Efficient and Noise Resilient Measurements for Quantum Chemistry on Near-Term Quantum Computers

Variational algorithms are a promising paradigm for utilizing near-term quantum devices for modeling molecular systems. However, previous bounds on the measurement time required have suggested that the application of these techniques to larger molecules might be infeasible. We present a measurement strategy based on a low rank factorization of the two-electron integral tensor. Our approach provides a cubic reduction in term groupings over prior state-of-the-art and enables measurement times four orders of magnitude smaller than those suggested by commonly referenced bounds for the largest systems we consider. Although our technique requires execution of a modest circuit prior to measurement, this is compensated for by eliminating challenges associated with sampling non-local Jordan-Wigner transformed operators in the presence of measurement error, while enabling a powerful form of error mitigation based on efficient postselection. We numerically characterize these benefits with noisy quantum circuit simulations of strongly correlated systems.


I. INTRODUCTION
Given the recent progress in quantum computing hardware, it is natural to ask where the first demonstration of a quantum advantage for a practical problem will occur.Since the first experimental demonstration by Peruzzo et al. [1], the variational quantum eigensolver (VQE) framework has offered a promising path towards utilizing small and noisy quantum devices for simulating quantum chemistry.The essence of the VQE approach is the use of the quantum device as a co-processor which prepares a parameterized quantum wavefunction and measures the expectation value of observables.In conjunction with a classical optimization algorithm, it is possible to then minimize the expectation value of the Hamiltonian as a function of the parameters, arriving at approximations for the wavefunction, energy, and other properties of the ground state [1][2][3][4][5][6][7][8].A growing body of work attempting to understand and ameliorate the challenges associated with using VQE to target non-trivial systems has emerged in recent years [9][10][11][12][13][14][15][16][17][18][19][20][21].In this article we shed more light on the challenge posed by the large number of circuit repetitions needed to perform accurate measurements and propose a new scheme that dramatically reduces this cost.Additionally, we explain how our approach to measurement has reduced sensitivity to readout errors and also enables a powerful form of error mitigation.
Within VQE, expectation values are typically estimated by Hamiltonian averaging, where the Hamilto-nian is decomposed into a sum of easily measured operators (such as Pauli strings), whose expectation values are sampled independently by repeated measurement.When measurements are distributed optimally between the easy-to-sample terms H , the total number of measurements M is upper bounded by , where is the Hamiltonian whose expectation value we estimate as ω H , the ω are scalars, and is the target precision [3,22].Most prior works assessing the viability of VQE apply this bound and conclude that chemistry applications require "a number of measurements which is astronomically large" (quoting from Ref. 3).
Several recent proposals attempt to address this obstacle by developing more sophisticated strategies for partitioning the Hamiltonian into sets of simultaneously measurable operators [16][17][18][19][20].We summarize their key findings in Table I.This work has a similar aim, but we take an approach rooted in a decomposition of the twoelectron integral tensor rather than focusing on properties of Pauli strings.We quantify the performance of our proposal by numerically simulating the variances of our term groupings to more accurately determine the number of circuit repetitions required for accurate measurement of the ground state energy.This contrasts with the analysis in other recent papers that have instead focused on using the number of separate terms which must be measured as a proxy for this quantity.By that metric, our approach requires a number of term groupings that is linear in the number of qubits -a quartic improvement over the naive strategy and a cubic improvement relative to these recent papers.However, we argue that the number Ref.
Partitioning Method Circuit Description # of Partitions Gate Count Depth Connectivity Diagonal [2] commuting Pauli heuristic -O(N 4 ) ---- [5] compatible Pauli heuristic single rotations O(N Here N represents the number of spin-orbitals in the basis.Gate counts and depths are given in terms of arbitrary 1-or 2-qubit gates restricted to the geometry of 2-qubit gates specified in the connectivity column.What we mean by "compatible" Pauli groupings is that the terms can be measured at the same time with only single qubit rotations prior to measurement.We report whether terms are measured in a diagonal representation as this is important for enabling strategies of error-mitigation by postselection.The number of partitions refers to the number of unique term groupings which can each be measured with a single circuit -thus, this reflects the number of unique circuits required to generate at least one sample of each term in the Hamiltonian.However, we caution that one cannot infer the total number of measurements required from the number of partitions, and often this metric is highly misleading.The overall number of measurements required is also critically determined by the variance of the estimator of the energy.As explained in the first entry of this table, when terms are measured simultaneously one must also consider the covariance of those terms.In some cases, a grouping strategy can decrease the number of partitions but increase the total number of measurements required by grouping terms with positive covariances.Alternatively, strategies such as the third entry in this table actually increase the number of partitions while reducing the number of measurements required overall by lowering the variance of the estimator.
of distinct term groupings alone is not generally predictive of the total number of circuit repetitions required, because it does not consider how the covariances of the different terms in these groupings can collude to either reduce or increase the overall variance.We will show below that our approach benefits from having these covariances conspire in our favor; for the systems considered here our approach gives up to four orders of magnitude reduction in the total number of measurements, while also providing an empirically observed asymptotic improvement.
Although there are a variety of approaches to simulating indistinguishable fermions with distinguishable qubits [23][24][25], the Jordan-Wigner transformation is the most widely used, due to its simplicity and to the fact that it allows for the explicit construction of a number of useful circuit primitives [26][27][28][29].However, a significant downside to using the Jordan-Wigner transformation is the fact that it maps operators acting on a constant number of fermionic modes to qubit operators with support on up to all N qubits.It has recently been shown that techniques based on fermionic swap networks can avoid this disadvantage in a variety of contexts [27][28][29], but the impact of this loss of locality on measurement has not yet been addressed.Under a simple model of readout error such as a symmetric bitflip channel, a Pauli string with support on N qubits has N opportunities for an error that reverses the sign of the measured value.This leads to expectation value measurements that are driven to zero by a multiplicative factor which is exponentially vanishing in N .Our work will avoid this challenge without leaving the Jordan-Wigner framework, allowing estimation of 1-and 2-particle fermionic operator expectation values by the measurement of only 1-local and 2-local qubit operators, respectively.
Besides this reduction in locality, our work offers an additional opportunity for mitigating errors.It has been observed that when one is interested in states with a definite eigenvalue of a symmetry operator, such as the total particle number, η, or the z-component of spin, S z , it can be useful to have a method which removes the components of some experimentally prepared state with support on the wrong symmetry manifold [8][9][10][11].Two basic strategies to accomplish this have been proposed.The first of these strategies is to directly and "non-destructively" measure the symmetry operator and discard those outcomes where the undesired eigenvalue is observed, projecting into the proper symmetry sector by postselection.The difficulty in performing these measurements efficiently has restricted the application of this strategy.Prior work has focused instead on measuring the parities of η and S z using non-local controlled operations.However these usually require O(N ) depth, which may induce further errors during their implementation.A related body of work builds upon the foundation of Ref. 14 and uses additional measurements together with classical post-processing to calculate expectation values of the projected state without requiring additional circuit depth [8][9][10], a procedure which can be efficiently applied to the parity of the number operator in each spin sector.In this work, we show how we can mitigate errors by postselecting directly on the proper eigenvalues of the operators η and S z , rather than on their parities, without any modification to our strategy for more efficiently performing the desired measurements.

A. Using Hamiltonian Factorization for Measurements
The crux of our strategy for improving the efficiency and error resilience of Hamiltonian averaging is the application of tensor factorization techniques to the measurement problem.Using a representation discussed in the context of quantum computing in Refs.29-31, we begin with the factorized form of the electronic structure Hamiltonian in second quantization: where the values g p and g ( ) pq are scalars, n p = a † p a p , L = O(N ) for the case of arbitrary basis quantum chemistry [29], L = 1 for the plane wave basis or dual basis of Ref. 26, and the U are unitary operators which implement a single particle change of orbital basis, e.g., where [e κ ] pq is the p, q entry of the matrix exponential of the anti-Hermitian matrix κ that characterizes U .Numerous approaches which accomplish this goal exist, including the density fitting approximation [32,33], and a double factorization which begins with a Cholesky decomposition or eigendecomposition of the two-electron integral tensor [29,[33][34][35][36][37].In this work, we use an eigendecomposition and refer readers to Ref. 29 for further details.The eigendecomposition step permits discarding small eigenvalues to yield a controllable approximation to the original Hamiltonian.While such low rank truncations are not central to our approach and would not significantly reduce the number of measurements, doing so would asymptotically reduce L (and thus ultimately, the number of distinct measurement term groupings).
Our measurement strategy, which we shall refer to as "Basis Rotation Grouping," is simply to recognize that if we apply the U circuit directly to the quantum state prior to measurement, we are then able to simultaneously sample all of the n p and n p n q expectation values in the rotated basis.We can then estimate the energy as where the subscript on the expectation values denotes that they are sampled after applying the basis transformation U .The reason that the n p and n p n q expectation values can be sampled simultaneously is because under the Jordan-Wigner transformation, n p = (1 1 + Z p )/2, which is a diagonal qubit operator.Thus, our approach is able to sample all terms in the Hamiltonian with only L + 1 = O(N ) distinct term groups.
Fortunately, the U are exceptionally efficient to implement, even on hardware with minimal connectivity.Following the strategy described in Ref. 28, and assuming that the system is an eigenstate of the total spin operator, any change of single-particle basis can be performed using N 2 /4−N/2 two qubit gates and gate depth of exactly N , even with the connectivity of only a linear array of qubits [28].This gate depth can actually be improved to N/2 by further parallelizing the approach of [28], making using ideas that are explained in the context of multiport interferometry in [38].In fact, a further optimization is possible by performing the second matrix factorization discussed in Ref. 29.This would result in only O(log 2 N ) distict values of the g ( ) pq and a gate complexity for implementing the U which is reduced to O(N log N ); however, we note that this scaling is only realized in fairly large systems when N is growing towards the thermodynamic (large system) rather than continuum (large basis) limit.
The first objective of our measurement strategy is to reduce the time required to measure the energy to within a fixed accuracy.We shall present data that demonstrates the effectiveness of our Basis Rotation Grouping approach in comparison to three other measurement strategies as well as to two estimates derived from Eq. ( 1).All calculations were performed using the open source software packages OpenFermion and Psi4 [39,40].Specifically, we use exact calculations of the variance of expectation values for the full configuration interaction (FCI) ground state to determine the number of circuit repetitions required.In the case of the largest Nitrogen dimer calculation we use configuration interaction singles and doubles (CISD) to approximate the ground state instead and present values of the variance for this.
We calculate the variance of the estimator of the expectation value of the energy as if measurements were distributed optimally according to the prescription of Refs.3, 22, i.e., so that (in the notation of Eq. ( 1)) each term H is measured a fraction of the time f equal to ( In practice, the expectation values in the above expression are not known ahead of time and so the optimal measurement fractions f cannot be efficiently and exactly determined a priori.However, we make the assumption that an adaptive measurement scheme which schedules additional measurements based on the observed sample variance can approximate the ideal partitioning of measurement time.This is particularly realistic for the central strategy of this paper, owing to the small number of separate term groupings.Thus, for simplicity we only present numbers based on the ideal partitioning.We perform these calculations for symmetrically stretched Hydrogen chains with various bond lengths and numbers of The cost of our proposed measurement strategy appears to have a lower asymptotic scaling than any other method we consider and obtains a speedup of more than an order of magnitude compared to the next best approach for a number of systems.
atoms, for a symmetrically stretched water molecule, and for a stretched Nitrogen dimer, all in multiple basis sets.
In Figure 1 we plot the number of circuit repetitions for our proposed "Basis Rotation Grouping" measurement approach, together with three other measurement strategies and the two upper bounds based on Eq. ( 1).The first and most basic alternative strategy is simply to apply no term groupings and measure each Pauli string independently, a strategy we refer to as "Separate Measurements."A more sophisticated approach, similar to the one described in Ref. 16, is to partition the Pauli strings into groups of terms that can be measured simultaneously.In the context of a near-term device we consider two Pauli strings H j and H k simultaneously measurable if and only if they act with the same Pauli operator on all qubits on which they both act non-trivially.In order to efficiently partition the Pauli strings into groups we choose to take all of the terms which only contain Z operators as one partition and then account for the remaining Pauli words heuristically by adding them at random to a group until no more valid choices remain before beginning a new group.We refer to this approach as "Pauli Word Grouping."The final strategy we compare with preprocesses the Hamiltonian by applying the techniques based on the fermionic marginal (RDM) constraints described in Ref. 22, before applying the Jordan-Wigner transformation and using the same heuristic grouping strategy to group simultaneously measurable Pauli strings together [41].We call this latter strategy "Pauli Word Grouping, RDM Constraints" (see legend of Figure 1).To make these plots we computed variances with respect to the FCI (CISD in the case of the largest N 2 calculation) ground state of the systems being studied.This is justified since most variational algorithms for chemistry attempt to optimize ansatze that are already initialized near the ground state.
We refer to the bound of Eq. ( 1) as being based on the L1 norm and calculate it from both the fermion operator Hamiltonian (meaning that the ω in Eq. ( 1) are the coefficients of the terms a † p a q or a † p a † q a r a s ) as well as the qubit operator Hamiltonian (meaning that the ω in Eq. ( 1) are the coefficients of Pauli strings).These two bounds are plotted in Figure 1 using pink and yellow circles respectively.While one would not measure the fermion operators directly, it is surprising that these bounds would be significantly different.Consequently, most prior papers on the topic have computed the bounds from the fermion operator representation for simplicity.However, we find that the L1 norms of the non-constant terms in the Jordan-Wigner transformed molecular Hamiltonians show a significant amount of cancellation.As a result, the bounds on measurement times derived from the qubit operators are smaller by as much as an order of magnitude or more than the corresponding bounds calculated directly from the fermionic operators.
Considering first the Hydrogenic systems in Figure 1 (left panel, A), we note that our Basis Rotation Grouping approach consistently outperforms the other strategies for molecules represented by more than four qubits, requiring significantly fewer measurements.Interestingly, while the L1 bounds and other three methods appear to have relative performances that are stable across a variety of system sizes, the Basis Rotation Grouping method appears to have a different asymptotic scaling, at least for Hydrogen chains of increasing length and basis set size.This is likely due to large scale effects that only manifest when approaching a system's thermodynamic limit (which one approaches particularly quickly for Hydrogen chains) [42].In Table II we quantify this asymptotic scaling by assuming that the dependence of the variance on the number of qubits N in the Hydrogen chain's Hamiltonian can be modeled by the functional form aN b for some constants a and b which we fit using a Bayesian analysis described in the table caption [43].By contrast, the data from the minimal basis water molecule (panel B in Figure 1) shows no benefit in measurement time from our method compared to the heuristic grouping strategies.However, the advantage of our approach becomes significant for even that system in larger basis sets, a trend which is also apparent to a lesser extent for the Nitrogen dimer (panel C).
Interestingly, the Pauli Word Grouping strategy with the RDM constraints of Ref. 22 (green symbols in Figure 1) does not show a significant advantage compared to our simpler heuristic grouping strategy.One possible explanation for this is that these constraints are applied in the fermionic Hilbert space to minimize the L1 norm; we have shown that in the context of the L1 bounds, the norm of the Jordan-Wigner transformed operators are surprisingly different from those of the fermionic operators.Performing this comparison again after adapting these techniques to use the tighter and more appropriate bounds formulated in the qubit Hilbert space may show different results.In principle, this would be straightforward to perform; however, it would require significant modification to the code that is currently available in OpenFermion, and is beyond the scope of this work.
Earlier we explained that the data presented in Figure 1 was calculated by optimally distributing the measurements between different term groupings according to the variance of each term.This was based on the assumption that an adaptive scheme for scheduling measurements would approximate this ideal partitioning.However, one could also imagine calculating approximate measurement fractions ahead of time using a classically tractable approximation to the state (such as truncated For each of the systems and measurement techniques presented in this work we computed the variances of the ground state energy estimation using the configuration interaction singles and doubles (CISD) approximation.We plot the variance calculated using the full configuration interaction (FCI) ground state on the x axis and the approximate variance on the y axis together with a dotted red y = x line.The variances derived from CISD agree well with those derived from FCI.This justifies our suggestion that one use expectation values from the CISD state in order to determine an initial guess of the measurement fractions f from Eq. ( 5).
configuration interaction) to estimate the H , and to evaluate f either as a starting point, or as a replacement for an adaptive approach.To support this idea, we present data on the effectiveness of approximating the variance of the ground state energy estimator using the CISD approximation to the FCI ground state in Figure 2. The approximate variances calculated using the CISD approximation to the ground state closely match those derived from the true FCI wavefunction for all systems we consider in this work.
Overall, Figure 1 speaks for itself in showing that in most cases there is a very significant reduction in the number of measurements required when using our strategy -sometimes by up to four orders of magnitude for even modest sized systems.Furthermore, these improvements become more significant as system size grows.

B. Error Mitigation by Ancilla-Free Postselection
Beyond the reduction in measurement time, our approach has the additional benefits of reducing susceptibility to readout errors as well as enabling a powerful form of error mitigation by postselection.Both properties stem from measuring the Hamiltonian only in terms of density operators in different basis sets.
The additional robustness to readout errors is a consequence of only needing to measure expectation values of operators that have support on at most two qubits, instead of at most N qubits, as would be the case if one attempts to directly measure the Jordan-Wigner transformed Pauli operators arising from the fermionic terms with four distinct indices.To see clearly how this helps to mitigate errors, we consider a simple model of measurement error: the independent, single-qubit symmetric bitflip channel.When estimating the expectation value of a Pauli string H acting on K qubits with a bitflip error rate of p, one would find that which means that noise channel will bias the estimator of the expectation value towards zero in a fashion that vanishes exponentially in K. Thus, the determination of expectation values is highly sensitive to the locality of the H , a behavior that we expect to persist under more realistic models of readout errors.
The other benefit we obtain by measuring the Hamiltonian purely in terms of density operators is that each measurement we prescribe is also simultaneously a measurement of the total particle number operator, η, and of the z component of spin, S z .Because of this, we can postselect our measurements conditioned on observing a desired combination of quantum numbers for both of these operators.Let P denote the projector onto the corresponding subspace and let ρ denote the density matrix of our state.We can then directly evaluate the projected expectation value as Tr (P ρH) This postselection procedure requires more measurements by a factor of 1/Tr(P ρ) to achieve a fixed statistical error if a fixed variance is assumed.However, discarding measurements with the wrong particle number is likely to lead to a lower observed variance owing to the large energetic effects of adding or removing particles, suggesting an additional route by which our Basis Rotation Grouping scheme will reduce the number of measurements in practice.This ability to directly perform postselection allows us to mitigate the impact of a wide range of error channels when using an ansatz which targets the appropriate symmetry sectors.It also allows us to improve the base performance of an ansatz which does not inherently respect the desired symmetries.Several recent works have proposed error mitigation strategies which allow for the targeting of specific symmetry sectors.We review these here in order to place our work in context.One class of strategies focuses on directly and non-destructively measuring one or more symmetry operators [9,11].After performing the measurements and conditioning on the desired eigenvalues the post-measurement state becomes P ρP/Tr(P ρ) and the usual Hamiltonian averaging can be performed.These approaches share some features with our strategy in that they also require an additional number of measurements that scale as 1/Tr(P ρ) and an increased circuit depth.However, they also have some drawbacks that we avoid.Because they separate the measurement of the symmetry operator from the measurement of the Hamiltonian they require the implementation of relatively complicated non-destructive measurements.As a consequence, existing proposals focus on measuring only the parity of the η and S z operators, leading to a strictly less powerful form of error mitigation.Additionally, most errors that occur during or after the symmetry operator measurement are undetectable, including errors incurred during readout.
A different class of approaches avoids the need for additional circuit depth at the expense of requiring more measurements [8][9][10].To understand this, let Π denote the fermionic parity operator and P = (1 + Π)/2 the projector onto the +1 parity subspace.Then, Tr (P ρH) To construct the projected energy it then suffices to measure the expectation values of the Hamiltonian, the parity operator, and the product of the Hamiltonian and the parity operator.A stochastic sampling scheme and a careful analysis of the cost of such an approach reveals that it is possible to use post-processing to estimate the projection onto the subspace with the correct particle number parity in each spin sector at a cost of roughly 1/Tr(P ↑ P ↓ ρ) 2 (where P ↑ and P ↓ are the parity projectors for the two spin sectors) [10].Unlike our approach, this class of error mitigation techniques does not easily allow for the projection onto the correct eigenvalues of η and S z , owing to the large number of terms required to construct these projection operators.Furthermore, the scaling in the number of additional measurements we described above, already more costly than our approach, is also too generous.This is because the product of the parity operators and the Hamiltonian will contain a larger number non-simultaneously measurable terms than the same Hamiltonian on its own.Maximum efficiency may require grouping schemes that consider this larger number of term groupings.
The most significant drawback of our method in the context of error mitigation is that the additional time and gates required for the basis transformation circuit lead to additional opportunities for errors.We therefore focus on comparing the performance of our strategy with a quantum subspace error mitigation approach that effectively projects onto the correct parity of the number operator on each spin sector [9,10].In order to do so, we use the open source software package Cirq to simulate the performance of both strategies for measuring the ground state energy of a chain of six Hydrogen atoms symmetrically stretched to 1.3 Å [45] in an STO-3G basis.We take an error model consisting of i) applying a single qubit depolarizing channel with some probability to both qubits following each two qubit gate and ii) applying a bitflip channel during the measurement process with some other probability.Here we do not consider the effect of a finite number of measurements and instead report the exact expectation values from the final density matrix.
In order to perform a fair comparison, it is necessary to include the impact of errors and error mitigation on the state preparation as well as measurement procedures.In FIG. 3. The absolute error in millihartrees of ground state measurements of a stretched chain of six Hydrogen atoms under an error model composed of single qubit dephasing noise applied after every two qubit gate together with a symmetric bitflip channel during readout.From left to right: A) The error incurred by a "Pauli Grouping" measurement strategy involving simultaneously measuring compatible Pauli words in the usual molecular orbital basis.B) The error when using our "Basis Rotation Grouping" scheme which performs a change of single-particle basis before measurement.C) The errors using the same Pauli word grouping strategy together with additional measurements and post-processing which effectively project the measured state onto a manifold with the correct parities of the total particle number and Sz operators.D) Those found when using our basis rotation strategy and postselecting on outcomes where the correct particle number and Sz were observed.In all panels, for the purpose of approximating a realistic ansatz circuit, three random Givens rotation networks which compose to the identity were simulated acting on the ground state prior to measurement.lieu of using a specific ansatz circuit, we instead take the FCI ground state wavefunction under the Jordan-Wigner encoding as the input to our circuit simulations and apply three random basis transformation circuits which compose to the identity.Approximating the state preparation procedure in this way is motivated both by the assumption that low-depth circuits will be required for successful application of VQE, and by the desire to quantify the impact of our additional measurement circuits in a regime where the cost of these is large compared to that of the state preparation procedure.When using our Basis Rotation Grouping strategy, a fourth Givens rotation circuit which applies the basis transformation indicated by the tensor factorization is required.We show the data for the symmetry protection error-mitigated Basis Rotation Grouping (far right panel) and Pauli Word Grouping (third panel) approaches alongside the expectation values for both measurement strategies without error mitigation (left panels) in Figure 3.
Figure 3 shows that both the Pauli Word Grouping and Basis Rotation Grouping approaches to measurement dramatically benefit from their respective error mitigation strategies.Despite the fact that our proposed Basis Rotation Grouping technique requires circuits that are a third again as deep as those used for the Pauli Word Grouping approach, we see that the errors remaining after mitigation are comparable in many regimes and are dramatically lower for our strategy when noise during measurement is the dominant error channel (compare the bottom right corners of the two rightmost panels).Even without applying postselection, where our proposed technique benefits from a strictly more powerful form of error mitigation, the locality of our Jordan-Wigner transformed operators leads to some benefit in suppressing the impact of readout errors.
We note that the absolute errors we find even at the lowest noise levels we consider are larger than the usual target of "chemical accuracy".In practice, an experimental implementation of VQE on non-trivial systems will require the combination of multiple forms of error mitigation.Prior work has shown that error mitigation by symmetry projection combines favorably with proposals to extrapolate expectation values to the zero noise limit and such an extrapolation would likely significantly improve the numbers we present here [11].Other avenues are also available.For example, one could rely on the error mitigation and efficiency provided by our measurement strategy during the outer loop optimization procedure before utilizing a richer quantum subspace expansion in an attempt to reduce errors in the ground state energy after determining the optimal ansatz parameters.

III. DISCUSSION
We have presented an improved strategy for measuring the expectation value of the quantum chemical Hamiltonian on near-term quantum computers.Our approach makes use of well studied factorizations of the twoelectron integral tensor, in order to rewrite the Hamiltonian in a form which is especially convenient for measuring under the Jordan-Wigner transformation.By doing so, we obtain O(N ) distinct sets of terms which must be measured separately, instead of the O(N 4 ) required by a naive approach.More practically, we require a much smaller number of repetitions to measure the ground state energy to within a fixed accuracy target.For example, assuming an experimental repetition rate of 10 kHz (consistent with the capabilities of commercial superconducting qubit platforms), a commonly referenced bound based on the Hamiltonian L1 norm suggests that approximately 1.6 years are required to estimate the ground state energy to of a symmetrically stretched chain of 6 Hydrogen atoms encoded as a wavefunction on 24 qubits to within chemical accuracy while our approach requires only 44 minutes.Our proposed method also removes the susceptibility to readout error caused by long Jordan-Wigner strings and allows for postselection by simultaneously measuring the total particle number and S z operators with each measurement shot.
Furthermore, the tensor factorization we used to realize our measurement strategy is only one of a family of such factorizations.Future work might explore the use of different factorizations, or even tailor the choice of single particle bases for measurement to a particular system by choosing them with some knowledge of the variances and covariances between terms in the Hamiltonian.As a more concrete direction for future work, the data we have shown regarding the difference between the L1 bounds when calculated directly from the fermionic operators and the same approach applied to the Jordan-Wigner transformed operators suggests that the cost estimates for error-corrected quantum algorithms should be recalculated using the qubit Hamiltonian.
For the largest systems we consider in this work, our numerical results indicate that using our approach results in a speedup of more than an order of magnitude (as well as an asymptotic advantage) when compared to recent state-of-the-art measurement strategies, and more than four orders of magnitude compared to the L1 bounds commonly used to perform estimates in the literature.We have also performed detailed circuit simulations that show that reduction in readout errors combined with the error mitigation enabled by our work largely balances out the requirement for deeper circuits, even when compared against a moderately expensive error mitigation strategy based on the quantum subspace expansion.We believe that the balance of reduced measurement time and efficient error mitigation provided by our approach will be useful in the application of variational quantum algorithms to more complex molecular systems.
And finally, we note that these techniques will generally be useful for quantum simulating any fermionic system, even those for which the tensor factorization cannot be truncated, such as the Sachdev-Ye-Kitaev model [46,47].In that case, our scheme will require O(N 2 ) partitions.Likewise, if the goal is to use the basis rotation grouping technique to estimate the fermionic 2-particle reduced density matrix rather than just the energy, one would need to measure in all O(N 2 ) bases.

FIG. 1 .
FIG.1.The number of circuit repetitions required to estimate the ground state energy of various Hydrogen chains, a water molecule, and a Nitrogen dimer with each of the six measurement strategies indicated in the legend.A target precision corresponding to a 2σ error bar of 1.0 millihartree is assumed.Calculations performed on systems which require the same number of qubits (spin-orbitals) are plotted together in columns and spread slightly horizontally for visibility.The Hydrogen chains contain between 2 and 12 atoms at a symmetric interatomic spacing between 0.6 and 1.3 Å in STO-3G, 6-31G, or cc-pVDZ basis sets.The bonds in the water molecule are symmetrically stretched between 0.8 and 1.5 Å while the spacing between the Nitrogen atoms ranges from 0.9 to 1.6 Å.The Nitrogen and water calculations are performed in STO-3G basis sets with and without the 1s orbitals on the Nitrogen/Oxygen molecules frozen and in the 6-31G basis set only with the core orbitals frozen.The cost of our proposed measurement strategy appears to have a lower asymptotic scaling than any other method we consider and obtains a speedup of more than an order of magnitude compared to the next best approach for a number of systems.
Bounds and uncertainties result from Bayesian inference using a Monte-Carlo approximation with 10 6 particles for all Hydrogen FCI data[43].We assume log(Nmeas) = log(a) + x + b log(N ) where x ∼ N (0, 0.1)[44].The prior distributions are uniform for log(a) and b over [−15, 1] and[1,20] respectively.Here σ(b) is the posterior standard deviation for b and ∆(a) is the posterior standard deviation of log(a) + x.

TABLE I .
A history of ideas reducing the measurements required for estimating the energy of arbitrary basis chemistry Hamiltonians with the variational quantum eigensolver.