Revising the measurement process in the variational quantum eigensolver: is it possible to reduce the number of separately measured operators?

We have introduced two approaches to reduce the number of separately measured terms in molecular Hamiltonians within the Variational Quantum Eigensolver (VQE) technique for solving the electronic structure problem.


Introduction
One of the most practical schemes for solving the electronic structure problem of current and near-future universal quantum computers is the variational quantum eigensolver (VQE) method. [1][2][3][4][5] This approach involves the following steps: (1) reformulating the electronic Hamiltonian (Ĥ e ) in the second quantized form, (2) transformingĤ e to the qubit form (Ĥ q ) by applying iso-spectral fermion-spin transformations such as Jordan-Wigner (JW) 6,7 or more resource-efficient Bravyi-Kitaev (BK), [8][9][10][11][12] (3) solving the eigenvalue problem forĤ q by variational optimization of unitary transformations for a qubit wavefunction. The last step uses a hybrid quantum-classical technique where a classical computer suggests a trial unitary transformation U, and its quantum counterpart provides an energy expectation value of E U ¼ hJ 0 |U †Ĥ q U|J 0 i, here |J 0 i is an initial qubit wavefunction (it is frequently taken as an uncorrelated product of all spin-up states of individual qubits). The two steps, on classical and quantum computers, are iterated till convergence. The VQE was successfully implemented on several quantum computers and used for few small molecules up to BeH 2 . 13 One of the big problems of the VQE is that to calculate E U , the quantum computer measures parts of H q rather than the whole H q on the U|J 0 i wavefunction. This stems from technological restrictions of what can be currently measured on available architectures. Dramatic consequences of this restriction can be easily understood with the following simple example. Let us assume thatĤ q ¼Â +B, whereÂ andB are measurable components ofĤ q and [Â,B] s 0, otherwise they could be measured at the same time at least in principle. The actual hardware restrictions on measurable components are somewhat different and will be discussed later, for this illustration these differences are not important. Even if one has an exact eigenstate ofĤ q , U|J 0 i, measuring it onÂ orB would not give a certain result becauseÂ andB do not commute withĤ q . Thus, one would not be able to distinguish the exact eigenstate from other states by its zero variance. The origin of the discrepancy between quantum uncertainty given by the variance (Var) ofĤ q (true uncertainty) and by the sum of variances forÂ andB is neglect of covariances (Cov) Var(Ĥ q ) ¼ Var(Â) + Var(B) + Cov(Â,B) + Cov(B,Â), (1) Cov(Â,B) ¼ hÂBi À hÂihBi.
Thus, even though theĤ q average is equal to averages ofÂ andB, the true quantum uncertainty ofĤ q is overestimated by a sum of variances forÂ andB. Moreover, the number of measurements to sampleÂ andB is twice as many as that forĤ q if the eigenstate nature of U|J 0 i is not known a priori.
The variance of any Hamiltonian depends only on the Hamiltonian and the wavefunction, but if one approximates the variance using only variances of Hamiltonian parts and neglects covariances between the parts, the result of such an approximation will depend on the partitioning. Importantly, the sum of variances for the Hamiltonian parts can either under-or overestimate the true Hamiltonian variance. To see how ignoring covariances can erroneously make estimates of the uncertainty arbitrarily small consider an articial example, where the Hamiltonian variance is measured as n independent measurements of itsĤ q /n identical parts. Due to the linear scaling of the variance sum with n and the inverse quadratic scaling of variances of individual terms with n, the overall scaling of the variance is inversely proportional to n and can be made arbitrarily small by choosing large enough n. This follows from a wrong assumption that parts (Ĥ q /n) are independent and covariances between them are zero.
Generally, the number of non-commuting terms inĤ q grows with the size of the original molecular problem, and the total uncertainty from the measurement of individual terms will increase. This increase raises the standard deviation of the total measurement process and leads to a large number of measurements to reach convergence in the energy expectation value. The question we would like to address is whether it is possible to reduce the number of theĤ q terms that needs to be measured separately.
In this paper we introduce a new systematic approach to decreasing uncertainty of the expectation energy measurement. We substitute the conventional measurement partitioning of the Hamiltonian with groups of qubit-wise commuting operators 13,14 by partitioning to terms whose eigenstates can be found exactly using the mean-eld procedure. Owing to a more general structure of such terms the Hamiltonian can be split into a fewer number of them. Interestingly, the general operator conditions on such mean-eld terms have not been found in the literature and have been derived in this work for the rst time.
To decrease the number of these terms even further, we augment the mean-eld treatment with few-qubit unitary transformations that allow us to measure few-qubit entangled terms. Measurement of newly introduced terms requires the scheme appearing in the cluster-state quantum computing, 15,16 it is qubit-wise measurement with use of previous measurement results to dene what single-qubit operators to measure next.

Qubit Hamiltonian
In order to formulate the electronic structure problem for a quantum computer that operates with qubits (two-level systems), the electronic Hamiltonian needs to be transformed iso-spectrally to its qubit form. This is done in two steps. First, the second quantized form ofĤ e is obtainedĤ whereâ † p (â p ) are fermionic creation (annihilation) operators, h pq and g pqrs are one-and two-electron integrals in a spin-orbital basis. 17 This step has polynomial complexity and is carried out on a classical computer. Then, using the JW 6,7 or more resourceefficient BK transformation, [8][9][10][11][12] the electronic Hamiltonian is converted iso-spectrally to a qubit form where C I are numerical coefficients, andP I are Pauli "words", products of Pauli operators of different qubitŝ s^i (I) is one of thex,ŷ,ẑ Pauli operators for the i th qubit. The number of qubits N is equal to the number of spin-orbitals used in the second quantized form [eqn (4)]. Since every fermionic operator is substituted by a product of Pauli operators in both JW and BK transformations, the total number of Pauli words in H q scales as N 4 .

Conventional measurement
In the conventional VQE scheme theĤ q is separated into sums of qubit-wise commuting (QWC) terms, A n ¼ ] ¼ 0. The opposite is not true, a simple example is [x 1x2 ,ŷ 1ŷ2 ] ¼ 0 but [x 1x2 ,ŷ 1ŷ2 ] qw s 0. We will not be using non-zero results of the qubit-wise commutator and therefore their exact values are not important, but it is assumed that [.,.] qw is bi-linear for both operators.
Partitioning of the H q in eqn (7) allows one to measure all Pauli words within eachÂ n term in a single set of N one-qubit measurements. For every qubit, it is known from the form of A n , what Pauli operator needs to be measured. The advantage of this scheme is that it requires only single-qubit measurements, which are technically easier than multi-qubit measurements. The disadvantage of this scheme is that the Hamiltonian may require measuring too manyÂ n terms separately.
A natural extension of partitioning in eqn (7) is to sum more general termsĤ with the condition thatĤ (MF) n eigenstates can be presented in a single-product form of single-qubit wavefunctions. In other words, the eigenstates of theĤ (MF) and can be obtained using a mean-eld procedure. This condition would allow measurement of eachĤ (MF) n fragment qubit aer a qubit. However, to perform the new splitting we need an exact denition of the mean-eld (MF) Hamiltonian so that we can recognize these new blocks within the total Hamiltonian.

Mean-eld Hamiltonians
What is the most general form of a qubit Hamiltonian whose eigenstates can be presented as single factorized products of one-qubit wavefunctions? Note that the well-known example of such Hamiltonians, separable operators, are a particular class that does not provide the most general form. In other words, there are many more Hamiltonians that are not separable but are still in the MF class, one simple example isĤ which does not follow the form of eqn (10) but whose eigenstates, |+ z i 1 |AE x+y i 2 and |Àzi 1 |AE xÀy i 2 , ‡ are unentangled products. We formulate the general criterion for a Hamiltonian H(1,.N) to be in the MF class as follows. There should exist N one-particle operators {Ô k (k)} N k ¼ 1 § that commute [Ô k ,Ĥ NÀk+1 ] ¼ 0 with the system of N Hamiltonians {Ĥ NÀk+1 } N k ¼ 1 constructed in the following way that we will refer as a reductive chain: whereÔ k |f k i ¼ l k |f k i. The nal operator in this chain is a oneparticle operator that commutes with itself and denesÔ N 1 H 1 . The proof of this criterion can be found in Appendix A. It is easy to see that jJi ¼ Q N k¼1 jf k i is an eigenfunction ofĤ. Clearly, separable Hamiltonians are in the MF class because for them, O k 's can be taken asĥ k (k) from eqn (10). However, note that because the system ofÔ k operators is required to commute not withĤ but with the reduced set of Hamiltonians, the formulated criterion goes beyond separable Hamiltonians. A general procedure to determine whether a particular qubit HamiltonianĤ is in the MF class or not requires nding all N one-particle operatorsÔ k . The procedure starts with a check whether there is at least one qubit k for which can be achieved by choosing a non-zero vector (a, b, c). Once the rst operatorÔ 1 (k) ¼ ax k + bŷ k + cẑ k is found its eigenstates can be integrated out to generateĤ NÀ1 , and the procedure can be repeated to ndÔ 2 that commutes withĤ NÀ1 .

Measurement of mean-eld Hamiltonians
Measuring an N-qubit mean-eld Hamiltonian can be done by performing a single set of sequential N one-qubit measurements. Each qubit projective measurement in this set will collapse the measured wavefunction to an eigenstate of the corresponding single qubit operator. The single qubit operators that need to be measured areÔ k 's operators. The denition of one particle operators may depend on the result of the previous measurement. Let us consider the mean-eld Hamiltonian in eqn (11):Ô 1 (1) ¼ẑ 1 , andÔ 2 (2) ¼x 2 AEŷ 2 , where AE is determined by the eigenfunction chosen from theÔ 1 spectrum to generate theĤ 1 ¼ hf 1 AE |Ĥ MF |f 1 AE i in the chain of eqn (12). This ambiguity does not allow one to presentĤ MF as an operator with all qubitwise commuting components. An attempt on this can be done by inserting the projectors on the eigenstates ofẑ 1 instead of the operator: i, and even though the projectors onto the |f 1 AE i eigenstates commute, the (x 2 AEŷ 2 ) parts do not. Therefore, the scheme for measuring theĤ MF will be as shown in Fig. 1. Note that no matter how entangled the initial wavefunction is, measuringĤ MF does not require measuringx 2 andẑ 1ŷ2 separately as was done in the regular VQE scheme.
In practice, qubit-wise measurements using previous measurement results to dene what single-qubit operators to measure next, or feedforward measurements, have been implemented in quantum computers based on superconductor and photonic qubit architectures. 18,19 The essential feasibility condition for the feedforward measurement is that the delay introduced by measurements is much shorter than the qubit coherence time. For superconducting (photonics) qubit architectures this condition has been achieved with typical timescales for a measurement and coherence as 2 ms (ref. 20

Mean-eld partitioning
Even though regular molecular qubit Hamiltonians are not guaranteed to be in the MF class, it is always possible to split any N-qubit Hamiltonian into a sum of MF Hamiltonians. To see this, we will present a heuristic partitioning scheme that guarantees the MF partitioning. Our scheme uses ranking of all qubits k ¼ 1,.,N based on a geometrical characteristic l(k), which is dened as follows. For an arbitrary qubit k, the total Hamiltonian can be written aŝ H ¼ĥ xxk +ĥ yŷk +ĥ zẑk +ĥ e (16) whereĥ x,y,z,e are the residual operators that do not contain Pauli matrices for the k th qubit. Assembling coefficients of Pauli words in operatorsĥ x,y,z into vectors, . Evaluating S k is equivalent to obtaining the overlap between three vectors h x,y,z assuming the orthogonal basis, while the dimensionality of its kernel is the number of its zero eigenvalues.
l(k) allows one to answer a question on whether there is a transformation involving only the k th qubit that can presentĤ in one of the two forms: k are operators containing only the k th qubit, and h,ĥ 0 ,ĥ 00 are the complementary operators that exclude the k th qubit. The positive answers in the forms of eqn (17) and (18) correspond to l(k) ¼ 2 and l(k) ¼ 1, respectively. l(k) ¼ 2 is equivalent to the MF condition of eqn (13), withÔ k ¼ ax k + bŷ k + cẑ k . For l(k) ¼ 1, the MF treatment of the k th qubit is not possible but using eqn (18) the k th qubit dependence in the Hamiltonian can be somewhat compactied. Coefficients forÔ k ;Ô 0 k ;Ô 00 k and h,ĥ 0 ,ĥ 00 operators can be found from non-zero eigenvectors of S k (this process is detailed in Appendix B). The negative answer to the question leavesĤ in the original form of eqn (16) and is equivalent to l(k) ¼ 0.
The question about possible compactication of the k th qubit dependence in the Hamiltonian has a simple geometric interpretation in terms of arrangement of the three vectors h x,y,z . These multi-dimensional vectors can be linearly independent (eqn (16)), located within some plane (eqn (18)), or collinear to each other (eqn (17)), Fig. 2 illustrates all three cases.
Using a set of l(k)'s for a given Hamiltonian one can decide how many qubits can be treated using the MF procedure, these will be all qubits with l(k) ¼ 2. Once all of such qubits have been considered, the MF partitioning of l(k) ¼ 1 qubits begins. For l(k) ¼ 1, the Hamiltonian can be split for any of such qubits into two parts:Ĥ ð1Þ ¼ĥ 0Ô0 k andĤ ð2Þ ¼ĥ 00Ô00 k þĥ e . In both parts the k th qubit can be treated using the MF treatment, which allows one to continue the consideration forĥ 0 ,ĥ 00 andĥ e . Finally, if only qubits with l(k) ¼ 0 are le, thenĤ needs to be partitioned to three HamiltoniansĤ (1) ¼ĥ xxk ,Ĥ (2) ¼ĥ yŷk , andĤ (3) ¼ĥ zẑk + h e , where at least the k th qubit can be treated using MF. Aer this separation one can apply the reduction chain to each of the three operators. Fig. 3 illustrates the partitioning for a three qubit case detailed in Appendix B. In the case when reducing the k th qubit does not produce a Hamiltonian with reducible qubits the partitioning needs to be repeated, as in Fig. 3 when transforming qubit 1 led to h(2,3) where none of the qubits can be reduced.
Our scheme can be considered as an example of a greedy algorithm because at every step it tries to nd locally the most optimal reduction, a qubit with the highest l(k). The reduction is only possible if there is linear dependency between complementary vectorsh x;y;z . The lower the dimensionality of the linear space, where these vectors are located, the more probable such linear dependence. Thus, treating qubits with the highest l(k) rst is justied by the reduction of the space dimensionality along the reductive scheme. In the example of Fig. 3 treatment of qubits 2 and 3 in the beginning would require partitioning of the Hamiltonian to two branches for each of them, while leaving the 3 rd qubit to the end did not generate any new terms for it.
It is possible that more than one qubit will have the highest l(k). To do more optimal selection in this case, one would need to consider maxima of l(k) functions on qubits that enter complementary Hamiltoniansĥ for different reduction candidates. This consideration makes the partitioning computationally costly and was not performed in this work.
Applying the partitioning scheme guarantees to result in a sum of MF Hamiltonians that can be measured in N-qubit one-particle measurements. Since any linear combination of QWC terms form a MF Hamiltonian, this partitioning scheme cannot produce more terms than those used in the regular VQE measuring scheme.

Unitary transformations generating mean-elds
Partitioning the non-MF blocks in the Hamiltonian to obtain more MF terms leads to growth of the terms needed to be measured. An alternative treatment of non-MF groups is to search for multi-qubit operators that commute with them. Finding such operators may lead to unitary transformations that can transform non-MF Hamiltonians into Hamiltonians where qubits shared with the commuting operator can be treated using the mean-eld procedure. Similar search for multi-qubit operators commuting with the system Hamiltonian was used recently by Bravyi and coworkers to reduce the qubit count in the conventional VQE scheme. 23 Let us consider an example where an N-qubit non-MF HamiltonianĤ has a two-qubit operatorÔ (2) (1,2) commuting with it (without loss of generality we can assume thatÔ (2) acts on the rst two qubits). Then, under certain conditions detailed in Appendix A,Ĥ allows for its eigenstates J to be written as J(1,.N) ¼ F(1,2)j(3,.N), where F(1,2) is an eigenstate ofÔ (2) . One can always write F(1,2) ¼Û(1,2)f 1 (1)f 2 (2), whereÛ(1,2) is an operator entangling the product state f 1 (1)f 2 (2) into F(1,2). Using this unitary operator, one can obtain the Hamiltonian H 12 ¼Û(1,2) †ĤÛ (1,2) that has an eigenstate J 12 (1,.N) ¼ f 1 (1) f 2 (2)j(3,.N) where qubits 1 and 2 are unentangled. Therefore, there should be one-particle operators of qubits 1 and 2 that commute withĤ 12 and its MF-reduced counterpart. Finding these operators and their eigenfunctions f 1 (1) and f 2 (2) allows us to integrate out qubits 1 and 2 Search for one-or multi-qubit operators commuting witĥ H NÀ2 can be continued. The procedure to nd commuting operators with increasing number of qubits requires exponentially increasing number of variables parametrizing such operators. Indeed, a k-qubit operator requires a 3 k coefficient for all Pauli words in commutation equations similar to eqn (13), also the number of different k-qubit operators among N qubits is C N k $ N k . Potentially, such operators always exist (e.g., projectors on eigenstates of the Hamiltonian) but the amount of resources needed for their search can exceed what is available. Thus we recommend interchanging this search with the partitioning described above if the multi-qubit search requires going beyond 2-qubit operators.
To illustrate the complete scheme involving multi-qubit transformations, let us assume that we can continue the reduction chain forĤ ¼Ĥ N by generating the set of Hamiltonians {Ĥ N ,Ĥ NÀ2 ,.,Ĥ k } using qubit unitary transformations {U(1,2), U(3,4,5),.,U(N À k,.N)} and integrating out variables from N to k. To take advantage of this reduction chain in measuring an expectation value of an arbitrary wavefunction c(1,.N) onĤ, such a measurement should be substituted by the following set of conditional measurements: Step 1: rst two qubits are measured usingĤ 12 and the unitary transformed function |Û(1,2) † ci because Depending on the results of these measurements the oper-atorĤ NÀ2 is formulated and its unitary transformation U(3,4,5) is found. U (3,4,5) gives rise to the transformed Hamiltonian H 35 ¼Û(3,4,5) †Ĥ NÀ2Û (3,4,5). The wavefunction aer measuring qubits 1 and 2 is denoted as |c 12 i.
Step 2: qubits 3-5 are measured onĤ 35 sequentially using the transformed wavefunctionÛ(3,4,5) † |c 12 i. Results of these measurements will dene the next reduction step and the wavefunction that should be unitarily transformed for the next measurement.
These steps can be continued until all qubits have been measured. If resources allow for nding corresponding multiqubit unitary transformations, theĤ Hamiltonian can be measured in N single-qubit measurements.

Numerical studies and discussion
To assess our developments we apply them to the Hamiltonians of the H 2 and LiH molecules obtained within the STO-3G basis and used to illustrate the performance of quantum computing techniques previously. 13,24,25
where some of the C i 's are equal, but it is not going to be important for us (the details of generating this Hamiltonian are given in Appendix C). ClearlyĤ H 2 contains three groups of QWC terms, the rst three lines form one group, and the two last terms fall into two other groups.Ĥ H 2 is not a MF Hamiltonian, only qubits 1 and 3 have one-particle operators commuting with the Hamiltonian, while aer their reduction the reduced Hamiltonian does not commute with any one-particle operator where D i 's are constants. Partitioning ofĤ 24 to three terms using qubit 2 or 4 would not be more efficient than partitioninĝ H H 2 in 3 groups of QWC terms from the beginning. However, there is the two-particle operatorẑ 2ẑ4 that commutes withĤ 24 where E i 's are some constants and the rst one-particle commuting operator isÔ 1 (4) ¼ŷ 4 . Aer integrating outÔ 1 's eigenfunction,Ô 2 (2) is a linear combination ofẑ 2 andŷ 2 .
To illustrate the superiority of the scheme with the use of U(2,4) and measurements of the MF Hamiltonian over the regular approach with splittingĤ H 2 to three groups of QWC operators, Table 1 presents variances for the Hamiltonian expectation value for two wavefunctions, the exact eigenfunction (J QCC ) of and the mean-eld approximation (J QMF ) to the ground state of the H 2 problem at R(H-H) ¼ 1.5Å. 25 The exact solution measured in the new scheme (MF-partitioning 2p) gives only one value with zero variance, while the regular schemes give three distributions for each non-commuting term.
In the approximate wavefunction case, the true variance obtained from the Hamiltonian is larger than that of the conventional approach. This is a consequence of ignoring covariances in the conventional approach. The MF partitioning 2p variance is equal to the exact one, since it is obtained from measuring a single term (the MF Hamiltonian in eqn (23)) and thus does not neglect any covariances.

LiH molecule
We will consider the LiH molecule at R(Li-H) ¼ 3.2Å, it has a 6qubit Hamiltonian containing 118 Pauli words (see Appendix C for details). This qubit Hamiltonian has 3 rd and 6 th stationary qubits, which allow one to replace the correspondingẑ operators by their eigenvalues, AE1, thus dening the different "sectors" of the original Hamiltonian. Each of these sectors is characterized by its own 4-qubit effective Hamiltonian. The ground state lies in the z 3 ¼ À1, z 6 ¼ 1 sector; the corresponding 4-qubit effective Hamiltonian (Ĥ LiH ) has 100 Pauli terms. Integrating out 3 rd and 6 th qubits can be done in the MF framework. The MF treatment ofĤ LiH is not possible without its partitioning.
Before discussing partitioning ofĤ LiH it is worth noting that there are two 2-qubit operators commuting withĤ (4) (we reenumerate qubits aer the reduction from 6 to 4 qubits in the Hamiltonian)Ô 1 (2) ¼ Àẑ 1 +ẑ 2 Àẑ 1ẑ2 (24) Unfortunately, both operators have degenerate spectra with a single non-degenerate eigenstate and three degenerate states. Moreover, these degeneracies do not satisfy the factorability condition introduced in Appendix A thus proving it impossible to nd 2-qubit unitary transformation that would factorize qubits 1 and 2 or 3 and 4. Table 1 summarizes results of partitioning forĤ LiH and variances calculated for different wavefunctions and partitioning schemes. The partitioning involving only one-qubit transformations (MF-partitioning 1p) reduces the number of QWC terms by half. Involving the two-qubit transformations at the  step before the last one in the MF partitioning reduces the number of terms to only 5 (MF-partitioning 2p), which is a vefold reduction compared to the conventional QWC form. Alternative pathways in the MP partitioning scheme related to different choices of partitioned qubits with the same value of l(k) generated not more than 15 and 9 terms for MF partitioning 1p and 2p, respectively. As discussed previously, the qubit mean-eld (J QMF ) and qubit coupled cluster (J QCC ) wavefunctions are considered, with the only difference that J QCC is a very accurate but not exact ground state wavefunction for LiH (thus there is a small but non-zero variance of theĤ LiH on J QCC ). Details on the generation of these functions can be found in ref. 25. Variances across different partitionings do not differ appreciably and the main advantage of the MFpartitioning schemes is in the reduction of the number of terms that need to be measured.

Conclusions
We have introduced and studied a new method for partitioning of the qubit Hamiltonian in the VQE approach to the electronic structure problem. The main idea of our approach is to nd Hamiltonian fragments that have eigenstates consisting of single products of one-and two-qubit wavefunctions. The most general criterion for identifying such Hamiltonian fragments was derived for the rst time. Once such fragments are found the total wavefunction of the system can be measured on a fragment Hamiltonian in a single pass of N single-qubit measurements intertwined with one-and two-qubit rotations that are dened on-the-y from results of previous qubit measurements. The main gain from such a reformulation is a decrease of separately measured Hamiltonian fragments. Indeed, illustrations on simple molecular systems (H 2 and LiH) show three-and ve-fold reductions of the number of terms that are needed to be measured with respect to the conventional scheme.
In the process of deriving our partitioning procedure, we discovered criteria for eigenstate factorability for an arbitrary Hamiltonian acting on N distinguishable particles. Our criteria involve search for few-body operators commuting with the Hamiltonian of interest. Even though the criteria for factorability are exact, realistic molecular Hamiltonians do not satisfy them in general. Therefore, we needed to introduce a heuristic partitioning procedure (greedy algorithm) that splits the system Hamiltonian to fragments that have factorable eigenstates. Even though the procedure does not guarantee the absolutely optimal partitioning to the smallest number of terms, it does not produce more terms than the number of qubit-wise commuting sub-sets.
Interestingly, when one is restricted with single-qubit measurements, the commutation property of two multi-qubit operatorsÂ andB has nothing to do with the ability to measure them together (see Table 2). This seeming contradiction with the laws of quantum mechanics arises purely from a hardware restriction that one can measure a single qubit at a time. On the other hand, qubit-wise commutativity is still a sufficient but not necessary condition for single-qubit measurability. Removing the single-qubit measurement restriction in the near future will not make our scheme obsolete but rather would allow us to skip the single-particle level. For example, if two-qubit measurements will be available, one can look for two-qubit operators commuting with the Hamiltonian and integrate out pairs of qubits to dene next measurable twoqubit operators.
The current approach can address difficulties arising in the exploration of the excited state via minimization of variance One of the largest practical difficulties is in an increasing number of terms that are required to be measured in eqn (26). Combining some of these terms using the current methodology can reduce the number of needed measurements.
A similar problem with a growing number of terms arises if one would like to obtain the true quantum uncertainty of the measurements for a partitioned Hamiltonian, it requires measuring all covariances between all parts. Ignoring covariances by assuming measurement independence can lead to incorrect estimation of the true uncertainty, both under-and over-estimation are possible.
From the hardware standpoint, the new scheme requires modication of the single-qubit measurement protocol, where measurement results for some qubits will dene unitary rotations of other qubits before their measurement, so-called feedforward measurement. This type of measurement has already been implemented in quantum computers based on superconducting 27 and photonic 19,28,29 qubit architectures in the context of measurement-based quantum computing. 15,16 Thus we hope that the new method will become the method of choice for quantum chemistry on a quantum computer in the near future.

Appendix A: factorization conditions for the Hamiltonian eigenstates
Here we prove that the condition given in the main text for a Nqubit Hamiltonian to be in the MF class is actually a necessary and sufficient condition, and hence is a criterion. We will split the proof into two parts: (1) If the Hamiltonian has N oneparticle operators satisfying the reduction chain, its eigenfunctions can be written as products (sufficiency); (2) if all the Hamiltonian eigenfunctions are in a product form then it will have N commuting one-particle operators dened by the reduction scheme (necessity).
(1) Proof of sufficiency: if there exist N one-particle operators commuting with a set of reduced Hamiltonians it is straightforward to check that a product of eigenstates of these operators is an eigenstate of the Hamiltonian. Note that any nontrivial one-qubit operator has a non-degenerate spectrum, therefore, there is no degree of freedom related to rotation within a degenerate subspace. The choice of the rst eigenstate of the rst operator (Ô 1 ) can dene the form of next one-particle operators and their eigenstates.
(2) Proof of necessity: for the N-particle eigenstate J(1,.N) to have a product form it is necessary for the Hamiltonian to have eigenstates of the f 1 (1)F(2,.N) form, where f 1 (1) and F(2,.N) are some arbitrary functions from Hilbert spaces of qubit 1 and N À 1 qubits. The latter form is an eigenstate of an operator of the formÔ 1 5 I NÀ1 , where I NÀ1 is an identity operator andÔ 1 is an operator for which f 1 (1) is an eigenfunction. Then, if the Hamiltonian andÔ 1 5 I NÀ1 share the eigenstates they must commute. This commutation is equivalent to [Ĥ,Ô 1 ] ¼ 0. The same logic can be applied to F(2,.N) because the next necessary condition for the total eigenfunction of the Hamiltonian to be in a product form is that F(2,.N) ¼ f 2 (2)F(3,.N), this gives rise to another commuting operatorÔ 2 whose eigenfunction is f 2 . It is important to note though that O 2 does not need to commute withĤ but only with its reduced version H NÀ1 ¼ hf 1 |Ĥ|f 1 i. This chain can be continued until we reach the end of the variable list.

Many-particle commuting operator extension
Similarly if we can nd an M-particle operatorÔ commuting withĤ then, because of the theorem on commuting operators, there is a common set of eigenfunctions. With multi-qubit operators one needs to be careful because they can have a degenerate spectrum. In the case of the non-degenerate spectrum ofÔ the common eigenstates have the factorized form J(1,.N) ¼ F(1,.M)c(M + 1,.N), which serves as a solid ground for the discussion in the main text. In the degenerate case, the most general form of a common eigenstate is Jð1; .NÞ ¼ P or not? To answer this question one needs to construct a reduced matrix operator within the degenerate subspace where h IJ are elements of a constant matrix andĤ (NÀM) is a single reduced operator acting on N À M variables. Note that for doing this analysis one needs to be able to obtain only eigenstates ofÔ. This is presumably an easier procedure since M < N.
Thus, in the degenerate case, having a product form is not guaranteed and therefore, one may be able to obtain the unitary transformation unentangling qubits only in the described two cases. Yet, nding the commuting operatorÔ is a necessary condition for the existence of an unentangling unitary transformation.
The complexity of a single step of the MF partitioning procedure is polynomial with the number of qubits. In each step we need to evaluate the l(k) function for each of the qubits present. Evaluation of the l(k) function requires building the corresponding overlap matrix S k , which involves inner products between columns of A k matrices. Since the length of A k columns ( h x,y,z ) scales as N 4 at most (this is the scaling of the total number of terms in the Hamiltonian), the construction of S k scales as N 4 as well. Thus funding l(k) functions for all qubits in general has O(N 5 ) scaling.
Notes and references ‡ Here, we use the notation |AE s i n for the n th qubit eigenstates of a s one-particle operator with AE1 eigenvalues. § To simplify the notation we use freedom in qubit enumeration and assume that we work with the qubit enumeration that follows the described reductive sequence.