# Eigenvalue estimation of differential operators with a quantum algorithm

## Abstract

We demonstrate how linear differential operators could be emulated by a quantum processor, should one ever be built, using the Abrams-Lloyd algorithm. Given a linear differential operator of order 2S, acting on functions {psi}(x{sub 1},x{sub 2},...,x{sub D}) with D arguments, the computational cost required to estimate a low order eigenvalue to accuracy {theta}(1/N{sup 2}) is {theta}((2(S+1)(1+1/{nu})+D)ln N) qubits and O(N{sup 2(S+1)(1+1/{nu})}ln{sup c} N{sup D}) gate operations, where N is the number of points to which each argument is discretized, {nu} and c are implementation dependent constants of O(1). Optimal classical methods require {theta}(N{sup D}) bits and {omega}(N{sup D}) gate operations to perform the same eigenvalue estimation. The Abrams-Lloyd algorithm thereby leads to exponential reduction in memory and polynomial reduction in gate operations, provided the domain has sufficiently large dimension D>2(S+1)(1+1/{nu}). In the case of Schroedinger's equation, ground state energy estimation of two or more particles can in principle be performed with fewer quantum mechanical gates than classical gates.

- Authors:

- Department of Electrical Engineering, University of California Los Angeles, Los Angeles, California 90095 (United States)
- (United States)

- Publication Date:

- OSTI Identifier:
- 20786281

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Physical Review. A; Journal Volume: 72; Journal Issue: 6; Other Information: DOI: 10.1103/PhysRevA.72.062318; (c) 2005 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; ALGORITHMS; EIGENFUNCTIONS; EIGENVALUES; GATING CIRCUITS; GROUND STATES; INFORMATION THEORY; POLYNOMIALS; QUANTUM MECHANICS; QUANTUM OPERATORS; QUBITS; SCHROEDINGER EQUATION

### Citation Formats

```
Szkopek, Thomas, Roychowdhury, Vwani, Yablonovitch, Eli, Abrams, Daniel S., and Luminescent Technologies, Inc., Mountain View, California 94041.
```*Eigenvalue estimation of differential operators with a quantum algorithm*. United States: N. p., 2005.
Web. doi:10.1103/PHYSREVA.72.0.

```
Szkopek, Thomas, Roychowdhury, Vwani, Yablonovitch, Eli, Abrams, Daniel S., & Luminescent Technologies, Inc., Mountain View, California 94041.
```*Eigenvalue estimation of differential operators with a quantum algorithm*. United States. doi:10.1103/PHYSREVA.72.0.

```
Szkopek, Thomas, Roychowdhury, Vwani, Yablonovitch, Eli, Abrams, Daniel S., and Luminescent Technologies, Inc., Mountain View, California 94041. Thu .
"Eigenvalue estimation of differential operators with a quantum algorithm". United States.
doi:10.1103/PHYSREVA.72.0.
```

```
@article{osti_20786281,
```

title = {Eigenvalue estimation of differential operators with a quantum algorithm},

author = {Szkopek, Thomas and Roychowdhury, Vwani and Yablonovitch, Eli and Abrams, Daniel S. and Luminescent Technologies, Inc., Mountain View, California 94041},

abstractNote = {We demonstrate how linear differential operators could be emulated by a quantum processor, should one ever be built, using the Abrams-Lloyd algorithm. Given a linear differential operator of order 2S, acting on functions {psi}(x{sub 1},x{sub 2},...,x{sub D}) with D arguments, the computational cost required to estimate a low order eigenvalue to accuracy {theta}(1/N{sup 2}) is {theta}((2(S+1)(1+1/{nu})+D)ln N) qubits and O(N{sup 2(S+1)(1+1/{nu})}ln{sup c} N{sup D}) gate operations, where N is the number of points to which each argument is discretized, {nu} and c are implementation dependent constants of O(1). Optimal classical methods require {theta}(N{sup D}) bits and {omega}(N{sup D}) gate operations to perform the same eigenvalue estimation. The Abrams-Lloyd algorithm thereby leads to exponential reduction in memory and polynomial reduction in gate operations, provided the domain has sufficiently large dimension D>2(S+1)(1+1/{nu}). In the case of Schroedinger's equation, ground state energy estimation of two or more particles can in principle be performed with fewer quantum mechanical gates than classical gates.},

doi = {10.1103/PHYSREVA.72.0},

journal = {Physical Review. A},

number = 6,

volume = 72,

place = {United States},

year = {Thu Dec 15 00:00:00 EST 2005},

month = {Thu Dec 15 00:00:00 EST 2005}

}