**
Abstract:
**

In principle, we can
use quantum mechanics to
*
exactly
*
describe any system of quantum particles—from simple molecules to unwieldy
proteins (and beyond, see figure)—but in practice this is impossible as the
number of equations grows exponentially with the number of particles.
Recognising this, Richard Feynman suggested that quantum systems be used to
model quantum problems [1]. For example, the fundamental problem faced in
quantum chemistry is the calculation of molecular properties, which are of
practical importance in fields ranging from materials science to biochemistry.
Within chemical precision, the total energy of a molecule as well as most other
properties, can be calculated by solving the Schrödinger equation. However, the
computational resources required increase exponentially with the number of
atoms involved [1, 2].

In the late 1990’s an efficient algorithm was proposed to enable a quantum processor to calculate molecular energies using resources that increase only polynomially in the molecular size [2–4]. Despite the many different physical architectures that have been explored experimentally since that time—including ions, atoms, superconducting circuits, and photons—this appealing algorithm was not demonstrated until last year.

I will discuss how we have taken advantage of recent advances in
photonic quantum computing [5] to present an optical implementation of the
smallest quantum chemistry problem: obtaining the energies of H
_{
2
}
,
the hydrogen molecule, in a minimal basis [6]. We perform a key algorithmic
step—the iterative phase estimation algorithm [7–10]—in full, achieving a high
level of precision and robustness to error. I’ll also report on our recent
results in simulating quantum systems in material science—phase transitions in
topological insulators—and in biology—light-harvesting molecules in
photosynthesis. Together this body of work represents early experimental
progress towards the long term goal of exploiting quantum information to speed
up calculations in biology, chemistry and physics.

[1] R. P. Feynman,
*
International Journal of Theoretical Physics
*
**
21
**
, 467 (1982).

[2] S. Lloyd,
*
Science
*
**
273
**
, 1073 (1996).

[3] D. Abrams and S. Lloyd,
*
Physical Review Letters
*
**
79
**
,
2586 (1997).

[4]
C. Zalka,
*
Proceedings of the Royal
Society of London
A
*
**
454
**
, 313 (1998).

[5] B. P. Lanyon, M. Barbieri, M. P. Almeida, et al.,
*
Nature Physics
*
**
5
**
, 134 (2009).

[6] B. P. Lanyon, J. D.
Whitfield, et al.,
*
Nature Chemistry
*
**
2
**
, 106 (2010).

[7] D. A. Lidar and H. Wang,
*
Physical Review E
*
**
59
**
,
2429 (1999).

[8]
A. Aspuru-Guzik, A. Dutoi, et al.,
*
Science
*
**
309
**
, 1704 (2005).

[9] K. R. Brown, R. J. Clark, and I.
L. Chuang,
*
Physical Review Letters
*
**
97
**
, 050504 (2006).

[10] C. R. Clark, K. R. Brown, et al., arXiv:0810.5626 (2008).