We provide a new approach to adiabatic state preparation that uses coherent control and measurement to average different adiabatic evolutions in ways that cause their diabatic errors to cancel, allowing highly accurate state preparations using less time than conventional approaches. We show that this new model for adiabatic state preparation is polynomially equivalent to conventional adiabatic quantum computation by providing upper bounds on the cost of simulating such evolutions on a circuit-based quantum computer. Finally, we show that this approach is robust to small errors in the quantum control register and that the system remains protected against noise on the adiabatic register by the spectral gap.
We provide a new approach to adiabatic state preparation that uses coherent control and measurement to average different adiabatic evolutions in ways that cause their diabatic errors to cancel, allowing highly accurate state preparations using less time than conventional approaches. We show that this new model for adiabatic state preparation is polynomially equivalent to conventional adiabatic quantum computation by providing upper bounds on the cost of simulating such evolutions on a circuit-based quantum computer. Finally, we show that this approach is robust to small errors in the quantum control register and that the system remains protected against noise on the adiabatic register by the spectral gap.
A major challenge facing adiabatic quantum computing is that algorithm design and error correction can be difficult for adiabatic quantum computing. Recent work has considered addressing his challenge by using coherently controlled adiabatic evolutions in the place of classically controlled evolution. An important question remains: what is the relative power of controlled adiabatic evolution to traditional adiabatic evolutions? We address this by showing that coherent control and measurement provides a way to average different adiabatic evolutions in ways that cause their diabatic errors to cancel, allowing for adiabatic evolutions to combine the best characteristics of existing adiabatic optimizations strategies that are mutually exclusive in conventional adiabatic QIP. This result shows that coherent control and measurement can provide advantages for adiabatic state preparation. We also provide upper bounds on the complexity of simulating such evolutions on a circuit based quantum computer and provide sufficiency conditions for the equivalence of controlled adiabatic evolutions to adiabatic quantum computing.
We present a quantum algorithm for adiabatic state preparation on a gate-based quantum computer, with complexity polylogarithmic in the inverse error. Our algorithm digitally simulates the adiabatic evolution between two self-adjoint operators H0 and H1, exponentially suppressing the diabatic error by harnessing the theoretical concept of quasi-adiabatic continuation as an algorithmic tool. Given an upper bound α on ∥H0∥ and ∥H1∥ along with the promise that the kth eigenstate ∣ψk(s)⟩ of H(s)≡(1−s)H0+sH1 is separated from the rest of the spectrum by a gap of at least γ>0 for all s∈[0,1], this algorithm implements an operator U such that ∥∣ψk(1)⟩−U∣ψk(s)⟩∥≤ε using O(α2/γ2)polylog(α/γε) queries to block-encodings of H0 and H1. In addition, we develop an algorithm that is applicable only to ground states and requires multiple queries to an oracle that prepares ∣ψ0(0)⟩, but has slightly better scaling in all parameters. We also show that the costs of both algorithms can be further reduced under certain reasonable conditions, such as when ∥H1−H0∥ is small compared to α, or when more information about the gap of H(s) is available. For certain problems, the scaling can even be improved to linear in ∥H1−H0∥/γ up to polylogarithmic factors.
Adiabatic quantum computing is a framework for quantum computing that is superficially very different to the standard circuit model. However, it can be shown that the two models are computationally equivalent. The key to the proof is a mapping of a quantum circuit to an adiabatic evolution, and then showing that the minimum spectral gap of the adiabatic Hamiltonian is at least inverse polynomial in the number of computational steps L. In this paper we provide two simplified proofs that the gap is inverse polynomial. Both proofs result in the same lower bound for the minimum gap, which for $L\ensuremath{\gg}1$ is ${min}_{s}{\mathrm{\ensuremath{\Delta}}}_{0}\ensuremath{\gtrsim}{\ensuremath{\pi}}^{2}/[8{(L+1)}^{2}]$, an improvement over previous bounds. Our first method is a direct approach based on an eigenstate ansatz, while the second uses Weyl's theorem to leverage known exact results into a bound for the gap. Our results suggest that it may be possible to use these methods to find bounds for spectral gaps of Hamiltonians in other scenarios.
Adiabatic state engineering is a powerful technique in quantum information and quantum control. However, its performance is limited by the adiabatic theorem of quantum mechanics. In this scenario, shortcuts to adiabaticity, such as provided by the superadiabatic theory, constitute a valuable tool to speed up the adiabatic quantum behavior. Here, we propose a superadiabatic route to implement universal quantum computation. Our method is based on the realization of piecewise controlled superadiabatic evolutions. Remarkably, they can be obtained by simple time-independent counter-diabatic Hamiltonians. In particular, we discuss the implementation of fast rotation gates and arbitrary n-qubit controlled gates, which can be used to design different sets of universal quantum gates. Concerning the energy cost of the superadiabatic implementation, we show that it is dictated by the quantum speed limit, providing an upper bound for the corresponding adiabatic counterparts.
Many problems in science could be solved by preparing the low-energy quantum state (or any eigenstate) of a Hamiltonian. A common example is the Boolean satisfiability problem, where each clause can be mapped to the energy of an interacting many-body system, and the problem reduces to minimizing the energy. In quantum computing, adiabatic quantum state transformations (ASTs) provide a tool for preparing the quantum state. ASTs are conventionally implemented via slow or adiabatic perturbations to the Hamiltonian, relying on the quantum adiabatic theorem. Nevertheless, more efficient implementations of ASTs exist. In this talk I will review recently developed methods for ASTs that are more efficient and require less assumptions on the Hamiltonians than the conventional implementation [1,2]. Such methods involve measurements of the states along the evolution path and have a best-case implementation cost of L/G, where L is the length of the (evolved) state path and G is a lower bound to the spectral gap of the Hamiltonians. I will show that this cost is optimal [3] and comment on results of the gap amplification problem, where the goal is to reduce the cost by increasing G [4].
Quantum state preparation by adiabatic evolution is currently rendered ineffective by the long implementation times of the underlying quantum circuits, comparable to the decoherence time of present and near-term quantum devices. These implementation times can be significantly reduced by realizing these circuits with custom gates. Using classical computing, we model the output of a realistic two-qubit processor implementing the adiabatic evolution of a two-spin system by means of custom gates. This modeled output is then compared with the results of quantum simulations solving the same problem on IBM Quantum (IBMQ) systems. When used to emulate the behavior of the IBMQ quantum circuit, our realistic model yields state fidelities ranging from 65% to 85%, similar to the actual performance of a diverse set of IBMQ devices. When we reduced the implementation time by using custom gates, however, the loss of fidelity was reduced by at least a factor of 4, allowing us to accurately extract the energy of the target state. This improvement is enough to render adiabatic evolution useful for quantum state preparation for small systems or as a preconditioner for other state preparation methods.