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# Unitary circuit synthesis for tomography of generalized coherent states

## Abstract

We present a method that outputs a sequence of simple unitary operations to prepare a given quantum state that is a generalized coherent state. Our method takes as inputs the expectation values of some relevant observables. Such expectations can be estimated by performing projective measurements on $$\mathcal{O}$$(M ^{3}log(M/δ)/ε ^{2}) copies of the state, where M is the dimension of an associated Lie algebra, ε is a precision parameter, and 1 - δ is the required confidence level. The method can be implemented on a classical computer and runs in time $$\mathcal{O}$$(M ^{4}log(M/ε)). It provides $$\mathcal{O}$$(Mlog(M/ε)) simple unitaries that form the sequence. The overall complexity is then polynomial in M, being very efficient in cases where M is significantly smaller than the Hilbert space dimension, as for some fermion algebras. When the algebra of relevant observables is given by certain Pauli matrices, each simple unitary may be easily decomposed into two-qubit gates. We discuss applications to efficient quantum state tomography and classical simulations of quantum circuits. $$\mathcal{O}$$

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1574752

- Alternate Identifier(s):
- OSTI ID: 1573100

- Report Number(s):
- LA-UR-18-30016

Journal ID: ISSN 0022-2488

- Grant/Contract Number:
- 89233218CNA000001

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 60; Journal Issue: 11; Journal ID: ISSN 0022-2488

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Computer Science; Information Science; Mathematics; Quantum tomography, quantum information, quantum computing

### Citation Formats

```
Somma, Rolando Diego. Unitary circuit synthesis for tomography of generalized coherent states. United States: N. p., 2019.
Web. doi:10.1063/1.5121549.
```

```
Somma, Rolando Diego. Unitary circuit synthesis for tomography of generalized coherent states. United States. doi:10.1063/1.5121549.
```

```
Somma, Rolando Diego. Mon .
"Unitary circuit synthesis for tomography of generalized coherent states". United States. doi:10.1063/1.5121549.
```

```
@article{osti_1574752,
```

title = {Unitary circuit synthesis for tomography of generalized coherent states},

author = {Somma, Rolando Diego},

abstractNote = {We present a method that outputs a sequence of simple unitary operations to prepare a given quantum state that is a generalized coherent state. Our method takes as inputs the expectation values of some relevant observables. Such expectations can be estimated by performing projective measurements on $\mathcal{O}$(M3log(M/δ)/ε2) copies of the state, where M is the dimension of an associated Lie algebra, ε is a precision parameter, and 1 - δ is the required confidence level. The method can be implemented on a classical computer and runs in time $\mathcal{O}$(M4log(M/ε)). It provides $\mathcal{O}$(Mlog(M/ε)) simple unitaries that form the sequence. The overall complexity is then polynomial in M, being very efficient in cases where M is significantly smaller than the Hilbert space dimension, as for some fermion algebras. When the algebra of relevant observables is given by certain Pauli matrices, each simple unitary may be easily decomposed into two-qubit gates. We discuss applications to efficient quantum state tomography and classical simulations of quantum circuits. $\mathcal{O}$},

doi = {10.1063/1.5121549},

journal = {Journal of Mathematical Physics},

number = 11,

volume = 60,

place = {United States},

year = {2019},

month = {11}

}

Works referenced in this record:

##
Efficient quantum state tomography

journal, December 2010

- Cramer, Marcus; Plenio, Martin B.; Flammia, Steven T.
- Nature Communications, Vol. 1, Issue 1

##
Permutationally Invariant Quantum Tomography

journal, December 2010

- Tóth, G.; Wieczorek, W.; Gross, D.
- Physical Review Letters, Vol. 105, Issue 25

##
Unconditionally verifiable blind quantum computation

journal, July 2017

- Fitzsimons, Joseph F.; Kashefi, Elham
- Physical Review A, Vol. 96, Issue 1

##
Quantum Simulations of Physics Problems

journal, June 2003

- Somma, Rolando; Ortiz, Gerardo; Knill, Emanuel
- International Journal of Quantum Information, Vol. 01, Issue 02

##
Diagonalization in Compact Lie Algebras and a New Proof of a Theorem of Kostant

journal, October 1993

- Wildberger, N. J.
- Proceedings of the American Mathematical Society, Vol. 119, Issue 2

##
Efficient Solvability of Hamiltonians and Limits on the Power of Some Quantum Computational Models

journal, November 2006

- Somma, Rolando; Barnum, Howard; Ortiz, Gerardo
- Physical Review Letters, Vol. 97, Issue 19

##
Coherent states: Theory and some applications

journal, October 1990

- Zhang, Wei-Min; Feng, Da Hsuan; Gilmore, Robert
- Reviews of Modern Physics, Vol. 62, Issue 4

##
Nature and measure of entanglement in quantum phase transitions

journal, October 2004

- Somma, Rolando; Ortiz, Gerardo; Barnum, Howard
- Physical Review A, Vol. 70, Issue 4

##
Generalizations of entanglement based on coherent states and convex sets

journal, September 2003

- Barnum, Howard; Knill, Emanuel; Ortiz, Gerardo
- Physical Review A, Vol. 68, Issue 3

##
Lower bounds for the fidelity of entangled-state preparation

journal, November 2006

- Somma, Rolando D.; Chiaverini, John; Berkeland, Dana J.
- Physical Review A, Vol. 74, Issue 5

##
Probability Inequalities for Sums of Bounded Random Variables

journal, March 1963

- Hoeffding, Wassily
- Journal of the American Statistical Association, Vol. 58, Issue 301