simulation of electronic structure hamiltonians using

Simulation of Electronic StructureHamiltonians Using Quantum Computers

The Harvard community has made thisarticle openly available. Please share howthis access benefits you. Your story matters

Citation Whitfield, James D., Jacob Biamonte, and Alán Aspuru-Guzik. 2011.Simulation of electronic structure Hamiltonians using quantumcomputers. Molecular Physics 109(5): 735-750.

Published Version doi:10.1080/00268976.2011.552441

Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:8403540

Terms of Use This article was downloaded from Harvard University’s DASHrepository, and is made available under the terms and conditionsapplicable to Open Access Policy Articles, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAP

http://osc.hul.harvard.edu/dash/open-access-feedback?handle=&title=Simulation%20of%20Electronic%20Structure%20Hamiltonians%20Using%20Quantum%20Computers&community=1/1&collection=1/2&owningCollection1/2&harvardAuthors=d694414f99898da66b926a3c7a4b87f3&departmentChemistry%20and%20Chemical%20Biology

http://nrs.harvard.edu/urn-3:HUL.InstRepos:8403540

http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAP



December 21, 2010 1:24 arXiv:1001.3855 h2.arxiv2

arXiv:1001.3855 1

Simulation of Electronic Structure Hamiltonians Using

Quantum Computers

James D. Whitfielda, Jacob Biamontea,b, and Alan Aspuru-Guzika

aHarvard University, Department of Chemistry and Chemical Biology, 12 Oxford

St., Cambridge, MA, 02138, USA;bOxford University Computing Laboratory, Oxford OX1 3QD, UK

()

Over the last century, a large number of physical and mathematical developments pairedwith rapidly advancing technology have allowed the field of quantum chemistry to ad-vance dramatically. However, the lack of computationally efficient methods for the exactsimulation of quantum systems on classical computers presents a limitation of currentcomputational approaches. We report, in detail, how a set of pre-computed molecularintegrals can be used to explicitly create a quantum circuit, i.e. a sequence of elementaryquantum operations, that, when run on a quantum computer, to obtain the energy ofa molecular system with fixed nuclear geometry using the quantum phase estimationalgorithm. We extend several known results related to this idea and discuss the adia-batic state preparation procedure for preparing the input states used in the algorithm.With current and near future quantum devices in mind, we provide a complete exampleusing the hydrogen molecule, of how a chemical Hamiltonian can be simulated using aquantum computer.

Keywords: electronic structure, quantum computing

1. Introduction

Theoretical and computational chemistry involves solving the equations ofmotion that govern quantum systems by analytical and numerical methods [1,2]. Except in standard cases such as, the harmonic oscillator or the hydrogenatom, analytic solutions are not known and computational methods have beendeveloped.

Although classical computers have tremendously aided our understanding ofchemical systems and their processes, the computational cost of the numericalmethods for solving Schrodinger’s equation grows rapidly with increases inthe quality of the description. Research is ongoing to improve computationalmethods, but large molecules and large basis sets have remained a consistentproblem despite the exponential growth of computational power of classicalcomputers [3].

Theoretical computer science suggests that these limitations are not mereshortcomings of the programmers but could stem from the inherent difficultlyof simulating quantum systems. Extensions of computer science using quantummechanics led to the exploitation of the novel effects of quantum mechanics

Email: [email protected]



arX

iv:1

001.

3855

v3 [

quan

t-ph

] 1

9 D

ec 2

010


2 Whitfield et al.

for computational purposes resulting in several proposals for quantum com-puters [4]. Quantum simulation is the idea of using quantum computationaldevices for more efficient simulation [5, 6]. Since the dynamics are simulatedby a quantum system rather than calculated by a classical system, quantumsimulation often offers exponential advantage over classical simulation for cal-culation of electronic energies [7], reaction rates [8, 9], correlation functions [10]and molecular properties [11]. Recently, a review of these techniques and otherapplications of quantum computing to chemistry has appeared [12].

This paper does not consider the effect of errors, however it is an importantconsideration that needs to be taken into account. Quantum error correctionmethods have been developed to counteract the unwanted effect of quantumnoise, however fault tolerant constructions require redundant qubits and onlyallow a discrete set of gates to be used [13, 14]. This is not a serious cause forconcern as the conversion from a continuous set of gates to a discrete set ofgates only requires a poly-logarithmic overhead [15]. Clark et al. [16] estimatedthe resources required to compute the ground state of a one dimensional trans-verse Ising model and found, using experimental parameters from a proposedion trap quantum computing implementation, that the fault tolerant construc-tions would be too costly for straight-forward applications to simulation. Thissuggests that quantum simulation without quantum error correction is morefeasible for the near future.

The state of the art in experimental realizations of quantum simulation forchemistry is represented by calculations of the energy spectrum of molecularhydrogen first by Lanyon et al. [17] using an optical quantum computer. Verysoon after, Du et al. [18] used NMR technology to demonstrate the adiabaticstate preparation procedure suggested by Ref. [7] as well as reproduce theenergy to higher accuracy.

The key limitation of both experimental algorithms was the representationof the simulated system’s propagator. Both experiments, relied on the lowdimensionality of the propagator for the minimal basis H2 model considered.The unitary propagator for a two-level system can be decomposed using thereal angles α, β, γ:

U = eiαRy(β)Rz(γ)Ry(−β)

Due to this decomposition, longer propagation times corresponding to higherpowers of unitary evolution operator, U j , can be achieved by changing α tojα and γ to jγ, thereby avoiding the need of further decomposition of theunitary operator. Beyond the two dimensional case, this decomposition is notavailable.

The objective of this paper is to provide a general decomposition for elec-tronic Hamiltonians and demonstrate this method with an explicit quantumcircuit for a single Trotter time step of the minimal basis hydrogen molecule.This is an extension of the supplementary material from Lanyon et al. [17].

2. Overview of the quantum algorithm

The use of the Fourier transform of correlation functions in computationalchemistry allows for the extraction of information about many properties suchas transport coefficients [19] and molecular spectra [20]. For the application ofquantum computation to molecular electronic energy calculations, the spec-tral solution of the Schrodinger equation [21] is pursued. The key idea of the


arXiv:1001.3855 3

H ≡∑hi

Trotter−−−−−−−−−−−→Decomposition

U ≈ (∏i e−ihi∆t)

t∆t

|ψ0〉−−−→ λ = e−iE0t

= e2πi(φ+k)

H ⇒ ↓ ↑ ⇒ E0

Jordan-WignerTransform

→ PhaseEstimation

Figure 1. An algorithmic overview of the steps taken to simulate a chemical Hamiltonian on aquantum computer. The time independent Hamiltonian of a molecular system (to the left of the box)will be decomposed into a sum of Hermitian matrices, (hi) and by means of a Trotter decompositionthe unitary propagator U can be constructed. The Jordan-Wigner mapping is used to convert thepropagator into a sequence of quantum gates. Phase estimation algorithms are used to evaluate theeigenvalue of a correctly prepared stationary state |ψ0〉.

approach is that the Fourier transform, F [f(t)] =∫

exp(iωt)f(t)dt of theautocorrelation of the time evolving state, P (t) = 〈ψ(0)|ψ(t)〉, has peaks atthe eigenenergies. Typically, the semiclassical propagator is used for examin-ing vibrational structure e.g. [22]. Since quantum computers can simulate thequantum evolution of electrons efficiently, the Fourier transform of the elec-tronic autocorrelation function can be calculated directly. The measurementtechnique presented here collapses an input wave function into an eigenstateand returns the frequency of the autocorrelation of the time-evolving eigen-state by employing the spectral method mentioned before to the extractionof molecular electronic energies. In this paper, we focus on the extraction ofeigenvalues given input states using the time evolution of the state.

The construction of the general quantum circuit to simulate the evolutionof the molecular system is performed in three steps:

(1) Write the Hamiltonian as a sum over products of Pauli spin operatorsacting on different qubits. This is described in Section 3 and made possibleby the Jordan-Wigner transformation.(2) Convert each of the operators defined in step (1) into unitary gatessuch that their sequential execution on a quantum computer can be madeto recover an approximation to the unitary propagator, exp(−iHt). Thisis detailed in Section 4.(3) The phase estimation algorithm, as described in Section 5, is then usedto approximate the eigenvalue of an input eigenstate using the quantumFourier transform of the time domain propagation. Section 6 discusseseigenstate preparation.

To demonstrate these steps, the construction is applied to the example ofthe hydrogen molecule in Section 7. The key components of the simulationprocedure are depicted in Fig. 1. The next section provides a basic reviewof the fundamental concepts and notations of molecular quantum chemistryfor the benefit of quantum information scientists and to establish the nota-tion. A detailed account of electronic structure methods can be found in themonographs [1, 2].

3. The Electronic Hamiltonian

The Born-Oppenheimer approximation assumes that the wave function of amolecule can be expressed as a product of the nuclear wave function andelectronic wave function (parameterized by the nuclear coordinates) due tothe difference in mass between electrons and nuclei. This approximation allowsfor the solution of the time independent Schrodinger equation of the electronicwave function for a given nuclear geometry.


4 Whitfield et al.

Table 1. An overview of second quantization for Fermionic particles. For quantum chemistry, the annihilation and creation oper-

ators correspond to removing (adding) an electron into a particular orbital, {χk}. The anti-symmetry is enforced by the canonical

commutation relations, and the N-electron wave function is expanded over the configuration state functions of the Fock space.

Second quantization

Creation operator a†i |j1, . . . , 0i, . . . , jn〉 = Γji|j1, ..., 1i, ..., jn〉 with Γj

i =∏i−1n=1(−1)jn

a†i |j1, . . . , 1i, . . . , jn〉 = 0

Annihilation operator ai|j1, . . . , 1i, . . . , jn〉 = Γji|j1, ..., 0i, ..., jn〉

ai|j1, . . . , 0i, . . . , jn〉 = 0

Canonical commutation relations {a†i , aj} = δij1{ai, aj} = 0

Fock space

Basis vectors,configuration state functions

|j〉 = |j1, j2, ..., jN 〉=

∏Np=1

(a†p)jp|vac〉 where ji = 0, 1

Inner product 〈j|k〉 =∏Np=1 δjp,kp

Vacuum state 〈vac|vac〉 = 1ai|vac〉 = 0

Single electron operator, A(x) Aija†iaj where Aij =

∫χi(x1)A(x1)χj(x1)dx1

is a single electron operatorand {χk} corresponds to {ak}

The molecular electronic Hamiltonian1 in second quantized form is given by[1, 2]:

H =∑p,q

hpqa†paq +

1

2

∑p,q,r,s

hpqrsa†pa†qaras, (1)

where the sum is over the single particle basis set described below. The annihi-

lation {aj} and creation operators {a†j} obey the Fermionic anti-commutation

relations (see Table 1):

[ap, aq]+ = 0, [ap, a†q]+ = δpq1, (2)

where the anti-commutator of operators A and B is defined as [A,B]+ ≡AB+BA and 1 is the identity operator. The annihilation (creation) operatorscorrespond to a set of orbitals, {χi}, where each orbital is a single-particlewave function composed of a spin and a spatial function, denoted σi and ϕi,respectively. As the Hamiltonian considered here commutes with the electronspin operators, σi is restricted to be one of two orthogonal functions of a spinvariable ω that we denote α(ω) and β(ω). Similar Hamiltonians can be foundin many physics problems involving Fermionic particles.

Although any basis can be used, the molecular orbitals are particularly con-venient for state preparation reasons discussed below. The molecular orbitals,in turn, are formed as a linear combinations of atomic basis functions [23, 24].The coefficients of this expansion are obtained by solving the set of Hartree-Fock equations which arise from the variational minimization of the energyusing a single determinant wave function. Due to its restriction to a singledeterminant, the Hartree-Fock solution is a mean field solution and the dif-ference between the Hartree-Fock solution using an infinite basis of atomic

1Throughout this paper, atomic units are used: ~ (1.054 · 10−34 J s), the mass of the electron(9.109 · 10−31 kg), the elementary charge (1.602 · 10−19 C), and the electrostatic force constant(1/4πε0 = 8.988 · 109 N m2 C−2) are set to unity.


arXiv:1001.3855 5

orbitals and the exact (correlated) solution defines the electron correlationenergy.

The matrix elements {hpq} and {hpqrs} in Eq. (1) denote the set of one- andtwo-electron integrals that must be evaluated using a known set of basis func-tions (the basis set) during the Hartree-Fock procedure. Ideally, the number ofbasis functions used would be infinite, but in practice only a finite number ofbasis functions are used. By selecting Gaussian single-particle basis functions,these integrals are efficiently computable. Next, to further establish notation,we develop the explicit form of the integrals hpq and hpqrs.

We denote the set of single-particle spatial functions which constitute themolecular orbitals {ϕk(x)}Mk=1. Finally, define the set of spin orbitals as{χp(x)}2Mp=1 with χp = ϕiσi and x = (x, ω) where σi is a spin function. Theone-electron integrals involving the electronic kinetic energy and the electron-nuclear attraction terms are:

hpq ≡∫

dxχ∗p(x)

(−1

2∇2 −

∑α

Zαrα,x

)χq(x) (3)

and the two-electron integrals involving the electron-electron interaction, 1/r12

are:

hpqrs ≡∫

dx1dx2

χ∗p(x1)χ∗q(x2)χr(x2)χs(x1)

r1,2(4)

In Eq. (3), ∇2 is the Laplacian with respect to the electronic spatial coor-dinates. The positive valued scalars rα,x and r1,2 are the Euclidean distancebetween the αth nucleus and the electron and the Euclidean distance betweenthe two electrons. In both Eqs. (3) and (4), the spin of the electron can be in-tegrated out resulting in integrals over the spatial components however sincethe focus is on spin orbitals, the electron spin will be included in the definitionof the integrals.

3.1. Representing the molecular Hamiltonian in terms of quantum bits

Just as classical computation is based on the notion of a bit, the basic unitof quantum information is the quantum bit (qubit). Qubits, being quantum,are described by a wave function instead of a probability distribution as in thecase of classical bits. The use of the wave function description allows for thesuperposition of states and for entanglement between different qubits. More-over, since qubits themselves are described using wave functions, quantumstates can be efficiently stored.

In principle, any two-level quantum mechanical system can be considereda qubit. Examples include photons, ions, and super conducting loops. Practi-cal requirements for qubits and their manipulation was originally outlined byDiVincenzo [25] and experimental progress towards satisfying the DiVincenzocriteria were recently reviewed [4]. Since two-level systems can be describedas spin-half particles, the relevant (Pauli) matrices are:

σx =

[0 11 0

]σy =

[0 −ii 0

]σz =

[1 00 −1

].


6 Whitfield et al.

Notice that we have defined the Pauli matrices with eigenvalues ±1 insteadof ±~/2. Together with the identify matrix, 1, the Pauli matrices form anoperator basis for two-level systems. The eigenvectors of σz are labeled as|0〉 and |1〉 corresponding to the eigenvalues +1 and -1 respectively. There areseveral computationally equivalent models of describing quantum computationbut here we focus on the circuit model of quantum computation. A morecomprehensive introduction to quantum computation can be found in Ref. [13].

The quantum circuit model uses a set of elementary gates to reproduce theaction of arbitrary unitary transforms. If the elementary gates are capable ofreproducing any desired unitary transform to arbitrary precision, the set ofgates is called universal. A universal gate set requires single qubit gates andany two-qubit entangling gate. The two-qubit CNOT gate leaves one qubitspace unchanged and acts with σx on the second qubit when the first qubit isin the state |1〉. The gate is given as CNOT= |1〉〈1| ⊗ σx + |0〉〈0| ⊗ 1. The set,Rx, Ry, and Rz gates can generate any single qubit gate where Rn is definedas exp[−iσnθ/2] for real θ.

For experimental addressability, the qubits must, in general, be distinguish-able. However, the electrons of the molecular system are indistinguishable.The Jordan-Wigner transform is used to circumvent this issue by expressingFermionic operators in terms of the Pauli spin operators {σx, σy, σz,1} thatcorrespond to the algebra of distinguishable spin 1/2 particles [10, 26]. TheJordan-Wigner transform is given by:

aj ⇔ 1⊗j−1 ⊗ σ+ ⊗ σz⊗N−j−1 =

[1 00 1

]⊗j−1

⊗[

0 10 0

]⊗[

1 00 −1

]⊗N−j−1

(5a)

a†j ⇔ 1⊗j−1 ⊗ σ− ⊗ σz⊗N−j−1 =

[1 00 1

]⊗j−1

⊗[

0 01 0

]⊗[

1 00 −1

]⊗N−j−1

(5b)

where σ+ ≡ σx+iσy

2 = |0〉〈1| and σ− ≡ σx−iσy2 = |1〉〈0|. The qubit state |0 . . . 0〉

corresponds to the vacuum state and the string of σz operators, preserve thecommutation relations in Eq. (2) since σz and σ± anti-commute. The spinvariable representation of relevant operators after the Jordan-Wigner trans-formation is given in Table A2.

4. Efficient approximations of the unitary propagator by a Trotterdecomposition

As mentioned in the introduction, the idea of using a quantum computerfor obtaining molecular energies requires efficiently approximating the uni-tary propagator, exp(−iHt), for a time sufficiently long to resolve the Fourierfrequency to a desired precision. The present section continues by describingthe Trotter decomposition for non-commuting Hamiltonian terms and thenpresents the quantum circuits for the relevant exponentials.

4.1. Trotter decomposition

Using the second-quantized representation allows for a straightforward decom-position of the exponential map of each term of the Hamiltonian. However, theterms of this decomposition do not always commute. The goal of the Trotterdecomposition is to approximate the time evolution operator of a set of non-


arXiv:1001.3855 7

commuting operators. The operators are exponentiated individually for smalltime steps and the procedure is repeated such that their product provides areasonable approximation to the exponentiation of the sum. Using this approx-imation, the construction of the time propagator can be efficiently carried outon a quantum computer provided that the Hamiltonian can be decomposedinto a sum of local Hamiltonians [6]. The first-order Trotter decomposition isgiven by:

e−iHt =(e−ih1∆te−ih2∆t · · · e−ihN∆t

) t

∆t

+O(t∆t), (6)

where t/∆t is the Trotter number [27]. As the Trotter number tends to infinity,or equivalently ∆t→ 0, the error in the approximation vanishes. If t/∆t is notan integer, the remainder is simulated as another Trotter time slice. There existhigher order approximates (Suzuki-Trotter formulas) which reduce the errorof approximation even further. For instance, the second order approximationsis given by:

e−iHt ≈((e−ih1

∆t

2 · · · e−ihN−1∆t

2

)e−ihN∆t

(e−ihN−1

∆t

2 · · · e−ih1∆t

2

)) t

∆t

+O(t(∆t)2).

(7)Higher order approximations take increasingly more complicated forms [27]

and were first studied in the context of black box quantum simulation of sparseHamiltonians by Berry et al. [28]. They considered Hamiltonians composed ofm efficiently implementable terms and showed that the number of exponen-tials cannot scale better than linear in the time desired and the maximumfrequency of the full Hamiltonian. The proof shows that if sublinear simula-tion of arbitrary Hamiltonians were possible, bounds for the power of quantumcomputation proven in Ref. [29] could be violated leading to a contradiction.

4.2. Quantum circuit primitives

Each term of Eq. (1) can be exponentiated using the universal gate set de-scribed in Subsection 3.1 after performing the Jordan-Wigner transformationto Pauli spin matrices. We will outline the procedure for generating quantumcircuits for chemical systems and summarise the results in Table A1. The con-struction of quantum circuits for general Fermionic Hamiltonians is furtherdiscussed in Ref. [10, 30, 31].

To understand the exponential map of the product of Pauli spin matrices,first consider the exponential map of two σz operators. To create the unitarygate exp[−i θ2(σz⊗σz)], the CNOT gate can be used to first entangle two qubits,then the Rz gate is applied, and followed by a second CNOT gate [13].

• •�� Rz ��

This construction can be generalized to more qubits by using additional CNOTgates. For example, the circuit for the three-body operator involving three


8 Whitfield et al.

qubits, exp[−i θ2(σz ⊗ σz ⊗ σz)], is simulated by the following quantum circuit:

• •�� • • ��

�� Rz ��

As seen from the three-qubit example above, this construction can be readilyextended for n-fold products of σz operators.

4.2.1. Construction of different Pauli matrix tensor products

If one requires a different tensor product of Pauli matrices besides the prod-uct of σz as described in Section 4.2, a change of basis can be accomplishedusing the appropriate unitary transformation: Hadamard transformation (de-noted H) changes between σx basis and σz basis, and Y = Rx(−π/2) =exp(iσxπ/4) transforms from σz basis to σy basis and Y† from σy to σz. Inmatrix form,

H =1√2

[1 11 −1

]Y =

1√2

[1 ii 1

].

Circuits of this form are the basis for the construction of the molecular unitarypropagator as illustrated in Table A1 where the circuit representations aregiven.

5. The phase estimation algorithm

In this section, we describe how to obtain molecular energies given the timeevolution of the molecular Hamiltonian described above. The key idea is toFourier transform the oscillating phase, 〈ψ(0)|ψ(t)〉 = exp(−iEt), to obtainthe electronic energy. The eigenenergy is converted into relative phases. Therelative phase can then be measured using the quantum phase estimationalgorithm (PEA). As the phase is measured the input state partially collapsesto the set of states consistent with the measurements obtained up to thatpoint.

To determine an eigenvalue associated with an eigenstate, consider the phaseof an eigenstate of the Hamiltonian, H, evolving dependent on a register qubit:

|0〉|ψn〉+ e−iHt|1〉|ψn〉 = |0〉|ψn〉+ e−iEnt|1〉|ψn〉. (8)

By letting En = 2π(φ − K)/t where 0 ≤ φ < 1 and K is an unobservableinteger, the unknown eigenvalue becomes encoded in the relative phase of theregister qubit quantum state as |0〉 + e−2πi(φ−K)|1〉 [7, 32, 33]. Then, φ isestimated using the phase estimation technique detailed in Subsection 5.1.

Once φ has been estimated, it must be inverted to obtain the energy. Givenbounds for the energy eigenvalue, [Emin, Emax), the time of evolution is se-lected as t = 2π/(Emax − Emin) = 2π/ω and an energy shift of Es is used toensure K = (Es − Emin)/ω is an integer. The energy shift, Es, is effected bya gate on the register qubit which applies a phase to the qubit if it is in state|1〉 and does nothing otherwise. Using these parameters the measured value ofφ corresponds to the value of the energy, Eφ = ω(φ−K) + Es.


arXiv:1001.3855 9

The PEA can be accomplished in several different ways depending on thetechnology available and in the recent experimental implementations, the it-erative phase estimation algorithm [32, 34–36] was used. We present it herefor completeness.

5.1. Iterative Phase Estimation

The goal of the phase estimation algorithm is to obtain the best L bit estimateof a phase φ. The readout will be values of 0 or 1 from the fractional binaryexpansion of φ:

φ = 0.j0j1j2 . . . jL−1 =

(j02

)+

(j14

)+

(j28

)+ · · ·+

(jL−1

2L

). (9)

Each jk is zero or one. The relative phase generated by the unitary prop-agator U(t) = exp(−iHt) applied as in (8) is defined as exp(2πiφ). Per-forming U(2kt) results in a relative phase exp(2πi(2kφ)). Observing that2kφ = j0j1 . . . jk−1.jkjk+1 . . . we can finite binary expansion of length L de-terministically.

First, U(2Lt) is implemented resulting in relative phaseexp(2πi j0 . . . jL−1.jL) = exp(2πi jL/2). Since jL is 0 or 1, the relativephase is either +1 or -1, respectively. The Hadamard transform describedin Section 4.2.1 distinguishes |0〉 + |1〉 from |0〉 − |1〉 allowing a projectivemeasurement to identify jL.

To obtain jL−1 and more significant bits, the iterative method uses the gateSk to deterministically obtaining jk. Since data is read from the least significantdigit first, the counter-rotation Sk is a function of previously obtained bits. Theform of the Sk gate is:

Sk =

[1 00 Φk

], with Φk = exp

[2πi

L−k+1∑l=2

jk+l−1

2l

].

This gate removes the L−k−1 least significant digits so the state of quantumcomputer becomes (|0〉 + exp[−iπjk]|1〉)|ψn〉 where jk is zero or one. Finally,effecting the Hadamard transformation leads to |jk〉|ψn〉 and measurement ofthe register in the {|0〉, |1〉} basis yields the value of jk. When the binaryexpansion of φ is length L, the measurements are deterministic, otherwise theremainder causes some randomness but does not significantly affect the results.A complete error analysis can be found in Refs. [13, 35].

As noted in Ref. [37], in phase estimation algorithm the number of uses ofU(t0) for a fixed time t0 scales exponentially with the number of bits desiredfrom φ. This is a consequence of the Fourier uncertainty principal; the moreinformation required in the frequency domain the longer the propagation time.When the unitary is decomposed into gates, this means an exponential increasein the gates is required for an exponential increase in the precision of themeasurement.

To produce the state indicated in (8), the evolution of the eigenstate must bedependent on the register qubit requiring that the construction of the unitaryevolution operator described in Section 4.2 be modified. The constructionslisted in Table A1 only need to be slightly modified; since the underlyingconstructions rely on Rz gates, changing these rotations into controlled Rz


10 Whitfield et al.

|0〉 H • Sk H NM jk

|ψ〉 U2k |ψ〉

Figure 2. Iterative phase estimation circuit for obtaining the kth bit. The phase is represented usinga binary decimal expansion φ = 0.j0j1j2j3 · · · jn. To gain precision in the algorithm, one adjusts theparameterized gate Sk according to the values of all previous obtained bits. This U is a representationof the propagator of the system being simulated. Acting on the eigenstate with this operator advancesthe phase of the state allowing bit jk to be read out deterministically.

rotations (|1〉〈1| ⊗ Rz + |0〉〈0| ⊗ 1) is sufficient to make the entire unitarydependent on the readout qubit.

To obtain ground state energies, PEA relies on the assumption that theinput wave function has signification overlap with the ground state. Since eachqubits represents the occupancy of molecular orbitals in the N -electron wavefunction, the HF guess for the ground state |ψHF0 〉 requires no superpositionof states and is thus straightforward to prepare. For |〈ψHF0 |ψFCI0 〉| = 1 − ε,where ε is small, the phase estimation algorithm can be applied to retrieve anestimate of the ground state energy. Simultaneously, the state of the systemwill collapse to |ψFCI0 〉 when measured in the HFCI basis (via PEA) with highprobability [32, 33]. If the Hartree-Fock guess is insufficient, more sophisticatedstate preparation procedures exist and these were reviewed in Ref. [12]. Theadiabatic scheme for state preparation [7] is analyzed in the following section.

6. Adiabatic state preparation

When the initial state prepared as input has low overlap with the groundstate, the initial state must be improved so that PEA will collapse the inputto the correct state with greater efficiency. This section explains the method ofpreparing an input state to the simulation algorithm using adiabatic quantumcomputation [7, 38, 39]. First, consider the Hartree-Fock wave function as anapproximation to the FCI wave function. This is the output of an classicalalgorithm returning, in polynomial time, a computational basis state wherethe qubits which correspond to occupied molecular orbitals are in the state|1〉 with the remaining qubits in state |0〉. To increase overlap of the wavefunction, after the system is prepared in state |ψHF0 〉, the Hamiltonian HFCI isslowly applied and the actual ground state is recovered by adiabatic evolution.Consider a smooth one-parameter family of adiabatic path Hamiltonians,

H(s) = (1− s)HHF + sHFCI , (10)

for monotonic s ∈ [0, 1]. This was the adiabatic path originally proposed byus in Ref. [7]. Other paths may be used as in Ref. [40] but in this study werestrict our attention to evolution of the form in Eq. (10).

Let the instantaneous energies of H(s) be given by the sequence,

E0(s) < E1(s) ≤ · · · ≤ EN−1(s), (11)

then the adiabatic state preparation procedure is efficient whenever the total


arXiv:1001.3855 11

run time, T , satisfies the following:

T � min0≤s≤1

(E1(s)− E0(s))−2 , (12)

according to known results relating the adiabatic theorem to complexity the-ory [38]. After adiabatic evolution, the state of the system is |ψFCI0 〉, whichis the ground state of the molecular Hamiltonian HFCI

0 . Modified versionsof this procedure exist using decoherence to achieve faster evolution and arediscussed in Refs. [41, 42].

Assume that the adiabatic evolution induced transitions into higher energystates and so the un-normalized state of the system is |ψFCI0 〉 + λ|k〉, where|k〉 ∈ H = 1− |ψFCI0 〉〈ψFCI0 |. While the error in the wave function is linear inλ < 1, the overestimate of the energy in the expectation value 〈HFCI

0 〉 is onlyquadratic.

It is unclear how this method will scale. It is possible to prepare a desiredground state efficiently provided that the gap between the ground and excitedstates is sufficiently large [38]. This depends on the adiabatic path taken.Finding the ground state energy of a random Hamiltonian, even for simplemodels, is known to be complete for the quantum analogue of the class NP [43,44].

There are other ways to perform state preparation, for example, by goingbeyond Hartree-Fock theory [45, 46]. A broader discussion of state preparationfor quantum simulation can be found in Refs. [12, 47]. Recently, in Ref. [48],the effects of initial states for the phase estimated quantum simulation CH2

molecules were studied for a variety of geometries and eigenstates using initialguesses obtained via multi-configuration approaches.

7. Simulating the hydrogen molecule

To illustrate the algorithmic details of a scalable simulation of quantum sys-tems, the hydrogen molecule in a minimal basis is used as an instructive exam-ple. The minimal basis is the minimum number of spatial-functions needed todescribe the system and in the case of H2, one spatial-function is needed peratom denoted ϕH1 and ϕH2. In this simple case, the Hartree-Fock procedureis not necessary as the molecular spatial-orbitals are determined by symmetryand are given by ϕg = ϕH1 + ϕH2 and ϕu = ϕH1 − ϕH2. These two spatialfunctions correspond to four orbitals that will be identified as:

|χ1〉 = |ϕg〉|α〉, |χ2〉 = |ϕg〉|β〉, |χ3〉 = |ϕu〉|α〉, |χ4〉 = |ϕu〉|β〉. (13)

The form of the spatial function is determined by the basis set used. TheSTO-3G basis is a common minimal basis that approximates a single electronspatial Slater type orbitals (STO), with a contraction of three real Gaussianfunctions [1]. Using this orbital basis, the spatial integrals of the Hamiltonianwere evaluated in Table 2 for H2 at bond distance 1.401000 atomic units(7.414·10−11 m).

Considering H from Eq. (1) as H(1) +H(2) we have,

H(1) = h11a†1a1 + h22a

†2a2 + h33a

†3a3 + h44a

†4a4 (14)


12 Whitfield et al.

Table 2. The one-electron and two-electron integrals defined in

Eqs. (3) and (4) are evaluated using the molecular spatial orbitals

obtained from a restricted Hartree-Fock calculation at an internu-

clear distance of 1.401000 atomic units (7.414·10−11 m) [49].

Integrals Value (a.u.)

h11 = h22 -1.252477h33 = h44 -0.475934

h1221 = h2112 0.674493

h3443 = h4334 0.697397

h1331 = h1441 = h2332 = h4224

= h3113 = h3223 = h4114 = h42240.663472

h1313 = h2424 = h3241 = h1423 = h1243 0.181287

The following circuit applies the single-electron propagator for a time t:

PEA • • • •

χ1 T(h11t)

χ2 T(h22t)

χ3 T(h33t)

χ4 T(h33t)

The gate T is defined as:

T(θ) =

[1e−iθ

]. (15)

The two electron Hamiltonian can also be expanded. As electrons are indis-tinguishable,

hpqrs =

∫dx1dx2

χp(x1)χq(x2)χr(x2)χs(x1)

r12=

∫dx2dx1

χp(x2)χq(x1)χr(x1)χs(x2)

r12= hqpsr,

and a†pa†qaras = a†qa

†pasar, the two electron Hamiltonian can be simplified as:

H(2) = h1221a†1a†2a2a1 + h3443a

†3a†4a4a3 + h1441a

†1a†4a4a1 + h2332a

†2a†3a3a2

+ (h1331 − h1313) a†1a†3a3a1 + (h2442 − h2424) a†2a

†4a4a2

+<(h1423)(a†1a†4a2a3 + a†3a

†2a4a1) + <(h1243)(a†1a

†2a4a3 + a†3a

†4a2a1)

+=(h1423)(a†1a†4a2a3 − a†3a

†2a4a1) + =(h1243)(a†1a

†2a4a3 − a†3a

†4a2a1)(16)

The first six terms,

h1221a†1a†2a2a1 + h3443a

†3a†4a4a3 + h1441a

†1a†4a4a1

+ h2332a†2a†3a3a2 + (h1331 − h1313) a†1a

†3a3a1 + (h2442 − h2424) a†2a

†4a4a2

can be simulated using the system Hamiltonian that employs only commut-ing two-local terms described in Section 4. Notice after the Jordan-Wigner


arXiv:1001.3855 13

transform of the relevant operator we have:

∑p<q

(hpqqp − hpqpqδσpσq)a†pa†qaqap =

(1

4

∑p<q

(hpqqp − hpqpqδσpσq)

)1

−1

4

∑q

∑p6=q

(hpqqp − hpqpqδσpσq)

σzq

+∑p<q

(hpqqp − hpqpqδσpσq)4

σzpσzq . (17)

The factor of 1/2 is accounted for because the indistinguishably of the electronsreduces the summation in Eq. (1).

Following Eq. (17), let Θ ≡ (1/4)∑

p<q(hpqqp − hpqpqδσpσq) and θp ≡∑q:p 6=q(hpqqp− hpqpqδσpσq). Then following circuit illustrates the one and two-

local interactions required to implement Eq. (17):

PEA T(Θt) • • • •

χ1 Rz(θ1t)

χ2 Rz(θ2t)

χ3 Rz(θ3t)

χ4 Rz(θ4t)

Defining ηpq ≡ 14(hpqqp − hpqpqδσpσq), the three local interactions can be de-

picted as:

PEA • • • • • •

χ1 • • • • • •

χ2 • • • • �� Rz(η12t)��

χ3 • • �� Rz(η23t)�� Rz(η13t)

�� χ4 �� Rz(η34t)

�� Rz(η24t)�� Rz(η14t)

�� Each term commutes thus can be realized in any order.

The remaining terms are strictly real leaving:

h1423(a†1a†4a2a3 + a†3a

†2a4a1) + (h1243)(a†1a

†2a4a3 + a†3a

†4a2a1).

Since the orbitals are real, the integrals h1243 = h1423 are equivalent. Therefore,

we are left with the task of simulating 2h1423(a†1a†4a2a3 + a†3a

†2a4a1).

Consider the general term a†pa†qaras+a

†sa†raqap. Due to the anti-commutation

rules, all sets of operators corresponding to a set of four distinct spin-orbitals,(p, q, r, s), are simulated using the same circuit. This is due to the fact thatthe Jordan-Wigner of each operator generates same set of operators (namely,the eight combinations involving an even number of σx and σy operators).However, depending on if σ+ or σ− is used each term of spin operators willhave a different sign. If we define:

h(1) = (hpqrsδσpσsδσqσr − hqprsδσpσrδσqσs) (18)

h(2) = (hpsqrδσpσrδσqσs − hspqrδσpσqδσrσs) (19)

h(3) = (hprsqδσpσqδσrσs − hprqsδσpσsδσqσr) (20)


14 Whitfield et al.

(a) Plot of gates to simulate the H2

versus time step used in the first orderTrotter approximation.

(b) Plot of relative error of approxima-tion as a function of gates used.

Figure 3. The unitary propagator corresponding to this Hamiltonian is approxi-mated using a first order Trotter decomposition and these graphs provide analysisof the Trotter error and the number of gates used at each Trotter number, Tn. Theunitary propagator is simulated by applying each small term of the second quantizedHamiltonian for small time steps dt and repeating the process Tn = t/dt times. Asdt decreases the error in the approximation decreases at the expense of more gates.Zero error represents the eigenvalue of the Hamiltonian of H2 in the minimal basis ata separation of 1.4 atomic units. The horizontal line of (b) represents the thresholdfor energy error of 10−4 atomic units.

then

1

2

∑hpqrsa

†pa†qaras + a†sa

†raqap =

1

8

q−1⊗k=p+1

s−1⊗k=r+1

σzk

(21)

(σxpσ

xq σ

xrσ

xs + σypσ

yqσ

yrσ

ys )(−h(1) − h(2) + h(3))

+(σxpσxq σ

yrσ

ys + σypσ

yqσxrσ

xs )(+h(1) − h(2) + h(3))

+(σypσxq σyrσxs + σxpσ

yqσxrσ

ys )(−h(1) − h(2) − h(3))

+(σypσxq σxrσ

ys + σxpσ

yqσ

yrσxs )(−h(1) + h(2) + h(3))

.

Applying this to the hydrogen molecule, observe that h(1) = −h(2) and h(3) =0 indicating that only the terms {σxσxσyσy, σyσyσxσx, σyσxσxσy, σxσyσyσx}must be considered. The resulting quantum circuit is illustrated in Table A3.

To assess the Trotter error, we simulated this circuit using a classical com-puter using the first-order Trotter decomposition. The pseudo-code for the H2

simulation is given in Appendix A and the results are summarized in Fig. 3.Although the gates increase with the Trotter number, reducing the Trottererror of the dynamics is only practical if the measurement is precise enoughto detect such errors. Thus, in practice, there is a balance between the Trotternumber selected and the number of bits to be obtained by the measurementprocedure. Finally, as noted in section 4.1, higher-order Trotter decomposi-tions also allow for more precise simulation with larger time steps. In fact,given the largest eigenvalue of the simulated system as λ the number of gatesfor simulation time t is O(tλ) using the appropriate order of Trotter decom-position [28].

8. Conclusions

In this paper, we mapped the full configuration interaction (FCI) methodfrom quantum chemistry into a quantum algorithm. We reviewed the elec-tronic structure problem, techniques of creating the simulated propagator,


arXiv:1001.3855 15

and explicitly illustrated this construction for H2 for a single time step of afirst-order Trotter expansion.

Applicability of quantum simulation comes down to the ability to propagatethe simulated system with a specified error tolerance. Since phase estimationis essentially a Fourier transform of the frequency of phase oscillations (whichare proportional to the eigenenergy) to obtain more precise determination ofthe frequency, a longer propagation time is necessary. The longer simulationtime requires more manipulations of the computational system.

In the coming future, small scale experiments such as the simulation ofthe circuits we have presented for H2 on a quantum computer will likely bepossible. Experimental realizations of quantum chemistry on quantum deviceshave only recently been achieved [17, 18]. We hope that the present paper willcontinue the interest by giving an example of a scalable construction of theunitary propagator for the H2 molecule in an explicit form, which poses thenext logical challenge for experimental realization.

Acknowledgements

We thank H. Wang, A. Dutoi, P. Love, M. Mohseni, B. Lanyon and A. White. This

work received funding from the Faculty of Arts and Sciences at Harvard University,

Engineering and Physical Sciences Research Council grant EP/G003017/1 (JDB),

Defense Advance Research Projects Agency under the Young Faculty Award (N66001-

09-1-2101) (AA-G), the Camille and Henry Dreyfus Foundation (AA-G), the Sloan

Foundation (AA-G), NSF CCI 1037992-CHE, and the Army Research Office under

contract W911-NF-07-0304 (JDW, AA-G).


16 Whitfield et al.

Appendix A. Example program for a quantum computer to calculateelectronic energies for the hydrogen molecule

The time evolution of input states is measured to extract eigenvalues of thesystem. A state preparation procedure must be used if a particular state isdesired (e.g. the ground state). Time evolution is implemented in such a waythat a measurable relative phase is generated on a register qubit. This registerqubit is then measured to obtain a bit of information about the phase. The timeevolution is then effected for half as long and again the register is measuredto obtain the next digit. This process is repeated until all digits are read outand then the phase can be converted to an energy eigenvalue following theprescription of Section 5.

In this appendix, we explicitly spell out the gates that would be necessaryfor effecting a single time step of the evolution generated by the Hydrogenmolecule’s Hamiltonian. The time step idea is to alternatively apply each ofthe non-commuting unitary operators for a fraction of the total time andrepeat the time steps until the total time has been reached. As the lengthof the time steps goes to zero the error of the approximation vanishes. Forconstructing long time evolution the time steps should be composed followingthe Trotter prescription 4.1

The decomposition of the propagator of H2 Hamiltonian is given usingstandard quantum gates. The entire unitary operator is applied if the reg-ister qubit is in state |1〉. Qubits are named: Register, Q1, Q2, Q3, Q4.The integrals hij and hijkl are parametrized by the nuclear configurationand are given in Table 2 for the for the equilibrium geometry. Addition-ally, we define nij = (hijji − hjiji)/4. Note only h1313 and h2424 are theonly non-zero terms due to spin orthogonality. Following the main text, letΘ ≡ (1/4)

∑p<q(hpqqp − hpqpqδσpσq) and θp ≡

∑q:p 6=q(hpqqp − hpqpqδσpσq).


arXiv:1001.3855 17

Gate Target qubit Control qubit Parameter

Hadamard Register

Single Electron OperatorscPhase Q1 Register h11tcPhase Q2 Register h22tcPhase Q3 Register h33tcPhase Q4 Register h44t

Two Electron Operators: number-number operatorPhase Register ΘtcRz Q1 Register -θ1tcRz Q2 Register -θ2tcRz Q3 Register -θ3tcRz Q4 Register -θ4tcNot Q4 Q3cRz Q4 Register 2n34tcNot Q4 Q3cNot Q4 Q2cRz Q4 Register 2n24tcNot Q4 Q2cNot Q4 Q1cRz Q4 Register 2n14tcNot Q4 Q1cNot Q3 Q2cRz Q3 Register 2n23tcNot Q3 Q2cNot Q3 Q1cRz Q3 Register 2n13tcNot Q3 Q1cNot Q2 Q1cRz Q2 Register 2n12tcNot Q2 Q1

Two Electron Operators: excitation-excitation operatorXXYYHadamard Q1Hadamard Q2Rx Q3 -π/2Rx Q4 -π/2cNot Q2 Q1cNot Q3 Q2cNot Q4 Q3cRz Q4 Register -t(h1423 + h1243)/4cNot Q4 Q3cNot Q3 Q2cNot Q2 Q1Rx Q4 π/2Rx Q3 π/2Hadamard Q2

continued on next page


18 Whitfield et al.

Gate Target qubit Control qubit Parameter

Hadamard Q1YYXXRx Q1 -π/2Rx Q2 -π/2Hadamard Q3Hadamard Q4cNot Q2 Q1cNot Q3 Q2cNot Q4 Q3cRz Q4 Register -t(h1423 + h1243)/4cNot Q4 Q3cNot Q3 Q2cNot Q2 Q1Hadamard Q4Hadamard Q3Rx Q2 π/2Rx Q1 π/2XYYXHadamard Q1Rx Q2 -π/2Rx Q3 -π/2Hadamard Q4cNot Q2 Q1cNot Q3 Q2cNot Q4 Q3cRz Q4 Register t(h1423 + h1243)/4cNot Q4 Q3cNot Q3 Q2cNot Q2 Q1Hadamard Q4Hadamard Q3Rx Q2 π/2Rx Q1 π/2YXXYRx Q1 π/2Hadamard Q2Hadamard Q3Rx Q4 π/2cNot Q2 Q1cNot Q3 Q2cNot Q4 Q3cRz Q4 Register t(h1423 + h1243)/4cNot Q4 Q3cNot Q3 Q2cNot Q2 Q1Rx Q1 −π/2Hadamard Q2Hadamard Q3Rx Q4 −π/2


arXiv:1001.3855 19

References

[1]A. Szabo and N. Ostlund, Modern Quantum Chemistry: Introduction to Advanced ElectronicStructure Theory (Dover Publications, Mineola, NY, 1996).

[2]T. Helgaker, P. Jorgensen and J. Olsen, Molecular Electronic-Structure Theory (John Wiley andSons, Chichester, UK, 2000).

[3]C. Sherrill, J. Chem. Phys. 132, 110902 (2010).[4]T.D. Ladd, F. Jelezko, R. Laflamme, Y. Nakamura, C. Monroe and J.L. O’Brien, Nature 464,

45 (2010).[5]R. Feynman, Optics News (now OPN) 11 (11), 11–22 (1982).[6]S. Lloyd, Science 273, 1073 (1996).[7]A. Aspuru-Guzik, A. Dutoi, P. Love and M. Head-Gordon, Science 309, 1704 (2005).[8]D. Lidar and H. Wang, Phys. Rev. E 59, 2429 (2008).[9]I. Kassal, S.P. Jordan, P.J. Love, M. Mohseni and A. Aspuru-Guzik, Proc. Natl. Acad. Sci. 105,

18681 (2008).[10]G. Ortiz, J. Gubernatis, E. Knill and R. Laflamme, Phys. Rev. A 64, 022319 (2001).[11]I. Kassal and A. Aspuru-Guzik, J. Chem. Phys. 131, 224102 (2009).[12]I. Kassal, J. Whitfield, A. Perdomo-Ortiz, M.H. Yung and A. Aspuru-Guzik, preprint,

arXiv:1007.2648 (2010). <http://arxiv.org/abs/1007.2648>.[13]M. Nielsen and I. Chuang, Quantum Computation and Quantum Information (Cambridge Uni-

versity Press, Cambridge, UK, 2001).[14]D. Gottesman, preprint, arXiv:0904.2557 (2009). <http://arxiv.org/abs/0904.2557>.[15]A.Y. Kitaev, Russian Math. Surveys 52, 1191 (1997).[16]C.R. Clark, T.S. Metodi, S.D. Gasster and K.R. Brown, Phys. Rev. A 79, 062314 (2009).[17]B.P. Lanyon, J.D. Whitfield, G.G. Gillet, M.E. Goggin, M.P. Almeida, I. Kassal, J.D. Biamonte,

M. Mohseni, B.J. Powell, M. Barbieri, A. Aspuru-Guzik and A.G. White, Nature Chem. 2, 106(2010).

[18]J. Du, N. Xu, X. Peng, P. Wang, S. Wu and D. Lu, Phys. Rev. Lett 104, 030502 (2010).[19]R. Zwanzig, Time-Correlation Functions and Transport Coefficients in Statistical MechanicsAnn.

Rev. Phys. Chem. 16, 67–102 (1965).[20]E. Heller, The Semiclassical Way to Molecular SpectroscopyAccounts of Chemical Research, Acc.

Chem. Res. 14, 368–375 (1981).[21]M. Feit, J. J.A. Fleck and A. Steiger, Journal of Computational Physics, J. Comp. Phys. 47,

412–433 (1982).[22]S.A. M. Ceotto, G.F. Tantardini and A. Aspuru-Guzik, Multiple coherent states for first-

principles semiclassical initial value representation molecular dynamicsThe Journal ChemicalPhysics, J. Chem. Phys. 130, 234113 (2009).

[23]D. Feller, J. Comp. Chem. 17, 1571 (1996).[24]K.L. Schuchardt, B.T. Didier, T. Elsethagen, L. Sun, V. Gurumoorthi, J. Chase, J. Li and T.L.

Windus, J. Chem. Inf. Model. 47, 1042 (2007).[25]D.P. DiVincenzo, Fortschr. Phys. 48, 771 (2000), also arXiv:quant-ph/0002077.[26]P. Jordan and E. Wigner, Z. Phys. A. 47, 631 (1928).[27]N. Hatano and M. Suzuki, in Quantum Annealing and Other Optimization Methods, edited by

A. Das and B.K. Chakrabarti (Springer, Berlin, Germany, 2005), Lectures Notes in Physics, pp.37–68.

[28]D.W. Berry, G. Ahokas, R. Cleve and B.C. Sanders, Commun. Math. Phys. 270, 359–371 (2007).[29]R. Beals, H. Buhrman, R. Cleve, M. Mosca and R. de Wolf, Quantum Lower Bounds by Poly-

nomials. in Proceedings of FOCS’ 98. <also see, http://arxiv.org/abs/quant-ph/9802049>, pp.352–361.

[30]R. Somma, G. Ortiz, J.E. Gubernatis, E. Knill and R. Laflamme, Phys. Rev. A 65, 042323 (2002).[31]E. Ovrum and M. Hjorth-Jensen, preprint, arXiv:0705.1928 (2007).

<http://arxiv.org/abs/0705.1928>.[32]S. Parker and M. Plenio, Phys. Rev. Lett. 85, 3049–52 (2000).[33]A. Kitaev, preprint, arXiv:quant-ph/9511026 (1995). <http://arxiv.org/abs/quant-

ph/9511026>.[34]A. Tomita and K. Nakamura, Int. J. Quantum Information 2, 119 (2004).[35]M. Dobsıcek, G. Johansson, V. Shumeiko and G. Wendin, Phys. Rev. A 76, 030306(R) (2007).[36]L. Xiu-Mei, L. Jun and S. Xian-Ping, Chinese Phys. Lett. 24, 3316 (2007).[37]K.R. Brown, R.J. Clark and I.L. Chuang, Phys. Rev. Lett. 97, 050504 (2006).[38]E. Farhi, J. Goldstone, S. Gutmann and M. Sipser, Science 292, 472 (2000).[39]L.A. Wu, M.S. Byrd and D.A. Lidar, Phys. Rev. Lett. 89, 057904 (2002).[40]A. Perdomo-Ortiz, S.E. Venegas-Andraca and A. Aspuru-Guzik, Quantum Information Process-

ing pp. 1–20 (2010).[41]S. Boixo, E. Knill and R.D. Somma, Quant. Inf. Comp. 9, 0833 (2009).[42]P. Wocjan and A. Abeyesinghe, Phys. Rev. A 78, 042336 (2008).[43]J. Kempe, A. Kitaev and O. Regev, SIAM J. Computing 35 (5), 1070–1097 (2006).[44]R. Oliveira and B. Terhal, Quant. Inf. Comp. 8, 0900 (2008).[45]H. Wang, S. Kais, A. Aspuru-Guzik and M. Hoffmann, Phys. Chem. Chem. Phys. 10, 5388

(2008).[46]H. Wang, S. Ashhab and F. Nori, Phys. Rev. A 79, 042335 (2009).[47]N. Ward, I. Kassal and A. Aspuru-Guzik, J. Chem. Phys. 130, 194105 (2009).[48]L. Veis and J. Pittner, preprint, arXiv:1008.3451 (2010). <http://arxiv.org/abs/1008.3451>.[49]R.P. Muller, Python Quantum Chemistry (PyQuante) program, version 1.6 (Sandia National

Laboratories, Albuquerque, NM, 2007).


20 Whitfield et. al

Table A1. The quantum circuits corresponding to evolution of the listed Hermitian second-quantized operators. Here, p, q,

r, and s are orbital indices corresponding to qubits such that qubit state |1〉 indicates an occupied orbital and |0〉 indicates

unoccupied. It is assumed that the orbital indices satisfy p > q > r > s. These circuits were found by performing the Jordan-

Wigner transformation given in Eqs. (5b) and (5a) and then propagating the obtained Pauli spin variables [30]. In each

circuit, θ = θ(h) where h is the integral preceding the operator. Gate T(θ) is defined by T|0〉 = |0〉 and T|1〉 = exp(−iθ)|1〉,G is the global phase gate given by exp(−iφ)1, and the change-of-basis gate Y is defined as Rx(−π/2). Gate H refers to the

Hadamard gate. For the number-excitation operator, both M = Y and M = H must be implemented in succession. Similarly,

for the double excitation operator each of the 8 quadruplets must be implemented in succession. The global phase gate must

be included due to the phase-estimation procedure. Phase estimation requires controlled versions of these operators which

can be accomplished by changing all gates with θ-dependence into controlled gates dependent on register qubits.

Second quantized operators Circuit

Numberoperator

hppa†pap T(θ)

Excitationoperator

hpq(a†paq+a†qap) p H •

��

•

��

H Y •

��

•

��

Y

/

q H �� Rz(θ) �� H Y �� Rz(θ) �� Y

Coulomb andexchange operators

hpqqpa†pa†qaqap

p G(θ) Rz(θ) • •

/

q Rz(θ) �� Rz(θ) ��Number-excitationa

operatorhpqqr (a†pa

†qaqar

+a†ra†qaqap)

p M •

��

•

��

M

/

q + 1 �� • • ��q

q − 1 �� •��

•

��

/

r M �� Rz(θ) �� M

where M={H,Y}

Double excitationoperator

hpqrs (a†pa†qaras

+a†sa†raqap)

p M1 •

��

•

��

M†1

/

q M2�� • • �� M†2

/

r M3�� •

��

•

�� M†3

/

s M4�� Rz(θ) �� M†4

where (M1,M2,M3,M4)={(H,H,H,H), (Y,Y,Y,Y),(H,Y,H,Y),(Y,H,Y,H),(Y,Y,H,H),(H,H,Y,Y),(Y,H,H,Y), (H,Y,Y,H)}

Notation: •��

•��

• •�� • • ��≡ �� • • ��

aThe spin variable representation of this operator depends on whether q lies in the range p to r or outside of it.


arXiv:1001.3855 21

Table

A2.

The

spin

vari

able

repre

senta

tion

of

second

quanti

zed

Herm

itia

nop

era

tors

aft

er

perf

orm

ing

the

Jord

an-W

igner

transf

orm

ati

on

toyie

ldte

nso

rpro

ducts

of

Pauli

sigm

am

atr

ices

(spin

vari

able

s)th

at

have

the

pro

per

anti

-com

muta

tion

rela

tions.

See

Eqs.

(5b)

and

(5a)

for

the

form

of

the

transf

orm

ati

on

use

d.

The

subsc

ripts

lab

el

the

qubit

and

the

mole

cula

rsp

inorb

ital

that

corr

esp

onds

toth

at

qubit

.T

he

pre

-facto

rshij

andhijkl

are

one-

and

two-

ele

ctr

on

inte

gra

lsgiv

en

inE

qs.

(3)

and

(4).

Inour

alg

ori

thm

,th

ese

are

calc

ula

ted

usi

ng

acla

ssic

al

com

pute

r.R

are

ly

isth

isth

elim

itin

gst

ep

since

the

basi

sse

tsare

typic

ally

chose

nfo

rease

of

inte

gra

tion.

Inth

ista

ble

,w

epro

vid

efo

rth

ecase

that

the

one-

and

two-e

lectr

on

inte

gra

lsare

com

ple

x.

The

inte

gra

ls

hpqqp

andhpqpq

are

refe

rred

toas

the

Coulo

mb

and

exch

ange

inte

gra

ls,

resp

ecti

vely

.T

he

exch

ange

op

era

tor

has

the

sam

ere

pre

senta

tion

(wit

hopp

osi

tesi

gn)

as

the

Coulo

mb

inte

gra

ldue

toth

e

com

muta

tion

rela

tions.

The

cir

cuit

repre

senta

tion

of

the

exp

onenti

ati

on

of

each

of

these

term

isgiv

en

inT

able

A1.

Des

crip

tion

Sec

ond

Qu

anti

zati

ona

Pau

lire

pre

senta

tion

Nu

mb

erO

per

ator

hppa† pap

hpp

2(1p−σz p)

Exci

tati

onO

per

ator

hpqa† paq

+hqpa† qap

1 2

p−

1 ⊗k=q+

1

σz k

( <{h

pq}(σx qσx p

+σy qσy p)

+={h

pq}(σy qσx p−σx qσy p)

)C

oulo

mb

Op

erat

ors

hpqqpa† pa† qaqap

hpqqp

4

( 1−σz p−σz q

+σz pσz q

)N

um

ber

wit

hb

Exci

tati

onO

per

ator

hpqqra† pa† qaqar

+hrqqpa† ra† qaqap

( p−1 ⊗

k=r+

1

σz k

) [(<{h

pqqr}(σx rσx p

+σy rσy p)

+={h

pqqr}(σy rσx p−σx rσy p)

) −σz q

( <{h

pqqr}(σx rσx p

+σy rσy p)

+={h

pqqr}(σy rσx p−σx rσy p)

)]

Dou

ble

Exci

tati

onO

per

ator

hpqrsa† pa† qaras

+hsrqpa† sa† raqap

( r−1 ⊗

k=s+

1

σz k

) p−

1 ⊗k=q+

1

σz k

<

{hpqrs}

8

σx sσx rσx qσx p−σx sσx rσy qσy p

+σx sσy rσx qσy p

+σy sσx rσx qσy p

+σy sσx rσy qσx p−σy sσy rσx qσx p

+σx sσy rσy qσx p

+σy sσy rσy qσy p

+={hpqrs}

8

σy sσx rσx qσx p

+σx sσy rσx qσx p−σx sσx rσy qσx p

−σx sσy rσy qσy p−σy sσx rσy qσy p

+σy sσy rσx qσy p

+σy sσy rσx qσy p

+σy sσy rσy qσx p

aIt

isass

um

edth

atp>q>r>s

for

all

case

slist

ed.

bT

he

spin

vari

ab

lere

pre

senta

tion

of

this

op

erato

rd

epen

ds

ifq

isan

orb

ital

inth

era

ngep

tor

or

ifit

isou

tsid

eth

isra

nge.


22 Whitfield et. al

Table

A3.

Acir

cuit

that

imple

ments

the

unit

ary

pro

pagato

rass

ocia

ted

wit

hth

etw

oele

ctr

on

op

era

tors

of

the

H2

Ham

ilto

nia

nin

the

min

imal

basi

s.

The

cir

cuit

isderi

ved

from

the

two

non-v

anis

hin

gre

al

inte

gra

lsof

the

two-e

lectr

on

inte

racti

on

and

the

corr

esp

ondin

gop

era

tors

.In

the

cir

cuit

,θ≡

h1423

+h

1243

=.3

6257.

InE

q.

22

two

of

the

four

pre

-facto

rsvanis

hdue

toth

esp

inort

hogonality

how

ever,

ingenera

l,th

ere

would

be

eig

ht

term

sto

sim

ula

te.

PEA

••

••

χ1

H•

•H

Y•

•Y†

H•

•H

Y•

•Y†

χ2

H�� •

•��

HY

�� ••

��Y†

Y�� •

•��

Y†

H�� •

•��

H

χ3

Y�� •

•��

Y†

H�� •

•��

HY

�� ••

��Y†

H�� •

•��

H

χ4

Y��

Rz(−

θ 4t)

��Y†

H��

Rz(−

θ 4t)

��H

H��

Rz(θ 4t)

��H

Y��

Rz(θ 4t)

��Y†

simulation of electronic structure hamiltonians using

Documents