Reducing Unitary Coupled Cluster Circuit Depth by Classical Stochastic Amplitude Pre-Screening

Unitary Coupled Cluster (UCC) approaches are an appealing route to utilising quantum hardware to perform quantum chemistry calculations, as quantum computers can in principle perform UCC calculations in a polynomially scaling fashion, as compared to the exponential scaling required on classical computers. Current noisy intermediate scale quantum (NISQ) computers are limited by both hardware capacity in number of logical qubits and the noise introduced by the deep circuits required for UCC calculations using the Variational Quantum Eigensolver (VQE) approach. We present a combined classical--quantum approach where a stochastic classical UCC pre-processing step is used to determine the important excitations in the UCC ansatz. The reduced number of selected excitations are then used in a UCC-based VQE calculation. This approach gives a systematically improvable approximation, and we show that significant reductions in quantum resources can be achieved, with simulations on the CH$_2$, N$_2$ and N$_2$H$_2$ molecules giving sub-milliHartree errors.

Unitary Coupled Cluster (UCC) approaches are an appealing route to utilising quantum hardware to perform quantum chemistry calculations, as quantum computers can in principle perform UCC calculations in a polynomially scaling fashion, as compared to the exponential scaling required on classical computers. Current noisy intermediate scale quantum (NISQ) computers are limited by both hardware capacity in number of logical qubits and the noise introduced by the deep circuits required for UCC calculations using the Variational Quantum Eigensolver (VQE) approach. We present a combined classical-quantum approach where a stochastic classical UCC pre-processing step is used to determine the important excitations in the UCC ansatz. The reduced number of selected excitations are then used in a UCC-based VQE calculation. This approach gives a systematically improvable approximation, and we show that significant reductions in quantum resources can be achieved, with simulations on the CH 2 , N 2 and N 2 H 2 molecules giving sub-milliHartree errors.

I. INTRODUCTION
Quantum chemistry, which often concerns itself with solutions to problems with exponentially scaling Hilbert spaces, has long been identified as a good target for quantum computation, which would allow the encoding of such problems in a linear number of qubits. For example, algorithms such as Quantum Phase Estimation [1,2] (QPE) have been suggested as means of efficiently computing the ground state energies of molecular systems. [3] However, this approach would require fault-tolerant quantum computers. In the current noisy intermediatescale quantum (NISQ) regime, the field of quantum chemistry calculations on quantum computers is dominated by hybrid quantum-classical approaches and in particular the Variational Quantum Eigensolver [4] (VQE), which requires a parameterised wavefunction. Various parameterised ansatz schemes have been proposed, such as the hardware efficient ansatz [5], the hardware variational ansatz [6], the symmetry preserving ansatz [7] and the unitary coupled cluster (UCC) ansatz [8][9][10][11]. UCC has seen a resurgence as a convenient parameterisation for VQE, [4,12] due to its ability to be easily encoded into a quantum circuit. While its physical origin leads to a series of appealing features, including a relatively well-behaved energy landscape, the qubit encoding of the fermionic operators in the UCC ansatz produces very deep circuits, which are challenging for currently available quantum architectures. Work has been done to decrease the depth of these circuits * maf63@cam.ac.uk † nathan.fitzpatrick@cambridgequantum.com ‡ ajwt3@cam.ac.uk while maintaining the physicality of the ansatz, for example by using Moller-Plesset [13] (MP2) theory results to screen amplitudes [14,15] or using adaptive ansätze such as ADAPT-VQE. [16] While the latter does generate shorter, highly accurate ansätze, it does so at the cost of an increased number of measurements relative to the standard UCC approach. Other approaches which select contributing amplitudes based on their energy gradient have been developed, such as Qubit Coupled Cluster (QCC) [17,18] or energy-sorted UCC, [19] but like ADAPT-VQE they require additional quantum computation to obtain the screened set of operators.
In this paper, we propose using the newly developed unitary coupled cluster Monte Carlo [20] (UCCMC) approach to screen amplitudes for a UCC-based VQE calculation. Quantum Monte Carlo (QMC) methods [21,22] take advantage of the sparsity of most chemical Hamiltonians to generate compact wavefunctions, thereby lowering memory requirements relative to the corresponding classical algorithms. Additionally, the Monte Carlo approach naturally samples "important" contributions to the wavefunction -determinants with large Hamiltonian coupling terms to the reference determinant -first and therefore, while fully converging a QMC calculation may be time-consuming, short runs may be used to quickly identify the most important contributions to a given wavefunction. [23] We use this property of the UC-CMC method and in particular its trotterized approximation to provide an initial set of amplitudes for a VQE calculation. We screen amplitudes based on these initial values and assess the effect of using the screened parameter sets on the accuracy of the obtained energy, comparing with the equivalent result from MP2 screening.
In Section 2, we review the underlying theory of both UCC-based VQE and the UCCMC method. Section 3 comprises results and discussion for a range of small molecules in a variety of scenarios and Section 4 presents our conclusions.

II. THEORY
Coupled cluster theory [24,25] has become well established as the "gold-standard" of ab initio quantum chemistry methods. The exponential ansatz whereT = iT i andT i is composed of all valid excitation operators of order i, naturally maintains sizeconsistency when the operatorT is truncated at some excitation level, with the most commonly employed truncations being coupled cluster singles and doubles (CCSD) and coupled cluster singles, doubles and triples (CCSDT). Such methods are polynomially scaling with system size and systematically improvable, both highly desirable properties. However, the exponential operator in this form is non-unitary and therefore it cannot be directly implemented on a quantum computer.

A. The UCC ansatz
We can construct the anti-Hermitian operatorT −T † , the exponential of which gives a unitary operator. The UCC wavefunction is therefore and its energy can be found variationally as Although e −(T −T † )Ĥ eT −T † has a non terminating Baker-Campbell-Hausdorff (BCH) expansion, which makes the implementation of UCC costly on a classical computer, the unitary operator eT −T † can be decomposed as a series of universal quantum gates on a quantum computer. We start by writing the cluster operator in terms of individual excitations.
where τ n represents a fermionic excitation operator and t n its corresponding cluster amplitude. We can take a Trotter-Suzuki expansion [26,27] of U (t) such that with equality achieved as ρ → ∞. We can take ρ = 1 which has been shown [12] to be a good approximation, leading to negligible errors in the resulting energy, to obtain also known as the disentangled UCC ansatz. [28]For the case of UCCSD the singles are represented bŷ and doubles are represented bŷ where the Latin indices represent occupied spin orbitals and the Greek indices represent virtual spin orbitals. This can be easily implemented on a quantum computer via an appropriate transformation of the fermionic operators to the unitary bosonic qubit operators using schemes like Jordan-Wigner [29] or Bravyi-Kitaev [30], where algebraic compilation strategies based on the ZX calculus have been proposed [31].
B. Quantum computing and the UCC ansatz NISQ hardware is characterized by small qubit counts and high noise levels, due mainly to short qubit coherence times, quantum gate errors, faulty readout operations and cross-talk between qubits during operation. These limitations prevent the use of error-correction protocols that would enable large quantum computations. Thus, it is advantageous to reduce the amount of quantum computation to a minimum. As a result, there is a strong research effort dedicated to the development of hybrid quantum-classical algorithms. One of the main algorithmic workhorses of this field is the Variational Quantum Eigensolver [4] , where a minimizer run on a classical machine optimizes a cost function evaluated by the quantum computer. In chemistry problems, this corresponds to finding the expected value of the energy with respect to a parametrized ansatz wavefunction. Alternative approaches have appeared in recent years; we highlight methods based on imaginary time propagation of a trial wavefunction, either via a variational principle [32] or approximating the nonunitary evolution with an appropriate quantum circuit [33], and techniques to variationally optimise the reduced density matrix of the system [34].
The UCC ansatz is attractive for this kind of quantum algorithm as it has a reduced number of parameters to optimize and a more favourable energy-search landscape compared with hardware-tailored ansätze. [5] It also naturally conserves the number of electrons and the M z spin projection quantum number during the calculation, thus helping to prevent convergence to unwanted states, or avoiding the barren plateau problem observed for hardware-efficient ansätze. [35] However, these advantages are obtained at the cost of large circuit depths. For example, in the Jordan-Wigner mapping, fermionic creation and annihilation operators may be expressed as Therefore a single fermionic excitation operator becomes while a double excitation is given by Therefore, each excitation included in the ansatz contributes with a series of Pauli gadgets (see Fig. 1), requiring a large number of two-qubit gates. These gates are responsible for most of the noise produced during a quantum computation on NISQ devices, so it is very important to reduce their number. Being able to identify which excitations have a negligible contribution to the total energy would enable their elimination in the ansatz and reduce the overall circuit depth of the state preparation step in variational quantum algorithms. Successful screening of excitations aims to bring the circuit execution time within the coherence time of near term hardware with maximal fidelity with the unscreened ansatz, incurring a minimum energy penalty.

C. QMC algorithms
Monte Carlo algorithms have become popular approaches to more effectively use classical computational resources for quantum chemistry. Real-space approaches such as Diffusion Monte Carlo (DMC) [36,37] suffer from the so-called "sign problem", which causes them to naturally converge to bosonic solutions. This can be somewhat mitigated by using arbitrary nodal surfaces, however this introduces an uncontrolled approximation. Recently, Hilbert space quantum Monte Carlo methods have been developed that naturally avoid the DMC sign problem. Full Configuration Quantum Monte Carlo (FCIQMC) [21] encodes a stochastic solution to the full configuration interaction (FCI) equation. The FCI wavefunction is expressed as a linear combination of a reference determinant (usually the Hartree-Fock (HF) wavefunction) and all possible excited determinants starting from it.
i,j,α,β C αβ ijâ αβ ij + ... and a α i ,â αβ ij are excitation operators. The coefficients C can be optimised by minimising the energy with respect to them. This gives the following set of equations: where |D i span the full Hilbert space of the system. Equivalently, which can be written in an iterative form as This equation can be solved stochastically by sampling the population dynamics of a set of walkers ('psips') in the Hilbert space of the system, which undergo the following processes: [21] • spawning from |D i to |D j with probability • death with probability • annihilation of particles of opposite sign on the same determinant.
In the death step, the shift S replaces the unknown exact energy E. This acts as a population control parameter and, once the system has reached a steady state, converges to the true energy. Another estimator for E is the projected energy A set of equations similar to Eq. 14 holds for the CC wavefunction given in Eq. 1. Since D i |Ψ CC = t i + O(t 2 ), we can write This can be described stochastically by the same three processes considered for FCIQMC, leading to an algorithm know as Coupled Cluster Monte Carlo (CCMC). [22] However, one must also take into consideration contributions from composite clusters. For example, and therefore any of these three terms may contribute to death on t αβ ij or to spawning onto some excitor coupled to it by the Hamiltonian. The selection process is therefore somewhat more complicated than for FCIQMC, originally consisting of the steps below: [22] 1. a cluster size s is selected with probability 2. a particular cluster of s distinct excitors is selected with probability where N ex is the total population on excitors. The total selection probability is therefore Improvements have since been made to this selection algorithm to better importance-sample the wavefunction. [38] Recently, some of us have implemented a stochastic version of unitary coupled cluster and its trotterized approximation, [20] the details of which are expanded upon in the following section.

(p)UCCMC
While the UCC wavefunction is generally found by variationally optimising the parameters, it is also possible [39,40] to solve a set of projected UCC (pUCC) equations, This projective method is naturally more amenable to the QMC algorithms described above, as it leads to similar population dynamics to those in Eq. 20. However, the presence of de-excitation operatorsT † in the full UCC ansatz substantially changes the structure of the allowed clusters, removing the constraint that cluster sizes must be at most equal to the maximum excitation level considered in the calculation. However, the expansion can be truncated to a finite cluster size, which, if large enough, does not significantly affect the accuracy of the obtained answer. The selection scheme for UCCMC is then as follows: 1. select a cluster size s with probability p(s) = 1 2 s+1 .
2. for all but the first excitor in the cluster, decide with probability 1 2 whether it will be an excitation or de-excitation operator 3. a particular cluster is selected as before, with probability given by Eq. (23) Having selected the cluster, it undergoes stochastic spawning and death as before. Overall this constitutes the projected Unitary Coupled Cluster Monte Carlo (pUCCMC) algorithm. The final aspect one must be careful of is the projection of the wavefunction onto the HF reference, as this includes contributions beyond the reference population. This projection may be sampled stochastically during the course of the calculation, concurrently with the reference population and the terms.
This ansatz now depends on the order of excitors in the product, with different orders leading to different cluster amplitudes and potentially different energy values. [41] Recent work by Evangelista et al. [28] has defined an optimal ordering that guarantees complete wavefunction expressibility in this framework. Applying e ti(τi−τ † i ) to an arbitrary single determinant wavefunction |Ψ leads to three possibilities: where |Ψ i is the result of applying the excitation to |Ψ .

2.τ
where |Ψ i is the result of applying the deexcitation to |Ψ .
To translate this into a stochastic algorithm, for each excitor present in the wavefunction, the algorithm assesses which of the cases listed above is appropriate. If the excitor cannot be applied, the next excitor is considered. If the excitor can be applied, this is done with probability (30) and the cluster amplitude is multiplied by ± sin(t). With probability 1 − p excit , the operator is not applied and the cluster amplitude is multiplied by cos(t). The cluster then undergoes the same spawning, death and annihilation steps as in traditional CCMC, leading to a trotterized pUCCMC approach (tpUCCMC). As in the case of full pUCCMC, this ansatz modifies the projection of the wavefunction onto the reference, making it different from the reference population. Depending on the ordering of the excitors, a closed form for this projection may be found, but in general it can easily be sampled during the stochastic propagation.

D. Technical details
The VQE calculations have been performed using a minimal (STO-3G) basis set for all molecular species considered. Molecular integrals and molecular orbital coefficients have been obtained using the PySCF package. [42] The wavefunction anstaz has been encoded into quantum circuits on a qubit register using the Jordan-Wigner scheme in the EUMEN program. For the optimization loop in the VQE procedure, the L-BFGS method has been applied. The state vector simulator Qulacs [43] has been used to evaluate the quantum circuits.
All QMC calculations have been carried out in a development version of HANDE-QMC. [44] Where directly compared, VQE and tpUCCMC calculations use the same ordering of excitors, applying all single (de)excitation operators ahead of the doubles. For triplet states, restricted open-shell HF (ROHF) reference states were used for all Monte Carlo, MP2 and VQE calculations. All screened VQE UCCSD calculations used the relevant tpUCCMCSD/MP2 amplitudes as starting values for the parameters.

III. RESULTS
In this section, we assess the viability of the tpUCCSD method as a screening technique for UCCSD-based VQE. Given one method is based on a projective approach, while the other is variational, we first ascertain whether the wavefunctions obtained by the two approaches are sufficiently similar for tpUCCSD amplitudes to be a good predictor of VQE UCCSD amplitudes. To assess this, we turn our attention to the LiH molecule. In the STO-3G basis, this system consists of 4 electrons in 12 spin-orbitals, which can easily be treated by both VQE and tpUCCMCSD. The system can be further simplified by freezing the core Li 1s electrons. As seen in Fig. 2, in both cases the energies obtained by the two methods agree within 50µE h for both the expectation value and the projected energy for tpUCCMCSD.
However, for tpUCCMCSD predictions to be useful as a starting guess or screening for VQE UCCSD amplitudes, the methods must agree not only in energy, but also in individual cluster amplitudes. In order for this agreement to be achieved, care must be taken that the same ordering is used in both ansätze. While for such small systems any arrangement of the excitors in the trotterized UCC wavefunction will be able to describe the ground state wavefunction, the individual amplitudes may vary extensively, particularly in the more highly cor-related regime. [45] For frozen core LiH, the tpUCCMCSD amplitudes agree with their deterministic counterparts within the 1σ-error bars.
Having convinced ourselves that the VQE UCCSD and tpUCCMCSD results are compatible, we investigate the latter's potential as a screening technique for the former. Table I gives the number of amplitudes in the LiH wavefunction of different orders of magnitude in a stochastic snapshot of the tpUCCMCSD expansion. We find that the lowest threshold (t > 0.001), recovers more than 99.9% of the correlation energy at all geometries, while decreasing the size of the considered Hilbert space by 14 -43%.  Even t > 0.01 generally recovers more than 97% of the correlation energy in all cases, despite only using 8 parameters.
Following on from these results, we investigate the ap-plicability of tpUCCMC screened VQE to a series of small molecules in different scenarios: CH 2 , which has a triplet ground state, N 2 which becomes increasingly strongly correlated as the bond between the nitrogen atoms is broken and N 2 H 2 , where there is a crossing of diabatic states as it rotates from a trans to a cis geometry. In all cases, we use a snapshot of the amplitudes at the end of a short tpUCCMCSD run for screening.
For CH 2 , considered with frozen 1s electrons on the C atom, we once again observe good agreement between tpUCCMCSD and VQE energies and wavefunctions, provided the same excitor ordering is used throughout. [45] In this study, we consider both the symmetric stretch at an angle ∠HCH = 135 • and the bend with r CH = 1.08Å, close to the experimental equilibrium geometry (∠HCH = 133.9 • , r CH = 1.075Å) [47] In this case, the tpUCCMCSD amplitudes are less spread out in magnitude and therefore the screening is less efficient at decreasing the number of VQE parameters to be considered (see Table II for an example). As expected from this distribution of amplitudes, screening at either t > 0.01 or t > 0.001 gives errors of less than 1 milliHartree relative to the full UCCSD VQE calculation, for both singlet and triplet CH 2 . Using MP2 as a screening method is competitive for the singlet state (see Figs. 3 and 4), while for the triplet, the MP2 screened results exhibit a systematic error of more than 10 milliHartree which persists across all screening thresholds. Even when considering relative energies, MP2 screened calculations overestimate the energy of the triplet as the C-H bonds are stretched, as seen in Fig. 4 and generally underestimate the singlet-triplet gap. Considering quantum resources, we note that the tpUCCMC screening described above leads to significant depth reduction with relatively little correlation loss, as can be seen in Fig. 5, which also highlights the unsuitable quality of MP2-screened results independently of circuit depth.
The N 2 molecule as it approaches dissociation is one of the archetypal examples of static correlation and poses significant challenges to methods such as coupled cluster, which are based around the assumption that a single determinant dominates the wavefunction. For example, for r NN > 3.6a 0 CCSD(T) [48] overestimates the correlation energy and diverges as the bond length is increased. [49] As can be seen in Fig. 6, the unitary, variationally optimised ansatz avoids the failure of the traditional CCSD ansatz, with the energy consistently above the FCI value and generally lower errors. In this case, both MP2 and tpUCCMCSD screening provide similar quality results, but the system highlights a few interesting differences between the two methods. First, in the left panel of Fig. 6 1.06 1.08 1.10 1.12 1.14 1. 16   we observe that, while in MP2 there are consistently some amplitudes allowed by the screening, tpUCCMCSD does not have any amplitudes above 0.1 for half of the geometries considered. As such, the screened method cannot provide any improvement over Hartree-Fock. Obviously, this threshold is too high and the results obtained with either MP2 or tpUCCMCSD screening all have significant systematic and non-parallelity errors (10 -500 mil-liHartree), making it inappropriate for the treatment of this system. Secondly, in the previous examples we have generally found that tpUCCMCSD screening converges slightly faster with screening threshold than MP2, or in-deed MP2 fails to converge at all. In this case, we find that tpUCCMCSD converges faster than MP2 for compressed bonds, but they are comparable in the stretched regime, as is further highlighted in Fig. 7, which shows that once again screening can be used to meaningfully reduce circuit depth without significant accuracy loss. The behaviour can be correlated to the number of amplitudes in the ansatz at each geometry, with tpUCCMCSD including more parameters at shorter bond lengths than MP2 (see Table III). Both of these features emphasize the fact that a constant amplitude threshold does not necessarily generate a consistent number of parameters across all geometries of a given system. Where such consistency is desirable, which will often be the case when constructing a description of a physical system, a threshold based directly on the number of parameters included may be preferable,although it is not without its own challenges, as we discuss below.
Finally, the energy landscape of the cis-trans interconversion of N 2 H 2 allows us to investigate another highlycorrelated regime around the transition state of this transformation. For simplicity, we investigate the rotation about the nitrogen-nitrogen bond starting from the equilibrium geometry of trans N 2 H 2 [50], without allowing any relaxation of other geometrical parameters. At the Hartree-Fock level, the singlet state of this system   [45]. This introduces an unphysical discontinuity in the energy surface, which often remains even after the application of further correlated methods. [51] In order to be able to treat this system using VQE, we freeze the 8 lowest-energy electrons. The resulting system ex-hibits an additional interesting feature as two FCI states of different spatial symmetry cross, as can be seen in Figure 8. The conservation of various symmetries -spin, particle number, point group -in quantum circuits is an interesting problem and in this system it turns out to be crucial for the correct description of the energy surface. Performance is comparable between the two methods, with tpUCCMC exhibiting slightly faster convergence in terms of both threshold and depth.
Not enforcing point group symmetry leads to a very compelling yet unphysical state which transitions smoothly from the A to the B state as the system approaches a 90 • rotation.
Once point group symmetry constraints are considered, VQE based on the UCCSD ansatz with the excitations symmetry filtered correctly reproduces the be-haviour observed in the exact surface, however starting from an m s = 0 open-shell Hartree-Fock reference or an m s = 1 reference leads to different energies for the B state. For the rest of our analysis we focus on the m s = 0 A state and the m s = 1 B state, which correspond to the two lowest-energy states in the system. The order of magnitude of amplitudes in a short tpUCCMCSD run for this system at various rotation angles are given in Table IV. This system exhibits many of the behaviours we noted in smaller examples. For the singlet state, both MP2 and tpUCCMCSD screened UCCSD VQE converge to the correct UCCSD energy with increasing number of amplitudes, but for the triplet state MP2 fails to converge, retaining an error of at least 30 milliHartree even for the lowest threshold -see Fig. 8. As there is no systematic error in the singlet state, this causes MP2 screening to predict the wrong ordering of the singlet and triplet states for rotation angles around 90 • . Fig. 10 shows the convergence of VQE UCCSD energy as a function of amplitude threshold and circuit depth for the A state at 0 • and the B state at 90 • . This highlights the same trend as before, with tpUCCMCSD screening only slightly more efficient than MP2 as a function of screening threshold for the singlet state, but clearly superior for the triplet. In both cases, using tpUCCMCSD it is possible to recover more than 90% of the correlation energy with a circuit that is only half as deep as the full VQE UCCSD implementation. In Fig. 9 we consider the correlation energy for the trans geometry of N 2 H 2 and the transition state at 90 • directly as a function of the number of parameters in the UCCSD expansion, using the truncation order from tpUCCMCSD. Due to the change in point-group symmetry from C 2 h(C 2 v) at the trans(cis)-geometry to C 2 along the rest of the binding curve, the number of possible parameters is halved at the ends. Therefore, where the symmetry of the system changes, computations based on a constant number of amplitudes are non-trivial to implement.  As a final point, we consider the scaling of such screening methods with basis set size. While the VQE method, even simulated on a classical machine, is currently limited to a minimal basis for the systems considered here, the sparse tpUCCMC approach can tackle more basis functions. Given results from this method have been shown to agree well with VQE, we use them to predict its expected behaviour. We consider the N 2 molecule in the cc-pVDZ, cc-pVTZ and cc-pVQZ basis sets. We start by running a tpUCCMCSD calculation for these systems to convergence. We then truncate the resulting parameter sets and rerun the calculation in this limited Hilbert space. Results are given in Table V. In all cases, the correlation energy increases with the number of amplitudes included, tpUCCMCSD (orange and red crosses) for screening. We consider two states, a closed-shell singlet state of A symmetry in the C2 point group and a triplet state of B symmetry. The A state is the ground state around the cis and trans geometries of N 2 H 2 , but the triplet becomes favoured around a rotation angle of 90 • . From left to right, we consider coefficients greater than 0.1, 0.01 and 0.001 respectively. In all cases, the energies in the top panel are quoted relative to the energy of the singlet state in the trans geometry, computed with the same method. For the A state, as with all singlet molecules so far, both MP2 and tpUCCMCSD converge towards the true UCCSD energy, although in this case at 0.001 MP2 is noticeably worse than tpUCCMCSD. For the B state however, MP2 is unable to capture the correlation, and does not improve with added amplitudes. This gives unphysical results, with the B state higher in energy than the A state for all geometries except ∠HNNH = 90 • even at the lowest threshold considered.
but we find that more than 85% of it is accounted for at a threshold of |t| > 0.001 in all cases. While at this value the number of amplitudes required increases with basis set size, the proportion of the Hilbert space decreases, suggesting larger relative resource reductions are possible for larger basis sets.

IV. CONCLUSION
We have shown that UCCMC can be used successfully as a screening method for UCCSD-based VQE, with short, potentially unconverged stochastic runs providing a set of amplitudes that may be truncated at a given coefficient threshold before being further optimised by VQE. We expect the method would also be compatible with the newly developed Projected Quantum Eigensolver. [52] Importantly, the classical screening approach is entirely a pre-processing step, so requires no additional quantum resources.
Standard noise models such as amplitude and phase damping [53] predict exponential decay of the quantum signal with time, controlled by the relaxation time T 1 and dephasing time T 2 [54]. Significantly reducing the depth, and therefore the computation time, of the VQE circuit, as we have seen is possible for N 2 H 2 , where more than 90% of the correlation energy could be recovered with half the circuit depth, would therefore lead to a major reduction in the error rates observed on real hardware. Furthermore, two-qubit gates such as CNOTs are responsible for most gate noise on current devices. The number of such gates is reduced proportionally to the number of parameters removed in screened calculations. Errors in the single qubit gates should also be reduced, as larger values for the angles in the R z gates involved in the UCC ansatz are easier to implement with high fidelity than lower values.
We have compared tpUCCMC screening with the preexistent MP2-based screening approach. In the case of singlet states, the results using UCCMC for screening are seen to converge slightly faster with screening threshold than the corresponding MP2-screened values. For triplets, we have found that the MP2-screened calculations fail to converge to the true ground state at all. This highlights the fact that tpUCCMC screening can be easily applied to any molecular system, while more care is needed when considering whether MP2 is suitable for a particular application. Furthermore, tUCC amplitudes are often highly dependent on operator ordering in the cluster expansion. While tpUCCMC can be adapted to  1  46  90  30  185  1  60  170  60  185  1  71  166  90  185  1  65  175  120  185  1  65  161  150  185  1  56  172  180  100  1  42  97   B   0  83  1  40  77  30  160  0  44  140  60  160  0  51  147  90  160  0  49  150  120  160  0  51  149  150  160  0  45  147  180  83  1 36 76 necessary. tpUCCMCSD results for N 2 in increasingly large basis sets confirm our expectation that the benefits of such screening methods get more significant as one considers larger systems. While the Hilbert spaces required for UCC calculations will scale with high-order polynomials of the system size, a smaller fraction of the amplitudes in this space will have large coefficients and therefore contribute significantly to the energy, at least in the largely single-reference, dynamically correlated areas of the landscape where coupled cluster methods perform well. Furthermore, it has been shown that linear scaling coupled cluster can be achieved by careful screening of amplitudes by distance [55][56][57][58][59]. We expect that the trotterized UCC approach and its stochastic counterpart will, in principle, be able to take advantage of this, especially on systems of multiple molecules and propagate these gains forward into screened VQE calculations.