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Approximating ground and excited state energies on a quantum computer

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Abstract

Approximating ground and a fixed number of excited state energies, or equivalently low-order Hamiltonian eigenvalues, is an important but computationally hard problem. Typically, the cost of classical deterministic algorithms grows exponentially with the number of degrees of freedom. Under general conditions, and using a perturbation approach, we provide a quantum algorithm that produces estimates of a constant number \(j\) of different low-order eigenvalues. The algorithm relies on a set of trial eigenvectors, whose construction depends on the particular Hamiltonian properties. We illustrate our results by considering a special case of the time-independent Schrödinger equation with \(d\) degrees of freedom. Our algorithm computes estimates of a constant number \(j\) of different low-order eigenvalues with error \(O(\varepsilon )\) and success probability at least \(\frac{3}{4}\), with cost polynomial in \(\frac{1}{\varepsilon }\) and \(d\). This extends our earlier results on algorithms for estimating the ground state energy. The technique we present is sufficiently general to apply to problems beyond the application studied in this paper.

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Notes

  1. For functions \(f,g\ge 0\) defined on \({\mathbb R}_+\), the notation \(f(\varepsilon )=\omega (g(\varepsilon ))\) means that for any \(M>0\), arbitrarily large, we have \(f(\varepsilon )\ge M g(\varepsilon )\) for sufficiently small \(\varepsilon \).

  2. We give an explicit construction for \(B\) in Eq. (17).

  3. Here and elsewhere, by reasonable overlap we mean that the magnitude of the projection is not exponentially small in \(d\).

  4. This follows immediately for \(m=O(1)\) from the bound \(\left( {\begin{array}{c}d\\ m\end{array}}\right) \le \frac{d^m}{m!} = poly(d)\).

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Acknowledgments

The authors would like to thank Joseph F. Traub for useful comments and suggestions. This research has been supported in part by NSF/DMS.

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  1. Department of Computer Science, Columbia University, New York, NY, 10027, USA

    Stuart Hadfield & Anargyros Papageorgiou

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  1. Stuart Hadfield
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  2. Anargyros Papageorgiou
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Correspondence to Stuart Hadfield.

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Hadfield, S., Papageorgiou, A. Approximating ground and excited state energies on a quantum computer. Quantum Inf Process 14, 1151–1178 (2015). https://doi.org/10.1007/s11128-015-0927-y

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