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Register a new userDuality approach to quantum annealing of the 3-XORSAT problem
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Classical models with complex energy landscapes represent a perspective avenue for the near-term application of quantum simulators. Until now, many theoretical works studied the performance of quantum algorithms for models with a unique ground state. However, when the classical problem is in a so-called clustering phase, the ground state manifold is highly degenerate. As an example, we consider a 3-XORSAT model defined on simple hypergraphs. The degeneracy of classical ground state manifold translates into the emergence of an extensive number of $Z_2$ symmetries, which remain intact even in the presence of a quantum transverse magnetic field. We establish a general duality approach that restricts the quantum problem to a given sector of conserved $Z_2$ charges and use it to study how the outcome of the quantum adiabatic algorithm depends on the hypergraph geometry. We show that the tree hypergraph which corresponds to a classically solvable instance of the 3-XORSAT problem features a constant gap, whereas the closed hypergraph encounters a second-order phase transition with a gap vanishing as a power-law in the problem size. The duality developed in this work provides a practical tool for studies of quantum models with classically degenerate energy manifold and reveals potential connections between glasses and gauge theories.
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The solution space of many classical optimization problems breaks up into clusters which are extensively distant from one another in the Hamming metric. Here, we show that an analogous quantum clustering phenomenon takes place in the ground state subspace of a certain quantum optimization problem. This involves extending the notion of clustering to Hilbert space, where the classical Hamming distance is not immediately useful. Quantum clusters correspond to macroscopically distinct subspaces of the full quantum ground state space which grow with the system size. We explicitly demonstrate that such clusters arise in the solution space of random quantum satisfiability (3-QSAT) at its satisfiability transition. We estimate both the number of these clusters and their internal entropy. The former are given by the number of hardcore dimer coverings of the core of the interaction graph, while the latter is related to the underconstrained degrees of freedom not touched by the dimers. We additionally provide new numerical evidence suggesting that the 3-QSAT satisfiability transition may coincide with the product satisfiability transition, which would imply the absence of an intermediate entangled satisfiable phase.
L0-regularization-based compressed sensing (L0-RBCS) is capable of outperforming L1-RBCS, but it is difficult to solve an optimization problem for L0-RBCS that cannot be formulated as a convex optimization. To achieve the optimization for L0-RBCS, we propose a quantum-classical hybrid system consisting of a quantum machine and a classical digital processor. Because forming a densely-connected network on a quantum machine is required for solving this problem, the coherent Ising machine (CIM) is one of suitable quantum machines for composing this hybrid system. To evaluate theoretically the performance of the CIM-classical hybrid system, a truncated Wigner stochastic differential equation (W-SDE) is obtained from the master equation for the density operator of the network of degenerate optical parametric oscillators, and macroscopic equations are derived from the W-SDE using statistical mechanics. We show that the system performance in principle approaches the theoretical limit of compressed sensing and in practical situations this hybrid system can exceed L1-RBCSs estimation accuracy.
New annealing schedules for quantum annealing are proposed based on the adiabatic theorem. These schedules exhibit faster decrease of the excitation probability than a linear schedule. To derive this conclusion, the asymptotic form of the excitation probability for quantum annealing is explicitly obtained in the limit of long annealing time. Its first-order term, which is inversely proportional to the square of the annealing time, is shown to be determined only by the information at the initial and final times. Our annealing schedules make it possible to drop this term, thus leading to a higher order (smaller) excitation probability. We verify these results by solving numerically the time-dependent Schrodinger equation for small size systems
Finding the global minimum in a rugged potential landscape is a computationally hard task, often equivalent to relevant optimization problems. Simulated annealing is a computational technique which explores the configuration space by mimicking thermal noise. By slow cooling, it freezes the system in a low-energy configuration, but the algorithm often gets stuck in local minima. In quantum annealing, the thermal noise is replaced by controllable quantum fluctuations, and the technique can be implemented in modern quantum simulators. However, quantum-adiabatic schemes become prohibitively slow in the presence of quasidegeneracies. Here we propose a strategy which combines ideas from simulated annealing and quantum annealing. In such hybrid algorithm, the outcome of a quantum simulator is processed on a classical device. While the quantum simulator explores the configuration space by repeatedly applying quantum fluctuations and performing projective measurements, the classical computer evaluates each configuration and enforces a lowering of the energy. We have simulated this algorithm for small instances of the random energy model, showing that it potentially outperforms both simulated thermal annealing and adiabatic quantum annealing. It becomes most efficient for problems involving many quasi-degenerate ground states.
The extension of many-body quantum dynamics to the non-unitary domain has led to a series of exciting developments, including new out-of-equilibrium entanglement phases and phase transitions. We show how a duality transformation between space and time on one hand, and unitarity and non-unitarity on the other, can be used to realize steady state phases of non-unitary dynamics that exhibit a rich variety of behavior in their entanglement scaling with subsystem size -- from logarithmic to extensive to emph{fractal}. We show how these outcomes in non-unitary circuits (that are spacetime-dual to unitary circuits) relate to the growth of entanglement in time in the corresponding unitary circuits, and how they differ, through an exact mapping to a problem of unitary evolution with boundary decoherence, in which information gets radiated away from one edge of the system. In spacetime-duals of chaotic unitary circuits, this mapping allows us to uncover a non-thermal volume-law entangled phase with a logarithmic correction to the entropy distinct from other known examples. Most notably, we also find novel steady state phases with emph{fractal} entanglement scaling, $S(ell) sim ell^{alpha}$ with tunable $0 < alpha < 1$ for subsystems of size $ell$ in one dimension. These fractally entangled states add a qualitatively new entry to the families of many-body quantum states that have been studied as energy eigenstates or dynamical steady states, whose entropy almost always displays either area-law, volume-law or logarithmic scaling. We also present an experimental protocol for preparing these novel steady states with only a very limited amount of postselection via a type of teleportation between spacelike and timelike slices of quantum circuits.
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