Explicit examples include disordered classical Ising and Heisenberg models, insulating and metallic random quantum magnets, and the disordered contact process. ization) as well as its corresponding perturbative value. We demonstrate the qualitative similarity between classical and quantum versions of this problem. In generic classical equilibrium systems, they lead to an essential singularity, the so-called Griffiths singularity, of the free energy in the vicinity of the phase transition. This is qualitatively different than any usual power law scaling predicted for pure systems by the Kibble-Zurek mechanism. Using perturbation expansion, we derive an analytical formula that irrespective of the nature of decay of these interactions along the chain. Using parameters obtained from a realistic superconducting With the right choice of spin-reversal transformation, a nonstoquastic Hamiltonian with YY-interaction can outperform stoquastic Hamiltonians with similar parameters. longitudinal magnetic field. The main argument we use in this paper to assess the performance of a QA algorithm is the presence or absence of an anti-crossing during quantum evolution. We prove that for a constant range of values for the transverse field, the spectral gap is exponentially small in the sector length. We introduce a transformation that maps every non-stoquastic adiabatic path ending in a classical Hamiltonian to a corresponding stoquastic adiabatic path by appropriately adjusting the phase of each matrix entry in the computational basis.


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