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arXiv:2206.14269 · Physical Review Applied

Demonstration of Error-Suppressed Quantum Annealing Via Boundary Cancellation

Humberto Munoz-Bauza, Lorenzo Campos Venuti, Daniel A. Lidar

Published Mar 29, 2023• 4 citations
arXivHTMLPDF

Abstract

The boundary-cancellation theorem for open systems extends the standard quantum adiabatic theorem: assuming the gap of the Liouvillian does not vanish, the distance between a state prepared by a boundary-cancelling adiabatic protocol and the steady state of the system shrinks as a power of the number of vanishing time derivatives of the Hamiltonian at the end of the preparation. Here we generalize the boundary-cancellation theorem so that it applies also to the case where the Liouvillian gap vanishes, and consider the effect of dynamical freezing of the evolution. We experimentally test the predictions of the boundary-cancellation theorem using quantum annealing hardware, and find qualitative agreement with the predicted error suppression despite using annealing schedules that only approximate the required smooth schedules. Performance is further improved by using quantum annealing correction, and we demonstrate that the boundary-cancellation protocol is significantly more robust to parameter variations than protocols that employ pausing to enhance the probability of finding the ground state.

DOI
10.1103/physrevapplied.19.034095

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The boundary-cancellation theorem for open systems extends the standard quantum adiabatic theorem: assuming the gap of the Liouvillian does not vanish, the distance between a state prepared by a boundary-cancelling adiabatic protocol and the steady state of the system shrinks as a power of the number of vanishing time derivatives of the Hamiltonian at the end of the preparation. Here we generalize the boundary-cancellation theorem so that it applies also to the case where the Liouvillian gap vanishes, and consider the effect of dynamical freezing of the evolution. We experimentally test the predictions of the boundary-cancellation theorem using quantum annealing hardware, and find qualitative agreement with the predicted error suppression despite using annealing schedules that only approximate the required smooth schedules. Performance is further improved by using quantum annealing correction, and we demonstrate that the boundary-cancellation protocol is significantly more robust to parameter variations than protocols that employ pausing to enhance the probability of finding the ground state.
The adiabatic theorem of quantum mechanics states that the error between an instantaneous eigenstate of a time-dependent Hamiltonian and the state given by quantum evolution of duration $\ensuremath{\tau}$ is upper bounded by $C/\ensuremath{\tau}$ for some positive constant CCC. It has been known for decades that this error can be reduced to ${C}_{k}/{\ensuremath{\tau}}^{k+1}$ if the Hamiltonian has vanishing derivatives up to order kkk at the beginning and end of the evolution. Here we extend this result to open systems described by a time-dependent Liouvillian superoperator. We find that the same results holds provided the Liouvillian has vanishing derivatives up to order kkk only at the end of the evolution. This asymmetry is ascribable to the arrow of time inherent in open system evolution. We further investigate whether it is possible to satisfy the required assumptions by controlling only the system, as required for realistic implementations. Surprisingly, we find the answer to be affirmative. We establish this rigorously in the setting of the Davies-Lindblad adiabatic master equation, and numerically in the setting of two different time-dependent Redfield-type master equations we derive. The results are shown to be stable with respect to imperfections in the preparation. Finally, we prove that the results hold also in a fully Hamiltonian model.
We provide a rigorous generalization of the quantum adiabatic theorem for open systems described by a Markovian master equation with time-dependent Liouvillian L(t)\mathcal{L}(t)L(t). We focus on the finite system case relevant for adiabatic quantum computing and quantum annealing. Adiabaticity is defined in terms of closeness to the instantaneous steady state. While the general result is conceptually similar to the closed-system case, there are important differences. Namely, a system initialized in the zero-eigenvalue eigenspace of L(t)\mathcal{L}(t)L(t) will remain in this eigenspace with a deviation that is inversely proportional to the total evolution time TTT. In the case of a finite number of level crossings, the scaling becomes ${T}^{\ensuremath{-}\ensuremath{\eta}}$ with an exponent $\ensuremath{\eta}$ that we relate to the rate of the gap closing. For master equations that describe relaxation to thermal equilibrium, we show that the evolution time TTT should be long compared to the corresponding minimum inverse gap squared of L(t)\mathcal{L}(t)L(t). Our results are illustrated with several examples.
Abstract Quantum annealing (QA) is a continuous-time heuristic quantum algorithm for solving or approximately solving classical optimization problems. The algorithm uses a schedule to interpolate between a driver Hamiltonian with an easy-to-prepare ground state and a problem Hamiltonian whose ground state encodes solutions to an optimization problem. The standard implementation relies on the evolution being adiabatic: keeping the system in the instantaneous ground state with high probability and requiring a time scale inversely related to the minimum energy gap between the instantaneous ground and excited states. However, adiabatic evolution can lead to evolution times that scale exponentially with the system size, even for computationally simple problems. Here, we study whether non-adiabatic evolutions with optimized annealing schedules can bypass this exponential slowdown for one such class of problems called the frustrated ring model. For sufficiently optimized annealing schedules and system sizes of up to 39 qubits, we provide numerical evidence that we can avoid the exponential slowdown. Our work highlights the potential of highly-controllable QA to circumvent bottlenecks associated with the standard implementation of QA.
Adiabatic quantum annealers encounter scalability challenges due to exponentially fast diminishing energy gaps between ground and excited states with qubit-count increase. This introduces errors in identifying ground states compounded by a thermal noise. We propose a novel algorithmic scheme called statistical qubit freezing (SQF) that selectively fixes the state of statistically deterministic qubit in the annealing Hamiltonian model of the given problem. Applying freezing repeatedly, SQF significantly enhances the spectral gap between of an adiabatic process, as an example, by up to 60\% compared to traditional annealing methods in the standard D-Wave's quantum Ising machine solution, effectively overcoming the fundamental limitations.
We derive a version of the adiabatic theorem that is especially suited for applications in adiabatic quantum computation, where it is reasonable to assume that the adiabatic interpolation between the initial and final Hamiltonians is controllable. Assuming that the Hamiltonian is analytic in a finite strip around the real time axis, that some number of its time-derivatives vanish at the initial and final times, and that the target adiabatic eigenstate is non-degenerate and separated by a gap from the rest of the spectrum, we show that one can obtain an error between the final adiabatic eigenstate and the actual time-evolved state which is exponentially small in the evolution time, where this time itself scales as the square of the norm of the time-derivative of the Hamiltonian, divided by the cube of the minimal gap.
In the computational model of quantum annealing, the size of the minimum gap between the ground state and the first excited state of the system is of particular importance, since it is inversely proportional to the running time of the algorithm. Thus, it is desirable to keep the gap as large as possible during the annealing process, since it allows the computation to remain under the protection of the adiabatic theorem while staying efficient. We propose steered quantum annealing as a new method to enlarge the gap throughout the process, in the case of diagonal final Hamiltonians, based on the exploitation of some assumptions we can make about the particular problem instance. In order to introduce this information, we propose beginning the anneal from a biased Hamiltonian that incorporates reliable assumptions about the final ground state. Our simulations show that this method yields a larger average gap throughout the whole computation, which results in an increased robustness of the overall annealing process.