Efficient eigenvalue determination for arbitrary Pauli products based on generalized spin-spin interactions

Abstract

Effective spin-spin interactions between $N+1$ qubits enable the determination of the eigenvalue of an arbitrary Pauli product of dimension N with a constant, small number of multi-qubit gates that is independent of N and encodes the eigenvalue in the measurement basis states of an extra ancilla qubit. Such interactions are available whenever qubits can be coupled to a shared harmonic oscillator, a situation that can be realized in many physical qubit implementations. For example, suitable interactions have already been realized for up to 14 qubits in ion traps. It should be possible to implement stabilizer codes for quantum error correction with a constant number of multi-qubit gates, in contrast to typical constructions with a number of two-qubit gates that increases as a function of N. The special case of finding the parity of N qubits only requires a small number of operations that is independent of N. This compares favorably to algorithms for computing the parity on conventional machines, which implies a genuine quantum advantage.

Keywords:

• Quantum information

Acknowledgements

This work was mostly developed in the environment of the NIST Ion Storage Group, which therefore owns a great deal of the credit. We would like to thank Jim Bergquist, John Bollinger, James Chou, David Hume, Wayne Itano, David Leibrandt and Andrew Wilson as well as all the post-docs, grad students and summer students and the administrative and technical staff of the Time and Frequency Division at NIST. In addition, we gratefully acknowledge numerous discussions and useful advice from Manny Knill and the members of his group. This paper is a contribution of NIST and is not subject to US copyright.

Notes

No potential conflict of interest was reported by the authors.

We dedicate this work to Danny Segal, scientist, community builder and friend. Your bold and cheerful way of leaving the trodden paths and pursuing your own goals have made our field richer. We will not be able to fill your void, but we will carry on.

Additional information

Funding

This work was supported by the Office of the Director of National Intelligence (ODNI), the Intelligence Advanced Research Projects Activity (IARPA) and the NIST Quantum Information Program.