## Wednesday, December 30, 2015

### Scalable Quantum Simulation of Molecular Energies

P. J. J. O’Malley, R. Babbush, I. D. Kivlichan, J. Romero, J. R. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. Ding, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Jeffrey, A. Megrant, J. Y. Mutus, C. Neill, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, P. V. Coveney, P. J. Love, H. Neven, A. Aspuru-Guzik, and J. M. Martinis (2015)
Contributed by Jan Jensen

This paper describes the first electronic structure calculation, the PES of H$_2$, performed on a quantum computer without "exponentially costly precompilation".  I know very little of quantum computing so bear with me as I try to explain this paper to myself in terms even I can begin to understand.

From Wikipedia
A classical computer has a memory made up of bits, where each bit is represented by either a one or a zero. A quantum computer maintains a sequence of qubits. A single qubit can represent a one, a zero, or any quantum superposition of those two qubit states; a pair of qubits can be in any quantum superposition of 4 states, and three qubits in any superposition of 8 states. In general, a quantum computer with $n$ qubits can be in an arbitrary superposition of up to 2$^n$ different states simultaneously (this compares to a normal computer that can only be in one of these 2$^n$ states at any one time). A quantum computer operates by setting the qubits in a controlled initial state that represents the problem at hand and by manipulating those qubits with a fixed sequence of quantum logic gates. The sequence of gates to be applied is called a quantum algorithm. The calculation ends with a measurement, collapsing the system of qubits into one of the 2$^n$ pure states, where each qubit is zero or one, decomposing into a classical state. The outcome can therefore be at most $n$ classical bits of information. Quantum algorithms are often non-deterministic, in that they provide the correct solution only with a certain known probability.
The paper describes two experiments and I'll focus on the first one. The experiment uses two Xmon transmon qubits, which is a kind of superconducting charge qubit where the two qubit states are uncharged and charged superconducting islands (essentially circuits made of superconducting aluminum film deposited on a sapphire substrate).

The molecular Hamiltonian of H$_2$ within a minimal basis set is rewritten as
$$H = g_0 \textbf{1} + g_1Z_0 + g_2Z_1 + g_3Z_0Z_1 + g_4X_0X_1+g_5Y_0Y_1$$
{$X_i,Y_i,Z_i$} is measured (quantum computed) and {$g_i$} are numbers that can be (classically) computed and that depend on the H-H bond length ($R$).  Since there are only two orbitals the energy depends on one variational parameter ($\theta$) so the objective is to find the value of $\theta$ that minimizes the energy for a given value of $R$.

So the authors set up a quantum algorithm that outputs {$X_i,Y_i,Z_i$} given an initial state that depends on $\theta$.  This is done for a thousands different values of $\theta$ and for each value of $R$ the value of $\theta$ that minimizes the energy defined by Eq 1 is found by classical minimization, to yield a PES (see Figure 2 from the paper reproduced below)

The "exponentially costly precompilation" mentioned above refers to the fact that the conventional quantum algorithm approach (QPE) (source):
requires a large number of n-qubit quantum controlled operations to be performed in series—placing considerable demands on the number of components and coherence time—while the inherent parallelism of our scheme enables a small number of n-qubit gates to be exploited many times, drastically reducing these demands.