Article
Cite This: J. Phys. Chem. B XXXX, XXX, XXX-XXX pubs.acs.org/JPCB
Electronic Structure Calculations and the Ising Hamiltonian † † † ‡ § Rongxin Xia, Teng Bian, and Sabre Kais*, , , † Department of Physics, Purdue University, West Lafayette, Indiana 47907, United States ‡ Department of Chemistry and Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, United States § Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, United States
ABSTRACT: Obtaining exact solutions to the Schrödinger equation for atoms, molecules, and extended systems continues to be a “Holy Grail” problem which the fields of theoretical chemistry and physics have been striving to solve since inception. Recent breakthroughs have been made in the development of hardware-efficient quantum optimizers and coherent Ising machines capable of simulating hundreds of interacting spins with an Ising-type Hamiltonian. One of the most vital questions pertaining to these new devices is, “Can these machines be used to perform electronic structure calculations?” Within this work, we review the general procedure used by these devices and prove that there is an exact mapping between the electronic structure Hamiltonian and + the Ising Hamiltonian. Additionally, we provide simulation results of the transformed Ising Hamiltonian for H2 ,He2 , HeH , and LiH molecules, which match the exact numerical calculations. This demonstrates that one can map the molecular Hamiltonian to an Ising-type Hamiltonian which could easily be implemented on currently available quantum hardware. This is an early step in developing generalized methods on such devices for chemical physics.
̈ − etermining solutions to the Schrodinger equation is a H(s)=(1 s)HB + sHP (where s is a time evolution parameter, s D fundamentally challenging problem due to the exponential ∈ [0, 1]). Adiabatic evolution is governed by the Schrödinger increase in the dimensionality of the corresponding Hilbert space equation for the time-dependent Hamiltonian H(s(t)). Thus, with the number of particles in the system; this increase in permitting the system to evolve from HB to HP is a slow and computation system size requires a commensurate increase in smooth manner, such that the system remains in its ground computational resources. Modern quantum chemistryham- eigenstate during the perturbative evolution. pered by the difficulty associated with solving the Schrödinger Hitherto, the largest scale implementation of AQC is by D- equation to chemical accuracy (∼1 kcal/mol)has largely Wave Systems.16,17 In the case of the D-Wave device, the physical become an endeavor to find approximate methods and process acting as adiabatic evolution is more broadly called corrections. A few products of this effort include ab initio, quantum annealing (QA). The quantum processors manufac- density functional, density matrix, algebraic, quantum Monte − tured by D-Wave are essentially a transverse Ising model with 1 8 fi ffi ∑ Δ σi Carlo, and dimensional scaling methods. All methods devised tunable local elds and coupling coe cients: H = i i x + ∑ σi ∑ σi σj Δ to date, however, face the insurmountable challenge of i hi z + i,j Jij z z , where the parameters i , hi , and Jij are computational resource requirements as the calculation is physically tunable. The qubits (quantum bits) are connected in a extended either to higher accuracy or to larger systems. specified graph geometry; this design permits the embedding of Computational complexity in electronic structure calculations arbitrary graphs. Zoller and co-workers presented a scalable suggests that these restrictions are a result of an inherent architecture with full connectivity, yet it can only be 9−11 difficulty associated with simulating quantum systems. implemented with local interactions.18 The adiabatic evolution − ∑ σi Electronic structure algorithms developed for quantum is initialized at HB = h i x and evolves into the problem computers provide a promising new route to advance electronic Hamiltonian: H = ∑ h σi + ∑ J σi σj . This equation describes a 12−14 P i i z i,j ij z z structure calculations for large systems. Recently, there has classical Ising model whose ground state isin the worst case been an attempt at using an adiabatic quantum computing NP-complete. Therefore, any combinatorial optimization NP- model as implemented on the D-Wave machine to perform hard problem may be encoded into the parameter assignments electronic structure calculations.15 Fundamentally, the approach ({hi, Jij}) of HP and may exploit adiabatic evolution as a method behind the adiabatic quantum computing (AQC) method begins by defining a problem Hamiltonian, H , engineered to have its P Special Issue: Benjamin Widom Festschrift ground state encode the solution of a corresponding computa- tional problem. The system is initialized in the ground state of a Received: October 19, 2017 “ ” beginning Hamiltonian, HB , which is easily solved by classical Revised: November 2, 2017 methods. The system is then allowed to evolve adiabatically as Published: November 3, 2017
© XXXX American Chemical Society A DOI: 10.1021/acs.jpcb.7b10371 J. Phys. Chem. B XXXX, XXX, XXX−XXX The Journal of Physical Chemistry B Article for reaching the ground state of HP. More recently, an optically commonly used subdivisions, three- to two-body and k-body based coherent Ising machine was developed and is capable of gadgets.28 finding the ground state of an Ising Hamiltonian describing a set Here we present a universal method of mapping an n-qubit 19−21 σ σ σ of several hundred coupled spin 1/2 particles. These Hamiltonian, H, which depends on x, y, z to an rn-qubit ffi ′ σ challenging NP-hard problems are characterized by di culty in Hamiltonian, H , consisting of only products of z. In this devising a polynomial-time algorithm; therefore, solutions process, we increase the number of qubits from n to rn, where the cannot be easily found using classical numerical algorithms in a integer r plays the role of a “variational parameter” to achieve the − reasonable time for large system sizes (N).19 21 Such special desired accuracy in the final step of energy calculations. With purpose machines may help in finding the solutions to some of increasing r, which means the use of more qubits to calculate, we the most difficult computational problems faced today. can achieve more accurate results; please see the Appendix for more details. ■ RESULTS AND DISCUSSION We use two steps to complete this mapping. The first step is to The technical scheme for performing electronic structure generate a mapping between the eigenstates. In an original n- calculations on such an Ising-type machine can be summarized qubit Hilbert, H, space, a general eigenstate is given by ψ = ∑ ϕ in the following four steps: First, write down the electronic i ai i; we can map this to the space of the rn-qubit structure Hamiltonian in terms of creation and annihilation Hamiltonian, H′, which increases the space from n qubits to rn ϕ Fermionic operators, delivering the second quantization form. qubits. The mapping method is to repeat the basis ( i) bi times, fi Second, use either the Jordan−Wigner or the Bravyi−Kitaev where bi is determined by ai de ned through the relationship transformation to move from Fermionic operators to spin ≈ bSii() b ∑ 13,22 ai 2 , where i bi = r. S(bi) is a sign function associated operators. Third, reduce the spin Hamiltonian (which is ∑m bm 23 generally a k-local Hamiltonian) to a 2-local Hamiltonian. with bi,asai can be positive and negative and repeat time can only Finally, map this 2-local Hamiltonian to an Ising-type be positive. If ai is positive, we have S(bi) = 1, and if ai is negative, − Hamiltonian. we have S(bi)= 1. After this procedure, we get a new eigenstate |Φ⟩ ⊗ ⊗bi |ϕ ⟩ This general procedure begins with a second quantization i j=1 i . This procedure is represented in Figure 1. description of a Fermionic system in which N single-particle states can be either empty or occupied by a spineless Fermionic particle.4,24 One may then use the tensor product of each | ⟩ individual spin orbital, written as f 0, ..., f n to represent states in ∈ Fermionic systems where f j {0, 1} is the occupation number of orbital j. Any interaction within the Fermionic system can be expressed in terms of products of the creation and annihilation † ∈ operators aj and aj, for j {0, ..., N}. Thus, the molecular electronic Hamiltonian can be written as ̂ †††1 H =+∑∑haaij i j hijkl aaaa i j k l 2 σ 2 ff ij,,,,ijkl (1) Figure 1. Example of mapping x between di erent bases from a state 1 1 ffi |Φ⟩ = |000 ⟩ + | 010⟩ to state |000010⟩, where 0 represents spin The above coe cients hij and hijkl are the one- and two-electron 2 2 integrals which can be precomputed in a classical way used as up and 1 represents spin down. The spin operators act on the second fi inputs to the quantum simulation. The next step is to develop qubit in the original Hamiltonian basis and on the second and fth and employ a Pauli matrix representation of the creation and qubits in the mapped Hamiltonian basis. annihilation operators. This can be done by employing the − − Bravyi Kitaev transformation or the Jordan Wigner trans- The second step is the mapping between the Hamiltonians, H formation to map between the second quantization operators → H′. By mapping the eigenstate, we have a corresponding way σ σ σ 22,25 and the Pauli matrices { x, y, z}. The molecular σi σi σi 13 to map the spin operator (Ii , x , y , z) on the ith qubit in n- Hamiltonian now takes the general form qubit space to rn-qubit space by ii ij i j ijk i j k H =+hhσσσσσσ + h +... ijki i ki j ∑∑αα αβ α β ∑αβγ α β γ i 1 − σσz z i σσz − z α αβ αβγ σ → SjSk′′() ( ) σ →i SjSk′′() ( ) i, ij ijk (2) x 2 y 2 α where the index = x, y, z notes the Pauli matrix and the index i ijki i jki i σσz + z i 1 + σσz z designates the spin orbital. Now, after having developed a k-local σz → SjSk′′() ( ) I → SjSk′′() ( ) spin Hamiltonian (many-body interactions), one should use a 2 2 general procedure to reduce the Hamiltonian to a 2-local (two- (3) body interactions) spin Hamiltonian form;26,27 this is a σij means acting on the ith qubit in jth n qubits in |Ψ⟩ (or the spin requirement, since the proposed computational devices are operator σ acting on the [(j − 1)n + i] qubit). In addition, S′(j), typically limited to restricted forms of two-body interactions. S′(k) represents the sign of the jth, kth of the n qubits in the new Therefore, universal adiabatic quantum computation requires a state of the rn-qubit space. The introduction of S′(j) is the same method for approximating a quantum many-body Hamiltonian as the introduction of S(bi). Now, the newly transformed ′ σ up to an arbitrary spectral error using at most two-body Hamiltonian, H , includes only products of z and I, and is interactions. Hamiltonian gadgets, for example, offer a systematic therefore diagonal (details of the transformation are presented procedure to address this requirement. Recently, we have with examples in the Appendix). employed analytical techniques resulting in a reduction of the So far, we are able to transform our Hamiltonian to a k-local σ resource scaling as a function of spectral error for the most Hamiltonian including only products of z terms. It is then
B DOI: 10.1021/acs.jpcb.7b10371 J. Phys. Chem. B XXXX, XXX, XXX−XXX The Journal of Physical Chemistry B Article
Figure 2. Comparing the numerical results of ground state energy of the Ising Hamiltonian with the exact STO-6G calculations of the ground state of H2, + He2, HeH , and LiH molecules as one varies the internuclear distance R. straightforward to transform the Hamiltonian to 2-local by using ′σσσi ′ i j H′=∑∑hJiz+ ij z z ancilla qubits, as the following example illustrating the trans- i ij (5) formation from 3-local to 2-local Ising Hamiltonian. To estimate the efficiency of this proposed method, we here min(±=±+−−+xxx123 ) min( xxxxxxxxx 43 12 2 14 2 24 3 4 ) derive the number of qubits required by constructing the final 2- local Hamiltonian in eq 5. Suppose we start with an original xxxx1234,,,∈ {0,1} Hamiltonian in the second quantization representation contain- (4) ing n orbitals. The first step of the transformation into the Pauli 4 Here, we see how, by including x4, one can show that minimizing operator representation requires O(n) qubits and O(n ) terms. 3-local is equivalent to minimizing the sum of 2-local terms. We The second step of mapping the form containing all of the Pauli σ can apply this technique in our reduction from k-local to 2-local operators to the form containing only z requires O(rn) qubits terms, as shown in the Appendix.29 and O(2nr2n4) terms. The final step of reduction to 2-local Finally, we succeed in transforming our initial complex requires O(2nr2n7) qubits. Due to the computational costs of the electronic structure Hamiltonian from the second-quantization current method, it is feasibly useful for small and medium size form to an Ising-type Hamiltonian which can be solved using molecular systems. To push this method to a large size molecular existing quantum computing hardware.14,19,30,31 system, we propose to combine our method with a recent To illustrate the proposed method (details are in the approach by Chan and co-workers.32 Instead of encoding the Appendix), we present in Figure 2 the calculations for the wave function using n Gaussian orbitals, leading to Hamiltonians 4 hydrogen molecule (H2), the helium dimer (He2), the with O(n ) second-quantized terms, Chan et al. utilize a dual hydrogen−helium molecular cation (HeH+), and lithium form of the plane wave basis which diagonalizes the potential hydride (LiH). First, we used the Bravyi−Kitaev transformation operator, leading to a Hamiltonian representation with O(n2) and the Jordan−Wigner transformation to convert the diatomic second-quantized terms. Moreover, one can repeat our initial molecular Hamiltonian in the minimal basis set (STO-6G) to the calculations using the new procedure with O(n2) second- σ σ σ spin Hamiltonian of ( x, y, z). We then used our transformed quantized terms. This will clearly save computational resources. Hamiltonian in rn-qubit space to obtain a diagonal k-local There is of course much space for improvement, and one should σ Hamiltonian constructed of z terms. Finally, we reduced the continue to explore new ideas to further optimize the proposed locality to obtain a 2-local Ising Hamiltonian of the general form method.
C DOI: 10.1021/acs.jpcb.7b10371 J. Phys. Chem. B XXXX, XXX, XXX−XXX The Journal of Physical Chemistry B Article
Figure 3. Left: If the 2-qubit state is|ψ⟩=1 |00 ⟩+1 | 11⟩, then the 4-qubit state is |Ψ⟩ = |0011⟩ with S′(1) = S′(2) = 1, r =2,b = b = 1, and S(b )= 2 2 1 2 1 S(b ) = 1. Middle: If the 2-qubit state is |ψ⟩=1 |00 ⟩−1 | 11⟩, then the 4-qubit state is |Ψ⟩ = |0011⟩ with S′(1) = −S′(2) = 1, r =2,b = b = 1, and 2 2 2 1 2 S(b )=−S(b ) = 1. Right: If the 2-qubit state is|ψ⟩=1 |00 ⟩−1 | 11⟩, then the 8-qubit state is |Ψ⟩ = |00111111⟩ with S′(1) = S′(2) = S′(3) = −S′(4) 1 2 2 2 ′ ′ =1,r =4,b1 = b2 = 1, and S(b1)=S(b2) = 1. (S (3) and S (4) cancel out, and thus, S(b2) = 1). ■ CONCLUSION Notation 1: We designate the ith qubit within the kth n-qubit |Ψ⟩ We have developed a general procedure of mapping the original subspace of the rn state space of as ik. Notation 2: We use b(j) to represent the jth n-qubit in the complex electronic structure Hamiltonian to a diagonal |Ψ⟩ |ϕ ⟩ Hamiltonian in terms of a σz basis, which is experimentally space of , which is in the basis b(j) . implementable via today’s Ising-type machines. Presently, bi may only be non-negative, yet ai may be positive or negative. implementation of σz, σx, and σy is not possible on current For a negative ai, we will introduce a function S(bi) containing Ising machines nor while using quantum annealing as on the D- the sign information to account for bi being non-negative. The Wave machine. Our generalized procedure provides an mapping is described by the following rules: • ffi opportunity to begin to utilize the current generation of S(bi) is the sign associated with the bi coe cient which is quantum computing devicesof about 2000 connected negative if ai is negative and is positive if ai is positive. spinsfor important computational tasks in chemical physics. • For the rn-qubit state |Ψ⟩, we can use a function S′(i)to Incidentally, recent experimental results presented by the IBM record the sign associated with each n qubits in the rn- group on few-electron diatomic molecules have shown that a qubit space. S′(i) represents the sign of the ith n qubits in ffi |∑|ϕ ⟩ |ϕ ⟩ ′ | hardware-e cient optimizer implemented on a 6-qubit super- the rn-qubit space. Thus, bi = b(j) = i S (j) and S(bi)is conducting quantum processor is capable of reproducing the ∑|ϕ ⟩ |ϕ ⟩ ′ 30 the sign of b(j) = i S (j). potential energy surface for such systems. The development of • efficient quantum hardware and the possibility of mapping the As before, bi is the integer which approximates ai by bS() b bSii() b ≈ ii electronic structure problem into an Ising-type Hamiltonian may .Ifr is large enough, we can just view ai 2 . ∑ b 2 ∑ b grant efficient ways to obtain exact solutions to the Schrödinger m m m m • |Ψ⟩ |Ψ⟩ ⊗ ⊗bi |ϕ ⟩ ∑r | ′ | equation, thus providing a solution path to one of the most can be written as = i j=1 i and r = i=1 S (i) . daunting computational problems in both chemistry and physics. Figure 3 shows the details about the mapping. Moreover, this study might open the possibility to connect with Theorem 1: With the mapping between |ψ⟩ and |Ψ⟩ as the rich field of statistical mechanics of he Ising model describing fi 33−37 described above, we can nd a mapping between the phase transitions. Hamiltonian in the space of |ψ⟩ to the space of |Ψ⟩. i + σσj ik Theorem 2: ⟨ϕ | ⊗ I |ϕ ⟩ is equal to ⟨Ψ| ∏ 1 z z |Ψ⟩, ■ APPENDIX b(j) i i b(k) i 2 i σσj ik |ψ⟩ 1 + z z Here, we present a procedure to construct a diagonal which means Ii in the space of can be mapped to in the Hamiltonian with a minimum eigenvalue corresponding to the 2 space of |Ψ⟩. ground state of a given initial Hermitian Hamiltonian. |ψ⟩ | |ψ⟩ Proof: Clearly, Ii in the space of is to observe if ith digits of For a given initial Hermitian Hamiltonian, an eigenstate ϕ ⟩ |ϕ ⟩ |ϕ ⟩ |ψ⟩ ∑ |ϕ ⟩ b(j) and b(k) are the same or not. If they are the same, it yields can be expanded in a basis set i as = i ai i . This basis set i + σσj ik consists of different combinations of spin-up and -down qubits. 1; otherwise, 0. On the other hand, 1 z z in the space of |Ψ⟩ is ffi 2 First, we will assume that all expansion coe cients, ai, are |ϕ ⟩ to check if the ith digits of the jth n-qubit subspace ( b(j) ) and nonnegative and will map the state to a new state |Ψ⟩ according |ϕ ⟩ the kth n-qubit subspace ( b(k) ) are the same or not. If they are i to the following rules: j ik 1 − σσz z • |Ψ⟩ ⊗ ⊗bi |ϕ ⟩ ∑ the same, it yields 1; otherwise, 0. (For , we omit the |Ψ⟩ can be written as = i j=1 i and r = i bi. 2 • If the original state |ψ⟩ exists within an n-qubit subspace, operators for other digits which are the identity I.) the new state |Ψ⟩ should be in an rn-qubit space, where r is Thus, we get the number of times we must replicate the n qubits to 1 + σσijki ⟨ϕϕ|⊗I | ⟩=⟨Ψ|∏ z z |Ψ⟩ achieve an arbitrary designated accuracy. bj()i bk ( ) 2 • |ψ⟩ i (6) The number of times, bi, we repeat the basis in an rn- If and only if all digits of |ϕ ⟩ and |ϕ ⟩ are the same, the left |Ψ⟩ bi b(j) b(k) qubit state, , approximates ai by .Ifr is large ∑ b 2 and right results are equal to 1; otherwise, they are equal to 0. m m ⟨ϕ | ⊗ ⊗ σm ⊗ |ϕ ⟩ Theorem 3: b(j) i
D DOI: 10.1021/acs.jpcb.7b10371 J. Phys. Chem. B XXXX, XXX, XXX−XXX The Journal of Physical Chemistry B Article
Chart 1
σi |ψ⟩ jkjkr≠≤,, Proof: Clearly, x in the space of is to check if the ith digits |ϕ ⟩ |ϕ ⟩ of b(j) and b(k) are the same or not. If they are the same, it ⟨Ψ|∑ HSjSk(,jk ) ′() ′ ( ) |Ψ⟩ i − σσj ik jk, yields 0; otherwise, 1. Similarly, 1 z z in the space |Ψ⟩ is to 2 jkjkr≠≤,, |ϕ ⟩ verify if the ith digits of the jth n-qubit subspace ( b(j) ) and the =⟨||⟩′′∑ ϕϕHSjSk() ( ) |ϕ ⟩ bj() bk ( ) kth n-qubit subspace ( b(k) ) are identical. If they are the same, it jk, ij i − σσk n gives 0; otherwise, 1. (For 1 z z , we omit operators for other jkjk≠≤,, 2 2 =⟨||⟩ϕϕ digits which are the identity I.) ∑ bSjjkk() b b S ( b ) jk H Thus, we get jk, jkjk≠≤,, 2n ⟨ϕσϕ|⊗II ⊗m ⊗ | ⟩ 2 bj()im<>iximi bk ( ) =⟨||⟩∑∑baaHm jkϕϕjk m jk, (11) 1 + σσijki 1 + σm j σm k =⟨Ψ|∏ z z × z z × Also, in the same basis, we have im< 2 2
ijki ij ij ij ij − σσ 1 −−σσz z σz σz 1 z z XY(,)jj ==00(,)jj ==i ∏ |Ψ⟩ i 2 i 2 > 2 ij ij ij ij im (7) σσ++1 σσ ZII(,)jj ==z z σij (,)jj =z z = i i 2 z i 2 (12) − σσj ik σi and 1 z z have the same function in different space to check x 2 |ϕ ⟩ |ϕ ⟩ Thus, as before, we can also get the digits of b(j) and b(j) . i σσik − j jr≤ Also, σi and i z z have the same function in different spaces. y 2 ⟨Ψ|∑ HSjSj ′() ′ () |Ψ⟩ |ϕ ⟩ | (,)jj These operators are used to check the ith digits of b(j) and j i σσj + ik ≤ ϕ ⟩. Also, σi and z z have the same function in different jr b(j) z 2 =⟨||⟩′′ϕϕ |ϕ ⟩ |ϕ ⟩ ∑ bj()HSjSj bj () () () spaces to check the ith digits of b(j) and b(j) . This can be j easily verified by the above discussion. j≤2n Theorem 4: Any Hermitian Hamiltonian in the space of |ψ⟩ =⟨||⟩2 ϕϕ can be written in the form of Pauli and identity matrices, which ∑ bHj jj can be mapped to the space of |Ψ⟩ as described above. j n Notation 3: We denote the mapping between the jth n-qubit j≤2 i 2 2 − σσj ik =⟨||⟩∑∑baHϕϕ subspace and the kth n-qubit subspace in |Ψ⟩, 1 z z as X(j,k), m j jj 2 i m j (13) i ij i ik j ij i σσ+ k σσ− z + σσk z z as Z(j,k), i z as Y(j,k), and 1 z z as I(j,k). 2 i 2 i 2 i Combining the two together, we can get Proof: n If H can be written as jk, ≤≤ r jk,2 a b c ⟨Ψ|∑∑∑HSjSk ′() ′ ( ) |Ψ⟩ = b2 aaH ⟨ϕϕ | | ⟩ H =⊗a σσσx ⊗b y ⊗c z ⊗d Id (8) (,jk ) m jk jk jk, m jk, ′ We can write the mapped H(j,k) as (14) |Ψ⟩ H′ = XYZI(,jk ) (,jk ) (,jk ) (,jk ) We construct a matrix, C, in the space of , which has elements (,jk ) ∏∏∏∏a b c d ∑ 2 a b c d (9) m bm as 2 fi ⎛ 1 ± σk ⎞ It can be veri ed following the rules above that • ⎜⎟ni z CSi= ∑∑± ∏ = ′() ⎝ i ()k 1i 2 ⎠ ⟨Ψ| ′ |Ψ⟩ = ⟨ϕϕ | | ⟩ HH(,jk ) bj() bk ( ) (10) • ∑± over all combinations of positive and negative signs of Thus, if we add the sign functions S′(j) and S′(k), we achieve each digit in each ith of the n-qubit in space |Ψ⟩.
E DOI: 10.1021/acs.jpcb.7b10371 J. Phys. Chem. B XXXX, XXX, XXX−XXX The Journal of Physical Chemistry B Article
• |Ψ⟩ discuss the entanglement for H molecule.38 The general ∑i over all n-qubit collection in to check whether each 2 n qubits is in a certain state. Hamiltonian for such a system is given by • ∏ni |Ψ⟩ k=1 over each qubits of ith n qubits in . i JJ12 12 1 2 ± σk H =−+−−−−(1γσσ ) xx(1 γσσ ) yyBB σ z σ z • 1 z is to check whether kth qubits of ith n-qubit subspace 2 2 2 (16) + σk is in a certain state. 1 z is 1 when the kth qubits are 2 where γ is the degree of anisotropy. | ⟩ ± present in the basis 0 or 0 otherwise. is to go over In the {|00⟩, |10⟩, |01⟩, |11⟩} basis, the eigenvectors can be ∑ combination by ±. written as (here we just use the eigenvectors to show how we So far, we have established a mapping between |ψ⟩ and |Ψ⟩. The map the Hamiltonian, but in an actual calculation, we do not |ψ⟩ ∑j,k≤r ′ Hamiltonian H in the space of and (j,k) H(j,k) in the space of know the eigenvectors) |Ψ⟩ ∑ 2 . Also, we have constructed a matrix C to compute m bm corresponding to |Ψ⟩. Thus, we have the final results: 1 |⟩=ξ1 (10 | ⟩+| 01) ⟩ jk, ≤≤ r jk,2n 2 ⟨Ψ|∑∑∑HSjSk ′() ′ ( ) |Ψ⟩ = b2 aaH ⟨ϕϕ | | ⟩ 1 (,jk ) m jk jk |⟩=ξ (10 | ⟩−|⟩ 01) jk, m jk, 2 2 (15) α − α + ∑j,k≤r ′ 2BB2 Here, we present an algorithm combining (j,k) H(j,k) and C to |⟩=ξ3 |⟩+11 |⟩00 calculate the ground state of the initial Hamiltonian H. 2α 2α |ψ⟩ λ′ Notation 4: We mark the eigenvalue of for H as . α + 2BBα − 2 According to the relationship above, the eigenvalue of |Ψ⟩ for H′ |⟩=ξ4 |⟩−11 |⟩00 ∑ 2λ′ λ 2α 2α (17) is m bm . Thus, if we choose a and construct a Hamiltonian, H′ − λC. The eigenvalue of |Ψ⟩ for H′ − λC is ∑ b 2(λ′ − λ). m m 22 Theorem 5: The algorithm shown in Chart 1 converges to the where α =+4BJγ. fi |ψ⟩ |ξ ⟩ |Ψ⟩ | ⟩ ⊗ | ⟩ minimum eigenvalue of H by nite iterations. If we set r = 2, for example, if = 1 , = 10 01 with ′ ′ |ψ⟩ |ξ ⟩ |Ψ⟩ | ⟩ ⊗ | ⟩ ′ Proof: S (1) = 1 and S (2) = 1. If = 2 , = 10 01 with S (1) Monotonic Decreasing If we can find an eigenstate |Ψ⟩ of H′ = −1 and S′(2) = 1. − λ ∑ 2 λ′ − λ ∑ 2 ≥ C with eigenvalue m bm ( ) < 0. Because m bm 0, Abiding by the previous mapping, the mapped Hamiltonian H′ we get λ′ − λ < 0. This means we can find an eigenstate |ψ⟩ of H and matrix C can be written as (this is illustrated in Figure 4) with an eigenvalue λ′ and λ′ − λ < 0. Thus, each time λ decreases 12 monotonically. HBB(1,1)′ =−σσzz − The minimum eigenvalue: Here we prove that the minimum 34 eigenvalue of H is achievable and the loop will converge when we HBB(2,2)′ =−σσzz − obtain the minimum eigenvalue. According to Monotonic 13 24 Decreasing, each time the eigenvalue we get will decrease. J 1 −−σσzz1 σσ z z fi H(1,2)′ =−(1 +γ ) Because we have a nite number of eigenvalues, this means we 2 2 2 will finally come to the minimum eigenvalue. Also, if we set λ to J σσσσ1342−− ′ − λ ∑ 2 λ′ − λ ≥ −−γ zzzz be the minimum eigenvalue of H, H C = m bm ( ) 0 (1 ) ∑ 2 ≥ λ′ − λ ≥ λ′ 2 22 because m bm 0 and 0. Also, if is just the minimum σσ13++1 σσ 24 eigenvalue, we get λ′ − λ = 0 and the loop stops. − B zz zz Thus, we prove that the eigenvalue decreases and finally 2 2 σσ24++1 σσ 13 converges to the minimum eigenvalue of H. − B zz zz Theorem 6: To account for the sign, we just need to set i from 2 2 ⎢ ⎥ 0to⎣ r ⎦ and set signs of the first ith n-qubit to be negative and the HH′ = ′ 2 (2,1) (1,2) (18) others to be positive in |Ψ⟩. Proof: If we have n qubits in the |Ψ⟩ space with negative sign, ⎛ 12 12⎞2 where |Ψ⟩ has total inqubits with negative sign. If these n qubits 1 ++σσzz1 1 ++ σσ zz1 CS= ⎜ ′+(1) S′(2)⎟ are not in the first inqubits, we can rearrange them to the first in ⎝ 2 2 2 2 ⎠ qubits by exchanging them with n qubits in the first inqubits ⎛ ⎞2 which have positive sign. Thus, all combinations can be reduced 1 +−σσ121 1 +− σσ 121 + ⎜ zzSS′+(1) zz′(2)⎟ to the combination stated in Theorem 6. ⎝ 2 2 2 2 ⎠ Thus, we have established a transformation from an initial ⎛ ⎞2 Hermitian Hamiltonian to a diagonal Hamiltonian and presented 1 −+σσ121 1 −+ σσ 121 an algorithm to calculate the minimum eigenvalue of the initial + ⎜ zzSS′+(1) zz′(2)⎟ ⎝ ⎠ Hamiltonian using the diagonal Hamiltonian. 2 2 2 2 Example ⎛ 12 12⎞2 1 −−σσzz1 1 −− σσ zz1 To illustrate the above procedure, we give details of the + ⎜ SS′+(1) ′(2)⎟ ⎝ 2 2 2 2 ⎠ transformation for the simple model of two spin-1 electrons with 2 (19) an exchange coupling constant J in an effective transverse magnetic field of strength B. This simple model has been used to If S′(1) = S′(2) = 1, we have
F DOI: 10.1021/acs.jpcb.7b10371 J. Phys. Chem. B XXXX, XXX, XXX−XXX The Journal of Physical Chemistry B Article
Figure 4. Mapped Hamiltonian −+J (1γσσ ) 12between different 2 xx bases.
jk,2≤
H′= ∑ HSjSk(,′jk ) ′′() ( ) jk, = HHHH′ + ′ + ′ + ′ (1,1) (2,2) (1,2) (2,1) (20) Figure 5. Comparing the ground state energy from exact (atomic units) We can write matrix C as of the original Hamiltonian H, eq 16, as a function of the internuclear distance, R (solid line), with the results of the transformed Hamiltonian ⎡4000⎤ H′, eqs 20 and 21. ⎢ ⎥ ⎢0200⎥ C = ⎢0020⎥ ⎣⎢ ⎦⎥ 0004 Thus, the 3-local x1x2x3 can be transformed to 2-local by setting x4 = x1x2: If −S′(1) = S′(2) = 1, we have
jk,2≤ min(xxx123 )=+−−+ min( xx 43 xx 12 2 xx 14 2 xx 24 3 x 4 ) H′= ∑ HSjSk(,′jk ) ′′() ( ) xxxx1234,,,∈ {0,1} (24) jk, = HHHH′ + ′ − ′ − ′ (1,1) (2,2) (1,2) (2,1) (21) min(−=−+−−+xxx123 ) min( xxxxxxxxx 43 12 2 14 2 24 3 4 )
We can write C in the matrix format: xxxx1234,,,∈ {0,1} (25) ⎡0000⎤ ⎢ ⎥ We can prove that min(x1x2x3 + f(x)=g1(x)) = min(x4x3 + x1x2 − − ⎢0200⎥ 2x1x4 2x2x4 +3x4 + f(x)=g2(x, x4)), where f(x) is the C = ⎢ ⎥ polynomial of all variables (including x1, x2, x3, and other 0020 ′ ′ ⎣⎢ ⎦⎥ variables, excluding x4). If there exists x , this makes g1(x ) be the 0000 ′ ′ minimum, and we can always make g2(x , x4)=g1(x )by ″ ″ For −S′(1) = S′(2) = 1, we here display a procedural choosing x4 = x1x2. Then, if there exists x , this makes g2(x , x4) ″ ≤ ″ representation of our algorithm (we set B = 0.001, J = −0.1, be the minimum, and then g1(x ) g2(x ): γ ″ ″ − − and = 0): 1. If x4 = x1x2, g1(x )=x1x2x3 + f(x )=x4x3 + x1x2 2x1x4 ″ ″ 1. First, when we choose λ = 100, we get that the minimum 2x2x4 +3x4 + f(x )=g2(x ,x4). ′ − − |Ψ⟩ | ⟩ ≠ ″ − − eigenvalue of H 100C is 400 with = 0000 . Thus, 2. If x4 x1x2, g2(x , x4)=x4x3 + x1x2 2x1x4 2x2x4 +3x4 + ∑ 2 λ′ ″ ≥ ″ ≥ ″ ″ we get m bm = 4 and =0. f(x ) x4x3 +1+f(x ) x1x2x3 + f(x )=g1(x ) because λ − ≥− 2. When we set = 0, we get that the minimum eigenvalue of x4x3 x1x2x3 1. ′ − |Ψ⟩ | ⟩ ∑ 2 H +0C is 0.2 with = 0110 . Thus, we get m bm = ′ ′ ≥ ″ ≥ ″ ≥ ′ Thus, we have g1(x )=g2(x , x4) g2(x , x4) g1(x ) g1(x ). 2 and λ′ = −0.1. ′ ′ ″ ″ λ − Thus, we have g1(x )=g2(x , x4)=g2(x , x4)=g1(x ), or all x 3. When we set = 0.1, we get that the minimum makes g (x) minimum would also makes g (x,x ) minimum and ′ |Ψ⟩ | ⟩ 1 2 4 eigenvalue of H + 0.1C is 0 with = 0110 . We stop vice versa. − here and get that the minimum eigenvalue of H is 0.1. Thus, we obtain Here we present the result of mapping the above Hamiltonian, eq − 38 ⎛ n ⎞ ⎛ 12 n i − 5/2 2R γ 1 ++σσ1 1 + ∏ σz 12, with B = 0.001, J = 0.821R e , and =0 in Figure 5. min⎜σσ12∏ σi⎟ = min⎜ 8 zz i=3 ⎝ zz z⎠ ⎝ Reduce Locality of the Transformed Hamiltonian i=3 2 2 2 ′ 12 n i 12 1 n i 2 n i ⎞ Here we present the procedure to reduce the locality of H from a 1 +++σzz σ∏ σσσσz ++zz z∏ σσz + z ∏ σz − 8 i===3 i 3 i 3 ⎟ k-local to a 2-local Ising-type Hamiltonian. 8 ⎠ For x, y, z ∈ {0, 1},29 ⎛ n n+1 n ⎜ i 12n+ 1 i 1 i xy=−−+= zif xy 2 xz 2 yz 3 z 0 (22) =+−−++min⎜ 7∏∏∏σσz 3zz 3 σ 6 σ z 2 σσσz −z z ⎝ i=3 i=3 i=3 and n ⎞ 2 i 11n++ 21n 12⎟ −−−+σz ∏ σz 44 σσzz σσzz σσzz⎟ xy≠−−+> zif xy 2 xz 2 yz 3 z 0 (23) i=3 ⎠ (26)
G DOI: 10.1021/acs.jpcb.7b10371 J. Phys. Chem. B XXXX, XXX, XXX−XXX The Journal of Physical Chemistry B Article
Table 1. Comparing the Exact Ground Energy (au) and the Simulated Ground Energy (au) (Simulated by 4 × 2 qubits) as a Function of the Intermolecule Distance R (au)
Rg0 g1 g2 g3 g4 exact simulated 0.6 1.5943 0.5132 −1.1008 0.6598 0.0809 −0.5617 −0.5703 0.65 1.4193 0.5009 −1.0366 0.6548 0.0813 −0.6785 −0.6877 0.7 1.2668 0.4887 −0.9767 0.6496 0.0818 −0.7720 −0.7817 0.75 1.1329 0.4767 −0.9208 0.6444 0.0824 −0.8472 −0.8575 0.8 1.0144 0.465 −0.8685 0.639 0.0829 −0.9078 −0.9188 0.85 0.909 0.4535 −0.8197 0.6336 0.0835 −0.9569 −0.9685 0.9 0.8146 0.4422 −0.774 0.6282 0.084 −0.9974 −1.0088 0.95 0.7297 0.4313 −0.7312 0.6227 0.0846 −1.0317 −1.0415 1.0 0.6531 0.4207 −0.691 0.6172 0.0852 −1.0595 −1.0678 1.05 0.5836 0.4103 −0.6533 0.6117 0.0859 −1.0820 −1.0889 1.1 0.5204 0.4003 −0.6178 0.6061 0.0865 −1.0999 −1.1056 1.15 0.4626 0.3906 −0.5843 0.6006 0.0872 −1.1140 −1.1186 1.2 0.4098 0.3811 −0.5528 0.5951 0.0879 −1.1249 −1.1285 1.25 0.3613 0.37200 −0.523 0.5897 0.0886 −1.1330 −1.1358 1.3 0.3167 0.3631 −0.4949 0.5842 0.0893 −1.1389 −1.1409 1.35 0.2755 0.3546 −0.4683 0.5788 0.09 −1.1427 −1.1441 1.4 0.2376 0.3463 −0.4431 0.5734 0.0907 −1.1448 −1.1457 1.45 0.2024 0.3383 −0.4192 0.5681 0.0915 −1.1454 −1.1459 1.5 0.1699 0.3305 −0.3966 0.5628 0.0922 −1.1448 −1.145 1.55 0.1397 0.32299 −0.3751 0.5575 0.09300 −1.1431 −1.1432 1.6 0.1116 0.3157 −0.3548 0.5524 0.0938 −1.1404 −1.1405 1.65 0.0855 0.3087 −0.3354 0.5472 0.0946 −1.1370 −1.1371 1.7 0.0612 0.3018 −0.317 0.5422 0.0954 −1.1329 −1.1332 1.75 0.0385 0.2952 −0.2995 0.5371 0.0962 −1.1281 −1.1287 1.8 0.0173 0.2888 −0.2829 0.5322 0.09699 −1.1230 −1.1239 1.85 −0.0023 0.2826 −0.267 0.5273 0.0978 −1.1183 −1.1187 1.9 −0.0208 0.2766 −0.252 0.5225 0.0987 −1.1131 −1.1133 1.95 −0.0381 0.2707 −0.2376 0.5177 0.0995 −1.1076 −1.1077 2.0 −0.0543 0.2651 −0.2238 0.513 0.1004 −1.1018 −1.1019 2.05 −0.0694 0.2596 −0.2108 0.5084 0.1012 −1.0958 −1.0961 2.1 −0.0837 0.2542 −0.1983 0.5039 0.1021 −1.0895 −1.0901 2.15 −0.0969 0.249 −0.1863 0.4994 0.10300 −1.0831 −1.0842 2.2 −0.1095 0.244 −0.1749 0.495 0.1038 −1.0765 −1.0782 2.25 −0.1213 0.2391 −0.1639 0.4906 0.1047 −1.0699 −1.0723 2.3 −0.1323 0.2343 −0.1536 0.4864 0.1056 −1.0630 −1.0664 2.35 −0.1427 0.2297 −0.1436 0.4822 0.1064 −1.0581 −1.0605 2.4 −0.1524 0.2252 −0.1341 0.478 0.1073 −1.0533 −1.0548 2.45 −0.1616 0.2208 −0.125 0.474 0.1082 −1.0484 −1.0492 2.5 −0.1703 0.2165 −0.1162 0.47 0.109 −1.0433 −1.0437
⎛ n ⎞ ⎛ 12 n i 1 ++σσ1 1 + ∏ σz |Ψ11ss ⟩ + |Ψ 12 ⟩ min⎜−=−σσ12∏ σi⎟ min⎜ 8 zz i=3 |χα⟩=|Ψ⟩|⟩= |⟩ α ⎝ zz z⎠ ⎝ 2 2 2 1 g i=3 2(1+ S ) (28) 12 n i 12 1 n i 2 n i ⎞ 1 +++σzz σ∏ σσσσz ++zz z∏ σσz + z ∏ σz + 8 i===3 i 3 i 3 ⎟ 8 ⎠ |Ψ11ss ⟩ + |Ψ 12 ⟩ ⎛ n n+1 n |χβ2 ⟩=|Ψ⟩|⟩=g |⟩ β ⎜ i 12n+ 1 i 1 i 2(1+ S ) (29) =−−−+−min⎜ 5∏∏∏σσz zz σ 2 σ z 2 σσz +z σz ⎝ i=3 i=3 i=3 n ⎞ 2 i 11n++ 21n 12⎟ |Ψ ⟩ − |Ψ ⟩ +−−+σz ∏ σz 443 σσzz σσzz σσzz⎟ 11ss 12 |χα⟩=|Ψ⟩|⟩=u | α⟩ i=3 ⎠ (27) 3 2(1− S ) (30) σ By repeating this, we can reduce the k-local in z terms to a 2- |Ψ11ss ⟩ − |Ψ 12 ⟩ local Hamiltonian. |χβ4 ⟩=|Ψ⟩|⟩=u |⟩ β 2(1− S ) (31) Mapping the H2 Hamiltonian to an Ising-Type Hamiltonian |Ψ ⟩ |Ψ ⟩ Here, we treat the hydrogen molecule in a minimal basis STO- where 1s 1 and 1s 2 are the spatial functions for the two atoms, respectively, |α⟩ and |β⟩ are spin up and spin down, and S 6G. By considering the spin functions, the four molecular spin ⟨Ψ |Ψ ⟩ 39 = 1 1s 1s 2 is the overlap integral. The one- and two-electron orbitals in H2 are integrals are given by
H DOI: 10.1021/acs.jpcb.7b10371 J. Phys. Chem. B XXXX, XXX, XXX−XXX The Journal of Physical Chemistry B Article
⎛ 1 Z ⎞ By applying the mapping method described above, we can get χχ* ⎜⎟ hrrij = ∫ d()⃗ ij⃗ ⎝−∇− ⎠ ()r⃗ ′ σ 2 r (32) the Hamiltonian H consisting of only z (where i1 and i2 mean the 1 and 2 qubits of ith 2 qubits): →⎯→⎯ →⎯→⎯→⎯ ⎯ 1 → hrrrr= ddχχ** () () χχ()()rr i i i i ijkl ∫ 12ij 1 2 kl21 H′=∑ ggσσσσ1212 + + g r12 (33) 12z z 3z z i Thus, we can write the second-quantization Hamiltonian of H2: ⎡ i1 j1 i2 j2 ⎢ (σσz ++z )(1 σσz z ) †††† + ∑ gSiSj′′() () H =+++haa haa haa haa ⎢ 1 H00001111222233332 ij≠ ⎣ 4 †† †† †† +++h aaaa h aaaa h aaaa i (,2)j i j 0110 0 1 1 0 2332 2 3 3 2 0330 0 3 3 0 (σσ21++ )(1 σσ1 ) + z z z z ′′ †† †† gSiSj2 () () ++−h1221 aaaa 1 2 2 1() h 0220 h 0202 aaaa 0 2 2 0 4 i j i j +−()hhaaaa†† ()()σσσσ1 ++1 2 2 1331 1313 1 3 3 1 + gSiSjz z z z ′′() () †† †† 3 4 ++h0132() aaaa 0 1 3 2 aaaa 2 3 1 0 i11i i 2j2 †† †† (1++σσz z )(1 σσz z ) ++haaaaaaaa()(34) + gSiSj′′() () 0312 0 3 1 2 2 1 3 0 4 4 25 By using the Bravyi−Kitaev transformation, we have i i j i ⎤ ()()σσσσ11−−2 2 ⎥ + gSiSjz z z z ′′() () 1 1 4 ⎥ aiai† =−=+σσ31() σ 0 σ 0 σσ31() σ 0 σ 0 4 ⎦ 0 2 xx x y0 2 xx x y (40) † 1 310 31 1 310 31 aiai1 =−=+()σσσxxz σσ xy1 () σσσ xxz σσ xy According to the scheme for reducing locality, if we want to 2 2 reduce H′ − λC, we can reduce H′ and C separately. By applying 1 1 aiai† =−σσ32() σσ 21 =+ σσ32() σσ 21 the method for reducing locality, we can get a 2-local Ising-type 2 2 xx yz2 2 xx yz Hamiltonian, H″. Here we show the example of all signs being k † 1 1 positive, where a , k = 1, 2, 3, ..., 6, are the indices of the new aiai=−=+()σσσ321 σ 3 () σσσ321 σ 3 ij 3 2 xzz y3 2 xzz y qubits we introduce to reduce locality. (35) ⎡ ⎢ σσi1 + j1 σσi2 + j2 Thus, the Hamiltonian of H takes the following form: ″=σσσσi1212 +i +i i + z z + z z 2 H ∑∑gg12z z g 3z z ⎢ g 1 g 2 i ij≠ ⎣⎢ 44 0 1 2 01 02 H =+ff1 σσσσσσσ + f + f + f + f i j i j i j i j H2 01zzzzzzz2 3 1 4 ()()σσσσ1 ++1 2 2 1 ++ σσσσ1 1 2 2 + ggz z z z + z z z z ++++σσ13 σσσ012 σσσ012 σσσ012 344 4 ffff5 zz6 x zx6 yzy7 zzz ()()σσi12−−j,1) σσi j2 023 123 0123 + g z z z z +++σσσ σσσ σσσσ 4 fff4 zzz3 zzz6 xzxz 4 a 1 a 1 a 1 0123 0123 i2 j2 i1 j1 iji1 ijj1 ij i1 j1 ++σσσσ σσσσ 733644+−−+−σσσσσσσσσσσz z z z z z z −z z +z z ff6 yzyz7 zzzz (36) + g 4 4 fl +− We can utilize the symmetry that qubits 1 and 3 never ip to ()gg14 reduce the Hamiltonian to the following form which just acts on a 23a a 2 a 3 ++−σσi1 j1 σσi2 +j2 + σij + σ ij + σσσσi1 ij +j1 ij only two qubits 14z z 6(z z ) 6(z z ) 2(z z z z ) 4 0 1 01 01 01 H =+gg1 σ + g σ + g σσ + g σσ + g σσ +−()gg H2 01zzzzxxyy2 3 4 4 14 a 23a i2 j2 ij ij i1 j1 i2 j2 i2 j2 =+ −+4(σσσσz z )(z +−+z ) ( σσσσz z )(z ++z ) 2 σσz z gH01 0 (37) 4 == = =+= + g g0 fg0 1 222()2 fg1 2 fg3 3 f47 f g4 f6 2 a 45a a 4 a 5 i2 j2 i1 j1 ij iji2 ijj2 ij (38) 10−−−σσz z 2( σσz +z ) + 2( σz + σz ) − 2( σσσσz z +z z ) 4 45 ghhhhh=+1.000 0.5 0000 − 0.5 0022 + 1.0 0220 + 1.0 22 i j aija ij i j i j i j 0 −+4(σσσ1 1 )(z + σz ) ++ ( σσσσ1 1 )(2 ++2 ) 6 σσ1 1 + g z z z z z z z z ++0.5hR2222 1.0/ 2 4 1 616 aiji j a ija ija ij ghhhh=−1.0 − 0.5 + 0.5 − 1.0 73362+−−σσσσσσz 2 2 +z +z z 1 00 0000 0022 0220 + g z z 4 2 ⎤ gh=−−−0.50022 1.0 h 0220 1.0 h 22 0.5 h 2222 a 1 a 1 a 6 a 6 2 i2 ijj2 iji2 ijj2 ij i2 j2 −−σσz z σσz z −44 σσz z − σσz z + σσz z ⎥ + g ⎥ ghhhh=−1.0 − 0.5 + 0.5 − 1.0 4 2 3 00 0000 0022 0220 ⎦⎥ (41) gh= 0.5 4 0022 We can write the corresponding count term C as (39) 2 fi ⎛ i12i ⎞ where {gi} depends on the xed bond length of the molecule. In (1±±σσ )(1 ) C = ∑∑⎜ z z ⎟ Table 1, we present the numerical values of {gi} as a function of ⎝ 4 ⎠ the internuclear distance in the minimal basis set STO-6G. ± i (42)
I DOI: 10.1021/acs.jpcb.7b10371 J. Phys. Chem. B XXXX, XXX, XXX−XXX The Journal of Physical Chemistry B Article
Figure 6. Results of the simulated transformed Ising-type Hamiltonian with 2 × 2 qubits and 4 × 2 qubits compared with the exact numerical results for the ground state of the H2 molecule.
By applying the method of reducing locality, the 2-local The set of parameters f i is related to the one- and two-electron corresponding count term is integrals: ⎧ ⎪ 7 7 =+ − + + 1 i j i j a i a fhhhhh0 1.000 0.25 0000 0.5 0022 1.0 0220 1.0 22 C′=⎨∑ [7 +σσ2 2 − 3 σ1 − 3 σ1 + 6 σij − 4 σσ1 ij ⎪ z z z z z z z ++ 4 ⎩ ij≠ 0.25hR2222 4.0/ 7 7 fhh=−0.5 − 0.25 + 0.25 h − 0.5 h j1 aiji1 j1 a ij i2 j2 1 00 0000 0022 0220 −+++−−42(733σσz z σσz z σz σz σz a 818a a i a 7 j a 7 fh=−0.25 + 0.25 hhh − 0.5 − 0.5 ij ij ij2 ij2 ij 2 0000 0022 0220 00 ++62)2(σσσσσσσz z z +−−z z z z i a 8 j a 8 i j i j fh=−−−0.25 0.5 h 0.5 h 0.25 h 2 ij2 ij 2 2 1 1 3 0022 0220 22 2222 −−++++44σσz z σσz z σσz z )14 σz σz 91 i2 j2 aija ij 0 fh=−−−0.250022 0.5 h 0220 0.25 h 2222 0.5 h 22 −+++6(σσz z ) 6( σz σz ) 4 i a 9 j a 100i j a 91a 1 ij1 ij2 2 ij ij fh= 0.25 0000 ++−++2(σσz z σσz z ) 4( σz σz )( σz σz ) 5 i j i j i j i j 1 1 2 2 2 2 1 1 fh=−0.25 + 0.25 h −+()()21σσσσz z z ++z σσz z ++ σσz z 6 0022 0220 ⎫ i12i = ⎪ (1+++σσ ) (1 ) fh7 0.25 0220 i2 j2 ⎬ z z ++σσz z ] ∑ ⎪ 4 fh= 0.25 ⎭ i (43) 8 0220 fh= 0.25 As stated in the section “Reduce Locality of the Transformed 9 2222 ” Hamiltonian , a change in the locality would not change the state fh=−0.25 when calculating the ground state energy. Thus, we can still use C 10 0022 on certain qubits to calculate ∑ a 2, and the algorithm we present = i i fh11 0.25 0022 above can still be used for the reduced Hamiltonian. = In Figure 6, we show our results from the transformed Ising- fh12 0.25 0022 type Hamiltonian of 2 × 2to4× 2 qubits compared with the fh=−0.25 exact numerical values. By increasing the number of qubits via r = 13 0022 2, we increased the accuracy and the result matches very well with (45) the exact results. Table 1 shows the numerical results. We can also use the mapping and reduction of locality as before Mapping the Hamiltonian for the He2 Molecule to the to get the final Ising Hamiltonian. Here we just present the Ising-Type Hamiltonian σ 1 mapping result of some terms for illustration. For z , the As shown above for transforming the Hamiltonian associated Hamiltonian between different bases can be mapped as with the H2 molecule, we repeat the procedure for the helium i j i j i j j i molecule in a minimal basis STO-6G using the Jordan−Wigner (σσ1 ++1 )(1 σσ2 2 )(1 + σσ3 34 )(1 + σσ4 ) ∑ z z z z z z z z SiSj′′() () transformation. 16 The molecular spin Hamiltonian has the form ij≠ (46) H =+ff1 σσσσσσ1 + f2 + f3 + f4 + f12 σ 1σ 2 ff He2 01zzzzzz2 3 4 5 For z z , the Hamiltonian between di erent bases can be mapped as ++++σσ13 σσ14 σσ23 σσ24 ffff6 zz7 zz8 z z8 z z i j i j i j j i σσσσ1 ++++1 2 2 σσ3 34 σσ4 ++σσ34 σσσσ1234 + σσσσ1234 (z z )(z z )(1z z )(1z z ) ff9 zz10 xxyy f11 xyyx ∑ SiSj′′() () ij≠ 16 ++ffσσσσ1234 σσσσ1234 12 yx x y13 yy xx (44) (47)
J DOI: 10.1021/acs.jpcb.7b10371 J. Phys. Chem. B XXXX, XXX, XXX−XXX The Journal of Physical Chemistry B Article
σ 1σ 2σ 3σ 4 ff ⎛ For x x y y , the Hamiltonian between di erent bases can be ⎜ σσi1 + j1 ∑ z z mapped as ⎜ ij≠ ⎝ 4
i j i j i j j i a 1 2 a 1 2 1 1 2 2 3 34 4 i1 j i2 j ij aij i1 ij j aij (1−−σσ )(1 σσ )( σ −− σ )( σ σ ) 14++−σσ1 6( σσ +2 ) + 6( σz + σ ) + 2( σσσσz +1 ) z z z z z z z z SiSj′′() () + z z z z z z z z ∑ 4 ij≠ 16 a 12a ⎞ i2 j2 ij ij i1 j1 i2 j2 i2 j2 ⎟ (48) −+4(σσσσz z )(z +z ) −+ ( σσσσz z )(z ++z ) 2 σσz z + ⎟ 4 ⎠ By reducing locality, we can get the 2-local Ising-type (53) σ 1σ 2σ 3σ 4 fi Hamiltonian. However, even just for x x y y , the nal 2- local Ising Hamiltonian would have about 1000 terms. Mapping the Molecular Hamiltonian of LiH to an Ising Mapping the Hamiltonian of HeH+ to an Ising Hamiltonian Hamiltonian + Similar to H2 and other molecules, next we treat the LiH Similar to H2 and He2 molecules, next we treat the HeH molecule with four electrons in a minimal basis STO-6G and use − molecule in the minimal basis STO-6G using the Jordan the Jordan−Wigner transformation. Using the technique defined Wigner transformation. Using the technique defined above,40 we above,41 we can reduce the locality to a Hamiltonian with 558 can reduce the locality to terms on 8 qubits. We just use 16 qubits for the simulations. Computation Complexity and Error Analysis 1 2 1 12 12 + =+σσσσσσσ + + + + HHeH ff011 zzxzzxz f2 f3 f4 f5 Specifying the required amount of qubit resources can be derived throughout the multistep process of constructing the final +++fffσσ12 σσ12 σσ12 6 zx7 xx8 yy (49) Hamiltonian. As an example, let us consider a Hamiltonian with n total orbitals; transforming this Hamiltonian from second The set of parameters f i is related to the one- and two-electron quantization to Pauli operators requires O(n) qubits and a series integrals: of O(n4) expansion terms. This is due to the need to use two- body operator and four-body operator terms in formulating the fhhhhh=+1.0 0.25 + 0.5 ++ 1.0 0.25 second quantization Hamiltonian. Thus, the number of terms in 0 00 0000 0220 22 2222 the second quantization form of the Hamiltonian should be + 2.0/R O(n4), yielding O(n4) terms to be described in the Pauli operator =− + − representation of the Hamiltonian. fhh1 0.250000 0.5 0220 0.25 h 2222 Mapping the Hamiltonian from a Pauli operator representa- =+−− fh2 0.250000 0.5 h 00 0.25 h 2222 0.5 h 22 tion to a diagonal matrix requires an additional O(rn) factor of qubits, and thereby O(2nr2n4) qubits. This is due to the fact that, fh=−0.25 − 0.25 hh + 0.5 3 0002 0020 0222 for each term in the Pauli operator representation of the n 2 fhh=−0.5 − 0.25 + 0.5 hh + 0.25 Hamiltonian, we now have O(2 r ) terms in the diagonal 4 00 0000 22 2222 Hamiltonian. This process leads us to the O(r2n4) mapping, =− − − n 2 4 fh5 0.250002 0.25 hh 0020 0.5 0222 which does in fact have O(2 r n ) terms. Reducing the representation down to 2-local interactions =− n 2 7 fh6 1.0 0022 requires O(rn +2r n ) qubits. Within this form, we do not know r. Consider that the qubit requirement during the reduction from fh=+−0.25 0.25 h 0.5 h 7 0002 0020 0222 a k-local to 2-local is given by f(k). We have found that 2 log10 3 fh=++0.250002 0.25 h 0020 0.5 h 0222 f ()kfk< 10 which gives us f(k)=O(n 3 )orO(n ) by the 8 ()3 (50) master theorem. Thus, for all terms, we finally have O(rn + 2nr2n7) qubits. We can also use the mapping and reducing locality as before to However, we cannot clearly give a relationship between r and fi get the nal Ising Hamiltonian. Again, here we just present the the desired accuracy in the solution. Similarly to the variation mapping result of some terms for illustration: method trying to optimize results under particular conditions, σ 1σ 2 ff For z z , the Hamiltonian between di erent bases can be this mapping tries to approach the optimal value of the desired mapped as ground eigenstate via a new state a by repetition of terms r times. Thus, we cannot calculate the errors associated with our optimal
i1 j1 i2 j2 result compared to the exact result, because exact results are ()()σσσσz ++z z z ∑ SiSj′′() () unknown within the varational method. Through repeated scaled 4 ij≠ (51) calculation, we can see that in increasing r we obtain greater accuracy. This is likely due to the fact that by increasing r we σ 1σ 2 ff For z x , the Hamiltonian between di erent bases can be increase the number of repetitious terms providing the necessary mapped as corrections allowing bi to approach ai.
i j i j (σσ1 +−1 )(1 σσ2 2 ) ■ AUTHOR INFORMATION ∑ z z z z SiSj′′() () 4 Corresponding Author ij≠ (52) *E-mail: [email protected]. And if the coefficient of mapping term is positive, we can get the ORCID 2-local term as Sabre Kais: 0000-0003-0574-5346
K DOI: 10.1021/acs.jpcb.7b10371 J. Phys. Chem. B XXXX, XXX, XXX−XXX The Journal of Physical Chemistry B Article
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L DOI: 10.1021/acs.jpcb.7b10371 J. Phys. Chem. B XXXX, XXX, XXX−XXX