Quantum computational chemistry

In the paper

“However, the cost of storing the wavefunction scales exponentially with the number of electrons, N .”

In the paper

“The diagonalised 1-RDM is given by Diag (1.96588, 0.00611, 0.02104, 0.0002, 0.00001, 0.00314, 0.00314, 0.00016, 0.00016, 0.00016).”

In the paper

“We can see that the 5 th orbital has a NOON that is 20 times smaller than the next smallest NOON.”

In the paper

“We can see that the differences in energy between the three minima are considerably larger than chemical accuracy (1 . 6 mHartree).”

In the paper

“which can reproduce the ground state given by Eq. (98).”

In the paper

“The exact application of these laws leads to equations much too complicated to be soluble.”

What the sources say

• Not checked yetcites Dirac, 1929

In the paper

“With only single qubit operations and CNOT gates, it is possible to approximate an arbitrary multi-qubit gate to any desired accuracy (DiVincenzo, 1995).”

What the sources say

• Not checked yetcites DiVincenzo, 1995

In the paper

“This leaves classical computers unable to exactly simulate quantum systems in an efficient way.”

In the paper

“As quantum computers are able to efficiently simulate quantum systems”

What the sources say

• Not checked yetcites Feynman, 1982

In the paper

“The problem described by Dirac is that the complexity of the wavefunction of a quantum system grows exponentially with the number of particles.”

What the sources say

• Not checked yetcites Dirac, 1929

In the paper

“However, their ability to guide design is often precluded by the computational complexity of realistic models.”

In the paper

“Physical error rates below the surface code threshold of around 1 % (Fowler et al. , 2012b; Stephens, 2014; Wang et al. , 2011) have been achieved for trapped ion (Ballance et al. , 2016; Gaebler et al. , 2016) and superconducting (Arute et al. , 2019; Barends et al. , 2014) qubits.”

What the sources say

• Not checked yetcites Fowler et al., 2012b
• Not checked yetcites Ballance et al., 2016
• Not checked yetcites Arute et al., 2019

In the paper

“Moreover, it is important to note that solving the ground state problem for a completely general local Hamiltonian is QMA-complete (quantum Merlin-Arthur complete, the quantum analogue of NP-complete) (Cubitt and Montanaro, 2016; Kempe et al. , 2006).”

What the sources say

• Not checked yetcites Cubitt and Montanaro, 2016
• Not checked yetcites Kempe et al., 2006

In the paper

“This makes it classically intractable to simulate more than a few particles on a grid using a classical computer.”

In the paper

“If the energy is known to chemical accuracy, then the chemical reaction rate at room temperature can be predicted to within an order of magnitude using the Eyring equation (Evans and Polanyi, 1935; Eyring, 1935)”

What the sources say

• Not checked yetcites Evans and Polanyi, 1935
• Not checked yetcites Eyring, 1935

In the paper

“While there has been significant experimental progress in understanding the structure of the nitrogenase enzyme over the last 100 years, the reaction mechanism is not yet fully understood (Burgess and Lowe, 1996).”

What the sources say

• Not checked yetcites Burgess and Lowe, 1996

In the paper

“The ground state problem can also be tackled using variational algorithms (Peruzzo et al. , 2014), which we will discuss in detail in Sec. V.B.”

What the sources say

• Not checked yetcites Peruzzo et al., 2014

In the paper

“It has also been shown that time evolution of open and closed quantum systems can be simulated using variational approaches (Chen et al. , 2019; Endo et al. , 2018b; Heya et al. , 2019; Li and Benjamin, 2017).”

What the sources say

• Not checked yetcites Chen et al., 2019
• Not checked yetcites Endo et al., 2018b
• Not checked yetcites Li and Benjamin, 2017

In the paper

“However, with these error rates, we would require around 10 3 -10 4 physical qubits per logical qubit to perform interesting tasks in a fault-tolerant manner (Campbell et al. , 2017; Fowler et al. , 2012a; O'Gorman and Campbell, 2017).”

What the sources say

• Not checked yetcites Campbell et al., 2017
• Not checked yetcites Fowler et al., 2012a
• Not checked yetcites O'Gorman and Campbell, 2017

In the paper

“This property does not hold for grid based methods.”

In the paper

“However, these problems are typically artificially constructed examples, rather than real-world problems (Boixo et al. , 2018; Harrow and Montanaro, 2017).”

What the sources say

• Not checked yetcites Boixo et al., 2018
• Not checked yetcites Harrow and Montanaro, 2017

In the paper

“While more complex models are likely required to fully understand the mechanism of cuprate superconductivity, a complete understanding of even the Fermi-Hubbard model is still elusive.”

In the paper

“Problems in this complexity class are not believed to be efficiently solvable with either a classical or quantum computer.”

In the paper

“A complete analytic or computational understanding of these layers is beyond the capabilities of current classical techniques.”

In the paper

“The number of terms in the Hamiltonian scales polynomially with the number of particles for systems of interest in chemistry, due to the two-body nature of the Coulomb interactions between particles (Helgaker et al. , 2014).”

What the sources say

• Not checked yetcites Helgaker et al., 2014

In the paper

“However, many of these systems show strong correlation, often due to their openshell nature, and spatially degenerate states (Lyakh et al. , 2012).”

What the sources say

• Not checked yetcites Lyakh et al., 2012

In the paper

“Finding the low lying energy levels of a quantum Hamiltonian is in general an exponentially difficult problem for classical computers.”

In the paper

“This strong correlation often precludes their study in silico , forcing us to carry out expensive, trial-and-error based discovery -especially for transition metal based biological catalysts (Podewitz et al. , 2011).”

What the sources say

• Not checked yetcites Podewitz et al., 2011

In the paper

“In particular, models of fermions hopping on a 2D square lattice, such as the one-band Fermi-Hubbard model (Hubbard, 1963), and the related t-J model, appear to reproduce many experimental observations (Anderson, 2002; Lee et al. , 2006).”

What the sources say

• Not checked yetcites Anderson, 2002
• Not checked yetcites Lee et al., 2006

In the paper

“Mapping out these potential energy curves explicitly is exponentially costly in the degrees of freedom of the molecule, and that there are a variety of methods being developed to solve this difficult problem more efficiently (Christiansen, 2012).”

What the sources say

• Not checked yetcites Christiansen, 2012

In the paper

“To the best of our knowledge, only the work by Berry et al. (2019a) considers a chemical reaction dynamics calculation on a quantum computer using basis set methods.”

What the sources say

• Not checked yetcites Berry et al., 2019a

In the paper

“As mentioned in the previous section, mapping out these potential energy surfaces is exponentially costly.”

In the paper

“In contrast, basis set methods exploit this knowledge to reduce the resources needed to simulate chemical systems.”

In the paper

“If the electronelectron Coulomb interaction term in Eq. (17) is neglected, we obtain a new Hamiltonian which describes the behaviour of N independent electrons.”

In the paper

“Instead, a mean-field approximation (the HartreeFock procedure, to be described in Sec. III.C.1) can be used to obtain more suitable molecular orbitals.”

In the paper

“MCSCF and CASSCF calculations are among the most effective classical methods at treating systems with strong static correlation (Szalay et al. , 2012).”

What the sources say

• Not checked yetcites Szalay et al. , 2012

In the paper

“Consequently, the eigenstates of the true Hamiltonian can be expressed as linear combinations of these Slater determinants, written as”

What the sources say

• Not checked yetcites Helgaker et al. , 2014
• Not checked yetcites Szabo and Ostlund, 2012

In the paper

“As first quantised operators conserve the number of electrons, the second quantised operators must contain an equal number of creation and annihilation operators.”

In the paper

“Truncated CI calculations also fail to satisfy a related property, known as 'size consistency'.”

In the paper

“As they are eigenstates of a Hermitian operator, these Slater determinants form a complete basis of the problem Hilbert space.”

In the paper

“MCSCF can be considered the best approximation to the exact wavefunction for a given number of determinants (Wang et al. , 2008).”

What the sources say

• Not checked yetcites Wang et al. , 2008

In the paper

“Wavefunctions displaying dynamic correlation are often dominated by the Hartree-Fock determinant, and have small additional contributions from a (potentially large) number of excited state determinants.”

In the paper

“As a result, static correlation often dominates in many systems of scientific interest, such as excited states (Lischka et al. , 2018), or transition metals (Vogiatzis et al. , 2019).”

What the sources say

• Not checked yetcites Lischka et al. , 2018
• Not checked yetcites Vogiatzis et al. , 2019

In the paper

“It is not possible to perform a complete MCSCF calculation on all possible determinants for systems with more than a few electrons, as the number of determinants scales exponentially with the number of electrons.”

In the paper

“However, as low energy excitations dominate the ground state wavefunction in many physical systems, these truncations can still produce good approximations to the ground state energy (Helgaker et al. , 2014; Szabo and Ostlund, 2012).”

What the sources say

• Not checked yetcites Helgaker et al. , 2014
• Not checked yetcites Szabo and Ostlund, 2012

In the paper

“In this case, the Hartree-Fock method provides a poor approximation to the ground state wavefunction.”

In the paper

“Lyakh et al. (2012) note that, while it is sometimes possible to reduce the level of static correlation by manually enforcing correct spin symmetries, and using appropriately localised orbitals, it is in general not possible to avoid strong static correlation when considering the whole of the potential energy surface.”

What the sources say

• Not checked yetcites Lyakh et al. (2012)

In the paper

“Therefore, the expectation values of second quantised operators must be equivalent to the expectation values of the corresponding first quantised operators.”

In the paper

“These solutions are exact, provided that the molecular orbitals form a complete basis for the single particle states, and the N -electron wavefunction contains all of the determinants that these molecular orbitals can generate (Helgaker et al. , 2014; Szabo and Ostlund, 2012).”

What the sources say

• Not checked yetcites Helgaker et al. , 2014
• Not checked yetcites Szabo and Ostlund, 2012

In the paper

“The method converges slowly to the full configuration interaction wavefunction, as a result of its linear parametrization.”

In the paper

“The energy obtained from a truncated CI calculation is not size extensive.”

What the sources say

• Not checked yetcites Lyakh et al. , 2012

In the paper

“As a result, the Hamiltonian contains up to M 4 terms, depending on the basis functions used.”

In the paper

“The presence of strong static correlation can be evidenced by multiple near-degenerate solutions of the Hartree-Fock procedure.”

In the paper

“These methods create parametrized trial states, which can then be optimised to approach the ground state (to an accuracy determined by the approximations made).”

In the paper

“The Hamiltonian only contains terms with up to four creation and annihilation operators (two creation, two annihilation) because the Coulomb interaction between the electrons is a two-body interaction.”

In the paper

“These basis sets diagonalise the kinetic and potential operators, respectively.”

What the sources say

• Not checked yetcites Babbush et al. , 2018c

In the paper

“The string of Z operators means that a fermionic operator mapped to qubits generally has a weight of O ( M ) Pauli operators, each acting on a different qubit.”

In the paper

“However, to describe molecular systems, approximately 10 to 100 times as many plane wave basis functions are required as GTOs (Babbush et al. , 2018c).”

What the sources say

• Not checked yetcites Babbush et al. , 2018c

In the paper

“It is sometimes the case that certain orbitals are very likely to be either occupied or virtual in all Slater determinants in the wavefunction.”

In the paper

“Specifically, we are able to remove spin-orbitals from the calculation if their expected occupation number is close to 0 or 1. Our calculation is reduced to including only the most important (ambiguously occupied) orbitals. This is known as performing the calculation in a reduced active space.”

In the paper

“Most notably, the wavefunction generated by the canonical CC method does not obey the Rayleigh-Ritz variational principle (Helgaker et al. , 2014).”

What the sources say

• Not checked yetcites Helgaker et al. , 2014

In the paper

“Takeshita et al. (2019) showed how to re-integrate some of the energy contribution of the virtual orbitals into a quantum simulation, without requiring additional qubits to represent the virtual orbitals. Their method utilises an increased number of measurements, and classical post-processing.”

What the sources say

• Not checked yetcites Takeshita et al. (2019)

In the paper

“cc-PV n Z basis sets with higher values of n contain orbitals that better approximate the true atomic orbitals than those with lower n values.”

In the paper

“While the aforementioned basis sets have a long history of use in classical computational chemistry (and as a result, early work in quantum computational chemistry), they are not necessarily optimal basis sets for calculations performed on quantum computers.”

In the paper

“This limitation can be overcome by measuring the ground state energy in several different cc-PV n Z bases, and then extrapolating to the basis set limit.”

In the paper

“However, using a quantum computer, we can instead store the full configuration interaction (FCI) wavefunction using only M qubits (Aspuru-Guzik et al. , 2005).”

What the sources say

• Not checked yetcites Aspuru-Guzik et al. , 2005

In the paper

“It has been noted that the BK-tree mapping produces qubit operators with a greater weight than the standard BK mapping (Sung, 2018).”

What the sources say

• Not checked yetcites Sung, 2018

In the paper

“This in turn leads to a reduction in the asymptotic scaling of quantum chemistry algorithms to find the ground state energy of molecules and solid state systems”

In the paper

“These mappings typically require more qubits than the JW encoding.”

In the paper

“Consequently, grid based approaches are typically considered unsuitable for near-term quantum computers, which will have relatively few qubits.”

In the paper

“It is important to note that when carrying out a HF calculation in an STOn G basis, the true HF energy (i.e. the energy obtained by performing a HF calculation using a grid based method, on an infinitely precise grid) will not be obtained, as the STOn G basis sets only approximate the true HF orbitals.”

In the paper

“Applying the BK mapping to a fermionic operator results in a qubit operator with a Pauli weight of O (log 2 M ).”

In the paper

“However, even large ( n = 5) basis sets struggle to exactly represent the true HF orbitals of simple molecules such as N 2 (Helgaker et al. , 2014).”

What the sources say

• Not checked yetcites Helgaker et al. , 2014

In the paper

“As we will see in Sec. V, the number of terms in the Hamiltonian plays a key role in the cost of some quantum chemistry algorithms.”

In the paper

“Phase estimation (Kitaev, 1995) can be used to find the lowest energy eigenstate, | E 0 〉 , and excited states, | E i> 0 〉 , of a physical Hamiltonian (Abrams and Lloyd, 1999).”

What the sources say

• Not checked yetcites Kitaev, 1995
• Not checked yetcites Abrams and Lloyd, 1999

In the paper

“Even more problematically, McClean et al. (2014) showed that phase estimation can become exponentially costly, by considering the imperfect preparation of eigenstates of non-interacting subsystems.”

What the sources say

• Not checked yetcites McClean et al., 2014

In the paper

“Wiebe et al. (2011) showed that the accuracy of such simulations depends on the derivatives of the Hamiltonian (although this dependence may be alleviated by incorporating randomisation procedures (Poulin et al. , 2011)).”

What the sources say

• Not checked yetcites Wiebe et al., 2011
• Not checked yetcites Poulin et al., 2011

In the paper

“As discussed in Sec. III.A, the electronic structure Hamiltonian in the canonical molecular orbital basis contains O ( M 4 ) terms, where M is the number of spinorbitals in the molecule.”

In the paper

“This has caused second quantised basis set methods to become the de facto option for experimental demonstrations of quantum computational chemistry algorithms, due to the limits of current quantum hardware.”

In the paper

“Qubitization naturally implements a quantum walk operator with eigenvalues e -i arcsin( E k /α ) where E k is the k th eigenvalue of the Hamiltonian H and α is a normalisation factor.”

In the paper

“This highlights the necessity of developing state preparation routines which result in at worst a polynomially decreasing overlap with the FCI ground state, as the system size increases.”

In the paper

“However, the number of gates required to implement a fermionic operator will scale as O ( d ) where d is the degree of the graph.”

In the paper

“For any Hamiltonian H s , we can prepare a state | Ψ 〉 that is close to its ground state via adiabatic state preparation (Albash and Lidar, 2018).”

What the sources say

• Not checked yetcites Albash and Lidar, 2018

In the paper

“They result in a second quantised Hamiltonian with O ( M 4 ) terms.”

In the paper

“This low rank decomposition technique was also used by Huggins et al. (2019b), to reduce the number of circuit repetitions required for variational algorithms to find the ground state energy of chemical systems.”

What the sources say

• Not checked yetcites Huggins et al. (2019b)

In the paper

“We note however that if an initial state with sufficiently large overlap with the ground state is available, we may be able to forgo adiabatic state preparation entirely, and instead carry out phase estimation directly on that initial state.”

In the paper

“In general, it is difficult to optimally allocate resources between these error budgets in order to minimise the total error (Kivlichan et al. , 2019a; Reiher et al. , 2017).”

What the sources say

• Not checked yetcites Kivlichan et al., 2019a
• Not checked yetcites Reiher et al., 2017

In the paper

“The ultimate limit for the number of calls required to e -iH is π/ϵ PE (for a completely general Hamiltonian, H ), which is approximately obtained using Bayesian approaches (Berry et al. , 2009; Higgins et al. , 2007; Paesani et al. , 2017; Wiebe and Granade, 2016), or entanglement based approaches (Babbush et al. , 2018b).”

What the sources say

• Not checked yetcites Berry et al. , 2009
• Not checked yetcites Higgins et al. , 2007
• Not checked yetcites Babbush et al. , 2018b

In the paper

“In practice, to achieve accuracy ε , the number of Trotter steps S = O ( t 2 /ε ) should be large in order to suppress the errors in the approximation.”

In the paper

“The finite precision, ϵ PE , obtained in E 0 is not the only source of error in the algorithm.”

In the paper

“For the specific case of a Trotter decomposition of e -iH , Reiher et al. (2017) showed that the error in the energy eigenvalue obtained from phase estimation is upper bounded by ϵ PE + ϵ U + ϵ CS .”

What the sources say

• Not checked yetcites Reiher et al., 2017

In the paper

“In order to account for the fact that c 0 < 1, we must multiply the number of repetitions of phase estimation by 1 / | c 0 | 2 , on average, to obtain the ground state.”

In the paper

“Randomisation procedures (such as randomly ordering the terms in the Trotter sequence, or stochastically choosing which terms to include in the Hamiltonian), have been shown to improve the accuracy obtained using product formulae (Campbell, 2018; Childs et al. , 2018; Ouyang et al. , 2019).”

What the sources say

• Not checked yetcites Campbell, 2018
• Not checked yetcites Childs et al., 2018
• Not checked yetcites Ouyang et al., 2019

In the paper

“A series of improvements throughout these papers reduced the scaling of phase estimation for molecules from O ( M 11 ) (Wecker et al. , 2014) to (empirically) O ( M 5 ) (Babbush et al. , 2015).”

What the sources say

• Not checked yetcites Wecker et al., 2014
• Not checked yetcites Babbush et al., 2015

In the paper

“It is also possible to use higher order product formulae (Berry et al. , 2007; D¨ ur et al. , 2008; Suzuki, 1976), which scale better with respect to the simulation error than the first order method.”

What the sources say

• Not checked yetcites Berry et al., 2007
• Not checked yetcites Dür et al., 2008
• Not checked yetcites Suzuki, 1976

In the paper

“Initialising the qubit register in a state which has a sufficiently large overlap with the target eigenstate (typically the ground state), is a non-trivial problem.”

In the paper

“Here, we will describe the method by Bravyi et al. (2017), which utilises two such symmetries: conservation of electron number and spin. This method enables the systematic reduction of two qubits when using the parity, BK (with the caveat that the number of orbitals is a power of two), or BK-tree encoding.”

What the sources say

• Not checked yetcites Bravyi et al. (2017)

In the paper

“This view is supported by the numerical simulations of Veis and Pittner (2014), who found that annealing times for methylene (CH 2 ) could be reduced by up to four orders of magnitude by using an initial state with larger overlap with the true ground state.”

What the sources say

• Not checked yetcites Veis and Pittner, 2014

In the paper

“One possible method for reducing the annealing time required is to introduce additional 'driving' Hamiltonians, as was numerically investigated by Matsuura et al. (2018).”

What the sources say

• Not checked yetcites Matsuura et al., 2018

In the paper

“The efficiency of adiabatic state preparation depends on the gap between the ground state and the first excited state along the path between H 0 and H s .”

In the paper

“There are also errors arising from imperfect implementation of the controlled unitary evolutions applied to the main register, which we denote as ϵ U .”

In the paper

“A notable feature of these Trotter-based simulations is that rigorous bounds on the Trotter error are believed to be loose by several orders of magnitude, which may impact the scaling of these approaches.”

What the sources say

• Not checked yetcites Babbush et al., 2015
• Not checked yetcites Poulin et al., 2015

In the paper

“This is important, because a randomly chosen state would have an exponentially vanishing probability of collapsing to the desired ground state, as the system size increases.”

In the paper

“While the advanced Hamiltonian simulation methods described above are asymptotically more efficient than Trotterization, the Trotter error bounds appear to be loose by several orders of magnitude (Babbush et al. , 2015; Poulin et al. , 2015).”

What the sources say

• Not checked yetcites Babbush et al., 2015
• Not checked yetcites Poulin et al., 2015

In the paper

“The VQE has been experimentally demonstrated on most leading quantum architectures including: superconducting qubits (Colless et al. , 2018; Kandala et al. , 2017; O'Malley et al. , 2016), trapped ions (Hempel et al. , 2018; Kokail et al. , 2019; Nam et al. , 2019; Shen et al. , 2017), and photonic systems (Peruzzo et al. , 2014), and shows many desirable features.”

What the sources say

• Not checked yetcites Colless et al., 2018
• Not checked yetcites Kandala et al., 2017
• Not checked yetcites O'Malley et al., 2016

In the paper

“Babbush et al. (2018a) obtained a scaling of O ( N 4 / 3 M 2 / 3 + N 8 / 3 M 1 / 3 ), similar to the scaling of their interaction picture algorithm discussed above.”

What the sources say

• Not checked yetcites Babbush et al., 2018a

In the paper

“Consequently, alternative methods are required for near-future chemistry simulation.”

In the paper

“The scaling of the CI matrix based algorithm of Babbush et al. (2017) was reduced from O ( N 2 M 3 ) in a Gaussian basis to O ( M 2 . 67 ) in a plane wave basis (Babbush et al. , 2018c).”

What the sources say

• Not checked yetcites Babbush et al., 2017
• Not checked yetcites Babbush et al., 2018c

In the paper

“The canonical UCCSD implementation requires O ( M 3 N 2 ) gates when using the Jordan-Wigner mapping (where M is the number of spin-orbitals, and N is the number of electrons) (Romero et al. , 2019).”

What the sources say

• Not checked yetcites Romero et al., 2019

In the paper

“A study of spin Hamiltonians (Childs et al. , 2017) found that the asymptotic scaling of Trotter methods was much worse than the qubitization + quantum signal processing, and Taylor series methods.”

What the sources say

• Not checked yetcites Childs et al., 2017

In the paper

“Those authors obtained a scaling of O (( N/h 2 + N 2 ) t ) (excluding logarithmic factors), where h is the grid spacing and t is the simulation time.”

What the sources say

• Not checked yetcites Kivlichan et al., 2017

In the paper

“However, it was later noted that for certain functions f ( H ), qubitization could be used on its own to directly implement the unitary e -if ( H ) t without approximation errors (Berry et al. , 2018; Poulin et al. , 2018).”

What the sources say

• Not checked yetcites Berry et al., 2018
• Not checked yetcites Poulin et al., 2018

In the paper

“For 3D materials, such as diamond, the scaling of Trotter based approaches to phase estimation in a plane wave dual basis was reduced to around O ( M 2 ).”

In the paper

“Those authors obtained a gate scaling of O ( M ) for the Fermi-Hubbard model.”

What the sources say

• Not checked yetcites Babbush et al., 2018b

In the paper

“Their approach to finding the ground state required O ( M 3 ) gates.”

What the sources say

• Not checked yetcites Wecker et al., 2015b

In the paper

“When considering an intensive error in the energy, this approach outperforms the Trotter-based method of Kivlichan et al. (2019a).”

What the sources say

• Not checked yetcites Kivlichan et al., 2019a

In the paper

“The Hamiltonian has a sparsity of O ( N 2 M 2 ) in this basis.”

In the paper

“However, when numerical simulations were performed, Trotter methods required lower gate counts than the other methods.”

In the paper

“This was subsequently improved by Kivlichan et al. (2019a), who considered the case where an extensive error in the energy is targeted, and found that the cost of Fermi-Hubbard model simulations was effectively between O (1) and O ( M 1 / 2 ) using Trotterisation.”

What the sources say

• Not checked yetcites Kivlichan et al., 2019a

In the paper

“A promising algorithm for near-term quantum hardware is the variational quantum eigensolver (VQE), first proposed and experimentally realised by Peruzzo et al. (2014), and elaborated on by McClean et al. (2016).”

What the sources say

• Not checked yetcites Peruzzo et al., 2014
• Not checked yetcites McClean et al., 2016

In the paper

“The best product based formulae approaches described above scale as O ( M 1 . 67 N 1 . 83 ) in a plane wave basis (Babbush et al. , 2018c).”

What the sources say

• Not checked yetcites Babbush et al., 2018c

In the paper

“This exchanges the long coherence times needed for phase estimation for a polynomial overhead due to measurement repetitions and classical processing.”

In the paper

“Similar results were obtained by Childs and Su (2019), who proved that a k th order product formula can simulate time evolution of an M qubit lattice Hamiltonian using O (( Mt ) 1+ 1 k ) elementary operations.”

What the sources say

• Not checked yetcites Childs and Su, 2019

In the paper

“This is known as the Hamiltonian averaging method of calculating the energy (McClean et al. , 2014), and requires O (1 /ϵ 2 ) measurements to determine the energy to a precision ϵ .”

What the sources say

• Not checked yetcites McClean et al., 2014

In the paper

“This effect becomes exponentially more prominent as the number of qubits and circuit depth increases.”

In the paper

“This produced an algorithm with a scaling of O ( M 1 . 5 λ ), where λ is the 1-norm of the Hamiltonian.”

What the sources say

• Not checked yetcites Berry et al., 2019b

In the paper

“As a result, these methods are typically assumed to require fault-tolerance.”

In the paper

“A time dependent form of the Taylor series approach was used for simulation in the 'interaction picture', which enables more efficient time evolution in a plane wave dual basis, scaling as O ( M 2 ) (Low and Wiebe, 2018) (although this algorithm uses O ( M log 2 ( M )) qubits).”

What the sources say

• Not checked yetcites Low and Wiebe, 2018

In the paper

“Despite this progress, all of the methods discussed above require circuits with a large number of gates.”

In the paper

“As near-term quantum computers will not have enough physical qubits for error correction, the long gate sequences required by these algorithms make them unsuitable for near-term hardware.”

In the paper

“Most notably, the coupled cluster (CC) method discussed in Sec. III.C.4 can be extended to produce the unitary coupled cluster (UCC) ansatz (Bartlett et al. , 1989; Hoffmann and Simons, 1988).”

What the sources say

• Not checked yetcites Bartlett et al., 1989
• Not checked yetcites Hoffmann and Simons, 1988

In the paper

“Haah et al. (2018) made use of arguments about the speed of information propagation in lattice systems to obtain a simulation algorithm requiring O ( Mt ) gates to simulate time evolution for time t under lattice Hamiltonians.”

What the sources say

• Not checked yetcites Haah et al., 2018

In the paper

“This suggests that randomly initialised hardware efficient ans¨ atze are not a scalable solution for problems in quantum computational chemistry.”

In the paper

“Quantum circuits have been proposed to calculate the analytic gradient of the energy with respect to one of the parameters (Guerreschi and Smelyanskiy, 2017; Mitarai and Fujii, 2018; Schuld et al. , 2019). This avoids the use of finite difference formulae, which restrict the accuracy of gradient evaluation, as the finite difference considered is limited by the noise in the energy evaluation.”

What the sources say

• Not checked yetcites Guerreschi and Smelyanskiy, 2017
• Not checked yetcites Mitarai and Fujii, 2018
• Not checked yetcites Schuld et al., 2019

In the paper

“For example, Wecker et al. (2015a) found that around 10 13 samples would be required per energy evaluation for a 112 spin-orbital molecule such as Fe 2 S 2 .”

What the sources say

• Not checked yetcites Wecker et al. (2015a)

In the paper

“McClean et al. (2016) and Romero et al. (2019) showed that the number of measurements N m , required to estimate the energy to a precision ϵ , is bounded by”

What the sources say

• Not checked yetcites McClean et al. (2016)
• Not checked yetcites Romero et al. (2019)

In the paper

“For example, the LiH molecule in an STO-3G basis naively has around 200 excitation operators to consider. However, taking into account symmetries and a reduced active space, one can achieve accurate results while considering only around 12 excitation operators (Hempel et al. , 2018).”

What the sources say

• Not checked yetcites Hempel et al., 2018

In the paper

“The Hamiltonian variational ansatz appears particularly suitable for the Fermi-Hubbard model, which only has O ( M ) terms in its Hamiltonian.”

In the paper

“In contrast, canonical gradient descent based methods have struggled to find the ground state, due to the high levels of noise present in current quantum devices (Peruzzo et al. , 2014).”

What the sources say

• Not checked yetcites Peruzzo et al., 2014

In the paper

“Kivlichan et al. (2018) showed that it is possible to implement Trotter steps of the Fermi-Hubbard Hamiltonian with O ( √ M ) depth and O ( M 1 . 5 ) gates on a linear array of qubits with nearest-neighbour connectivity.”

What the sources say

• Not checked yetcites Kivlichan et al. (2018)

In the paper

“In comparisons of 'out-of-the-box' optimisers (McClean et al. , 2016; Romero et al. , 2019), the NelderMead simplex algorithm utilised in experimental studies was outperformed by both other direct search approaches, and gradient based approaches, such as LBFGS-B.”

What the sources say

• Not checked yetcites McClean et al., 2016
• Not checked yetcites Romero et al., 2019

In the paper

“or O ( M 3 ) - O ( M 4 ) gates using a low-rank decomposition of the UCCSD tensor (Motta et al. , 2018).”

What the sources say

• Not checked yetcites Motta et al., 2018

In the paper

“The efficiency of this ansatz is determined by the number of terms in the Hamiltonian.”

In the paper

“If we are only interested in measuring the energy expectation value, rather than the 1 and 2-RDMs, Huggins et al. (2019b) showed that it is possible to use the low rank Hamiltonian decomposition technique (discussed in Sec. IV.C.2) and orbital basis rotations to divide the Hamiltonian into O ( M ) measurement groups.”

What the sources say

• Not checked yetcites Huggins et al. (2019b)

In the paper

“This leads to a scaling of approximately O ( M 4 ) when considering a Gaussian molecular orbital basis (Wecker et al. , 2015a).”

What the sources say

• Not checked yetcites Wecker et al., 2015a

In the paper

“The number of parameters in this ansatz scales linearly with the number of Trotter steps, S .”

In the paper

“The electronic structure Hamiltonian in a plane wave dual basis only contains O ( M 2 ) terms.”

In the paper

“All methods were liable to becoming trapped in local minima, even for small systems.”

What the sources say

• Not checked yetcites McClean et al., 2016
• Not checked yetcites Romero et al., 2019

In the paper

“When taking into account covariances between commuting terms, the total number of measurements required for small molecules was reduced by several orders of magnitude.”

In the paper

“In practice, the gate scaling is typically better than the bounds given above, as many excitations are forbidden by the symmetry point groups of molecular orbitals.”

In the paper

“By grouping terms at a fermionic level, rather than a qubit level, Bonet-Monroig et al. (2019) devised a method for measuring the fermionic 2-RDM using O ( M 2 ) circuits (an additional gate depth of O ( M ) is required).”

What the sources say

• Not checked yetcites Bonet-Monroig et al. (2019)

In the paper

“In general, these techniques only require a multiplicative overhead in the number of measurements required, if the error rate is sufficiently low.”

In the paper

“The QSE has been experimentally demonstrated, using a two qubit superconducting system to measure the ground and excited state energies of H 2 (Colless et al. , 2018).”

What the sources say

• Not checked yetcites Colless et al. , 2018

In the paper

“The latter approach reduces the effect of errors due to state preparation and measurement (Endo et al. , 2017).”

What the sources say

• Not checked yetcites Endo et al. , 2017

In the paper

“Overlap based methods exploit the orthogonality of energy eigenstates. Once an eigenstate is found, we can find other eigenstates by ensuring that they are orthogonal to the original state (Endo et al. , 2018a; Higgott et al. , 2018).”

What the sources say

• Not checked yetcites Endo et al., 2018a
• Not checked yetcites Higgott et al., 2018

In the paper

“The probabilistic error cancellation method introduced by Temme et al. (2017) works by effectively realising the inverse of an error channel, N -1 , such that N -1 ( N ( ρ 0 )) = ρ 0 .”

What the sources say

• Not checked yetcites Temme et al. (2017)

In the paper

“In practice, it is not possible to exactly measure all of the possible terms resulting from errors if there are many gates in the circuit.”

In the paper

“The original formulation of the QSE used a polynomial number of additional measurements to find the excited states of a quantum system (McClean et al. , 2017a).”

What the sources say

• Not checked yetcites McClean et al., 2017a

In the paper

“Probabilistic error cancellation requires full knowledge of the noise model associated with each gate.”

In the paper

“It has been shown both theoretically (McClean et al. , 2016) and experimentally (O'Malley et al. , 2016) that the VQE can be inherently robust to some coherent errors, such as qubit over-rotation.”

What the sources say

• Not checked yetcites McClean et al., 2016
• Not checked yetcites O'Malley et al., 2016

In the paper

“The ground state of the updated Hamiltonian H ′ is no longer | E 0 〉 , but the first excited state | E 1 〉 of the original Hamiltonian H .”

In the paper

“However, small experimental demonstrations of the VQE have shown that noise can still prevent us from reaching the desired levels of accuracy (Hempel et al. , 2018; Kandala et al. , 2017).”

What the sources say

• Not checked yetcites Hempel et al., 2018
• Not checked yetcites Kandala et al., 2017

In the paper

“This is designed to target the low-lying excited states, which are assumed to differ from the ground state by a small number of excitations.”

In the paper

“The variance in our estimate of ¯ O 0 is increased by a factor of γ G , where γ is the overhead coefficient, and G is the number of gates in the circuit.”

In the paper

“It is also possible to mitigate some errors by using symmetry checks on a suitably constructed ansatz state (Bonet-Monroig et al. , 2018; McArdle et al. , 2018c).”

What the sources say

• Not checked yetcites Bonet-Monroig et al. , 2018
• Not checked yetcites McArdle et al. , 2018c

In the paper

“Takeshita et al. (2019) showed how the QSE can be used to recover the energy contribution of virtual orbitals, without requiring additional qubits to represent the virtual orbitals.”

What the sources say

• Not checked yetcites Takeshita et al. (2019)

In the paper

“If these errors are not dealt with, they will corrupt the results of our algorithms, rendering the calculations meaningless.”

In the paper

“The probabilistic error cancellation method described above has been shown to work for general Markovian noise (Endo et al. , 2017), and has also been extended to work with temporally correlated errors and low frequency noise (Huo and Li, 2018).”

What the sources say

• Not checked yetcites Endo et al. , 2017

In the paper

“The QSE was experimentally demonstrated in a two qubit superconducting system to find the ground and excited states of H 2 (Colless et al. , 2018).”

What the sources say

• Not checked yetcites Colless et al., 2018

In the paper

“Barkoutsos et al. (2018) introduced a transformation of the Hamiltonian, such that it only measures the energy of excitations above the Hartree-Fock state (the correlation energy).”

What the sources say

• Not checked yetcites Barkoutsos et al. (2018)

In the paper

“Exponential extrapolation was introduced by Endo et al. (2017) as a more appropriate extrapolation technique for large quantum circuits.”

What the sources say

• Not checked yetcites Endo et al. (2017)

In the paper

“The extrapolation method has been demonstrated for VQE experiments in both molecular chemistry (Kandala et al. , 2018; Ollitrault et al. , 2019), and nuclear physics (Shehab et al. , 2019).”

What the sources say

• Not checked yetcites Kandala et al., 2018
• Not checked yetcites Ollitrault et al., 2019
• Not checked yetcites Shehab et al., 2019

In the paper

“This was confirmed for the case of chemistry calculations by Sawaya et al. (2016), who used numerical simulations to show the impact that noise has on phase estimation based chemistry calculations.”

What the sources say

• Not checked yetcites Sawaya et al. (2016)

In the paper

“A key concern for the VQE is preserving particle number, as states with electron number far from the true value appear to have a larger energy variance than those with smaller particle number errors (Sawaya et al. , 2016).”

What the sources say

• Not checked yetcites Sawaya et al. , 2016

In the paper

“Ollitrault et al. (2019) showed that the QSE with single and double excitations can be understood as an approximation to the 'equation of motion' (EoM) method for finding excited states, which was originally introduced as a classical method in quantum chemistry by Rowe (1968).”

What the sources say

• Not checked yetcites Ollitrault et al. (2019)

In the paper

“As discussed in Sec. IV.B, quantum computers can store the FCI wavefunction of M spin-orbitals using only M qubits in second quantisation.”

In the paper

“Interestingly, many problems of scientific interest fit this description, including: transition metal catalysts (Podewitz et al. , 2011; Vogiatzis et al. , 2019) and the Fermi-Hubbard model (LeBlanc et al. , 2015).”

What the sources say

• Not checked yetcites Podewitz et al. , 2011
• Not checked yetcites Vogiatzis et al. , 2019
• Not checked yetcites LeBlanc et al. , 2015

In the paper

“Coupled cluster methods (often CCSD) are some of the most widely used high-accuracy methods. They are applicable to large systems (hundreds of spinorbitals (Bartlett and Musiaglyph[suppress] l, 2007)) which do not display strong static correlation.”

What the sources say

• Not checked yetcites Bartlett and Musiaglyph[suppress] l, 2007

In the paper

“This highlights that working in a suitably large basis set is crucial for obtaining accurate results.”

In the paper

“A related extension of the quantum subspace expansion was developed by McClean et al. (2019b), who effectively engineered additional symmetries in the system using an error correcting code.”

What the sources say

• Not checked yetcites McClean et al. (2019b)

In the paper

“It was shown by Romero et al. (2019) that all Pauli terms arising from the same excitation operators commute.”

What the sources say

• Not checked yetcites Romero et al. (2019)

In the paper

“Post-processing the results produces a more accurate expectation value than would be obtained without the mitigation technique.”

What the sources say

• Not checked yetcites Otten and Gray, 2018a

In the paper

“This would imply that these problems are among the best targets for quantum computers.”

In the paper

“We can see from the above discussion that there appears to be an untreated 'sweet spot', of systems with around 100 to 200 spin-orbitals, that require high accuracy calculations.”