Benchmarking Quantum Chemistry Computations with Variational, Imaginary Time Evolution, and Krylov Space Solver Algorithms
Kübra Yeter-Aydeniz, Bryan T. Gard, Jacek Jakowski, Swarnadeep Majumder, George S. Barron, George Siopsis, Travis Humble, Raphael C. Pooser
BBenchmarking Quantum Chemistry Computations with Variational, Imaginary TimeEvolution, and Krylov Space Solver Algorithms
K¨ubra Yeter-Aydeniz,
1, 2, ∗ Bryan T. Gard, † Jacek Jakowski, ‡ Swarnadeep Majumder, § George S. Barron, ¶ George Siopsis, ∗∗ Travis Humble, †† and Raphael C. Pooser ‡‡ Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Computational Sciences and Engineering Division,Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Department of Physics, Virginia Tech, Blacksburg, VA 24061, U.S.A Department of Electrical and Computer Engineering, Duke University, Durham, NC 27708 USA Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996-1200, USA Quantum Science Center, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6211 USA (Dated: February 18, 2021)The rapid progress of noisy intermediate-scale quantum (NISQ) computing underscores the needto test and evaluate new devices and applications. Quantum chemistry is a key application areafor these devices, and therefore serves as an important benchmark for current and future quantumcomputer performance. Previous benchmarks in this field have focused on variational methods forcomputing ground and excited states of various molecules, including a benchmarking suite focusedon performance of computing ground states for alkali-hydrides under an array of error mitigationmethods. Here, we outline state of the art methods to reach chemical accuracy in hybrid quantum-classical electronic structure calculations of alkali hydride molecules on NISQ devices from IBM.We demonstrate how to extend the reach of variational eigensolvers with new symmetry preservingAns¨atze. Next, we outline how to use quantum imaginary time evolution and Lanczos as a comple-mentary method to variational techniques, highlighting the advantages of each approach. Finally,we demonstrate a new error mitigation method which uses systematic error cancellation via hiddeninverse gate constructions, improving the performance of typical variational algorithms. These re-sults show that electronic structure calculations have advanced rapidly, to routine chemical accuracyfor simple molecules, from their inception on quantum computers a few short years ago, and theypoint to further rapid progress to larger molecules as the power of NISQ devices grows.
I. INTRODUCTION
Electronic structure calculation in quantum chemicalsystems is one of the most important applications of noisyintermediate scale quantum (NISQ) computers. Becauseof its important role as a key application, it has recentlyfilled the gap in application-level benchmarks of quantumcomputer performance. Computational chemistry haslong played a major role in benchmarking classical com-puters, starting with interpretation of molecular spectra This manuscript has been authored by UT-Battelle, LLC, underContract No. DE-AC0500OR22725 with the U.S. Departmentof Energy. The United States Government retains and the pub-lisher, by accepting the article for publication, acknowledges thatthe United States Government retains a non-exclusive, paid-up,irrevocable, world-wide license to publish or reproduce the pub-lished form of this manuscript, or allow others to do so, for theUnited States Government purposes. The Department of Energywill provide public access to these results of federally sponsoredresearch in accordance with the DOE Public Access Plan. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] ∗∗ [email protected] †† [email protected] ‡‡ [email protected] against quantum chemistry calculations. Its importancehas been recognized in Gerhard Hertzberg’s 1971 No-bel lecture awarded “ for his contributions to the knowl-edge of electronic structure and geometry of molecules ”.Today, a wide range recent electronic structure calcu-lations using variational methods in the literature rep-resent a promising paradigm within which to evaluateNISQ computers as well [1–6]. Recently these ideas werecombined into a benchmark suite that aimed to enumer-ate the potential of cloud NISQ devices under an arrayof configurations including active orbitals, Ansatz con-struction, and error mitigation methods [7]. Furtherresults in preparing and measuring quantum states forsimple molecules indicate a growing capability that mayeventually demonstrate quantum computational advan-tage [8], which makes tracking and validating perfor-mance of NISQ devices in this field all the more impor-tant.Notably, recent methods for benchmarking, includingquantum volume benchmarking [9], cross-entropy bench-marking [10], and randomized benchmarking [11], pro-vide good low-level detail about quantum gate perfor-mance, but they have lacked an application-specific con-text that relates to the role of computation as a toolfor discovery. This gap is compounded by the fact thatperformance metrics such as single and two qubit gatefidelities are not always good predictors of higher level a r X i v : . [ qu a n t - ph ] F e b application performance. This gap further motivates theuse of quantum chemistry as a key performance indi-cator in the NISQ era. Here, we present the currentprogress in developing this application area as a perfor-mance benchmark, and we outline the accuracy and per-formance that is achievable for electronic structure cal-culations on quantum computers today with the state ofthe art algorithms and error mitigation techniques. Wesubsequently estimate where progress in this field canmove to in the near term in light of these developments. II. QUANTUM CHEMISTRY ALGORITHMS
One of the most well-known NISQ algorithms isthe variational quantum eigensolver (VQE), which hasproven to be especially versatile for testing and tun-ing applications of quantum chemistry on NISQ devices.In particular, the VQE method offered the first experi-mental example of an electronic structure calculation byquantum computing on quantum photonic hardware [12].Upon its first implementation on superconducting hard-ware, the algorithm was shown to outperform time evo-lution via Trotterization and subsequent phase estima-tion [13]. The advantage of variational methods is thatlow-depth, parameterized circuits produce simple trialwavefunctions, minimizing errors due to decoherence,while classical optimization proved robust to the system-atic and depolarization errors found in NISQ devices.Several implementations that followed focused on theutility of unitary coupled cluster methods and variationsthereof [7, 14–17]. Other implementations demonstratedthat the trial Ansatz need not be in UCC form to obtainchemical accuracy [3]. Eventually, variational methodswere modified to be able to compute excited states aswell [4].The key limitation in time-evolution computation out-lined in [13] motivated a search other techniques to com-pute time-evolved quantities such as full eigenspectra.Quantum imaginary time evolution (QITE) algorithm tofind the ground-state energy of many-particle systems isone such method [18]. In principle, QITE is capable ofpreparing exact quantum states without variational op-timization. Direct computation of the eigenspectra with-out ancilla qubits using this algorithm was outlined in[19], and the method was further extended to a prac-tical application on NISQ devices in [20]. The quan-tum Lanczos method was used in tandem with QITE inthese demonstrations in order to recover higher excitedstates of molecules and nuclei, avoiding the problem ofhigh circuit depth associated with Trotterization and realtime evolution. Further demonstrations of the QITE al-gorithm applied to chemistry problems include [21, 22].These early examples were notably limited to two-qubitcomputations due to hardware noise. Details on the in-ner workings of these algorithms were included in a recentreview [23].Here, we start from this current state of the field and produce new benchmark techniques that extendthe performance of both variational methods andQITE/QLanczos algorithms beyond the state of the artin electronic structure calculations on NISQ devices. Inparticular, we apply and obtain experimental data for anAnsatz constructed with a symmetry preserving circuit(SPC) introduced in Refs. [24–26]. This Ansatz preservesrelevant chemical symmetries and has the benefit of min-imal parameter count and low depth. Since the mainlimitation of coupled cluster methods is noise associatedwith large circuit depth, particularly in the number ofCNOTs required, we find that VQE combined with SPCsvastly outperforms coupled cluster methods due to thecommensurately lower noise associated with short depth.Further, since these circuits identify target symmetryspaces, they are also capable of probing select excitedstates for each unique combination of symmetry eigenval-ues. We demonstrate this capability by directly calculat-ing eigenspectra for a range of alkali-hydride molecules.Importantly, we show that VQE+SPC enables chemicalaccuracy in all of the test molecules in our suite overthe cloud for the first time. We also compare and con-trast the strengths of VQE+SPC with QITE/QLanczosand demonstrate the use of QITE on 4 qubit depths forthe first time in chemistry problems. We make exten-sive use of error mitigation in our benchmarks, and herewe also outline a new error mitigation technique via theuse of hidden inverse gate constructions and demonstrateits use in variational algorithms. We find that this errormitigation technique substantially reduces the effects ofcoherent errors in simulation, further extending the reachof both variational and QITE algorithms by mitigatingcircuit noise.The rest of this report proceeds as follows. In Sec-tion III, we discuss the chemical Hamiltonians of the al-kali hydride molecules LiH, NaH, KH, RbH which areused in this paper as a suite of model chemical systemsfor benchmarking. In Section IV, we discuss three hybridquantum-classical algorithms that are used to calculatethe eigenvalues of the molecules of interest on a quan-tum computer. Specifically, in Section IV A, we discussSPCs in VQE and present our data obtained using IBM’squantum computers. We then discuss QITE and QLanc-zos algorithms in Section IV B and present our exper-imental data collected from IBM’s quantum hardware.In Section IV C, we introduce and demonstrate the hid-den inverse as an error mitigation strategy for variationalalgorithms and demonstrate its application on a noisyquantum computer simulator for LiH and NaH molecules.Finally, we conclude and present further predictions forthe state of the field in Section V.
III. CHEMISTRY MODEL
In this section, we introduce the underlying model ofchemistry used throughout this paper. We start by dis-cussing electronic structure of alkali hydrides moleculesLiH, NaH, KH, RbH. First, we discuss model of Hamil-tonian in the second quantization and sketch the embed-ding of the quantum computation kernel in Hartree-Fockwhich separates small active set of orbitals from the re-maining orbital treated with a mean-field approach. Ourmodel of the quantum region includes four spin orbitalsand requires four qubits when using the Jordan-Wignertransformation. Finally, we discuss eigenstates that cor-respond to our model Hamiltonian.The test molecular systems used in this work are alkalihydride molecules LiH, NaH, KH, RbH which, respec-tively, have 4, 12, 20, 38 electrons. The minimal STO-3Gbasis set is used and the corresponding Hilbert space isspanned by, respectively, 6, 10, 14, 23 spherical atomicorbitals and twice as many molecular spin-orbitals. Eventhough we use only a minimal basis set, the size of theHilbert space is too large for a direct computation ofelectronic structure on NISQ devices with the customaryJordan-Wigner transformation, as it requires that eachspin orbital is represented by an individual qubit. Fortu-itously, this problem can be simplified. In the majorityof chemical processes, such as bond breaking or forma-tion, only the highest energy, valence electrons play asignificant (active) role, whereas the tightly bound coreelectrons are largely (inactive) spectators whose role islimited to a mean-field screening of the Coulomb elec-trostatic field from the nuclei. The resulting effectiveHamiltonian in the second quantization language is givenby [7]: H = H + H + H , (1)with H given by H = E nucl + (cid:88) a (cid:0) h aa + (cid:88) b ¯ g abab (cid:1) , (2)where a and b run over inactive occupied spin-orbitalsof frozen-core. The E nucl describes Coulomb repulsionbetween bare nuclei cores and the second term describesthe effect of screening by the core electrons. Similarly,the 1-body term is given by H = (cid:88) p,q ˆ p † ˆ q · (cid:16) h pq + (cid:88) a ¯ g apaq (cid:17) (3)where the term h pq represents interaction of valence elec-trons with all core ions and the second term in the paren-thesis describes its screening by core electrons. Finally,the 2-body part is H = (cid:88) p,q,r,s ¯ g pqsr · ˆ p † ˆ q † ˆ r ˆ s. (4)where the indices p , q , r , s run over active spin orbitalswhereas indices a and b run over inactive spin-orbitalsof a frozen core. The symbols h pq and ¯ g pqsr denote, re-spectively, matrix elements of core Hamiltonian (kineticenergy of electrons plus Coulomb interaction with coreions) and an anti-symmetrized repulsion integral:¯ g pqsr = g pqsr − g pqr,s = (cid:104) p, q | s, r (cid:105) − (cid:104) p, q | r, s (cid:105) . (5) This Hamiltonian provides a powerful recipe for embed-ding of the quantum electronic structure calculation intoa classically computed environment obtained via Hartree-Fock (HF) [7, 28].For the model alkali hydride molecules studied here,only the 2 highest energy electrons are involved in chem-ical bonding. The selected active space includes onlythe highest occupied and lowest unoccupied molecularorbitals (HOMO and LUMO) and four molecular spinorbitals. The resulting Hilbert space spanned by theHOMO and LUMO is of O(2 ) size and can describe upto four electrons. The neutral case is described by plac-ing two electrons within the active space. The Hamil-tonian matrix has 16 ×
16 elements and is block diago-nal with each block corresponding to a different numberelectrons, with block size of 1, 4, 6, 4, and 1 for, respec-tively, 0, 1, 2, 3, and 4 electrons cases. In summary,the effective Hamiltonian applied allows for a significantreduction of quantum resources requiring only 4 qubitsfor the Jordan-Wigner transformed Hamiltonian. Thisconstruction is effectively the state of the art in quan-tum chemistry computations to date. While higher basissets have been demonstrated, our benchmark basis setand subsequent qubit numbers are well-matched to to-day’s hardware. Subsequent implementations of thesetechniques underway will use larger basis sets and enablebenchmarking of larger numbers of qubits as the coher-ence time of the larger cloud devices increases.For the 4-qubit problems we investigate here, thereare 2 = 16 eigenstates which we can group into distinctsymmetry subspaces. Denoting the number of electronswithin active space as N e and corresponding spin projec-tion as s z one can write parametrized vectors associatedwith each Hamiltonian block as labeled by their distin-guishable quantum numbers in general form: | N e = 1 , s z = 0 . (cid:105) = α | (cid:105) + α | (cid:105)| N e = 2 , s z = 0 (cid:105) = β | (cid:105) + β | (cid:105) + β | (cid:105) + β | (cid:105)| N e = 3 , s z = 0 . (cid:105) = γ | (cid:105) + γ | (cid:105) , (6)where we used spin block notation such that the first twobits in the ket occupation vector correspond to spin-uporbital whereas the last two bits in the ket correspond tospin-down orbitals. Next, we note that kets correspond-ing s z = − . N e = 0 ( | (cid:105) ) and N e = 4( | (cid:105) ) have been skipped. The second formula in Eq.6 for N e = 2 parametrizes three singlet states ( s =0) andone triplet state ( s =1) mixed. For the N e = 2 case, thesinglet states can be written as | N e = 2 , s = 0 , s z = 0 (cid:105) = β (cid:48) | (cid:105) + β (cid:48) | (cid:105) (7)+ β (cid:48) · √ | (cid:105) + | (cid:105) )whereas the triplet states are | N e = 2 , s = 1 , s z = 1 (cid:105) = | (cid:105)| N e = 2 , s = 1 , s z = 0 (cid:105) = 1 √ | (cid:105) − | (cid:105) ) | N e = 2 , s = 1 , s z = − (cid:105) = | (cid:105) . (8)Note that of these, only the one with s z = 0 is an en-tangled state, the remaining states are separable. Theground state of our model chemistry is a singlet statemodel described by Eq. 7. IV. QUANTUM VARIATIONAL ELECTRONICSTRUCTURE COMPUTATIONS
In this section we discuss application of various vari-ational quantum algorithms towards the estimation ofground and excited state electronic structure for modelchemical Hamiltonians discussed in Sec. III. First weexplain the symmetry preserving circuits in which trialstate vectors are constructed to preserve spin projectionduring the variational search. As an illustration, thesecircuits are used to explore ground and excited states ofour model systems on IBM Q hardware. Next, we discussimaginary time evolution and quantum Lanczos eigen-solvers and its application to alkali hydrides on IBM Qhardware. Finally, we illustrate the importance of noisemitigation and introduce the hidden inverse method asan approach to reduce noise in quantum computing.
A. Symmetry Preserving Circuits
In a standard VQE, parameterized Ans¨atze are chosenthat, ideally, are accurate, have a small amount of pa-rameters and a low enough circuit depth for NISQ hard-ware. Symmetry Preserving Circuits (SPC) [24–26] havevery efficient implementation on NISQ hardware, whichwe demonstrate here on metal alkali hydrides, experi-mentally for the first time. Since all of the moleculeswe consider can be mapped using the Jordan-Wignermapping onto 4 qubits, the methodology of construct-ing the SPC results in a fairly simple, compact set ofcircuits. We note that the qubit tapering methods us-ing symmetry information can also be used to improveperformance by reducing circuit width [27]. The SPCmethod is complementary in its focus on circuit depth.Moreover, SPC uses the minimum number of parame-ters needed to describe arbitrary states with constraintson symmetries. These circuits are constructed followingspecific rules to preserve natural symmetries which existin the problem Hamiltonian and are maintained throughthe Jordan-Wigner mapping. Specifically, in a system of2 n qubits, each qubit represents the occupation of a spin-orbital, and if we assume a block assignment of spins,then the n “top” qubits correspond to spin-up orbitals,while the n remaining “bottom” qubits correspond to | (cid:105) X | (cid:105) R † ( θ ) X R ( θ ) Z • R † ( θ ) X R ( θ ) •| (cid:105) X H H | (cid:105) X R † ( θ ) X R ( θ ) • FIG. 1: A simplified sample circuit for the case oftime-reversal symmetry with n = 4 , N e = 2 , s z = 0 spanningthe four-dimensional subspace with N e = 2 in Eq. (6) usinga minimal number of parameters (3). Here we define R ( θ ) ≡ R z ( π ) R y ( θ + π ) with R z ( φ ) = e − iφσ z / , R y ( θ ) = e − iθσ y / . | (cid:105) X | (cid:105) R † ( θ ) X R ( θ ) •| (cid:105)| (cid:105) FIG. 2: A circuit which generates exactly any state with N e = 1 in Eq. (6). As explained in the text, by usingparticle-hole symmetry we obtain a similar circuit for anystate with N e = 3 in Eq. (6). spin-down orbitals. The SPC methodology constructs acircuit with gates that preserve the number of electrons( N e ) and spin projection ( s z ). The primitive gate usedin the SPC construction is a parameterized SWAP-typegate (denoted as ASWAP), given in the computationalbasis by, A ( θ, φ ) = θ e iφ sin θ e − iφ sin θ − cos θ
00 0 0 1 . (9)Since a chemical Hamiltonian is real-valued and sym-metric (Hermitian) hence its energy eigenvalues are real(time-reversal symmetry). Therefore, we can safely set φ = 0 in Eq. (9). By inspection, we can also see that Eq.(9) naturally preserves particle number as it only actsnon-trivially on the subspace spanned by {| (cid:105) , | (cid:105)} ascontrolled by the parameter θ . Similarly, ASWAP alsopreserves spin projection if the gate is not placed in a waythat mixes the up-spin and down-spin subspaces, which isachieved by the construction described in Ref. [25]. Un-der the Jordan-Wigner mapping, total spin is a non-localquantity and therefore it is not simple to conserve us-ing local (two-qubit) gates. Consequently, we focus onlyon preserving the local quantities of particle number andspin projection, using circuits constructed using Eq. (9).In principle SPC are parameterized circuits con-structed of ASWAP gates that operate within a smallsubspace while preserving the number of particles, N e ,and spin projection, s z , but not the total spin. More-over, the triplet states in our 4-qubits test case are notparameterized. When we consider full VQE experiments,we therefore are only concerned with generating the sin-glet state with s z = 0. For completeness, the numericalparameters which generate the triplet state with s z = 0using the circuit in Fig. 1 are θ = − π/ θ = − π/ θ = 3 π/ N e = 2 in Eq. (6). For clarity we discusshow to simplify this circuit in the appendix and onlypresent the result here. In total this circuit is composedof 3 CNOT gates, 21 single qubit gates (when expressedin terms of R z , R y , X , H , though other choices are pos-sible), and the minimal number of parameters for thissymmetry subspace, which is 3. The resulting quantumcircuit is very similar to the recently proposed UCC-3 cir-cuit in Ref. [7]. Since the ground state of the moleculeswe investigate is of the form of a state with N e = 2 inEq. (6), the circuit given by Fig. 1 can efficiently findthese ground states.In Fig. 2, we similarly show a SPC which finds anystate with N e = 1 in Eq. (6). Since this symmetry spaceis smaller, the resulting circuit is simpler. In this case,the total number of CNOT gates is 1, single qubit gatesis 6 and the minimal parameter count is 1. Note that thestates with N e = 1 in Eq. (6) are degenerate with thosewith N e = 1 , s z = − .
5. However, we could also builda SPC which finds this degenerate state, since it has adifferent s z eigenvalue. We omit finding these degener-ate states in practice, but note that they are in principle,distinguishable with SPCs. By particle-hole symmetrywe can also view Fig. 2 as the case of a single hole beingexchanged (rather than a particle), therefore naturallyrepresenting the N e = 3 , s z = − . N e and N − N e particles cases only differ in the ini-tial single qubit gates. Therefore we have two primaryconfigurations of SPC, Fig. 1 and Fig. 2, for the case ofchemical Hamiltonians on 4-qubits.By employing direct search of energy minimum andmaximum within each SPC subspace one can find all ofthe 16 eigenstates except one of three singlet states whichis neither a minimum nor maximum energy within itsown symmetry subspace. That is, energy minimization(maximization) within the singlet state subspace leadsto the lowest (highest) energy state within this manifold.The third singlet state can be extracted by employingconstraint optimization in which the orthogonality to theground singlet state is enforced. We summarize the totalnumber of states within each symmetry subspace in Ta-ble I. As discussed, the SPC allows to efficiently reducesearch space while at the same time employing relativelysimple circuits.In Fig. 3 we show the dissociation of LiH, NaH, KH,and RbH molecules at five fixed interatomic distances.We plot the exact eigenenergies (found through diagonal-ization) in solid lines, while results from IBM quantumhardware are shown with markers. Specifically, we ran Symmetry Space N e = 1 , s = 0 . , s z = 0 . N e = 1 , s = 0 . , s z = − . N e = 2 , s = 0 , s z = 0 3 3 3 N e = 2 , s = 1 , s z = 0 1 3 0 N e = 3 , s = 0 . , s z = 0 . N e = 3 , s = 0 . , s z = − . experiments on IBM Q devices Valencia, Bogota, Santi-ago, and Vigo but note that results are consistent acrossthese devices. When executing SPC with three CNOTgates (blue, green and purple), we utilize Richardson ex-trapolation [29] and readout error mitigation strategies.For SPCs with only a single CNOT (red, orange, brown,and black), we chose to only use readout error mitiga-tion as the benefit from Richardson extrapolation is notsignificant. As discussed above for each symmetry sub-space, the SPCs can target two eigenenergies, one as aminimization and one as a maximization optimizationwithin a target subspace.In Fig. 4 we show the energy difference between theexact target eigenenergy and the energy obtained usingIBM hardware. Each marker is offset to reduce crowdingbut the same fixed interatomic distances are present asshown in Fig. 3. Error bars are included, representingone standard deviation. We note that many of the stan-dard deviations are of the same order as the energy errormean values. Even though we use the maximum allowednumber of shots (8192) this indicates that shot noise is amajor source of noise. We can also see that building ourSPCs for these three molecules leads to similar results,as well as many measurements within chemical accuracy. B. Quantum Imaginary Time Evolution andQuantum Lanczos Algorithms
In this section, we discuss the QITE and QLanczos al-gorithms which we used to calculate the energy levels forthe same suite of molecules as above. The QITE algo-rithm offers the quantum version of imaginary-time evo-lution for the non-unitary evolution operator U = e − βH .Starting with an appropriately chosen initial state | Ψ (cid:105) ,after n steps of imaginary-time evolution we obtain | Ψ( β ) (cid:105) = c n ( e − ∆ τH ) n | Ψ (cid:105) (10)where ∆ τ = βn and c n is the normalization constant suchthat c − n = (cid:104) Ψ |U | Ψ (cid:105) . β can be thought of as the in-verse temperature. As the temperature goes to zero, thesystem cools down to its ground state as long as the ini-tial state has a non-zero overlap with it. To representthis non-unitary imaginary-time evolution on a quantum − . − − . E n e r g y ( H a ) LiH − − − E n e r g y ( H a ) NaH1 2 3 − − − −
590 Bond Distance (Angstroms) E n e r g y ( H a ) KH N e = 2 , s z = 0 N e = 2 , s = 1 , s z = 0 N e = 1 , s z = 0 . N e = 3 , s z = 0 . N e = 2 , s z = 0 N e = 1 , s z = 0 . N e = 3 , s z = 0 . − , − , − , E n e r g y ( H a ) RbHFIG. 3: Dissociation curves of four molecules run on IBM Hardware (markers) at fixed interatomic distances of { . , . , . , . , . } Angstroms. For each molecule considered, the SPC can accurately probe seven of the eight interestingeigenstates (discussed in text). In all cases, we can see excellent agreement with the exact energies within each symmetrysubspace (solid lines). Since the SPC distinguish between different symmetry spaces, we label each curve by its symmetryeigenvalues throughout dissociation. computer without the requirement of any ancilla qubits,the QITE algorithm proposes to approximate these non-unitary operators with unitary updates so that the s thstep of evolution is | Ψ s (cid:105) = c s c s − e − ∆ τH | Ψ s − (cid:105) ≈ e − i ∆ τA [ s ] | Ψ s − (cid:105) (11)with c = 1 and the unitary update operator given interms of the Hermitian operator A [ s ] = (cid:88) i ,...,i Nq a [ s ] i ,...,i Nq σ i . . . σ i Nq . (12)where N q is the number of qubits and σ ∈ { X, Y, Z } arePauli matrices. Our next step is to calculate the coeffi-cients a [ s ] by solving Eq. (11) to second order, O (∆ τ ).This amounts to solving a linear system of equations,( S + S T ) · a = b where S I , I (cid:48) = (cid:104) σ i . . . σ i Nq σ i (cid:48) . . . σ i (cid:48) Nq (cid:105) (13)and b I = − i (cid:114) c s − c s (cid:104) σ i . . . σ i Nq H (cid:105) (14) with I = { i , . . . , i N q } and similarly for I (cid:48) . The expecta-tion values at a given step s are calculated with respectto the state in the previous QITE step, | Ψ s − (cid:105) . Thebottleneck in this algorithm is due to fact that the num-ber of measurements needed to calculate S (a 3 N q × N q matrix) and b (a 3 N q -dimensional vector) scales expo-nentially with the number of qubits N q in the system.Having a real Hamiltonian eliminates the terms with anodd number of Y Pauli matrices in b . Also, the fact thatthe matrix S + S T is symmetric reduces the requirednumber of measurements significantly. Nevertheless, theproblem of complexity persists, and reducing QITE com-plexity is an active field of research, just as it has beenfor reducing circuit depth in VQE (see, e.g., [30]).The cloud access to NISQ devices limits the number ofmeasurements that can be done in a reasonable amountof time. Additionally, the depth of quantum circuits islimited by the decoherence time of the qubits. Becauseof these limitations, we perform QITE algorithms on aNISQ device by following these steps: • To decide on the convergence of the energy ex-pectation value, we use exact QITE calculations.This allows us to validate the hardware results even − − − − E n e r g y E rr o r ( H a ) LiH 10 − − − − E n e r g y E rr o r ( H a ) NaH1 2 310 − − − − Bond Distance (Angstroms) E n e r g y E rr o r ( H a ) KH 1 2 3 10 − − − − Bond Distance (Angstroms) E n e r g y E rr o r ( H a ) RbH N e = 2 , s z = 0 N e = 2 , s = 1 , s z = 0 N e = 1 , s z = 0 . N e = 3 , s z = 0 . N e = 2 , s z = 0 N e = 1 , s z = 0 . N e = 3 , s z = 0 . FIG. 4: Energy differences from exact eigenenergies of each of the four molecules we consider. Markers are results from IBMhardware with error bars representing one standard deviation. Different symmetry spaces are artificially offset for easierviewing but still have the same fixed interatomic distances shown in Fig. 3. We show several results within chemical accuracy(dashed line). though the noise in quantum hardware might resultin deviations of energy values that are greater thanthe value that we choose for our convergence cri-terion. We set the latter to (cid:15) = 0 .
001 for energyexpectation values, and find the number of stepsrequired for convergence. • We then calculate the unitary updates in (12) atevery QITE step from (13) and (14) by performingthe energy expectation value calculations on a noisyquantum simulator which includes the noise modelof the quantum hardware to be used. • After choosing the initial state | Ψ (cid:105) using the sym-metry considerations for each of the reduced Hamil-tonian blocks, we find the state | Ψ s (cid:105) = e − i ∆ τA [ s ] e − i ∆ τA [ s − · · · e − i ∆ τA [1] | Ψ (cid:105) after application of s unitary updates. • We obtain the quantum circuit corresponding tothe state | Ψ s (cid:105) by using the isometry function inIBM Qiskit library. This function is based on [31]which decomposes a given state into single-qubitand Controlled-NOT (CNOT) gates with the aim of having the least number of CNOT gates. An exam-ple of this quantum circuit can be seen in Fig. 14.This quantum circuit has the same single-qubit U U U θ, φ, λ ) = (cid:18) cos θ − e iλ sin θ e iφ sin θ e i ( φ + λ ) cos θ (cid:19) . (15) • We then run the quantum circuits corresponding tothe states | Ψ s (cid:105) on quantum hardware and obtainthe energy expectation values at every QITE step.(The experimental results of these calculations canbe seen in Fig. 5).Next, we discuss how to implement the QLanczos algo-rithm.The QLanczos algorithm makes use of the QITEalgorithm to obtain the states in Krylov space K which is spanned by {| Φ (cid:105) , | Φ (cid:105) , . . . } , where | Φ l (cid:105) ∈{| Ψ (cid:105) , | Ψ (cid:105) , . . . } . After choosing the vectors in Krylovspace, we calculate the overlap ( T ) and Hamiltonian( H ) matrices from the QITE algorithm measurements − . − − . E n e r g y ( H a ) LiH − − − E n e r g y ( H a ) NaH0 . . . − − − −
590 Bond Distance (Angstroms) E n e r g y ( H a ) KH N e = 2 , s z = 0 N e = 2 , s = 1 , s z = 0 N e = 1 , s z = 0 . N e = 3 , s z = 0 . N e = 2 , s z = 0 N e = 1 , s z = 0 . N e = 3 , s z = 0 . . . . − , − , − , E n e r g y ( H a ) RbHFIG. 5: Energy vs. bond distance obtained using the reduced Hamiltonian blocks for molecules LiH, KH, RbH, NaH. Theexperimental data, notated with markers in the figure, were obtained using QITE and QLanczos algorithms. The quantumcircuits were run on IBM Q’s several quantum computers such as Casablanca, Manhattan, Vigo, Bogota, Rome. Theexperimental results are in very good agreement with the exact values shown with straight lines in figure. The error barsrepresent ± σ . on quantum hardware. The normalization constants arecalculated recursively using1 c r +1 = (cid:104) Φ r | e − τH | Φ r (cid:105) c r (16)Since we are using a small ∆ τ approximation, we cal-culate the expectation value by expanding to first orderin ∆ τ , (cid:104) Φ r | e − τH | Φ r (cid:105) = 1 − τ (cid:104) Φ r | H | Φ r (cid:105) + O (∆ τ ).Thus, we obtain the normalization constants using theenergy expectation values obtained from quantum hard-ware in the implementation of the QITE algorithm. Toincrease the accuracy of the calculations, one can usehigher-order approximations.The overlap and Hamiltonian matrix elements can beexpressed in terms of these constants and energy expec-tation values, T l,l (cid:48) = (cid:104) Φ l | Φ l (cid:48) (cid:105) = c l c l (cid:48) c r , (17)and H l,l (cid:48) = (cid:104) Φ l | H | Φ l (cid:48) (cid:105) = T l,l (cid:48) (cid:104) Φ r | H | Φ r (cid:105) , (18) where r = l + l (cid:48) , and l, l (cid:48) are even integers.The energy eigenvalues E are found by solving the gen-eralized eigenvalue equation H x = E T x , (19)The corresponding eigenvectors x = ( x , x , . . . ) deter-mine the eigenstates of the system Hamiltonian, | Ψ[ E ] (cid:105) = c E ( x | Φ (cid:105) + x | Φ (cid:105) + . . . ) (20)where c − E = (cid:107) (cid:80) l x l | Φ l (cid:105)(cid:107) .As explained in the previous section, the symmetryof each of these molecules leads to a simplification of thesystem which makes them good candidates for implemen-tation on NISQ devices. In the previous section we usedthis symmetry to simplify the quantum circuits. As asecond perspective, we present a simplification of the sys-tem by reducing the Hamiltonian into smaller blocks sothat we can study these molecules using a smaller num-ber of qubits on quantum hardware. For benchmarkingpurposes, we also present our QITE algorithm results forthe full LiH molecule Hamiltonian for comparison withthe results presented using the SPC method. In what − − − E n e r g y E rr o r ( H a ) LiH N e = 2 , s z = 0 N e = 2 , s = 1 , s z = 0 N e = 1 , s z = 0 . N e = 3 , s z = 0 . N e = 2 , s z = 0 N e = 1 , s z = 0 . N e = 3 , s z = 0 . − − − − − E n e r g y E rr o r ( H a ) NaH1 2 310 − − − − − Bond Distance (Angstroms) E n e r g y E rr o r ( H a ) KH 1 2 3 10 − − − − − − Bond Distance (Angstroms) E n e r g y E rr o r ( H a ) RbHFIG. 6: Error in energy vs. bond distance obtained using the reduced Hamiltonian blocks for molecules LiH, KH, RbH, NaH.The dashed line in each panel represents the chemical accuracy (1 . × − ). As seen in the figures we were able to obtainseveral energy eigenvalues within the chemical accuracy. follows, we explain the reduction of the system size usingthe symmetry of the Hamiltonian of various molecules.In Section III we discuss chemical Hamiltonian for ourbenchmark molecules. Some of the solutions are givenin Eqs. 6, 7 and 8, and also listed in Table I. In a 4-qubit molecular Hamiltonian, the matrix elements cor-responding to a no-electron state in the active space( N e = 0 , s z = 0) and a 4-electron state ( N e = 4 , s z = 0)are diagonal so no further computation of eigenvaluesand corresponding eigenstates is needed. The Hamilto-nian blocks formed by single ( N e = 1) and three ( N e = 3)electron states are 4 × × × s z = + and s z = − ). Each of the four 2 × N e = 2) form a 6 × × s z = 0(three singlet states and one triplet state) and can berepresented using a 2-qubit circuit. The remaining twostates represent triplet states with s z = 1+ and s z = − × molecular Hamiltonian using only two-qubit andsingle-qubit circuits. This analysis shows that decompo-sition of the full Hamiltonian into diagonal blocks based on the number of electron and spin symmetries is essen-tial making quantum computation compatible with NISQdevices.We start by calculating the energy expectation valuesof the system using the QITE algorithm which will give usthe ground state energies of the symmetry sector that theinitial state belongs to. To this end, as mentioned ear-lier, the Hamiltonian of the molecules are divided intosmaller blocks and the initial states are chosen accord-ingly using symmetry which results in a smaller numberof imaginary-time steps required for convergence. To beable to represent these reduced Hamiltonian blocks on aquantum computer we need to express them in terms ofPauli matrices. This can be done by writing the blockHamiltonian H block as H block = (cid:88) I c I σ I (21)where I = { i , . . . , i n q − } and σ I = σ i σ i . . . σ n q − , with σ ∈ { I, X, Y, Z } and n q the number of qubits used for thisblock Hamiltonian. The coefficients c I are found from c I = 12 n q Tr[ H block · σ I ] (22)Once the coefficients c I are determined, the Hamiltonian0 . . . . − . − − . E n e r g y ( H a ) N e = 2 , s z = 0 N e = 2 , s = 1 s z = 0 N e = 1 , s z = 0 . N e = 3 , s z = 0 . N e = 2 , s z = 0 N e = 1 , s z = 0 . N e = 3 , s z = 0 . FIG. 7: Energy vs. bond distance obtained using the 4-qubitLiH molecule Hamiltonian. The Richardson extrapolationapplied experimental data, notated with markers in thefigure, were obtained using QITE algorithm. The quantumcircuits were run on IBM Q’s Yorktown and Melbournedevice. The experimental results are in agreement with theexact values shown with straight lines in figure but most ofthem are not in chemical accuracy. The error bars represent ± σ . H block expressed in terms of Pauli matrices can be imple-mented on a quantum computer. We use H block in QITEand QLanczos algorithms to calculate the energy eigen-values of each molecular Hamiltonian with states shownin Table I.We use the information from symmetry considerationsof the reduced block Hamiltonians to choose different ini-tial states, | Ψ (cid:105) , so as we obtain all energy levels of thesystem Hamiltonian. QITE and QLanczos Results
To calculate the energy expectation values at eachQITE step for the molecules that we study, we first ranthe quantum circuits obtained using the isometry func-tion for each QITE step on IBM’s cloud accessible quan-tum computers. We did this for the reduced Hamiltonianblocks first and then for the 4-qubit full LiH moleculeHamiltonian. An example of 2-qubit quantum circuit forreduced block Hamiltonian of N e = 2 states can be foundin Fig. 14. At each QITE step the angles θ , θ and θ change but the quantum circuit depth remains fixed. Wethen used the measured energy expectation values to findthe energy eigenvalues of the systems of interest.The calculation of the energy eigenvalue from theQLanczos algorithm using the method described aboveis often numerically unstable due to the noise introducedby NISQ devices. To stabilize the results, we worked asfollows. We only used two-dimensional Krylov spaces for the T and H matrices in (19), which have been shownto yield more accurate results in NISQ devices [32]. TheHamiltonian matrices are diagonally dominant with thediagonal elements being a large negative number sincedue to the electron–nuclei Coulomb interaction. Eachreduced Hamiltonian block was shifted by appropriateconstant value, to ensure that all matrix elements are ofthe same order. While this did not affect the eigenstatesof the system, it improved the accuracy of the resultssignificantly. The accuracy of the eigenvalues can be im-proved further if one includes higher-order terms in theexpansion of e − ∆ τH , but this would require more mea-surements of higher moments of the Hamiltonian result-ing in longer runs on quantum hardware.See Appendix B for an example application of QITEalgorithm on IBM Q hardware for 2-qubit reduced Hamil-tonian blocks of molecules LiH and RbH where we alsodemonstrate the 2-qubit circuit obtained from isometry function of IBM’s Qiskit library.In Fig. 5, we present our experimental results obtainedusing QITE and QLanczos algorithms for the energy ofmolecules LiH, KH, RbH, NaH as a function of bondlength. We see an excellent agreement between the ex-perimental results and the values obtained from exactdiagonalization. In these plots, we present 7 of the 8eigenvalues that could not be readily obtained from theHamiltonian. The remaining eighth eigenvalue is the firstexcited-state energy of the N e = 2 symmetry subspacefor the molecules. It turns out to be exactly given by theanalytic expression √ ( | (cid:105) − | (cid:105) ). This is confirmed byimplementing the QITE algorithm on hardware startingwith the initial state | Ψ (cid:105) = √ ( | (cid:105) − | (cid:105) ). We showthe experimental results for this state in upper left panelof Fig. 5 for the LiH molecule.In the results presented here we used the QITE algo-rithm to find the minimum and maximum state energiescorresponding to N e = 1, N e = 2, and N e = 3 symme-try subspaces and the QLanczos algorithm to find theremaining excited-state energies.We also present the errors in the energy eigenvaluesas a function of the bond length as a result of using theQITE and QLanczos algorithms from reduced Hamilto-nian blocks of molecules LiH, RbH, NaH, KH with re-spect to chemical accuracy in Fig. 6. Although we onlyused readout error mitigation, we were able to obtainseveral energy eigenvalues within chemical accuracy.Although the Hamiltonian can be separated intosmaller blocks and these molecules can be studied insmaller quantum hardware sizes, for benchmarking pur-poses we also ran the QITE algorithm for the full 4-qubitLiH molecule Hamiltonian. Compared to the reducedsystem, the quantum circuit obtained using the IBM Qlibrary isometry function is deeper than before and re-quires connections between all qubits in the system. Mostof the IBM Q devices have linear layout with IBM QYorktown device having the most connections betweenqubits. Consequently, we ran the QITE algorithm on theIBM Q Yorktown and Melbourne device and the results1can be seen in Fig. 7. The Richardson extrapolation errormitigation strategy was applied to experimental energyvalues seen in Fig. 7. Although the experimental dataare in agreement with exact values, most of them are notchemically accurate. Due to this noise, the QLanczos al-gorithm is not numerically stable and cannot be used toimprove on QITE algorithm results in this case. C. Reducing noise with hidden inverse
Variational algorithms are designed to work in near-term quantum computers with imperfect gates and in anascent quantum error correction regime. It is knownthat the variational formalism has inherent robustnessto certain types of time independent systematic errors.In addition, as demonstrated in the previous two sec-tions, the “workhorse” error mitigation methods, readouterror mitigation and Richardson extrapolation, can im-prove the expectation value accuracy in many instances.These two methods have key limitations, however. Be-yond a certain circuit depth determined by overall hard-ware noise, Richardson extrapolation will not yield use-ful data [33]. Meanwhile, exact readout error mitigationis in principle unscalable, though scalable approximatemethods are under development. On the other hand Re-cent advancements in quantum characterization proce-dures [34] have demonstrated systematic error parame-ters to be time-dependent stochastic in the time scale ofmultiple circuits run within a given circuit list. One wayto model this type of error (in the context of variationalalgorithms) is with systematic errors that are drawn in-dependently from a stationary distribution in every iter-ation of the classical optimizer. This has a direct effecton the performance of variational algorithms. In particu-lar, mitigating such an error does not require exponentialresources, and it can be done while maintaining constantcircuit depth, unlike extrapolation methods.In this section, we test the effectiveness of a circuit levelerror mitigation technique called hidden inverse [35, 36]to mitigate precisely this type of error. The key ideabehind this error mitigation technique is that each self-inverse gate (such as CNOT) can be experimentally im-plemented in standard or inverted configuration. CNOTgates are not native gates in most hardware but are im-plemented by a combination of single qubit and two-qubitnative rotations. By carefully choosing either the stan-dard gate or the inverted gate within a circuit, it is pos-sible to mitigate some of the errors of native rotations.Here we simulate VQE circuits with UCC-3 Ansatz [7] forLiH and NaH molecules under a two qubit over-rotationerror noise model (See Appendix C for details on thesimulation such as noise model, Ansatz and classical op-timization).Fig. 8 and Fig. 9 display the comparison between er-rors in calculated ground state energy computed with thenative Ansatz and the hidden inverse optimized Ansatzas a function of the over-rotation error (cid:15) . These plots
FIG. 8: Average error in estimated ground state energy ofLiH at 0.5 interatomic distance plotted against over-rotationerror (cid:15) . Hidden inverse optimized Ansatz outperforms nativeUCC-3 Ansatz for the entire range of (cid:15) . Error bars representone standard deviation.FIG. 9: Average error in estimated ground state energy ofNaH at 0.5 interatomic distance plotted againstover-rotation error (cid:15) . Hidden inverse optimized Ansatzoutperforms native UCC-3 Ansatz for the entire range of (cid:15) .Error bars represent one standard deviation. are for LiH and NaH for a fixed interatomic distance of0.5 Angstroms. The range of (cid:15) is chosen to match real-istic noise parameters of current hardware. We can seefrom these two plots that when the over-rotation error in-creases, the average error in the calculated ground stateenergy with the native UCC-3 Ansatz (orange curve)tends to get worse. We also see large variability in theestimated energy as expected from the stochastic mod-eling of the error parameters. The hidden inverse opti-mized Ansatz (blue curve) is found to be robust againstthis stochastic over-rotation error as we see the error staylow (with small variability) across the error range.2
V. CONCLUSION
In this report, we have outlined the state of the artof the field devoted to benchmarking quantum chemistrycomputations on NISQ devices. This application is a keyindicator of NISQ capabilities, and it fills a performanceassessment gap left open by low level benchmarks to date.We outlined how to extend these benchmarks with vari-ational methods plus short depth Ans¨atze on one hand,and imaginary time evolution plus Lanczos on the other.We also outlined how to improve all of our algorithmswith scalable error mitigation techniques via the hiddeninverse.In doing so we have added QITE, Lanczos, and SPCs toour electronic structure calculation benchmark suite. Weused these algorithms to demonstrate that NISQ devicesare now capable of chemical accuracy over the cloud forminimal basis set computations. Notably, only four yearsago the first chemically accurate computations were per-formed by researchers with direct access to the hardwarewith intricate precision in analog quantum control, andchemical accuracy over the cloud was impossible. Mean-while, the benchmarks presented here are fully specifiedin QASM. When this suite of molecules was simulatedon quantum computers two years ago, it was found thatchemical accuracy was just out of reach. Here, manyof the same computations resulted in chemical accuracy,highlighting both the immense progress of quantum com-puters to date and the efficacy of application-level bench-marking in tracking this progress.While the algorithms we tested were quite different intheir approach and circuit constructions, we note thatboth were able to achieve chemically-accurate results forcertain configurations. While all NISQ era algorithmsnecessarily suffer drawbacks, notably the requirement ofvery short depths, we note here that VQE+SPC andQITE+Lanczos have complementary properties and thusare optimal for different scenarios.The SPC Ans¨atze are primarily designed to recoverground state and are optimal in circuit depth for groundstate energy computation. However, VQE+SPC can alsobe applied to extract some excited states, particularlythose which correspond to a minimum or maximum ofenergy within given symmetry Hamiltonian block. Onthe other hand, the QITE+Lanczos method is suitablenot only for ground state calculations but is also in-trinsically designed to recover excited states which arenot accessible to VQE+SPC. However, the scalability ofthe algorithm is at an earlier state of development com-pared to VQE+SPC. We look forward to further refiningthese algorithms, and combining them with advanced er-ror mitigation via the hidden inverse in hardware in thenear future. Such advanced error mitigation techniqueswill be critical for enabling simulation of chemical sys-tems beyond 4-qubits.Finally, we note that given the rapid progress of NISQdevices in chemistry, as outlined by performance in thisbenchmark suite, we expect expanded basis sets to be us- able for chemical accuracy in the very near future. Whilequantum computers with more than 50 qubits are nowavailable, large molecules with many active orbitals willrequire a consistent and concerted effort in coherencetime improvement in order to be simulatable. This ismostly due to the fact that the Hilbert space coveredby active orbitals spanning an encoding across 50 qubitscannot be fully explored within the coherence times oftoday’s devices. Even linear-depth circuits will require a5 to 10 × improvement in coherence time over today’s torun to completion before full decoherence. Nonetheless,given the current rate of progress, 10 qubit computationswith more advanced bases will be routine within a yearor two, and computations involving more than 30 qubitswill soon follow. In this mesoscale era, just at the bor-der of a potential quantum advantage, the pace will bebrisk and the potential for new science and new discoverycannot be overestimated. ACKNOWLEDGMENTS
The quantum circuits were drawn using Q-circuit pack-age [38]. This work was supported by the ASCR Quan-tum Testbed Pathfinder program at Oak Ridge NationalLaboratory under FWP number ERKJ332. This researchused quantum computing system resources of the OakRidge Leadership Computing Facility, which is a DOEOffice of Science User Facility supported under ContractDE-AC05-00OR22725. S.M. was supported through USDepartment of Energy grant DE-SC0019294 awarded toDuke and is funded in part by an NSF QISE-NET fellow-ship (1747426). G. B. and B. G. were supported throughUS Department of Energy grants awarded to VirginiaTech (Awards DE-SC0019318 and DE-SC0019199 re-spectively). G. S. acknowledges the Army Research Of-fice award W911NF-19-1-0397 and the National ScienceFoundation award OMA-1937008.
Appendix A: Details of SPC simplifications
The SPC presented in the main text are constructedusing the ASWAP gate as a primitive. Each ASWAPgate can be decomposed into three CNOTs and four sin-gle qubit gates according to Fig. 10. A ( θ, φ ) • = • R ( θ, φ ) † R ( θ, φ ) • FIG. 10: Decomposition of the A gate in terms of elementarysingle and two-qubit gates. R ( θ, φ ) = R z ( φ + π ) R y ( θ + π/ R z ( θ ) = exp( − iθσ z / R y ( φ ) = exp( − iφσ y / Building the SPC for the case of N e = 2 , s z = 0 imme-3 | (cid:105) X A ( θ , A ( θ , | (cid:105) A (0 , | (cid:105) A ( θ , | (cid:105) X FIG. 11: Following the SPC construction rules, we candirectly build the SPC for the n = 4 , m = 2 , s z = 0symmetry eigenvalues. This circuit can be significantlysimplified and the result is presented in the main text Fig. 1. | (cid:105) X | (cid:105) X | (cid:105) X | (cid:105) R ( θ ) † X R ( θ ) • FIG. 12: A circuit which exactly generates any state with N e = 3 in Eq. (6). This circuit can be thought of asswapping a single hole, while Fig. 2 swaps a single particle. diately results in the circuit shown in Fig. 11. Naively,this circuit contains 12 CNOT gates, but with furtherinvestigation we can see that we can significantly sim-plify the circuit. The first such simplification can be seensince the input state is fixed by placing an X on qubits q , q . In this case, the first two CNOT gates in bothof the first two ASWAP gates are unnecessary and canbe replaced by either an I (Identity) or X gate. Fur-thermore, by inspection of Eq. (9), we can see that theASWAP gate with zero arguments is locally equivalentto a controlled- Z gate. Additionally, another simplifica-tion can be performed since the final ASWAP( θ ) gatehas a ‘first’ CNOT gate which can be commuted throughthe controlled- Z gate and cancelled with the ‘last’ CNOTgate of ASWAP( θ ). We can then also replace the ‘sec-ond’ CNOT gate of ASWAP( θ ) with a single qubit X gate. In total, this then simplifies the SPC to only re-quire three CNOT gates. Similar simplifications can beapplied to arrive at the simplified Fig. 2.We can slightly modify the N e = 1 , s z = 0 . N e = 3 , s z = 0 . X gates on the q , q qubits and we also move theASWAP gate to act on the bottom two qubits, then weimmediately have a circuit which targets N e = 3 , s z = 0 . Appendix B: Example QITE Results
In Fig. 13 we present two examples of the measured en-ergy expectation values as a function of imaginary time obtained from the QITE algorithm. Since the QITEalgorithm always converges to the ground-state energyof the symmetry subspace that the initial state belongsto, in order to access some of the excited-state ener-gies, we ran the QITE algorithm using − H block , instead.Fig. 13(a) shows an example in which the energy expec-tation value converges to the third excited-state energyof the N e = 2 symmetry subspace block Hamiltonian ofthe LiH molecule where we started with the initial state | Ψ (cid:105) = | (cid:105) . Fig. 13(b) shows the convergence to theground state energy of the N e = 2 symmetry subspace ofRbH starting with the initial state | Ψ (cid:105) = | (cid:105) and using H block in the QITE algorithm.The readout-error-mitigated (ROEM) experimentaldata in Fig. 13 were obtained by running the quantumcircuits of the form shown in Fig. 14 on IBM Q 7-qubitCasablanca quantum hardware using the number of shots N shots = 8192 at each measurement. The experimentsin Fig. 13(a) (13(b)) were run N runs = 3 ( N runs = 2)times and the error bars represent one standard devia-tion ( ± σ ). Although in these two examples we used IBMQ’s Casablanca device, for the rest of the measurementson quantum hardware we used various devices such asManhattan, Vigo, Bogota, and Rome depending on theiravailabilities. Appendix C: Details of hidden inverse errormitigation
In order to demonstrate the effectiveness of the hiddeninverse protocol, we have decided to perform simulationunder a ion trap noise model. A CNOT gate in trappedion systems is synthesized using XX interaction paddedwith single qubit rotations as seen in Fig. 15 while theCNOT inverse is implemented using the configurationfound in Fig. 16. Fig. 17 shows the standard UCC-3Ansatz with three unknown parameters ( θ , θ , θ ). Wethen modified this Ansatz (shown in Fig. 18) where wecarefully replaced some of the CNOT gates with CNOTinverse gates to cancel out systematic errors. For oursimulation, we assumed single qubit gates are error freeand XX gates contain multiplicative over-rotation typeerrors as such we apply XX ( ± (1 + err ) ∗ π ). Erroramounts are drawn independently from err ∼ N (0 , (cid:15) ) ateach round of the classical optimizer. We used BOBYQA(Bound Optimization BY Quadratic Approximation) asimplemented in SKQuant-Opt [37], a standard optimizerpackage for near term hybrid quantum classical algo-rithms. Although we have simulated the results undera ion trap model, hidden inverse can be applied to othersystems too(as an example to reduce errors in ZX gatesfor a superconducting hardware).4 FIG. 13: The convergence of the energy expectation values as a function of imaginary-time, β , using QITE algorithm. (a)Convergence to the ground state of N e = 2 symmetry subspace for LiH molecule with initial state | Ψ (cid:105) = | (cid:105) and − H . Theconverged energy corresponds to the negative of the maximum (3rd excited state) energy of that symmetry subspace. (b)Convergence to the ground state of N e = 2 symmetry subspace for RbH molecule with initial state | Ψ (cid:105) = | (cid:105) and H . Thereadout error mitigated experimental data were collected on IBM Q 7-qubit Casablanca device and compared to the datacalculated using exact diagonalization. Error bars represent ± σ . geometry : ’ Na 0.0 0.0 0.0H 0.0 0.0 1.914388 ’frozen - spin - orbitals : [0 ,1 ,2 ,3 ,4 , 10 ,11 ,12 ,13 ,14]active - spin - orbitals : [5 ,9 ,15 ,19] The circuit for | Ψ i = | i is as seen below. | i U ( θ , , − π/ U ( θ , , | i U ( θ , π, − π ) • The circuit for | Ψ i = | i is as seen below. | i U (0 , , − π/ U ( θ , , U ( θ , , | i U ( θ , π, − π ) • FIG. 14: The quantum circuit of each QITE step for 2-qubitreduced Hamiltonian blocks. At every QITE step the angles θ , θ and θ change but the circuit depth is fixed. RY ( π/ XX ( π/ RX ( − π/ RY ( − π/ RX ( − π/ RY ( π/ RX ( π/ XX ( − π/ RY ( − π/ RX ( π/ CNOT † into native ion trapgates[1] Y. Cao, J. Romero, J. P. Olson, M. Degroote, P. D.Johnson, M. Kieferov´a, I.D. Kivlichan, T. Menke, B. Per-opadre, N. P. Sawaya, S. Sim, L. Veis, and A. Aspuru-Guzik “ Quantum chemistry in the age of quantum com-puting ”, Chem. Rev. , 10856–1091 (2019). [2] S. McArdle, S. Endo, A. Aspuru-Guzik, S. C.Benjamin,X. Yuan, “
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Q-circuit Tutorial ”,arXiv:quant-ph/0406003v2 (2004). | (cid:105) X RX ( − π/ CX CX RX ( π/ | (cid:105) H RZ ( θ ) CX CX H | (cid:105) X RX ( − π/ CX CX RX ( π/ H CX CX H | (cid:105) H RZ ( θ ) RZ ( θ ) H FIG. 17: UCC-3 Native Ansatz. The top qubit is control and the bottom qubit is target for all the CX gates in the Ansatz. | (cid:105) X RX ( − π/ CX CX † RX ( π/ | (cid:105) H RZ ( θ ) CX CX † H | (cid:105) X RX ( − π/ CX CX † RX ( π/ H CX CX † H | (cid:105) H RZ ( θ ) RZ ( θ ) H FIG. 18: UCC-3 Hidden inverse Ansatz. The top qubit is control and the bottom qubit is target for all the CX and CX ††