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arXiv:quant-ph/0510183 · Physical Review A

General error estimate for adiabatic quantum computing

Gernot Schaller, Sarah Mostame, Ralf Schützhold

Published Jun 7, 2006• 64 citations
arXivHTMLPDF

Abstract

Most investigations devoted to the conditions for adiabatic quantum computing are based on the first-order correction $⟨{\ensuremath{\Psi}}_{\mathrm{ground}}(t)\ensuremath{\mid}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}(t)\ensuremath{\mid}{\ensuremath{\Psi}}_{\mathrm{excited}}(t)⟩∕\mathrm{\ensuremath{\Delta}}{E}^{2}(t)⪡1$. However, it is demonstrated that this first-order correction does not yield a good estimate for the computational error. Therefore, a more general criterion is proposed, which includes higher-order corrections as well, and shows that the computational error can be made exponentially small---which facilitates significantly shorter evolution times than the above first-order estimate in certain situations. Based on this criterion and rather general arguments and assumptions, it can be demonstrated that a run-time $T$ of order of the inverse minimum energy gap $\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{min}}$ is sufficient and necessary, i.e., $T=O(\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{min}}^{\ensuremath{-}1})$. For some examples, these analytical investigations are confirmed by numerical simulations.

DOI
10.1103/physreva.73.062307

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Most investigations devoted to the conditions for adiabatic quantum computing are based on the first-order correction ⟨\ensuremathΨground(t)\ensuremath∣H\ifmmode˙\else˙\fi(t)\ensuremath∣\ensuremathΨexcited(t)⟩∕\ensuremathΔE2(t)⪡1⟨{\ensuremath{\Psi}}_{\mathrm{ground}}(t)\ensuremath{\mid}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{H}(t)\ensuremath{\mid}{\ensuremath{\Psi}}_{\mathrm{excited}}(t)⟩∕\mathrm{\ensuremath{\Delta}}{E}^{2}(t)⪡1⟨\ensuremathΨground​(t)\ensuremath∣H\ifmmode˙\else˙\fi​(t)\ensuremath∣\ensuremathΨexcited​(t)⟩∕\ensuremathΔE2(t)⪡1. However, it is demonstrated that this first-order correction does not yield a good estimate for the computational error. Therefore, a more general criterion is proposed, which includes higher-order corrections as well, and shows that the computational error can be made exponentially small---which facilitates significantly shorter evolution times than the above first-order estimate in certain situations. Based on this criterion and rather general arguments and assumptions, it can be demonstrated that a run-time TTT of order of the inverse minimum energy gap \ensuremathΔEmin\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{min}}\ensuremathΔEmin​ is sufficient and necessary, i.e., T=O(\ensuremathΔEmin\ensuremath−1)T=O(\mathrm{\ensuremath{\Delta}}{E}_{\mathrm{min}}^{\ensuremath{-}1})T=O(\ensuremathΔEmin\ensuremath−1​). For some examples, these analytical investigations are confirmed by numerical simulations.
This Thesis focuses on different aspects of quantum computation theory: adiabatic quantum algorithms, decoherence during the adiabatic evolution and quantum simulators. After an overview on the area of quantum computation and setting up the formal ground for the rest of the Thesis we derive a general error estimate for adiabatic quantum computing. We demonstrate that the first-order correction, which has frequently been used as a condition for adiabatic quantum computation, does not yield a good estimate for the computational error. Therefore, a more general criterion is proposed, which includes higher-order corrections and shows that the computational error can be made exponentially small – which facilitates significantly shorter evolution times than the first-order estimate in certain situations. Based on this criterion and rather general arguments and assumptions, it can be demonstrated that a run-time of order of the inverse minimum energy gap is sufficient and necessary. Furthermore, exploiting the similarity between adiabatic quantum algorithms and quantum phase transitions, we study the impact of decoherence on the sweep through a second-order quantum phase transition for the prototypical example of the Ising chain in a transverse field and compare it to the adiabatic version of Grover’s search algorithm. It turns out that (in contrast to first-order transitions) the impact of decoherence caused by a weak coupling to a rather general environment increases with system size (i.e., number of spins/qubits), which might limit the scalability of the system. Finally, we propose the use of electron systems to construct laboratory systems based on present-day technology which reproduce and thereby simulate the quantum dynamics of the Ising model and the O(3) nonlinear sigma model.
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.
Quantum adiabatic computation is a novel paradigm for the design of quantum algorithms, which is usually used to find the minimum of a classical function. In this paper, we show that if the initial hamiltonian of a quantum adiabatic evolution with a interpolation path is too simple, the minimal gap between the ground state and the first excited state of this quantum adiabatic evolution is an inverse exponential distance. Thus quantum adiabatic evolutions of this kind can't be used to design efficient quantum algorithms. Similarly, we show that a quantum adiabatic evolution with a simple final hamiltonian also has a long running time, which suggests that some functions can't be minimized efficiently by any quantum adiabatic evolution with a interpolation path.
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 nondegenerate 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.
We explore the relationship between two figures of merit for an adiabatic quantum computation process: the success probability P and the minimum gap Δ min between the ground and first excited states, investigating to what extent the success probability for an ensemble of problem Hamiltonians can be fitted by a function of Δ min and the computation time T . We study a generic adiabatic algorithm and show that a rich structure exists in the distribution of P and Δ min . In the case of two qubits, P is to a good approximation a function of Δ min , of the stage in the evolution at which the minimum occurs and of T . This structure persists in examples of larger systems.
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 LLL. 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.