Artur F.
Izmaylov
*^{ab},
Tzu-Ching
Yen
^{ab} and
Ilya G.
Ryabinkin
^{c}
^{a}Department of Physical and Environmental Sciences, University of Toronto Scarborough, Toronto, Ontario M1C 1A4, Canada
^{b}Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ontario M5S 3H6, Canada. E-mail: artur.izmaylov@utoronto.ca
^{c}OTI Lumionics Inc., 100 College Street 351, Toronto, Ontario M5G 1L5, Canada
First published on 12th February 2019
Current implementations of the Variational Quantum Eigensolver (VQE) technique for solving the electronic structure problem involve splitting the system qubit Hamiltonian into parts whose elements commute within their single qubit subspaces. The number of such parts rapidly grows with the size of the molecule. This increases the computational cost and can increase uncertainty in the measurement of the energy expectation value because elements from different parts need to be measured independently. To address this problem we introduce a more efficient partitioning of the qubit Hamiltonian using fewer parts that need to be measured separately. The new partitioning scheme is based on two ideas: (1) grouping terms into parts whose eigenstates have a single-qubit product structure, and (2) devising multi-qubit unitary transformations for the Hamiltonian or its parts to produce less entangled operators. The first condition allows the new parts to be measured in the number of involved qubit consequential one-particle measurements. Advantages of the new partitioning scheme resulting in severalfold reduction of separately measured terms are illustrated with regard to the H_{2} and LiH problems.
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|Ψ_{0}〉 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 that Ĥ_{q} = Â + , where Â and are measurable components of Ĥ_{q} and [Â, ] ≠ 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 of Ĥ_{q}, U|Ψ_{0}〉, measuring it on Â or would not give a certain result because Â and do not commute with Ĥ_{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) of Ĥ_{q} (true uncertainty) and by the sum of variances for Â and is neglect of covariances (Cov)
Var(Ĥ_{q}) = Var(Â) + Var() + Cov(Â, ) + Cov(, Â), | (1) |
Var(Â) = 〈Â^{2}〉 − 〈Â〉^{2}, | (2) |
Cov(Â, ) = 〈Â〉 − 〈Â〉〈〉. | (3) |
Thus, even though the Ĥ_{q} average is equal to averages of Â and , the true quantum uncertainty of Ĥ_{q} is overestimated by a sum of variances for Â and . Moreover, the number of measurements to sample Â and is twice as many as that for Ĥ_{q} if the eigenstate nature of U|Ψ_{0}〉 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 artificial example, where the Hamiltonian variance is measured as n independent measurements of its Ĥ_{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 in Ĥ_{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 the Ĥ_{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-field 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-field terms have not been found in the literature and have been derived in this work for the first time. To decrease the number of these terms even further, we augment the mean-field 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 define what single-qubit operators to measure next.
(4) |
(5) |
_{I} = ⋯_{2}^{(I)}_{1}^{(I)}, | (6) |
(7) |
(8) |
Partitioning of the H_{q} in eqn (7) allows one to measure all Pauli words within each Â_{n} term in a single set of N one-qubit measurements. For every qubit, it is known from the form of Â_{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 many Â_{n} terms separately.
A natural extension of partitioning in eqn (7) is to sum more general terms
(9) |
(10) |
Ĥ_{MF}(1,2) = _{2} + ẑ_{1}ŷ_{2}, | (11) |
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 = 1}§ that commute [Ô_{k}, Ĥ_{N−k+1}] = 0 with the system of N Hamiltonians {Ĥ_{N−k+1}}^{N}_{k = 1} constructed in the following way that we will refer as a reductive chain:
(12) |
A general procedure to determine whether a particular qubit Hamiltonian Ĥ is in the MF class or not requires finding all N one-particle operators Ô_{k}. The procedure starts with a check whether there is at least one qubit k for which
[Ĥ, (a_{k} + bŷ_{k} + cẑ_{k})] = 0 | (13) |
Ĥ_{MF} = (_{2} + ŷ_{2})|ϕ_{1}^{+}〉〈ϕ_{1}^{+}| + (_{2} − ŷ_{2})|ϕ_{1}^{−}〉〈ϕ_{1}^{−}| | (14) |
Ĥ_{MF} = [(_{2} + ŷ_{2})(1 + ẑ_{1}) + (_{2} − ŷ_{2})(1 − ẑ_{1})]/2, | (15) |
Therefore, the scheme for measuring the Ĥ_{MF} will be as shown in Fig. 1. Note that no matter how entangled the initial wavefunction is, measuring Ĥ_{MF} does not require measuring _{2} and ẑ_{1}ŷ_{2} separately as was done in the regular VQE scheme.
Fig. 1 Measurement where the second qubit is rotated by U_{2} depending on the result of the first qubit measurement. |
In practice, qubit-wise measurements using previous measurement results to define 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 μs (ref. 20) (150 ns (ref. 19)) and 40 μs (ref. 21) (100 ms (ref. 22)), respectively.
Our scheme uses ranking of all qubits k = 1,…,N based on a geometrical characteristic l(k), which is defined as follows. For an arbitrary qubit k, the total Hamiltonian can be written as
Ĥ = ĥ_{x}_{k} + ĥ_{y}ŷ_{k} + ĥ_{z}ẑ_{k} + ĥ_{e} | (16) |
l(k) allows one to answer a question on whether there is a transformation involving only the k^{th} qubit that can present Ĥ in one of the two forms:
Ĥ = ĥÔ_{k} + ĥ_{e}, | (17) |
(18) |
The question about possible compactification of the k^{th} qubit dependence in the Hamiltonian has a simple geometric interpretation in terms of arrangement of the three vectors _{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) = 2. Once all of such qubits have been considered, the MF partitioning of l(k) = 1 qubits begins. For l(k) = 1, the Hamiltonian can be split for any of such qubits into two parts: and . In both parts the k^{th} qubit can be treated using the MF treatment, which allows one to continue the consideration for ĥ′, ĥ′′ and ĥ_{e}. Finally, if only qubits with l(k) = 0 are left, then Ĥ needs to be partitioned to three Hamiltonians Ĥ^{(1)} = ĥ_{x}_{k}, Ĥ^{(2)} = ĥ_{y}ŷ_{k}, and Ĥ^{(3)} = ĥ_{z}ẑ_{k} + ĥ_{e}, where at least the k^{th} qubit can be treated using MF. After 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 find locally the most optimal reduction, a qubit with the highest l(k). The reduction is only possible if there is linear dependency between complementary vectors . 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) first is justified 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 Hamiltonians ĥ 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.
Let us consider an example where an N-qubit non-MF Hamiltonian Ĥ has a two-qubit operator Ô^{(2)}(1,2) commuting with it (without loss of generality we can assume that Ô^{(2)} acts on the first two qubits). Then, under certain conditions detailed in Appendix A, Ĥ allows for its eigenstates Ψ to be written as Ψ(1,…N) = Φ(1,2)ψ(3,…N), where Φ(1,2) is an eigenstate of Ô^{(2)}. One can always write Φ(1,2) = Û(1,2)ϕ_{1}(1)ϕ_{2}(2), where Û(1,2) is an operator entangling the product state ϕ_{1}(1)ϕ_{2}(2) into Φ(1,2). Using this unitary operator, one can obtain the Hamiltonian Ĥ_{12} = Û(1,2)^{†}ĤÛ(1,2) that has an eigenstate Ψ_{12}(1,…N) = ϕ_{1}(1)ϕ_{2}(2)ψ(3,…N) where qubits 1 and 2 are unentangled. Therefore, there should be one-particle operators of qubits 1 and 2 that commute with Ĥ_{12} and its MF-reduced counterpart. Finding these operators and their eigenfunctions ϕ_{1}(1) and ϕ_{2}(2) allows us to integrate out qubits 1 and 2
Ĥ_{N−2} = 〈ϕ_{1}ϕ_{2}|H_{12}|ϕ_{1}ϕ_{2}〉. | (19) |
Search for one- or multi-qubit operators commuting with Ĥ_{N−2} can be continued. The procedure to find 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 for Ĥ = Ĥ_{N} by generating the set of Hamiltonians {Ĥ_{N}, Ĥ_{N−2},…,Ĥ_{k}} using qubit unitary transformations {U(1,2), U(3,4,5),…,U(N − 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 χ(1,…N) on Ĥ, such a measurement should be substituted by the following set of conditional measurements:
Step 1: first two qubits are measured using Ĥ_{12} and the unitary transformed function |Û(1,2)^{†}χ〉 because
(20) |
Depending on the results of these measurements the operator Ĥ_{N−2} is formulated and its unitary transformation U(3,4,5) is found. U(3,4,5) gives rise to the transformed Hamiltonian Ĥ_{35} = Û(3,4,5)^{†}Ĥ_{N−2}Û(3,4,5). The wavefunction after measuring qubits 1 and 2 is denoted as |χ_{12}〉.
Step 2: qubits 3–5 are measured on Ĥ_{35} sequentially using the transformed wavefunction Û(3,4,5)^{†}|χ_{12}〉. Results of these measurements will define 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 finding corresponding multi-qubit unitary transformations, the Ĥ Hamiltonian can be measured in N single-qubit measurements.
Ĥ_{H2} = C_{0} + C_{1}ẑ_{2} + C_{2}ẑ_{3} + C_{3}ẑ_{4} + C_{4}ẑ_{1}ẑ_{3} + C_{5}ẑ_{2}ẑ_{4} + C_{6}ẑ_{3}ẑ_{4} + C_{7}ẑ_{1}ẑ_{2}ẑ_{3} + C_{8}(1 + ẑ_{1})ẑ_{2}ẑ_{3}ẑ_{4} + C_{9}ẑ_{1}ẑ_{2}ẑ_{4} + C_{10}(1 + ẑ_{1})ŷ_{2}ẑ_{3}ŷ_{4} + C_{11}(1 + ẑ_{1})_{2}ẑ_{3}_{4}. | (21) |
Ĥ_{24} = D_{0} + D_{1}ẑ_{2} + D_{2}ẑ_{4} + D_{3}ẑ_{2}ẑ_{4} + D_{4}_{2}_{4} + D_{5}ŷ_{2}ŷ_{4}, | (22) |
U(2,4)^{†}Ĥ_{24}U(2,4) = E_{0} + E_{1}ẑ_{2} + E_{2}ŷ_{2} + E_{3}ŷ_{4} + E_{4}ŷ_{2}ŷ_{4} + E_{5}ẑ_{2}ŷ_{4}, | (23) |
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 splitting Ĥ_{H2} to three groups of QWC operators, Table 1 presents variances for the Hamiltonian expectation value for two wavefunctions, the exact eigenfunction (Ψ_{QCC}) of and the mean-field approximation (Ψ_{QMF}) to the ground state of the H_{2} problem at R(H–H) = 1.5 Å.^{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.
Approach | Number of terms | Var (Ψ_{QCC}) | Var (Ψ_{QMF}) |
---|---|---|---|
a The number of terms corresponds to the number of separately measured N-qubit terms. For all partitionings, covariances have not been included in the Var estimates, which simulates practical estimation of the total variance. | |||
H _{ 2 } | |||
QWC-partitioning | 3 | 0.044 | 0.026 |
MF-partitioning 2p | 1 | 0 | 0.053 |
〈Ĥ_{H2}^{2}〉 − 〈Ĥ_{H2}〉^{2} | 1 | 0 | 0.053 |
LiH | |||
QWC-partitioning | 25 | 0.043 | 0.037 |
MF-partitioning 1p | 13 | 0.029 | 0.036 |
MF-partitioning 2p | 5 | 0.030 | 0.038 |
〈Ĥ_{LiH}^{2}〉 − 〈Ĥ_{LiH}〉^{2} | 1 | 5.6 × 10^{−4} | 0.027 |
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.
Before discussing partitioning of Ĥ_{LiH} it is worth noting that there are two 2-qubit operators commuting with Ĥ^{(4)} (we re-enumerate qubits after the reduction from 6 to 4 qubits in the Hamiltonian)
Ô_{1}^{(2)} = −ẑ_{1} + ẑ_{2} − ẑ_{1}ẑ_{2} | (24) |
Ô_{2}^{(2)} = −ẑ_{3} + ẑ_{4} + ẑ_{3}ẑ_{4}. | (25) |
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 find 2-qubit unitary transformation that would factorize qubits 1 and 2 or 3 and 4.
Table 1 summarizes results of partitioning for Ĥ_{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 fivefold 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-field (Ψ_{QMF}) and qubit coupled cluster (Ψ_{QCC}) wavefunctions are considered, with the only difference that Ψ_{QCC} is a very accurate but not exact ground state wavefunction for LiH (thus there is a small but non-zero variance of the Ĥ_{LiH} on Ψ_{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 MF-partitioning schemes is in the reduction of the number of terms that need to be measured.
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 operators Â and 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 define next measurable two-qubit operators.
Â | [Â, ] | SQM of (Â + ) | |
---|---|---|---|
ẑ _{1} ẑ _{2} | ẑ _{2} ẑ _{3} | 0 | Yes |
ẑ _{1} ẑ _{2} | _{1} _{2} | 0 | No |
ẑ _{1} ẑ _{3} | _{1} ẑ _{2} | ≠0 | Yes |
ẑ _{1} ẑ _{2} | _{1} ŷ _{2} | ≠0 | No |
The current approach can address difficulties arising in the exploration of the excited state via minimization of variance
(26) |
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 modification of the single-qubit measurement protocol, where measurement results for some qubits will define 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.
(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 first eigenstate of the first operator (Ô_{1}) can define the form of next one-particle operators and their eigenstates.
(2) Proof of necessity: for the N-particle eigenstate Ψ(1,…N) to have a product form it is necessary for the Hamiltonian to have eigenstates of the ϕ_{1}(1)Φ(2,…N) form, where ϕ_{1}(1) and Φ(2,…N) are some arbitrary functions from Hilbert spaces of qubit 1 and N − 1 qubits. The latter form is an eigenstate of an operator of the form Ô_{1} ⊗ I_{N−1}, where I_{N−1} is an identity operator and Ô_{1} is an operator for which ϕ_{1}(1) is an eigenfunction. Then, if the Hamiltonian and Ô_{1} ⊗ I_{N−1} share the eigenstates they must commute. This commutation is equivalent to [Ĥ, Ô_{1}] = 0. The same logic can be applied to Φ(2,…N) because the next necessary condition for the total eigenfunction of the Hamiltonian to be in a product form is that Φ(2,…N) = ϕ_{2}(2)(3,…N), this gives rise to another commuting operator Ô_{2} whose eigenfunction is ϕ_{2}. It is important to note though that Ô_{2} does not need to commute with Ĥ but only with its reduced version H_{N−1} = 〈ϕ_{1}|Ĥ|ϕ_{1}〉. This chain can be continued until we reach the end of the variable list.
Ĥ_{IJ}^{(N−M)} = 〈Φ_{I}|Ĥ|Φ_{J}〉, | (27) |
Ĥ_{IJ}^{(N−M)} = h_{IJ}Ĥ^{(N−M)}, | (28) |
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, finding the commuting operator Ô is a necessary condition for the existence of an unentangling unitary transformation.
Ĥ = 3_{1}_{2}_{3} + _{1}_{2}ŷ_{3} + 5_{1}_{2}ẑ_{3} + 5_{1}ŷ_{2}_{3} + 7_{1}ŷ_{2}ẑ_{3} + 3_{1}ẑ_{2}_{3} + _{1}ẑ_{2}ŷ_{3} + 5_{1}ẑ_{2}ẑ_{3} + 6ŷ_{1}_{2}_{3} + 2ŷ_{1}_{2}ŷ_{3} + 10ŷ_{1}_{2}ẑ_{3} + 10ŷ_{1}ŷ_{2}_{3} + 14ŷ_{1}ŷ_{2}ẑ_{3} + 6ŷ_{1}ẑ_{2}_{3} + 2ŷ_{1}ẑ_{2}ŷ_{3} + 10ŷ_{1}ẑ_{2}ẑ_{3} + 3ẑ_{1}_{2}_{3} + ẑ_{1}_{2}ŷ_{3} + 5ẑ_{1}_{2}ẑ_{3} + 5ẑ_{1}ŷ_{2}_{3} + 7ẑ_{1}ŷ_{2}ẑ_{3} + 3ẑ_{1}ẑ_{2}_{3} + ẑ_{1}ẑ_{2}ŷ_{3} + 5ẑ_{1}ẑ_{2}ẑ_{3} | (29) |
To assess whether the partitioning of Ĥ is possible based on qubit k = 1 we rewrite the Hamiltonian as
Ĥ = _{1}ĥ_{x} + ŷ_{1}ĥ_{y} + ẑ_{1}ĥ_{z}, | (30) |
ĥ_{x} = 3_{2}_{3} + _{2}ŷ_{3} + 5_{2}ẑ_{3} + 5ŷ_{2}_{3} + 7ŷ_{2}ẑ_{3} + 3ẑ_{2}_{3} + ẑ_{2}ŷ_{3} + 5ẑ_{2}ẑ_{3} | (31) |
ĥ_{y} = 6_{2}_{3} + 2_{2}ŷ_{3} + 10_{2}ẑ_{3} + 10ŷ_{2}_{3} + 14ŷ_{2}ẑ_{3} + 6ẑ_{2}_{3} + 2ẑ_{2}ŷ_{3} + 10ẑ_{2}ẑ_{3} | (32) |
ĥ_{z} = 3_{2}_{3} + _{2}ŷ_{3} + 5_{2}ẑ_{3} + 5ŷ_{2}_{3} + 7ŷ_{2}ẑ_{3} + 3ẑ_{2}_{3} + ẑ_{2}ŷ_{3} + 5ẑ_{2}ẑ_{3} | (33) |
Each ĥ_{x,y,z} is transformed into a vector. For example
(34) |
Ô_{1} = 0.408248_{1} + 0.816497ŷ_{1} + 0.408248ẑ_{1} | (35) |
ĥ(2,3) = 7.34847_{2}_{3} + 2.44949_{2}ŷ_{3} + 12.2474_{2}ẑ_{3} + 12.2474ŷ_{2}_{3} + 17.1464ŷ_{2}ẑ_{3} + 7.34847ẑ_{2}_{3} + 2.44949ẑ_{2}ŷ_{3} + 12.2474ẑ_{2}ẑ_{3} | (36) |
As the next step, we consider ĥ(2,3), it can be partitioned based on either qubit k = 2 or k = 3. Both qubits have the same values of l(k) = 1 and are in a single plane (Fig. 2b). Here, we choose arbitrarily k = 2, diagonalizing S_{2} leads to two non-zero eigenvalues (d_{1},d_{2}) and corresponding eigenvectors . Following the procedure, ĥ(2,3) decomposes to
(37) |
(38) |
ĥ′(3)=−1.08532_{3} + 2.48388ŷ_{3} + 0.467647ẑ_{3} | (39) |
(40) |
ĥ′′(3) = 16.0257_{3} + 2.41461ŷ_{3} + 24.3676ẑ_{3}. | (41) |
The single-qubit operators and their complements {ĥ′, ĥ′′} were obtained taking linear combinations of {_{2}, ŷ_{2}, ẑ_{2}} and {ĥ_{x}, ĥ_{y}, ĥ_{z}} with coefficients from the eigenvectors , respectively.
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 (_{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.
Footnotes |
† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8sc05592k |
‡ Here, we use the notation |±_{σ}〉_{n} for the n^{th} qubit eigenstates of a σ one-particle operator with ±1 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. |
This journal is © The Royal Society of Chemistry 2019 |