# Variational quantum state diagonalization

## Abstract

Variational hybrid quantum-classical algorithms are promising candidates for near-term implementation on quantum computers. In these algorithms, a quantum computer evaluates the cost of a gate sequence (with speedup over classical cost evaluation), and a classical computer uses this information to adjust the parameters of the gate sequence. We introduce such an algorithm for quantum state diagonalization. State diagonalization has applications in condensed matter physics (e.g., entanglement spectroscopy) as well as in machine learning (e.g., principal component analysis). For a quantum state ρ and gate sequence U, our cost function quantifies how far UρU ^{†} is from being diagonal. We introduce short-depth quantum circuits to quantify our cost. Minimizing this cost returns a gate sequence that approximately diagonalizes ρ. One can then read out approximations of the largest eigenvalues, and the associated eigenvectors, of ρ. As a proof-of-principle, we implement our algorithm on Rigetti’s quantum computer to diagonalize one-qubit states and on a simulator to find the entanglement spectrum of the Heisenberg model ground state.

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Michigan State Univ., East Lansing, MI (United States)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Imperial College, London (United Kingdom)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States); Quantum Information Science (QIS)

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Materials Sciences & Engineering Division; USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1544727

- Report Number(s):
- LA-UR-18-29266

Journal ID: ISSN 2056-6387

- Grant/Contract Number:
- 89233218CNA000001

- Resource Type:
- Accepted Manuscript

- Journal Name:
- npj Quantum Information

- Additional Journal Information:
- Journal Volume: 5; Journal Issue: 1; Journal ID: ISSN 2056-6387

- Publisher:
- Nature Partner Journals

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Quantum Information Science (QIS)

### Citation Formats

```
LaRose, Ryan, Tikku, Arkin, O’Neel-Judy, Étude, Cincio, Lukasz, and Coles, Patrick J. Variational quantum state diagonalization. United States: N. p., 2019.
Web. doi:10.1038/s41534-019-0167-6.
```

```
LaRose, Ryan, Tikku, Arkin, O’Neel-Judy, Étude, Cincio, Lukasz, & Coles, Patrick J. Variational quantum state diagonalization. United States. doi:10.1038/s41534-019-0167-6.
```

```
LaRose, Ryan, Tikku, Arkin, O’Neel-Judy, Étude, Cincio, Lukasz, and Coles, Patrick J. Wed .
"Variational quantum state diagonalization". United States. doi:10.1038/s41534-019-0167-6. https://www.osti.gov/servlets/purl/1544727.
```

```
@article{osti_1544727,
```

title = {Variational quantum state diagonalization},

author = {LaRose, Ryan and Tikku, Arkin and O’Neel-Judy, Étude and Cincio, Lukasz and Coles, Patrick J.},

abstractNote = {Variational hybrid quantum-classical algorithms are promising candidates for near-term implementation on quantum computers. In these algorithms, a quantum computer evaluates the cost of a gate sequence (with speedup over classical cost evaluation), and a classical computer uses this information to adjust the parameters of the gate sequence. We introduce such an algorithm for quantum state diagonalization. State diagonalization has applications in condensed matter physics (e.g., entanglement spectroscopy) as well as in machine learning (e.g., principal component analysis). For a quantum state ρ and gate sequence U, our cost function quantifies how far UρU† is from being diagonal. We introduce short-depth quantum circuits to quantify our cost. Minimizing this cost returns a gate sequence that approximately diagonalizes ρ. One can then read out approximations of the largest eigenvalues, and the associated eigenvectors, of ρ. As a proof-of-principle, we implement our algorithm on Rigetti’s quantum computer to diagonalize one-qubit states and on a simulator to find the entanglement spectrum of the Heisenberg model ground state.},

doi = {10.1038/s41534-019-0167-6},

journal = {npj Quantum Information},

number = 1,

volume = 5,

place = {United States},

year = {2019},

month = {6}

}

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