The Bravyi-Kitaev Transformation: Properties And Applications
WWW.Q-CHEM.ORGFULL PAPERThe Bravyi–Kitaev Transformation: Properties andApplicationsAndrew Tranter,[a,b] Sarah Sofia,[c,d] Jake Seeley,[d,e] Michael Kaicher,[f ] Jarrod McClean,[g]Ryan Babbush,[g] Peter V. Coveney,[b] Florian Mintert,[a] Frank Wilhelm,[f ]and Peter J. Love*[d]Quantum chemistry is an important area of application forquantum computation. In particular, quantum algorithmsapplied to the electronic structure problem promise exact, efficient methods for determination of the electronic energy ofatoms and molecules. The Bravyi–Kitaev transformation is amethod of mapping the occupation state of a fermionic system onto qubits. This transformation maps the Hamiltonian ofn interacting fermions to an Oðlog nÞ-local Hamiltonian of nqubits. This is an improvement in locality over the Jordan–Wigner transformation, which results in an O(n)-local qubitHamiltonian. We present the Bravyi–Kitaev transformation inIntroductionQuantum simulation was first proposed by Feynman[1] andallows for an exponential speedup over classical simulation ofsome quantum mechanical systems.[2–6] In the context of quantum chemistry, efficient algorithms have been developed forthe calculation of energy spectra,[7] reaction rates,[8,9] and reaction details.[10] Quantum computational schemes have beenextended into the study of relativistic quantum chemistry.[11]Crucially for this project, the quantum-phase estimation algorithm[12] allows for efficient calculation of molecular energies atan accuracy equivalent to that of classical full configurationinteraction calculations. There are three basic approaches to thequantum simulation of chemical systems.One approach—the so called “first quantization”approach—has been studied in the context of chemical reactive scattering.[10] Here, physical position space is discretized.The electronic wavefunction is then represented in the position representation by the state of the qubits. The chemicalHamiltonian is:X p 2 X qi qji H512Mi i j riji(1)where sums are over nuclei and electrons, pi is the momentumof the ith particle, Mi is the mass of the ith particle, qi is thecharge of the ith particle, and rij is the distance between particles i and j in atomic units. We can simulate the effect of thisHamiltonian by unitarily evolving the qubits through the propagator corresponding to the molecular Hamiltonian, approximated using the quantum split operator method of Zalka[4] orby quantum lattice gas methods.[5,6]detail, introducing the sets of qubits which must be acted onto change occupancy and parity of states in the occupationnumber basis. We give recursive definitions of these sets andof the transformation and inverse transformation matrices,which relate the occupation number basis and the Bravyi–Kitaev basis. We then compare the use of the Jordan–Wignerand Bravyi–Kitaev Hamiltonians for the quantum simulation ofC 2015 Wiley Periodicals, Inc.methane using the STO-6G basis. VDOI: 10.1002/qua.24969An alternative to grid-based first-quantized approaches isthe use of a second-quantized formalism. Here, the molecularHamiltonian is expressed in terms of creation and annihilationoperators acting on some basis of molecular orbitals. Thismethod is the main topic of this article and so discussed in[a] A. Tranter, F. MintertDepartment of Physics, Imperial College London, South KensingtonCampus, London SW7 2AZ, United Kingdom[b] A. Tranter, P. V. CoveneyCentre for Computational Science, University College London, 20 GordonStreet, London, WC1H 0AJ, United Kingdom[c] S. SofiaPhotovoltaics Research Laboratory, Massachusetts Institute of Technology,Cambridge, Massachusetts 02139[d] S. Sofia, J. Seeley, P. J. LoveDepartment of Physics, Haverford College, 370 Lancaster Ave., Haverford,Pennsylvania 19041E-mail: plove@haverford.edu[e] J. SeeleyEarth and Planetary Science, University of California, Berkeley, 307McCone Hall, Berkeley, California 94720-4767[f ] M. Kaicher, F. WilhelmTheoretical Physics, Saarland University, 66123 Saarbr ucken, Germany[g] J. McClean, R. BabbushDepartment of Chemistry and Chemical Biology, Harvard University,Cambridge, Massachusetts 02138Contract grant sponsor: NSF CCI center, Quantum Information for QuantumChemistry (QIQC); contract grant number: CHE-1037992.Contract grant sponsor: NSF award; contract grant number: PHY-0955518.Contract grant sponsor: AFOSR award; contract grant number: FA9550-121–0046.Contract grant sponsor: DOE Computational Science Graduate Fellowship;contract grant number: DE-FG02–97ER25308.Contract grant sponsor: EPSRC and UCLQ for support through a UCLQ visiting fellowship (P.J.L.).C 2015 Wiley Periodicals, Inc.VInternational Journal of Quantum Chemistry 2015, 115, 1431–14411431
FULL PAPERWWW.Q-CHEM.ORGgreater detail below. While this technique scales less efficientlythan the prior method in the asymptotic limit, smaller scalesimulations require substantially fewer resources. The reason isthat a molecular orbital basis is more efficient for the representation of localized chemical wavefunctions than a Cartesiangrid, and hence the first-quantized methods lead to wider, butshallower circuits. Details of resource requirements for suchfirst-quantized simulations for chemistry are given in [10].One of the differences between the first- and secondquantized approaches lies in whether the antisymmetricnature of the wavefunction is represented through propertiesof the state (first quantized) or the operators (second quantized). An alternative to grid-based methods in which thedynamics preserves an initially antisymmetric wavefunction isthe use of a basis of Slater determinants. In this case, the challenge for quantum algorithms is the evolution under the CImatrix representation of the Hamiltonian. Unlike the secondquantized case, this matrix has no natural expression as a sumof local terms, and no tensor product structure. However, theCI matrix is sparse, and hence quantum simulation techniquesfor sparse matrices may be applied to this problem. This yieldsmethods that both use an efficient molecular orbital representation of the wavefunction and have optimal asymptotic scaling. This also enables the use of sparse methods, which scalelogarithmically with the error. The penalty is that the molecularintegrals must be computed on the fly during the quantumcomputation.[13–15]The calculation of the energies of molecular Hydrogen andHelium-Hydride using minimal basis sets have been experimentally achieved using linear optical quantum, NMR, andNitrogen vacancy in diamond quantum computers.[16–19] Thefirst digital fermionic quantum simulation was recentlyachieved of a four-site Hubbard model in superconductinghardware.[20] These proofs of principle demonstrations arecomparable to early quantum chemical calculations carried outin the twentieth century.[21]The development and optimization of quantum algorithmsfor chemistry is ongoing. This work is driven by two goals.First is the desire to determine the true optimal asymptoticscaling of these algorithms for large quantum computers. Thesecond is to reduce the resource requirements of small examples to the point that they can be realized experimentally inthe near future. Recently, the possibility of using a small quantum computer of around a hundred qubits for the purposes ofquantum chemistry has been investigated in detail. Initialupper bounds on the cost indicated that large polynomialscaling would be impractical for such problems.[22] Furtheranalysis developing circuit improvements, tighter upperbounds, and numerical investigation of errors restricted to thechemical ground state resulted in tight and efficiently computable upper bounds on the resources required.[23–25] One mayalso improve these algorithms by exploiting locality.[26]The topic of this article is the Bravyi–Kitaev transformation,an alternative to the use of the Jordan–Wigner transformationto map fermions to spins.[27–29] This transformation wasdefined in [28] in the context of using fermions to performquantum computations. Its use for the simulation of fermions1432International Journal of Quantum Chemistry 2015, 115, 1431–1441by quantum computers, and in particular, its use for the quantum simulation of quantum chemistry, was introduced in [29].We describe the transformation in detail, and derive some newproperties of the transformation that are relevant to the specific case of second-quantized Hamiltonians defined in a basisof spin–orbitals. We give a new recursive definition for theinverse Bravyi–Kitaev transformation matrix, as well as recursive relationships for the update, parity, and flip sets (definedbelow) which facilitate the computation of these sets. We analyze the efficiency of the Bravyi–Kitaev method for the simulation of the methane molecule. We find that the Bravyi–Kitaevmapping leads to a small improvement, particularly in thenumber of nonlocal gates required for accurate simulation.The Second-Quantized HamiltonianAs in classical quantum chemistry, we invoke the BornOppenheimer approximation, fixing the nuclear coordinates,and calculating the electronic energy at a given geometry. Inthe second-quantized formalism previously mentioned, theelectronic Hamiltonian is given by:X1X†† † H5hij ai aj 1hijkl ai aj ak al2i;ji;j;k;l(2)where hij and hijkl are integrals, which can be efficiently classically precomputed.†The a and a operators in the Hamiltonian are creation andannihilation operators on a basis set of molecular orbitals, asdiscussed below. Note that here, the two-operator terms effectively correspond to single-electron terms, and the fouroperator terms effectively correspond to electron–electroninteraction terms.Because electrons are fermions, we require antisymmetry onexchange of particle index. This is enforced through the use ofanticommutator restrictions on the creation and annihilationoperators: n † †oaj ; ak 5 aj ; ak 50no†aj ; ak 5djk I(3)Our task, therefore, is effectively to find the lowest eigenvalue of this Hamiltonian. As the dimension of the Fock spacegrows exponentially with the number of basis orbitals, this isclassically intractable for systems of any reasonable size. However, a quantum computer could remove this problem usingquantum-phase estimation.[7]To achieve this, three steps must be taken. First, a mappingbetween the physical electronic states and qubit states in aquantum computer must be established. Second, a welldefined evolution operator equivalent to that of the molecularHamiltonian must be determined for the qubit basis. Thisnecessitates the derivation of qubit representations of theelectronic creation and annihilation operators. Finally, thephase estimation algorithm requires the preparation of a guiding state. A guiding state is an input state to the algorithm,WWW.CHEMISTRYVIEWS.ORG
FULL PAPERWWW.Q-CHEM.ORGwhich has an overlap with the true ground state, which decaysat worst as an inverse polynomial in the system size.In the worst case, Hamiltonians are known for which theproblem of finding the ground state is QMA-complete (thequantum equivalent of NP-complete).[30,31] Quantum computersare not believed to be capable of efficiently solving QMAcomplete problems in the worst case, just as classical computersare not believed to be capable of efficiently solving NP-completeproblems in the worst case. Assuming this is true, there existHamiltonians for which no efficiently preparable guiding state islikely to be available, and for which, the phase estimation algorithm is, therefore, incapable of finding the ground state.However, these worst case Hamiltonians rely on clock constructions so that their ground states are superpositions of quantum states corresponding to time slices of an arbitrary quantumcircuit of depth polynomial in the number of qubits.[30] Evenconstructions that show the QMA-completeness of specific physical models rely on geometrically complex interactions.[32] It is,therefore, a widely believed conjecture that typical physical Hamiltonians do not correspond to worst case instances, and therefore, have efficiently preparable ground states. Specificalgorithms for state preparation are considered in [7,33–36].One may also ask whether the requirement to prepare guidingstates may rely on features of physical Hamiltonians, which canalso be exploited for the development of classical algorithms.The requirement on a guiding state for a quantum computationof an energy eigenvalue is only that its overlap with the trueground state is bounded by an inverse polynomial in the systemsize. Recent consideration of Quantum Monte Carlo methods(which simulate quantum systems using conventional computers)showed that a much stronger guiding state was required tomake these methods efficient, even in the case of so-called stoquastic Hamiltonians where there is no fermion sign problem.[37]Qubit creation and annihilation operatorsIn this section, we describe three mappings of fermionic statesand operators to qubit states and operators. In each case, wemap the occupation number basis to the qubit basis. The occupation number configuration basis states are given by specifying the occupation fi 2 f0; 1g of every orbital. The fermioniccreation and annihilation operators, when acting on a system ofn orbitals with occupation state vector, jfn21 fn22 :::f1 f0 i yield:j21P†fsaj jfn21 :::fj11 0fj21 :::f1 f0 i5ð21Þs50 jfn21 :::fj11 1fj21 :::f1 f0 i†aj jfn21 :::fj11 1fj21 :::f1 f0 i50(4)(5)j21Pfsaj jfn21 :::fj11 1fj21 :::f1 f0 i5ð21Þs50 jfn21 :::fj11 0fj21 :::f1 f0 iaj jfn21 :::fj11 0fj21 :::f1 f0 i50valid from the point of view of realizing the anticommutationrelations, and is more common in the chemical literature.As can be seen in Eqs. (4) through (7), these operatorsdepend on both the occupation of orbital j as well as its parityPpj 5 j21s50 fs , as the phase shift in Eqs. (4) and (6) can be writtenin terms of the parity as ð21Þpj . If the parity is odd, the stateis multiplied by a factor of 21, and if it is even, there is nophase shift. Since the fermionic creation and annihilation operators change both occupation and parity, their qubit analogues also need to do so. Therefore, both the occupation andparity of each orbital must be stored when mapping from theoccupation basis state onto a qubit basis state.We consider three mappings where sums of fermionic occupations are stored in the qubit state. These are the Jordan–Wigner basis, the parity basis and the Bravyi–Kitaev basis. In allcases, it is helpful to define several subsets of the qubits,which contain the information needed to apply fermionicoperators to the state. These sets are defined below, and we use f i to indicate Boolean negation 051;150.1. The update set, U(i). This is the set of qubits, apart from ithat must be updated, when the occupancy fi changes.2. The parity set P(i). This is the set of qubits that deterPmines the parity pi 5 j i fj. Note that the occupancy fi isnot included in this sum.3. The flip set F(i). This is the set of qubits that determineswhether the qubit value is equal to the occupancy fi orits negation f i4. The remainder set R(i). The flip set is a subset of the parity set, and so it is convenient to define the complementof the flip set in the parity set—RðiÞ5PðiÞnFðiÞ.Note that all of the sets U(i), P(i), F(i) and R(i) are definedsuch that the reference qubit i is never a member of them.Our task is then to represent the electronic creation andannihilation operators as operators on the qubit space. Qubitcreation and annihilation operators can be defined in terms ofPauli operators as follows:11Q 5 j1ih0j 5 ðX2iY Þ212Q 5 j0ih1j 5 ðX1iY Þ2(8)These operators contain only Pauli operators acting on the qubitbeing created or annihilated. They, therefore, commute for differentqubits, and so clearly do not fulfil the anticommutation relationsrequired. We must combine these operators with actions on thesets defined above to obtain qubit creation and annihilation operators that satisfy the canonical fermionic commutation relations.(6)(7)We note that one is free to choose the ordering of the orbitals here. We have chosen an ordering in which the orbitals fswith s j determine the parity, but the choice s j is equallyThe Jordan–Wigner transformationIn the Jordan–Wigner transformation, we use the state of a qubitto denote whether or not a particular basis orbital is occupied—clearly, as electrons are fermionic, occupation numbers which arenot zero or one are impossible. The qubits directly store theInternational Journal of Quantum Chemistry 2015, 115, 1431–14411433
FULL PAPERWWW.Q-CHEM.ORGoccupation basis.[38] In this case, the update set is empty (recallthat qubit i is not a member of the update set).Parity information needed to correctly apply the creationand annihilation operators for orbital i is contained in allqubits j i. Hence the parity set is defined by PðiÞ5fjjj ig.This is the Jordan–Wigner transformation.[27] We consequentially have the qubit operators:parity nonlocally or vice versa, as is the case for the Jordan–Wignerand parity bases. In the Bravyi–Kitaev basis, for any index j, if j iseven, qubit j holds only the occupation state of orbital j, and if j isodd, qubit j holds a partial sum of the occupation state of a set oforbitals of index less than j. The Bravyi–Kitaev transformation thatmaps the fermionic occupation state vector to the qubit state,denoted bn for n orbitals such that bn fn 5b n , is given by:1†ai 5 ðXi 2iYi Þ Zi 5Q1i ZPðiÞ2j i1ai 5 ðXi 1iYi Þ Zi 5Q2i ZPðiÞ2j i(9)where ZPðiÞ means a Pauli Z operator acting on all qubits inthe set P(i). The fact that these operators obey the fermionicanticommutation relations follows from the fact that fZ; Q6 g50 and fQ1 ; Q2 g5I.Parity basisThe Jordan–Wigner transformation stored occupancy locally,and parity is nonlocal. The parity basis stores the paritylocally,[28] and the occupancy is nonlocal. The parity information of each orbital j is stored in the corresponding qubit j,jXqj 5pj 1fj 5fs :(10)s50Evidently, PðjÞ5fj21g in the parity basis. Whether qubit jstores fj or f j is determined by qubit j 2 1 in the parity basis.Hence, the flip set in this basis is equal to the parity set:FðjÞ5PðjÞ, and so the remainder set RðjÞ51. The update setU(j) is the set of qubits that must be updated when occupancyfj changes. Now fj appears in every qi such that i j, andso when fj changes every qubit i j must be updated. Hence,UðjÞ5 fiji jg for the parity basis. Given the definitions ofthese sets, we can now write the qubit creation and annihilation operators.†11aj 5ð Xi Þ ðP0FðjÞ Q2j 1PFðjÞ Qj Þ ZPðjÞi j(11)12aj 5ð Xi Þ ðP0FðjÞ Q1j 1PFðjÞ Qj Þ ZPðjÞi jwhere Pb 5jbihbj. Now, because PðjÞ5FðjÞ and because Px Z5ð21Þx Px we can write:1†11aj 5ð Xi Þ ðP0FðjÞ Q2j 2PFðjÞ Qj Þ5 ð Xi ÞðZj Zj21 2iYj Þ2 i ji j112aj 5ð Xi Þ ðP0FðjÞ Q1j 2PFðjÞ Qj Þ5 ð Xi ÞðZj Zj21 1iYj Þ2 i ji j(12)The number of nontrivial Pauli factors in these operatorsscales as O(n), just as for Jordan–Wigner. In this case, it is theupdate set whose size scales linearly with the number of qubits.Bravyi–Kitaev transformationThe Bravyi–Kitaev transformation stores both occupation and parity nonlocally, rather than storing the occupation state locally and1434International Journal of Quantum Chemistry 2015, 115, 1431–1441where1 ! indicates a row of ones in the bottom row.For example, for eight qubits, the fermion occupation statevector is mapped to the qubit basis state as shown in Eq. (14)(all sums in mod(2)):01BB1BBB0BBB1BBB0BBB0BBB0@0000 001000 000100 001110 000001 000001 100000 011 1111 111 01f0f0CCB C BCB C Bf1 1f00CCCB f1 C BCCB C BCCCBBf20 CB f2 C BCCCB C BCB f3 C Bf1f1f1f0C3210CCB C BCCB C5BCCCBBf40 CB f4 C BCCCB C BCB C Bf5 1f40CCCB f5 C BCCB C BCB f6 C Bf0C6AA@ A @f7f7 1f6 1f5 1f4 1f3 1f2 1f1 1f01010(14)From this definition, we proceed to obtain the update, parity, flip, and remainder sets.The update set, U(j), is the set of qubits that must beupdated when the occupation of some orbital j is changed.This is the set of qubits that hold partial sums that depend onthe occupation of orbital j. Because the transformation matrixbn is lower diagonal, only qubits with i j will be contained inU(j). We abuse notation to write U(j) j to indicate this. Sincequbits of even j hold only the occupation state of orbital j, theupdate set will only contain odd qubit indices, as only qubitswith odd j hold partial sums. From the Bravyi–Kitaev transformation matrix, given that any column j contains the vectorthat acts on occupation state vector entry j, the update set forchanging the occupation of orbital j is simply the set of qubitswith index greater than j and equal to the indices of the nonzero entries in column j.[29] The update sets for each orbitalfor systems of 1–8 orbitals are given in Table 1.The parity set, P(j), is the set of qubits needed to determinethe parity of the set of orbitals with index j. The parities aredetermined from the occupation number vector by the actionof a matrix p, which is defined by(½pn ij 51if i j0 otherwise(15)Note this is not the transformation matrix, which gives theparity basis, as pn has a zero diagonal, given that, it computesthe parity of all orbitals strictly less than i. For four orbitals,this matrix is given by:WWW.CHEMISTRYVIEWS.ORG
FULL PAPERWWW.Q-CHEM.ORGTable 1. Indices of qubits in the update set, U(j), which is the set of all qubits whose state must be updated when the occupation state of an orbital j ischanged, for systems of 1–8 orbitals.# Qubits22448888# }{1}{1, 3}{1, 3}{1, 3, 7}{1, 3,7}{1, 3,7}{1, 3,7}–1{3}{3}{3, 7}{3, 7}{3, 7}{3, 7}––{3}{3}{3, 7}{3, 7}{3, 7}{3, 7}–––1{7}{7}{7}{7}––––{5, 7}{5, 7}{5, 7}{5, –––––––100BB1Bp4 5BB1@10000101101C0CCC0CA(16)0number of these qubits scales as Oðlog ðjÞÞ Oðlog ðnÞÞ.[28,29]Since the fermionic occupation state vector fn is transformedinto the Bravyi–Kitaev basis, b n by bn fn 5b n , this transformationcan be reversed to get back to the fermionic occupation basis by b21n bn 5fn . We find that the parity transformation for theBravyi–Kitaev basis is pn b21n . For eight orbitals:and addition is taken modulo two in the matrix multiplication.This method stores the parity of orbital j in partial sums heldin several qubits of index less than or equal to j, where the00 000 000000 000100 000110 000001 000001 100001 0100 001 011BB1BBB0BBB021 B p 8 5p8 f8 5p8 b8 b8 BB0BBB0BBB0@Therefore, the parity set, P(j), is the set of qubits with indexequal to the nonzero entries of pn b21n in row j as these are thequbits whose sum gives the parity of orbital j.[29] The productpn b21is lower triangular because it is the product of twonlower triangular matrices. Hence, the parity set P(j) only contains indices i j, so P(j) j. This also implies that the intersection of parity and update sets is always empty. The parity setsfor each orbital for systems of 1 through 8 orbitals are givenin Table 2.Lastly, the flip set, F(j), is the set of qubits that determinewhether qubit j and orbital j are equal or opposite. The flip setis the set of qubits that hold the parity of the occupation ofthe orbitals with index j included in the partial sum held inqubit j. Note that this definition implies that the flip set is asubset of the parity set, and because the qubits hold partialsums, it is usually a proper subset of the parity set. As foreven j, qubit j holds the occupation of orbital j, the flip set isempty for all even j. However, for odd j, we need to find whichoccupation states are included in each partial sum to trans-010b01 010CCB C BCB C Bb00CCCB b 1 C BCCB C BCB b2 C Bb0C1CCB C BCCB C BB b3 C B b2 1b1 C0CCCB C BCCB C5BCB b4 C Bb0C3CCB C BCCB C BB b5 C B b4 1b3 C0CCCB C BCCB C BB b6 C B b5 1b3 C0CAA@ A @b7b6 1b5 1b30(17)form back to the fermionic occupation state. To do this, wecan look at the inverse transformation. For eight qubits,01 000 00100 00010 00111 00000 10000 11000 000 001 01BB1BBB0BBB021 Bb8 5BB0BBB0BBB0@0 01C0 0CCC0 0CCC0 0CCC0 0CCC0 0CCC1 0CA1 1(18) As b21n bn 5fn , the set of qubits whose states sum to theoccupation state of orbital j are those with indices equal tothe indices of nonzero entries in row j of b21n . Therefore, theflip set of orbital j is the set of these qubits with indices j, asthis is the set of qubits that hold the sum of all occupationInternational Journal of Quantum Chemistry 2015, 115, 1431–14411435
FULL PAPERWWW.Q-CHEM.ORGTable 2. Indices of qubits in the parity set, P(j), which is the set of qubits whose occupation is needed to determine the parity of the orbital j, for eachorbital in systems of 1–8 orbitals.# Qubits22448888# }{1}–––{1, 2}{1, 2}{1, 2}{1, 2}{1, 2}––––{3}{3}{3}{3}–––––{3, 4}{3, 4}{3, 4}––––––{3, 5}{3, 5}–––––––{3, 5, 6}states of orbitals with index j held in qubit j. The flip sets foreach orbital of systems of 1–8 orbitals are listed in Table 3.One final property of these sets is worth remarking upon.The Jordan–Wigner transformation finds application in condensed matter physics to map one-dimensional spin systemsto noninteracting spinless fermions, hence, enabling exactdiagonalization of several systems of interest.[39] Here, we areusing the Bravyi–Kitaev transformation to map fermions withspin to spin one-half systems. The fact that we are mappingspin-orbitals to qubits manifests itself in the pairing of spinorbitals that correspond to the same spatial orbital, and inextra structure present in our second-quantized Hamiltonian.We assume that even-numbered spin-orbitals correspond toone spin function and odd to the other, such that adjacentspin-orbitals correspond to the same spatial function. The factthat Bravyi–Kiatev treats even and odd orbitals differently isconvenient in this case. In particular, the fact that U(j) j andP(j) j gives rise to the intersections among even and odd parity and update sets shown in Table 4.Bravyi–Kitaev operatorsHaving defined the update, parity, and flip sets for the Bravyi–Kitaev transformation, we can define qubit creation and annihilation operators. For even indexed qubits, this is relativelysimple. Even indexed qubits only store their correspondingoccupation, performing operations requires only the actual6creation or annihilation operation (Q ), updating the updateset with a bit flip, and introducing a negative sign dependingon the parity of the parity set. Hence, the creation and annihilation operator equivalents for even indexed qubits are:1†aj 5XUðjÞ Q j ZPðjÞ 5 1 XUðjÞ Xj ZPðjÞ 2iXUðjÞ Yj ZPðjÞ2(19) 1 2aj 5XUðjÞ Q j ZPðjÞ 5 XUðjÞ Xj ZPðjÞ 1iXUðjÞ Yj ZPðjÞ2(20)where we know that U(j) j and P(j) j, so these operators acton disjoint sets of qubits.The qubit operators for qubits with odd index are morecomplicated. First, we note that where the flip set has nonzeroparity, the occupation of the qubit in question is flipped fromthat of the electronic state. Consequentially, in this case, thecreation operator must be applied to the qubit where theannihilation operator is applied to the electronic state, andvice versa. Therefore, defining projectors onto the even andodd states of a set, S, of qubits:1E S 5 ðI1ZS Þ21O S 5 ðI2ZS Þ2(21)We then have new creation and annihilation operators toexpress this behavior: 6 E FðjÞ 2Q 7 O FðjÞ 5 1 Xj ZFðjÞ 7iYj 6 5QPjjj2(22)Here, we have already implicitly accounted for the phase ofthe qubits in F(j). Thus, in determining whether a sign changemust be implemented, we must only additionally determinethe phase of the qubits in the parity set which are not in theTable 3. Indices of qubits in the flip set, F(j), which is the set of qubits that determine whether orbital j and qubit j have the same or flipped parity, forsystems of 1–8 orbitals.# Qubits224488881436# �{1, 2}{1, 2}{1, 2}{1, 2}{1, �–––11–––––––{3, 5, 6}International Journal of Quantum Chemistry 2015, 115, 1431–1441WWW.CHEMISTRYVIEWS.ORG
FULL PAPERWWW.Q-CHEM.ORGTable 4. Intersections between parity and update sets appearing in theBravyi–Kitaev transformation for adjacent odd and even orbital P(2i)Pð2i11ÞU(2i) Uð2i11Þ11 Uð2i11ÞUð2i11Þ1111P(2i) Pð2iÞ11 Pð2i11ÞPð2i11Þbottom left quadrant is all zero except the bottom, right-mostentry.We can verify this form for b21 directly. The equation forthe inverse Bravyi–Kitaev transformation matrix satisfies thecondition that bn b21n 5I. From Eqs. (13) and (28), we obtain:flip set. To do this, we make use of the remainder set, definedabove. This gives us the qubit representation of the electroniccreation and annihilation operators for odd indexed orbitals:1† ZRðjÞ 5aj 5XUðjÞ Pjaj 5XUðjÞ 2Pj 1 XUðjÞ Xj ZPðjÞ 2iXUðjÞ Yj ZRðjÞ2(23) 1 ZPðjÞ 5 XUðjÞ Xj ZPðjÞ 1iXUðjÞ Yj ZRðjÞ2(24)The only difference between these operators and those ofthe even indexed qubits, is the application of Z to the remainder set, rather than the parity set, in the second term. Thus,by defining a final set:(qðjÞ5PðjÞ;j evenRðjÞ;j odd 1 XUðjÞ Xj ZPðjÞ 2iXUðjÞ Yj ZqðjÞ2 1 aj 5 XUðjÞ Xj ZPðjÞ 1iXUðjÞ Yj ZqðjÞ2†(26)(27)These two expressions allow for general products—such asthose observed in the molecular Hamiltonian—to be built upthrough simple multiplication.[29]Inverse Bravyi–Kitaev Transformation MatrixWe have constructive definitions for the Bravyi–Kitaev transformation matrix for any number of qubits that is a power oftwo, and for the parity transformation matrix in the occupationstate basis. We do not have a constructive definition for theinverse Bravyi–Kitaev transformation matrix. The parity, update,and flip set are determined by these three matrices, so a constructive definition for b21would greatly simplify the processnof
mated using the quantum split operator method of Zalka[4] or by quantum lattice gas methods.[5,6] An alternative to grid-based first-quantized approaches is the use of a second-quantized formalism. Here, the molecular Hamiltonian is expressed in terms of creation and annihilation operators acting on some basis of molecular orbitals. This
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On an exceptional basis, Member States may request UNESCO to provide thé candidates with access to thé platform so they can complète thé form by themselves. Thèse requests must be addressed to esd rize unesco. or by 15 A ril 2021 UNESCO will provide thé nomineewith accessto thé platform via their émail address.
̶The leading indicator of employee engagement is based on the quality of the relationship between employee and supervisor Empower your managers! ̶Help them understand the impact on the organization ̶Share important changes, plan options, tasks, and deadlines ̶Provide key messages and talking points ̶Prepare them to answer employee questions
Dr. Sunita Bharatwal** Dr. Pawan Garga*** Abstract Customer satisfaction is derived from thè functionalities and values, a product or Service can provide. The current study aims to segregate thè dimensions of ordine Service quality and gather insights on its impact on web shopping. The trends of purchases have
Chính Văn.- Còn đức Thế tôn thì tuệ giác cực kỳ trong sạch 8: hiện hành bất nhị 9, đạt đến vô tướng 10, đứng vào chỗ đứng của các đức Thế tôn 11, thể hiện tính bình đẳng của các Ngài, đến chỗ không còn chướng ngại 12, giáo pháp không thể khuynh đảo, tâm thức không bị cản trở, cái được
Clifford algebra has been done in Refs. 32–34. In this article, we generalize the Kitaev model from the Pauli matrices to the Clifford algebra of andmatrices. For the 4 24 representation, we construct a model in a decorated square lattice with coordination number 5, which can be in-terpreted as a spin-3 2magnetic model with anisotropic inter-
Adventure Tourism has grown exponentially worldwide over the past years with tourists visiting destinations previously undiscovered. This allows for new destinations to market themselves as truly .
