Abstract Quantum annealing has been demonstrated with superconducting qubits. Such a quantum annealer has been used to solve combinational optimization problems. Moreover, it serves as a quantum simulator for investigating the properties of the quantum many-body systems. However, the coherence properties of actual devices provided by D-Wave Quantum Inc. have not been explored sufficiently. Here, we measure the energy relaxation of the excited state in quantum annealing with the D-Wave device. Specifically, we investigate the incoherent decay rate of the first excited states of a fully connected Ising model with a transverse field. We find that the decay rate of the excited states of the model is orders of magnitude smaller than that of the excited state of a single qubit, and we qualitatively explain this phenomenon by using theoretical methods. Since our numerical simulations show that the first excited state during QA for the model is entangled, our experimental results indicate that the long-lived entangled state can be generated during QA with the D-Wave machine.
Abstract Quantum annealing has been demonstrated with superconducting qubits. Such a quantum annealer has been used to solve combinational optimization problems. Moreover, it serves as a quantum simulator for investigating the properties of the quantum many-body systems. However, the coherence properties of actual devices provided by D-Wave Quantum Inc. have not been explored sufficiently. Here, we measure the energy relaxation of the excited state in quantum annealing with the D-Wave device. Specifically, we investigate the incoherent decay rate of the first excited states of a fully connected Ising model with a transverse field. We find that the decay rate of the excited states of the model is orders of magnitude smaller than that of the excited state of a single qubit, and we qualitatively explain this phenomenon by using theoretical methods. Since our numerical simulations show that the first excited state during QA for the model is entangled, our experimental results indicate that the long-lived entangled state can be generated during QA with the D-Wave machine.
Quantum annealing has been demonstrated with superconducting qubits. Such a quantum annealer has been used to solve combinatorial optimization problems. Moreover, it serves as a quantum simulator for investigating the properties of quantum many-body systems. However, the coherence properties of multi-qubit states provided by D-Wave Quantum Inc. have not been explored sufficiently. Here, using the D-Wave device, we measure the energy relaxation rate of superconducting qubits and find long-lived multi-qubit states. Specifically, we investigate the energy relaxation rate of the first excited states of a fully connected Ising model with a transverse field. We find that the decay rate of the excited states of such a system with four qubits is orders of magnitude smaller than that of the excited state of a single qubit, which demonstrates the existence of long-lived multi-qubit states. We elucidate the mechanism using an independent decoherence model that qualitatively describes the phenomenon. In addition, by using such a mechanism, we theoretically predict a long-lived entangled state whose energy relaxation rate is smaller than that of the separable states.
Quantum annealing is a way to prepare an eigenstate of the problem Hamiltonian. Starting from an eigenstate of a trivial Hamiltonian, we slowly change the Hamiltonian to the problem Hamiltonian, and the system remains in the eigenstate of the Hamiltonian as long as the so-called adiabatic condition is satisfied. By using devices provided by D-Wave Systems Inc., there were experimental demonstrations to prepare a ground state of the problem Hamiltonian. However, up to date, there are no demonstrations to prepare the excited state of the problem Hamiltonian with quantum annealing. Here, we demonstrate the excited-state search by using the D-wave processor. The key idea is to use the reverse quantum annealing with a hot start where the initial state is the excited state of the trivial Hamiltonian. During the reverse quantum annealing, we control not only the transverse field but also the longitudinal field and slowly change the Hamiltonian to the problem Hamiltonian so that we can obtain the desired excited state. As an example of the exited state search, we adopt a two-qubit Ising model as the problem Hamiltonian and succeed to prepare the excited state. Also, we solve the shortest vector problem where the solution is embedded into the first excited state of the Ising Hamiltonian. Our results pave the way for new applications of quantum annealers to use the excited states.
In D-Wave quantum annealers, various properties of the ground state have been clarified, whereas the correspondence between theory and experiment for energy relaxation rates in multiqubit excited states remains insufficiently understood. Here, we measured the energy relaxation rates of the first excited states in single-qubit systems and interacting two-qubit systems using a D-Wave quantum annealer. We analyzed the measured relaxation rates using a Gorini-Kossakowski-Sudarshan-Lindblad master equation with independent local σz-type noise channels. The calculated relaxation rates reproduced the overall trends observed experimentally, supporting the model at a qualitative level. We then used the relaxation rates measured for the uncoupled single-qubit systems to calibrate the local relaxation parameters and predict the relaxation rates of the interacting two-qubit systems. The predicted rates remained within a factor of approximately four of the measured rates. These results show that the relaxation measurements on individual qubits can provide a practical estimate of relaxation in small interacting quantum systems and may help clarify relaxation mechanisms in programmable quantum annealers.
We discuss quantum annealing of the two-dimensional transverse-field Ising model on a D-Wave device, encoded on L×L lattices with L≤32. Analyzing the residual energy and deviation from maximal magnetization in the final classical state, we find an optimal L dependent annealing rate v for which the two quantities are minimized. The results are well described by a phenomenological model with two powers of v and L-dependent prefactors to describe the competing effects of reduced quantum fluctuations (for which we see evidence of the Kibble-Zurek mechanism) and increasing noise impact when v is lowered. The same scaling form also describes results of numerical solutions of a transverse-field Ising model with the spins coupled to noise sources. We explain why the optimal annealing time is much longer than the coherence time of the individual qubits.
Abstract Quantum annealing is a type of analog computation that aims to use quantum mechanical fluctuations in search of optimal solutions of QUBO (quadratic unconstrained binary optimization) or, equivalently, Ising problems. Since NP-hard problems can in general be mapped to Ising and QUBO formulations, the quantum annealing paradigm has the potential to help solve various NP-hard problems. Current quantum annealers, such as those manufactured by D-Wave Systems, Inc. have various practical limitations including the size (number of qubits) of the problem that can be solved, the qubit connectivity, and error due to the environment or system calibration, which can reduce the quality of the solutions. Typically, for an arbitrary problem instance, the corresponding QUBO (or Ising) structure will not natively embed onto the available qubit architecture on the quantum chip. Thus, in these cases, a minor embedding of the problem structure onto the device is necessary. However, minor embeddings on these devices do not always make use of the full sparse chip hardware graph, and a large portion of the available qubits stay unused during quantum annealing. In this work, we embed a disjoint random QUBO on the unused parts of the chip alongside the QUBO to be solved, which acts as an indicator of the solution quality of the device over time. Using experiments on three different D-Wave quantum annealers, we demonstrate that (i) long term trends in solution quality exist on the D-Wave device, and (ii) the unused qubits can be used to measure the current level of noise of the quantum system.
Extensive experiments have demonstrated quantum behaviour in the long-time operation of the D-Wave quantum computer. The decoherence time of a single flux qubit is reported to be on the order of nanoseconds [1], which is much shorter than the time required to carry out a computation on the timescale of seconds [2, 3]. Previous judgements of whether the D-Wave device should be thought of as a quantum computer have been based on correlations of the input-output behaviour of the D-Wave machine with a quantum model, called simulated quantum annealing, or classical models, called simulated annealing and classical spin dynamics [4]. Explanations for a factor of 10 8 discrepancy between the single flux qubit decoherence time and the long-time coherent quantum behaviour of many integrated flux qubits of the D-Wave device have not been offered so far. In our contribution we investigate a model of four qubits with one qubit coupled to a phonon and (optionally) to environmental particles of high density of states, called gravonons. The calculations indicate that when no gravonons are present, the current in the qubit is flipped at some time and adiabatic evolution is discontinued. The time dependent wave functional becomes a non-correctable superposition of many excited states. The results demonstrate the possibility of effectively suppressing the current flip and allowing for continued adiabatic evolution when the entanglement to gravonons is included. This adiabatic evolution is, however, a coherent evolution in high dimensional spacetime and cannot be understood as a solution of Schrödinger's time dependent equation in 4 dimensional spacetime. Compared to Schrödinger's time development in 4D, the evolution is considerably slowed down, though still adiabatic. The properties of our model reflect correctly the experimentally found behaviour of the D-Wave machine and explain the factor of 10 8 discrepancy between decoherence time and quantum computation time. The observation and our explanation are in anology to the 108 discrepancy factor found, when comparing experimental results on adsorbate quantum diffusion rate with predictions of Schrödinger's time dependent equation, which can also be resolved in a model with the coupling to gravonons included.