INVERSIÓN TOTAL

The last step of quantum chemistry algorithms is extracting information about the system from the wave function. Such information includes the ground state energy, scattering am- plitudes, electronic charge density or k-particle correlations38_{. In principle, it is possible}

to estimate any physical quantity or observable that can be expressed through a low-depth quantum circuit and single-qubit measurements.

The most sought-for information in these algorithms is the ground state energy. Zalka41

sketched a method for preparing a known quantum state and introduced the “von Neumann trick” for extracting properties from the prepared wave function. Lidar and Wang44 _then

applied these techniques to develop an algorithm for determining the thermal rate constant of a chemical reaction. Soon after, Abrams and Lloyd38 _{further developed these techniques}

Product formulas 1996 Lloyd37

2001 Dodd et al.251

2002 Nielsen et al.252

2003 Aharonov and Ta-Shma253

2007 Berry et al.260

2010 Childs and Kothari257

2011 Wiebe et al.270

2012 Raeisi et al.312

Quantum walks 2010 Childs262

2011 Berry and Childs313

Linear combinations of unitaries 2012 Childs and Wiebe256

2013 Berry et al.187

2015 Berry et al.314

2015 Berry et al.188

2017 Novo and Berry315

Signal processing & qubitization 2016 Low and Chuang189

2017 Low and Chuang190

2018 Gily´en et al.266

Composite algorithms 2018 Low and Wiebe197

2018 Low265

1996 Zalka41

1996 Wiesner40

1997 Abrams and Lloyd99

2005 Aspuru-Guzik et al.45

2009 Kassal et al.75

2010 Whitfield et al.186

2013 Toloui and Love254

2013 Wecker et al.47 2014 McClean et al.306 2014 Hastings et al.48 2014 Poulin et al.49 2014 Babbush et al.50 2015 Babbush et al.191 2016 Babbush et al.192 2017 Babbush et al.301 2017 Kivlichan et al.76 2018 Kivlichan et al.302

2018 Low and Wiebe197

2018 Babbush et al.281

2018 Babbush et al.198

Figure 5: Chronological overview of quantum chemistry simulation algorithms. On the left-hand side we list quantum simulation algorithm grouped by the techniques they use. The right-hand side outlines the improvement for time evolution in quantum chemistry. We indicate the underlying simulation technique with an arrow from the left column to the right. Furthermore, we color-code the Hamiltonian representation – yellow (lighter color) for first

to specifically apply them towards calculating static properties of a quantum system. In 2005, Aspuru-Guzik et al.45 _{adapted these techniques for the electronic structure problem.}

This allowed the number of qubits in the QPEA ancilla register to be reduced from 20 to 4, enabling the study of quantum algorithms for electronic structure problems on a classical computer45_{. This work shows that even with modest quantum computers of 30-100 error-}

corrected qubits, ground state energy calculations of H2O and LiH could be carried out to

an accuracy beyond that of classical computers. The essential algorithm underlying this approach is quantum phase estimation and its modifications (see Section 4.1.2).

An iterative version of QPEA for ground state estimation was recently introduced by Berry et al.279_{. The authors assume that there is an upper bound on the ground state}

energy, say from a classical variational method, that is guaranteed to be lower than the energy of the first excited state. Given this assumption, one can perform QPEA gradually, measuring the ancilla qubits sequentially instead of postponing the measurement to the end of the circuit. If the outcome of the QPEA is likely to be a state above this threshold, i.e. not the ground state, it is possible to abort phase estimation and restart the algorithm.

QPEA can be also used for estimating the energies of excited states. In the simplest setting, one can use a state that is not an energy eigenstate and use it to sample multi- ple energies from the spectrum of the Hamiltonian. Santagati et al.316 _{introduced a more}

sophisticated technique for approximating excited state energies.

A quantum algorithm for ground state energy estimation (as discussed here) requires techniques from state preparation (Section 4.2.1) and as a subroutine simulation (Section 4.2.2). Combining the recent development all these areas, Babbush et al.281 _{give detailed}

resource estimates, or “pricing”, for classically intractable ground state energy calculations of diamond, graphite, silicon, metallic lithium, and crystalline lithium hydride. Incorporating state-of-the-art error correction methods, they show that the estimates for the number of gates (which is dominated by the number of T -gates) required to estimate the ground state energy of FeMoCo is millions of times smaller than the number needed in the methods used

earlier305_.

Without access to fault-tolerant quantum computers, only proof-of-principle quantum chemistry calculations have been demonstrated. In particular, few-qubit simulations, often without error correction, have been carried out in most major architectures used for quantum information. In 2010, Lanyon et al.234 _{demonstrated the use of the IPEA to measure the}

energy of molecular wave functions. In this case, the wave function of molecular hydrogen (H2) in a minimal basis set was encoded in a one-qubit state and the IPEA was realized

using a two-qubit photonic chip, calculating the molecular hydrogen spectrum to 20 bits of precision. A similar procedure was applied to H2 using nuclear magnetic resonance Lanyon

et al.234_{, Du et al.}235_{, Dolde et al.}236 _{and to helium hydride (HeH}+_{) using nitrogen vacancies}

in diamond236_.

Although these proof-of-principle experiments are groundbreaking, it is not clear how to scale them because of their reliance on Hamiltonians simplifications and tomography. The first scalable demonstration of the IPEA (and the variational quantum eigensolver algorithm, as discussed in Section 5) employed three superconducting qubits for simulating H2 in a

minimal basis and was carried out by a Google Research and Harvard collaboration55_{. We}

will explain hybrid quantum-classical algorithms better suitable for these devices in the next section.