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Exploring adiabatic quantum trajectories via optimal control

Constantin Brif, Matthew Grace, Mohan Sarovar, Kevin C Young

Published Aug 14, 2016• 16 citations
OpenAlex

Abstract

Abstract. Adiabatic quantum computation employs a slow change of a time-dependent control function (or functions) to interpolate between an initial and final Hamiltonian, which helps to keep the system in the instantaneous ground state. When the evolution time is finite, the degree of adiabaticity (quantified in this work as the average ground-state population during evolution) depends on the particulars of a dynamic trajectory associated with a given set of control functions. We use quantum optimal control theory with a composite objective functional to numerically search for controls that achieve the target final state with a high fidelity while simultaneously maximizing the degree of adiabaticity. Exploring properties of optimal adiabatic trajectories in model systems elucidates the dynamic mechanisms that suppress unwanted excitations from the ground state. Specifically, we discover that the use of multiple control functions makes it possible to access a rich set of dynamic trajectories, some of which attain a significantly improved performance (in terms of both fidelity and adiabaticity) through the increase of the energy gap during most of the evolution time. Submitted to: New J. Phys. † Author to whom any correspondence should be addressed. ar

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Exploring adiabatic quantum trajectories via optimal control

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Adiabatic quantum computation employs a slow change of a time-dependent control function (or functions) to interpolate between an initial and final Hamiltonian, which helps to keep the system in the instantaneous ground state. When the evolution time is finite, the degree of adiabaticity (quantified in this work as the average ground-state population during evolution) depends on the particulars of a dynamic trajectory associated with a given set of control functions. We use quantum optimal control theory with a composite objective functional to numerically search for controls that achieve the target final state with a high fidelity while simultaneously maximizing the degree of adiabaticity. Exploring the properties of optimal adiabatic trajectories in model systems elucidates the dynamic mechanisms that suppress unwanted excitations from the ground state. Specifically, we discover that the use of multiple control functions makes it possible to access a rich set of dynamic trajectories, some of which attain a significantly improved performance (in terms of both fidelity and adiabaticity) through the increase of the energy gap during most of the evolution time.

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Motivation• In adiabatic quantum computation (AQC), the system is kept in the instantaneous ground state |φ 0 (t) of a slowly varying Hamiltonian H(t), which provides inherent protection against dephasing and relaxation in the energy eigenstate basis.• However, this protection does not guarantee full fault tolerance, since there still exist some types of noise, which can drive undesirable transitions out of the ground state.• Therefore, it would be desirable to try decreasing the accumulation of errors by performing a computation in a shorter time.However, how to reconcile faster evolution with the requirement of adiabaticity?• Our approach: (1) consider Hamiltonians containing multiple control functions and (2) use quantum optimal control theory (QOCT) to find control sets which achieve the target state with a high fidelity while simultaneously maximizing the degree of adiabaticity.• Exploring properties of optimal adiabatic trajectories in few-qubit model systems elucidates the dynamic mechanisms which suppress unwanted excitations from the ground state.

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Abstract. Adiabatic quantum computation employs a slow change of a time-dependent control function (or functions) to interpolate between an initial and final Hamiltonian, which helps to keep the system in the instantaneous ground state. When the evolution time is finite, the degree of adiabaticity (quantified in this work as the average ground-state population during evolution) depends on the particulars of a dynamic trajectory associated with a given set of control functions. We use quantum optimal control theory with a composite objective functional to numerically search for controls that achieve the target final state with a high fidelity while simultaneously maximizing the degree of adiabaticity. Exploring properties of optimal adiabatic trajectories in model systems elucidates the dynamic mechanisms that suppress unwanted excitations from the ground state. Specifically, we discover that the use of multiple control functions makes it possible to access a rich set of dynamic trajectories, some of which attain a significantly improved performance (in terms of both fidelity and adiabaticity) through the increase of the energy gap during most of the evolution time. Submitted to: New J. Phys. † Author to whom any correspondence should be addressed. ar
Adiabatic quantum computation employs a slow change of a time-dependent control function (or functions) to interpolate between an initial and final Hamiltonian, which helps to keep the system in the instantaneous ground state. When the evolution time is finite, the degree of adiabaticity (quantified in this work as the average ground-state population during evolution) depends on the particulars of a dynamic trajectory associated with a given set of control functions. We use quantum optimal control theory with a composite objective functional to numerically search for controls that achieve the target final state with a high fidelity while simultaneously maximizing the degree of adiabaticity. Exploring the properties of optimal adiabatic trajectories in model systems elucidates the dynamic mechanisms that suppress unwanted excitations from the ground state. Specifically, we discover that the use of multiple control functions makes it possible to access a rich set of dynamic trajectories, some of which attain a significantly improved performance (in terms of both fidelity and adiabaticity) through the increase of the energy gap during most of the evolution time.
Motivation• In adiabatic quantum computation (AQC), the system is kept in the instantaneous ground state |φ 0 (t) of a slowly varying Hamiltonian H(t), which provides inherent protection against dephasing and relaxation in the energy eigenstate basis.• However, this protection does not guarantee full fault tolerance, since there still exist some types of noise, which can drive undesirable transitions out of the ground state.• Therefore, it would be desirable to try decreasing the accumulation of errors by performing a computation in a shorter time.However, how to reconcile faster evolution with the requirement of adiabaticity?• Our approach: (1) consider Hamiltonians containing multiple control functions and (2) use quantum optimal control theory (QOCT) to find control sets which achieve the target state with a high fidelity while simultaneously maximizing the degree of adiabaticity.• Exploring properties of optimal adiabatic trajectories in few-qubit model systems elucidates the dynamic mechanisms which suppress unwanted excitations from the ground state.
We describe a class of the Hamiltonians, for which the local adiabaticity condition is satisfied automatically. We consider the controllable adiabatic interpolations between the initial and final Hamiltonians. We show that the quantum search by the adiabatic quantum computation can be mapped into this class. Assuming that the interpolation is infinitely differentiable and the target adiabatic ground state is nondegenerate and separated by a gap from the rest of the spectrum, it is shown that one can obtain an exponentially small error in the algorithm run time T betweenthe finale adiabatic eigenstate and the actual state of the system. The optimal for quantum search scaling of T as square root of the system size is also satisfied.
Abstract Adiabatic quantum computing and optimization have garnered much attention recently as possible models for achieving a quantum advantage over classical approaches to optimization and other special purpose computations. Both techniques are probabilistic in nature and the minimum gap between the ground state and first excited state of the system during evolution is a major factor in determining the success probability. In this work we investigate a strategy for increasing the minimum gap and success probability by introducing intermediate Hamiltonians that modify the evolution path between initial and final Hamiltonians. We focus on an optimization problem relevant to recent hardware implementations and present numerical evidence for the existence of a purely local intermediate Hamiltonian that achieve the optimum performance in terms of pushing the minimum gap to one of the end points of the evolution. As a part of this study we develop a convex optimization formulation of the search for optimal adiabatic schedules that makes this computation more tractable, and which may be of independent interest. We further study the effectiveness of random intermediate Hamiltonians on the minimum gap and success probability, and empirically find that random Hamiltonians have a significant probability of increasing the success probability, but only by a modest amount.
A central problem in quantum computing is the finding of an unknown target state that encodes the solution of a certain computational task. To accomplish this goal, the evolution from a given initial state is performed with an associated total Hamiltonian, which is a time-dependent combination of two time-independent Hamiltonians: the problem-Hamiltonian, whose ground state is the unknown target, and a driving-Hamiltonian, whose ground state is the initial state. Here, we analyze this computational problem in the light of optimal control theory, considering each Hamiltonian modulated by an independent control function. For bounded controls, there is a minimum total evolution time beyond which the target state can be exactly prepared. We show that below this minimum time a possible optimal solution consists of both controls constantly tuned at their upper bound, provided the gradients of the associated control Hamiltonian with respect to the controls (the switching functions) are positive. We refer to this type of solution as the double-bang solution. We show that the double-bang solution is optimal for a teleportation protocol up to the minimum time. Additionally, we combine the double-bang solution and the adiabatic gate teleportation protocol to implement universal quantum computing. This approach to quantum computing is very appealing because of its simplicity and experimental feasibility. To corroborate our analytical results, we propose the use of a numerical quantum optimal control technique adapted to limit the amplitude of the controls, which converges to the double-bang solution when the final evolution time is shorter than the minimum time. We compare the fidelity of the teleported state obtained for the numerically optimized two-control functions with the usual one-control function scheme and with the quantum approximate optimization algorithm (QAOA). We find that the two-control approach has a better performance than the other approaches. Moreover, we investigate the energetic cost and the robustness against systematic errors in the teleportation protocol, considering different schemes.
Quantum control could be implemented by varying the system Hamiltonian. According to adiabatic theorem, a slowly changing Hamiltonian can approximately keep the system at the ground state during the evolution if the initial state is a ground state. In this paper we consider this process as an interpolation between the initial and final Hamiltonians. We use the mean value of a single operator to measure the distance between the final state and the ideal ground state. This measure resembles the excitation energy or excess work performed in thermodynamics, which can be taken as the error of adiabatic approximation. We prove that under certain conditions, this error can be estimated for an arbitrarily given interpolating function. This error estimation could be used as guideline to induce adiabatic evolution. According to our calculation, the adiabatic approximation error is not linearly proportional to the average speed of the variation of the system Hamiltonian and the inverse of the energy gaps in many cases. In particular, we apply this analysis to an example in which the applicability of the adiabatic theorem is questionable.