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arXiv:1710.05972 · Physical Review A

Robust quantum control for adiabatic quantum computation

Gregory Quiroz

Published Jun 7, 2019• 22 citations
arXivHTMLPDF

Abstract

Properly designed control has been shown to be particularly advantageous for improving adiabatic quantum computation (AQC) accuracy and time complexity scaling. Here, an in situ quantum control optimization protocol is developed to indirectly optimize state fidelity without knowledge of the instantaneous spectral gap or the computational solution. The protocol is shown to converge to analytically derived time-optimal controls for Grover's search algorithm. Furthermore, the protocol is utilized to explore optimized control trajectories for the maximum two-bit satisfiability problem, where appreciable improvement in fidelity and the minimum spectral gap over a linear schedule is observed. The approach is also shown to be robust against system model uncertainties (unitary control errors). This method is designed to enable robust control optimization on existing quantum annealing hardware and future AQC processors.

DOI
10.1103/physreva.99.062306

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•Aug 14, 2016•16 citations
Properly designed control has been shown to be particularly advantageous for improving adiabatic quantum computation (AQC) accuracy and time complexity scaling. Here, an in situ quantum control optimization protocol is developed to indirectly optimize state fidelity without knowledge of the instantaneous spectral gap or the computational solution. The protocol is shown to converge to analytically derived time-optimal controls for Grover's search algorithm. Furthermore, the protocol is utilized to explore optimized control trajectories for the maximum two-bit satisfiability problem, where appreciable improvement in fidelity and the minimum spectral gap over a linear schedule is observed. The approach is also shown to be robust against system model uncertainties (unitary control errors). This method is designed to enable robust control optimization on existing quantum annealing hardware and future AQC processors.
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.
Quantum control techniques are employed to perform adiabatic quantum computing in the presence of noise. First, we analyze the adiabatic entanglement protocol (AEP) for two qubits. In this case, we found that this protocol is very robust against noise. The reason behind this fact is related to the chosen Hamiltonians, where the ground state of the initial Hamiltonian is not affected by the noise. The optimal control solution, in this case, is to leave the system in its ground state and apply a fast pulse to entangle the qubits at the end of the time evolution. Secondly, we probe a system composed of three qubits, where the goal is to teleport the first qubit to the third qubit. In this case, the ground state of the system does not share the same robustness against noise as in the case of AEP. To improve the robustness against noise, we propose the inclusion of a local control field that can drive the system to an intermediate state, which is more robust against noise in comparison to other states. The target state is also achieved by a fast pulse at the final time. We found that this approach provides a significant gain in the fidelity and can improve the adiabatic quantum computing in the so-called Noisy Intermediate-Scale Quantum (NISQ) devices in a near future.
Adiabatic pulses are used extensively to enable robust control of quantum operations. We introduce an approach to adiabatic control that uses the superadiabatic quality factor as a performance metric to design robust, high-fidelity pulses. This approach permits the systematic design of quantum control schemes to maximize the adiabaticity of a unitary operation in a particular time interval given the available control resources. The interplay between adiabaticity, fidelity, and robustness of the resulting pulses is examined for the case of single-qubit inversion, and superadiabatic pulses are demonstrated to have improved robustness to control errors. A numerical search strategy is developed to find a broader class of adiabatic operations, including multiqubit adiabatic unitaries. We illustrate the utility of this search strategy by designing control waveforms that adiabatically implement a two-qubit entangling gate for a model NMR system.
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.
We demonstrate that with an optimally tuned scheduling function, adiabatic quantum computing (AQC) can readily solve a quantum linear system problem (QLSP) with O (κ poly(log (κ ε))) runtime, where κ is the condition number, and ε is the target accuracy. This is near optimal with respect to both κ and ε, and is achieved without relying on complicated amplitude amplification procedures that are difficult to implement. Our method is applicable to general non-Hermitian matrices, and the cost as well as the number of qubits can be reduced when restricted to Hermitian matrices, and further to Hermitian positive definite matrices. The success of the time-optimal AQC implies that the quantum approximate optimization algorithm (QAOA) with an optimal control protocol can also achieve the same complexity in terms of the runtime. Numerical results indicate that QAOA can yield the lowest runtime compared to the time-optimal AQC, vanilla AQC, and the recently proposed randomization method.
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