Cartan sub-algebra approach to efficient measurements of quantum observables

An arbitrary operator corresponding to a physical observable cannot be measured in a single measurement on currently available quantum hardware. To obtain the expectation value of the observable, one needs to partition its operator to measurable fragments. However, the observable and its fragments generally do not share any eigenstates, and thus the number of measurements needed to obtain the expectation value of the observable can grow rapidly even when the wavefunction prepared is close to an eigenstate of the observable. We provide a unified Lie algebraic framework for developing efficient measurement schemes for quantum observables, it is based on two elements: 1) embedding the observable operator in a Lie algebra and 2) transforming Lie algebra elements into those of a Cartan sub-algebra (CSA) using unitary operators. The CSA plays the central role because all its elements are mutually commutative and thus can be measured simultaneously. We illustrate the framework on measuring expectation values of Hamiltonians appearing in the Variational Quantum Eigensolver approach to quantum chemistry. The CSA approach puts many recently proposed methods for the measurement optimization within a single framework, and allows one not only to reduce the number of measurable fragments but also the total number of measurements.


I. INTRODUCTION
In digital quantum computing, one prepares a wavefunction of the simulated quantum system and any property of interest needs to be physically measured to obtain estimates that constitute the result of the computation. One popular example of this approach is the Variational Quantum Eigensolver (VQE) 1 that is used to solve an eigenvalue problem for the Hamiltonian of interest. Note that in contrast to analogue quantum computing, 2,3 the Hamiltonian of interest is not encoded in VQE but rather its expectation value is measured. The main problem of this measurement setup is that the entire Hamiltonian cannot be measured in a single measurement. This makes efficient partitioning of operators to measurable components one of the most important problems of digital quantum computing.
Any quantum observable is represented by an operator in some mathematical form. To obtain the expectation value of this operator in digital quantum computing, one needs to prepare the quantum system in a particular state corresponding to a wavefunction in the Hilbert space of N qubits (|Ψ ) and to represent the operator of interest O in that Hilbert spacê where c n are numeric coefficients, andP n are tensor products of single-qubit operatorsσ k , which are either Pauli spin operatorsx k ,ŷ k ,ẑ k or the identity1 k . Digital quantum computers can measure only single qubit polarization along the z-axis. This allows one to measure straightforwardly only 2 N − 1 operators for N qubits:ẑ i and all possible productsẑ i ⊗ẑ j ...⊗ẑ k . An arbitrary operatorÔ can be expressed as a linear combination of such products only after applying some multi-qubit unitary transformationÛ : where a i , b ij , ... are some constants. The problem of ob-tainingÛ is equivalent to solving the eigenvalue problem forÔ, and thus is hard to solve in general. However, one can take advantage of additivity of theÔ expectation value with respect toÔ components. A feasible alternative to representation in Eq. (2) is partitioningÔ into fragmentsÔ = nÔ n , where eachÔ n can be written as Eq.
(2) and has corresponding "diagonalizing"Û n . This raises a question: how to select such fragmentsÔ n ? Two main requirements for these fragments are that there is not a large number of them required to represent the wholeÔ, and thatÛ n is not difficult to obtain using a classical computer. Having these fragments allows one to measure the expectation value ofÔ on a wavefunction |Ψ as in Fig. 1(a). The overall efficiency of theÔ measurement will depend on how many measurements each fragment will require to obtain accurate estimates of their expectation values. This consideration makes fragment variances, Ô 2 n − Ô n 2 , to be key quantities for efficient measurement.
There are multiple ways to chooseÔ n 's considering the two conditions. For example, one can try to find exactly solvable models withinÔ asÔ n fragments, but some models are not straightforward to find withinÔ, and their sizes may be small. Also, the sum of two exactly solvable models may not necessarily be another exactly solvable model, which prevents combining such models into larger fragments. One of the most productive approaches so far has been trying to increase the size ofÔ n by collecting all terms ofÔ that mutually commute. The condition of mutual commutativity allows all commuting operators in eachÔ n to be simultaneously diagonalizable, or to have one unitary operator,Û n , transforming all of them into allẑ i form as in Eq. (2). This idea has been applied to the electronic Hamiltonian in both qubit [4][5][6][7][8][9] and fermionic 10,11 operator forms. It was found that working in the fermionic representation provided superior results compared to those obtained in the qubit case. 10,11 Even though all these methods stem from the same idea of finding commuting terms, there was no general framework that would encompass all mentioned methods developed so far. This work will provide such a framework with solid foundations in theory of Lie algebras. Moreover, the introduced framework allows us to propose an extension of previous approaches that results in lower numbers of measurements required to accurately obtain expectation value.
Alternative to a simple measurement scheme presented in Fig. 1(a), there were few more sophisticated schemes put forward for efficient measurement, see Figs. 1(b) and 1(c). The scheme on Fig. 1(b) exploits partitioning of the operator to a linear combination of unitaries that can be measured indirectly via measuring the orientation of an auxiliary qubit (the Hadamard test). 12 This scheme requires efficient partitioning of the operator to unitary components and implementing them in the controlled form. Such controlled unitaries are generally more expensive in terms of two-qubit gates than their non-controlled counterparts. The scheme on Fig. 1(c) uses results of previous measurements to adjust next measurements, this is a so-called feed-forward scheme. Recently, it became available not only in labs 13 but also in an actual quan-tum computer produced by Honeywell. 14 Efficiency of this scheme depends on partitioning of the operator into measurable fragments. Even though the definition of such fragments have been introduced, 15 there is still no systematic procedure to do the optimal partitioning of an arbitrary operator to such fragments.
Yet another class of measurement techniques that appeared recently are methods based on the shadow tomography. [16][17][18] It is based on stochastic sampling of unitary rotationsÛ n for optimal measurement of an operator setÔ n . Even though this technique was used for measurement of electronic Hamiltonians recently, 19-21 its careful comparison with deterministic partitions developed here goes beyond the scope of the current work and will be reported elsewhere. 22 The rest of the paper is organized as follows. Section II A shows how a quantum observable operatorÔ can be expressed using elements of different Lie algebras. Section II B details the role of the CSA for measurement ofÔ and CSA decomposition. Section III illustrates how application of the CSA decomposition to the fermionic and qubit algebras connects several existing approaches, and Section IV proposes schemes improving CSA decompositions in the number of measurements. Section V assesses the new approach and several existing methods on a set of molecular systems (H 2 , LiH, BeH 2 , H 2 O, NH 3 , and N 2 ). Section VI summarizes the main results and provides concluding remarks.

A. Operator embeddings
Any operator in quantum mechanics can be written as a polynomial expression of some elementary operators where {Â k } form a Lie algebra A with respect to the commutation operation ij are so-called structural constants from the number field K. 23 Note that products likeÂ kÂk and higher powers of A elements do not generally belong to the Lie algebra A, instead they are part of a universal enveloping algebra (UEA), E A , which is built as a direct sum of tensor powers of the Lie algebra where the Lie bracket operation is equivalent to the commutator. Thus, any operator in quantum mechanics is an element of some UEA.
Any operator for a physical observable is self-adjoint (i.e. hermitian), which allows us to only consider compact Lie algebras (i.e. all generators are hermitian) for expressing such operators. Any compact Lie algebra can be expressed as a direct sum of abelian and semisimple Lie algebras. 24 For considering the measurement problem within compact Lie algebras, it is convenient to introduce a notion of the Cartan sub-algebra C ⊂ A, which in this case, is a maximal abelian sub-algebra. 25 We will denote elements of C asĈ k 's. The UEA constructed from C, E C , is abelian as well. Thus, in principle, all elements of E C can be measured simultaneously. In practice, there is need for unitary transformations that transform E C elements of a particular CSA into those of allẑ qubit operators. For all algebras discussed in this work these transformations involve only standard fermion-qubit mappings.
One operator can be written as an element of different UEAs, where different UEAs are built from different Lie algebras. We will refer to expressions of the same operator using different Lie algebras as different embeddings. Various embeddings of the electronic Hamiltonian are illustrated in Appendix A.

B. Cartan sub-algebra approach
We will illustrate how elements of E C can be used for measuring operatorÔ that can be written using elements of UEA of a Lie algebra A up to quadratic terms,Â kÂk in Eq. (3). This case can be easily generalized to operators that contain higher powers ofÂ k . Also, we generally assume thatÂ kÂk do not belong to the Lie algebra A. To use elements of E C we need to find a minimum number of unitary transformations that allow us to present the operator of interest aŝ whereĈ l ∈ C, |C| is the CSA size, λ (1,α) l and λ (2,α) ll are some tensors.
What are possible candidates forÛ α ? Clearly, they should depend onÂ k 's and do not create complicated expressions when act on CSA elements. In this work we will consider two constructions ofÛ α 's, but generally one can search for other ways to constructÛ α 's as functions ofÂ k 's. The main guiding principle in this search can be a requirement thatÛ α transformation of any element of E C produces a low-degree polynomial number of terms from E.

Lie group unitaries
The first approach forÛ α 's is to take the elements of the corresponding Lie group where θ (α) k ∈ R, |A| is the Lie algebra size, andÂ k 's are assumed to be hermitian. Due to the closure in A we where c (α) k are some constants. Essentially, the Lie group elementsÛ α provide a set of automorphisms for the corresponding Lie algebra. Moreover, according to the maximal tori theorem for compact groups (all algebras involved in the Hamiltonian embeddings correspond to compact groups), 26 it is guaranteed that there exists a choice of θ for any values of c k . The maximal tori theorem provides a basis for finding a singleÛ that transforms the linear part ofÔ into a linear combination of the CSA terms. Amplitudes θ = {θ k } for thisÛ can be found numerically by solving the system of equations where c (l) k (θ) are some functions whose explicit form depends on the Lie algebra. Thus, in what follows we can focus on representation of the quadratic part ofÔ,Ô (2) .
SinceÛ α 's in Eq. (7) do not change the power ofÂ k 's after transformation (8), one can find θ (α) l , and λ (2,α) ll by minimization of the difference between the quadratic parts of Eqs. (3) and (6). To facilitate the process it is useful to introduce the appropriate basis in the UEA. Such bases are given by the Poincare-Birkhoff-Witt theorem: 27 where k 2 ≤ k 1 = 1, ..., |A|. Note that these bases can be continued to the higher polynomial functions ofÂ k 's but we do not need them beyond the quadratic terms. Both representations ofÔ [Eqs. (3) and (6)] can be transformed to a linear combination of the basis elements. Symmetric basis 2 is somewhat simpler to work with, and thus, we will use it here denoting {Â k1Âk2 } S = (Â k1Âk2 +Â k2Âk1 )/2. Assuming hermiticity of d kk , the quadratic part of Eq. (3) transforms tô TheÔ (2) part of Eq. (6) can be written aŝ Applying the unitary transformation to CSA elementŝ leads tô Term-wise comparison of Eqs. (13) and (16) gives equations on λ (2,α) and θ (α) where the only functions to derive are c (l) k (θ α ). This consideration can be extended to higher powers of Lie algebra elements beyond quadratic. However, for such extensions the number of equations will grow exponentially with the algebraic degree and become computationally overwhelming.

Number of terms conserving unitaries
An alternative that is more efficient for cases with higher powers ofÂ k 's is to useÛ α 's that conserve the number of terms in the transformation: The functional form ofÛ α =Û α ({Â k }) depends on associative algebraic properties of the used UEA, and cannot be given explicitly for a general UEA. For example, in the qubit embedding, every element of the Lie algebra is involutory,Â kÂk =1, which leads toÛ α 's to be elements of the Clifford group. Since all elements of E C commute among themselves, the resulting products ofÂ kj in Eq. (18) are also commutative. Thus, practically, to find a linear combination of operator terms that can be treated by a singleÛ α , one needs to group all mutually commuting terms. Generally, the commutation relation between operator terms can be represented by a graph whose edges connect commuting terms represented by vertices. Partitioning of theÔ expression to mutually commuting groups can be done in various ways, but finding the optimal partitioning to a minimum number of such groups is a standard NPhard problem in graph theory, the minimum clique cover problem. There are many heuristic polynomial algorithms to solve this problem. 4

III. APPLICATIONS
Here we will illustrate how novel algorithms and several previously developed methods can be derived from the CSA framework applied to the electronic HamiltonianĤ e as an example of a many-body operatorÔ. Fermionic and qubit embeddings will be used to illustrate the CSA decomposition ofĤ e . As shown in Appendix A, the fermionic embeddings contain low powers of Lie algebra elements and thus one can expect more benefits from using the Lie group unitaries [Eq. (7)], while the qubit embedding has the involutory associative multiplication property for the algebra elements and thus benefits more from the number of term conserving unitaries [Eq. (18)].

A. Fermionic algebras
To apply the Lie group unitaries [Eq. (7)] one needs operator embeddings involving compact Lie algebras. Even though we use generators of the non-compact gl(N ) algebra,Ê p q 's, this does not create any problem because in what followsÊ p q 's always appear as linear combinations i(Ê p q −Ê q p ) and (Ê p q +Ê q p ), which are generators of the compact Lie algebra u(N ). The origin of this compactness is the hermiticity of the system Hamiltonian that allows to rewrite the Hamiltonian in u(N ) generators. The CSA of gl(N ) and u(N ) are the same: N elementsÊ p p . E p q 's are isomorphic to real N by N matrices (E p q ) mn = δ mp δ nq , and there are N 2 such elements. This faithful representation ofÊ p q makes search forÛ α in Eq. (9) equivalent to a simple diagonalization of the hermitian matrix, h pq .
Unfortunately, the two-electron part of the fermionic Hamiltonian cannot be treated as easily as the oneelectron part. ProductsÊ p qÊ r s do not form a Lie algebra, and thus their representation as tensor products of pairs of N by N matrices do not lead to unitary transformations that we are looking for. However, diagonalization of 4-index tensors g pq,rs can provide some approximation to the CSA decomposition and is discussed as a Hamiltonian factorization.

Hamiltonian factorization
Using the maximal tori theorem one can employ the following heuristic approach to obtaining the expansion for the quadratic partÔ (2) in products of linear combinations ofÂ kÔ whered kk is an analogue of d kk from Eq.

Full rank optimization
Extension of the factorization approach to Eq. (6) requires substituting SVD to a more general decomposition. To arrive at Eq. (6), one can start with the symmetrized form ofĤ where g pq,rs = g rs,pq due to permutation symmetry in the two-electron integrals. The equivalent of Eq. (6) for this Hamiltonian iŝ wherê There is a homomorphism of these unitary operators onto unitary matrices that are obtained by substituting the excitation operatorsÊ t u by matrices E t u , which is the faithful representation of algebraic generators. This homomorphism allows us to perform the operator transformations in Eq.
by substituting the operators with corresponding matrices. This substitution gives coefficients c Using independence of the basis set elements {Ê p qÊ r s } S , the following system of algebraic equations can be written Its solution is found via gradient minimization of the norm for the vector of differences between the right and left hand sides using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm. 28

B. Qubit algebras
There are two compact, semisimple qubit algebras that can be used for the CSA decomposition of the qubit counterpart of the electronic Hamiltonian,Ĥ q (Appendix A): 1) UEA of S = ⊕ N k=1 su(2) k , E S , or 2) so(2 N ). One advantage of E S is a single-particle picture (elements of S) that can define a class of computationally feasible unitariesÛ α .

Single-qubit unitaries
The Lie group constructed by exponentiating the S algebra consists of the following elementŝ where τ k is an amplitude,n k is a unit vector on the Bloch sphere, andσ k = (x k ,ŷ k ,ẑ k ). The fragments that can be measured afterÛ QMF transformations are exactly solvable within the qubit mean-field (QMF) approach 29 whereĈ (Z) k are elements from the directly measurable CSA: {1,ẑ i ,ẑ iẑj , ...}. One-qubit rotations in the qubit space do not translate to one-electron fermionic transformations as was shown. 29 Therefore, the fragments in Eq. (29) are different from those in Eq. (23). The latter are also exactly solvable but with the fermionic one-particle transformations.
Identifying fragments of Eq. (29) is not straightforward and requires a tensor decomposition of the qubit Hamiltonian,Ĥ q . Note that sinceÛ QMF transformations are not necessarily in the Clifford group, 30 each of them can produce exponentially many Pauli products by transforming a single product. The number of terms in the directly measurable CSA in Eq. (29) can be also exponentially large. Thus, even though the total number ofĤ q terms is O(N 4 ), a general tensor decomposition 31,32 ofĤ q to fragments can have exponential computational cost and therefore is unfeasible on a classical computer.
To avoid exponential growth of terms, one can restrict U QMF to a Clifford subset of transformations, thus switching from the Lie group unitaries (Eq. (7)) to the unitaries conserving the number of terms (Eq. (18)). Then, the only fragments ofĤ q that will be transformed intoĈ (Z) k are the qubit-wise commuting (QWC) sets of Pauli products. 4 A practical way to find all QWC sets of Pauli products was found through mapping this problems to the minimum clique cover problem for a graph representing qubit-wise commutativity relation inĤ q .

Multi-qubit unitaries
Alternative consideration ofĤ q within the so(2 N ) algebra has an advantage that the maximal tori theorem guarantees existence of a singleÛ ∈ Spin(2 N ) 33 for the transformationÛĤ Unfortunately, this unitary requires an exponential number of algebra elements,Û = exp[i k c kPk ]. Therefore, the exponential size of so(2 N ) prevents us from taking advantage of the maximal tori theorem. For practical purpose, we restrictÛ to unitaries from the Clifford group that conserve the number of terms Eq. (18). In this case, a singleÛ will not be sufficient andĤ q decomposition involves several termŝ To findÛ α 's one can use the main property of the Clifford group that any Pauli product is transformed by an element of the Clifford group to another Pauli product. This means that any fragmentĤ (α) q consists of commuting Pauli products because their Clifford unitary imageŝ C (Z) k are also commuting. This consideration connects Eq. (31) with a partitioning method based on finding fully commuting (FC) sets described in Ref. 6. FC sets are also found by heuristic solutions of the minimum clique cover problem for a graph built forĤ q using the commutativity relation.

IV. MEASUREMENT OPTIMIZATION
In this section we review the standard approach to computing the number of measurements K needed to estimate the expectation value of the electronic Hamiltonian within error . For a decomposition of the Hamiltonian where the expectation value of each fragmentĤ (α) is sampled K (α) times, the total number of measurements is K = α K (α) . As shown previously, 9,34 the optimal choice of K (α) is Var Ψ (Ĥ (β) ), (33) where is the desired accuracy, |Ψ is the VQE trial wavefunction, and Var Ψ (Ĥ (α) ) = Ψ|Ĥ (α)2 |Ψ − Ψ|Ĥ (α) |Ψ 2 is the operator variance. Then the total number of measurements is where the square of the expression in the brackets can be seen as the variance of the estimators for Ψ|Ĥ |Ψ originating from the decomposition in Eq. (32). There are two main heuristics one could employ to reduce K. First, it is generally favorable to have an uneven distribution of Var(Ĥ α ) due to the square root functions. 9 One can achieve this by iteratively applying FRO with few M until convergence, where at each iteration one finds the fragment that minimizes the norm of the remaining coefficients using the "greedy" approach. This approach generally results in concentrated coefficients and thus larger variances in the first few fragments. Second, one can approximate K with K ϕ computed using wavefunction |ϕ calculated from classical methods (e.g., Hartree-Fock (HF)) and then search for partitioning that minimizes K ϕ . Based on these two approaches, we propose the following modifications of the full rank optimization (FRO) that reduce the total number of measurements: a. Greedy FRO (GFRO): This algorithm iteratively applies the FRO with M = 1. At the α-th iteration, GFRO finds the λ (2,α) , θ (α) , φ (α) of Eq. (27) that minimizes where parameters of f (β) pq,rs were fixed from previous iterations.
b. Variance-estimate Greedy FRO (VGFRO): This algorithm also uses FRO with M = 1 consecutively and fixes coefficients from previous iterations. Unlike GFRO, VGFRO minimizes at the α-th iteration, where and w is a fixed parameter. Note that one needs to choose a w small enough to encourage the optimization to reduce pq,rs = 0. In practice, the variances of the first few fragments obtained from GFROs are significantly larger than those of the fragments from subsequent iterations. Thus, for computational efficiency of VGFRO, we drop the second term of Eq. (36) after first µ fragments. This is equivalent to performing GFRO on the remaining d (µ) pq,rs tensor. c. Qubit-based algorithms: In qubit algebra, the greedy approach naturally leads to the Sorted Insertion (SI) algorithm, 9 which also groups the commuting Pauli products with large coefficients together in a few measurable fragments. However, further utilizing approximated variance in SI is not straightforward, since partitioning of the qubit Hamiltonian is not guided by gradients that are used to minimize the norm and variance in a continuous fashion in VGFRO.

V. RESULTS
To assess the CSA decomposition techniques in fermionic and qubit algebras, we applied them to a set of Hamiltonians previously used to demonstrate performance of similar measurement techniques 4,6,12 (Tables I and II). Details of these Hamiltonians are provided in Appendix C. QWC fragments were obtained by the largest first (LF) heuristic 4 and the sorted insertion (SI) algorithm. 9 Comparing the number of fragments in QWC-LF and QWC-SI shows slight advantage of QWC-LF (Table I). However, the number of measurements is almost factor of 2 lower for large Hamiltonians when fragments are defined by QWC-SI (Table II). This shows a general trend that the number of fragments does not always correlate with the number of measurements. Also, this illustrates advantage of the SI algorithm for the qubit term grouping, therefore for the FC method we used the SI algorithm as well.
All fermionic-algebra methods approximate the twoelectron integral tensor with a finite accuracy, which we judiciously chose to be 10 −5 in 1-norm of the difference between all g pq,rs before the symmetrized algebra generator products and their restored values. For the SVD factorization approach, the singular values arranged in the descending order and their eigenvectors were used to reconstruct the g pq,rs matrix until the 1-norm threshold was satisfied.
For the FRO based algorithms, it was found that real rotation generators inÛ α and real λ (2,α) are sufficient for decomposition of the two-electron part of the electronic Hamiltonian. Additional simplification came from the electron-spin symmetry in g pq,rs that allowed us to manipulate with orbitals (N/2) rather than spin-orbitals (N ). Therefore, we used the unitary operatorsÛ α generated by exponentiation of so(N/2) instead of u(N ), and ourÛ α 's were in the Spin(N/2) subgroup of the original U (N ). As discussed in Ref. 10,Û α can be efficiently implemented (N 2 /4 − N/2 two qubit gates and gate depth of exactly N ) on a quantum computer with a limited connectivity. Limiting λ (2,α) to real entries and accounting for its symmetric property (λ (2,α) tu = λ (2,α) ut ), the FRO procedures had N 2 /4 parameters in total for each fragment. For all molecules but H 2 parameters w (see Eq. (36)) and µ are set to 0.5 and 30, for H 2 , w = 0.5 and µ = 1. To obtain fragments variance estimates for the VGFRO optimization we used Hartree-Fock wavefunctions.
As expected, the QWC method results in one of the highest numbers of measurable groups and one of the highest numbers of needed measurements. The FRO method without any concern about fragment variances provides the minimum number of fragments (Table I). However, more measurable fragments does not necessarily mean more measurements. As shown in Tables I and II, the GFRO and VGFRO methods partition the Hamiltonians into considerably more measurable parts than SVD and FRO, yet they improve on SVD consistently by 10-30 percent in the number of measurements. This confirms the efficacy of the heuristic that groups operators with large coefficients in the same measurable fragment.
Interestingly, for the smaller molecules, FC-SI outperforms the fermionic-algebra methods. This suggests that the CSA decomposition based on commutativity between Pauli products in qubit-algebra is better at collecting large coefficients in few groups. However, since the number of measurable fragments in FC-SI and SVD correspondingly scale with O(N 3 ) and O(N 2 ), 8,10 we expect that as the size of the systems grows, the fermionic-algebra methods that have more flexible measurable parts will TABLE I. Number of measurable groups provided by different methods for Hamiltonians of several molecular systems [the number of spin-orbitals (N ) and the total number of Pauli products in the qubit form (Total)]: qubit-algebra methods based on qubit-wise and full commutativity (QWC and FC), fermionic-algebra methods based on the SVD factorization (SVD), 10 full rank optimization (FRO), greedy FRO (GFRO) and variance-estimate greedy FRO (VGFRO). The norm-1 accuracy for the fermionic-algebra methods is 10 −5 .

VI. CONCLUSIONS
In this work we provided a unifying framework for many recently suggested approaches of efficient partitioning of quantum operators into measurable fragments. The framework is based on identifying a Lie algebra that is used for the operator expression (embedding), analysis of the corresponding Lie group action, and the Cartan sub-algebra of the Lie algebra. The latter encodes the measurable fragments since it involves mutually commuting terms.
To obtain measurable fragments we suggested two types of unitary transformation. First, the unitaries that are elements of the Lie group corresponding to the Lie algebra, Eq. (7). These unitaries have the advantage of conserving the degree of the Lie algebra polynomials in the operator expression. Second, the unitaries that preserve the number of terms, Eq. (18). These unitaries use associative multiplication properties of algebraic operators and for qubit algebras correspond to the Clifford group. An intuitive rule to select between these choices of unitaries is the degree of the Lie algebraic operator polynomial expression: the Lie group unitaries are more efficient for lower degrees, and the number of term conserving unitaries are more useful for higher degrees.
Being able to embed a single operator in multiple Lie algebras opens directions for further search for efficient partitioning schemes. Here, we mainly focused on embedding of the electronic Hamiltonian in fermionic (u(N )) and qubit (su(2 N )) Lie algebras. While the qubit embedding provides a more efficient scheme for electronic Hamiltonians of fewer orbitals than those of fermionic embedding, this trend can change as the size of the system grows.
To minimize the total number of measurements, it is not enough to reduce the number of the measurable fragments because of possible increase of fragments variances. It was found that grouping algorithms that employ greedy techniques were advantageous for lowering of the overall variance of the expectation value estimator. This advantage can be attributed to lowering the total variance in cases when fragment variances are distributed nonuniformly. In addition, the fermionic CSA decomposition allows one to optimize partitioning by using fragment variance estimates and to lower the overall variance in VGFRO further than in the greedy approach (GFRO).
Research Council of Canada.
[Ê p q ,Ê r s ] =Ê p s δ qr −Ê r q δ ps , therefore we can consider two partsÊ Analogously, for theÊ r s part we will obtain the same result but with the minus sign, thus the commutator is zero.
Therefore, one can measure each k part of the twoelectronic Hamiltonian Considering the norm of Ω