Scattering in the Ising Model with the Quantum Lanczos Algorithm∗

Kubra Yeter-Aydeniz,1, 2, † George Siopsis,3, ‡ and Raphael C. Pooser2, 3, 1, §

1Computational Sciences and Engineering Division,Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA

2Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA3Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996-1200, USA

(Dated: September 11, 2020)

Time evolution and scattering simulation in phenomenological models are of great interest for testingand validating the potential for near-term quantum computers to simulate quantum field theories. Here,we simulate one-particle propagation and two-particle scattering in the one-dimensional transverse Isingmodel for 3 and 4 spatial sites with periodic boundary conditions on a quantum computer. We usethe quantum Lanczos algorithm to obtain all energy levels and corresponding eigenstates of the system.We simplify the quantum computation by taking advantage of the symmetries of the system. Theseresults enable us to compute one- and two-particle transition amplitudes, particle numbers for spatialsites, and the transverse magnetization as functions of time. The quantum circuits were executed onIBM 5-qubit superconducting hardware. The experimental results with readout error mitigation are invery good agreement with the values obtained using exact diagonalization.

INTRODUCTION

The Ising model is a quintessential spin system withinwhich one can simulate and study many-body interactions.The model allows for simulating spin-spin physics and thecalculation of properties such as magnetization and spin-frustration. It also serves as a useful arena for the study ofmore complex quantum field theories on a lattice. For ex-ample, scattering in a spin system on a lattice holds manyparallels with scattering between particles in high energyphysics experiments [1–5]. Computing scattering, transi-tion rates, and other physical quantities involving quantumfields are difficult tasks for classical computers. Quantumcomputers promise exponential speedup, however the ap-proach with quantum simulators often revolves around thecomputation of real-time evolution based on Trotterizationwhich is of limited utility on NISQ (noisy intermediate-scalequantum [6]) hardware [7]. Previous studies have simulatedreal-time dynamics of interactions [1, 3–5] and evolution ofdisordered Hamiltonians [8] with this method. In this typeof simulation, the number of gates grows linearly with thesystem size and the number of Trotter steps. Therefore, thenoise in the system grows as the system size grows. Othershave simulated the model both variationally [9] and via di-rect diagonalization within the quantum circuit [10]. Here,

∗ This manuscript has been authored by UT-Battelle, LLC, under Con-tract No. DE-AC0500OR22725 with the U.S. Department of En-ergy. The United States Government retains and the publisher, byaccepting the article for publication, acknowledges that the UnitedStates Government retains a non-exclusive, paid-up, irrevocable,world-wide license to publish or reproduce the published form ofthis manuscript, or allow others to do so, for the United States Gov-ernment purposes. The Department of Energy will provide publicaccess to these results of federally sponsored research in accordancewith the DOE Public Access Plan.† yeteraydenik@ornl.gov‡ siopsis@tennessee.edu§ pooserrc@ornl.gov

we use the Quantum Lanczos (QLanczos) algorithm [11] tosimulate transition probabilities and scattering in the onedimensional transverse Ising model. We use the quantumimaginary-time evolution algorithm (QITE) to supply a ba-sis for QLanczos. We tune the QITE step size, and thusthe total noise in the circuit, by using a hybrid quantum-classical approach to the algorithm. Using this techniquewe also demonstrate computation of the occupation num-ber and transverse magnetization.

Although the QITE algorithm in [11] has advantagessuch as not requiring optimization or ancilla qubits whenit comes to its implementation on NISQ devices, increas-ing circuit depth at each QITE step raises the impact ofnoise from short coherence time, cross-talk between qubits,etc. Recent efforts have sought to economize the circuitdepth in the QITE algorithm [12–14] to reduce the impactof these noise sources. We previously introduced a single-step method which replaces the unitary updates calculatedin each QITE step with a single update that takes the ini-tial state to the state at the particular QITE step, therebyreducing the gate depth and noise at each step. Here, wepresent a slightly different method while keeping the samelogic.

For Ns = 3 and Ns = 4 spatial sites in the Ising spinchain with periodic boundary conditions (PBC), we usedthe QLanczos algorithm to compute the eigenvalues andeigenstates of the system so that transition probability, oc-cupation number, and transverse magnetization could becalculated. We computed the energy expectation value asa function of imaginary-time on the IBM Q 5-qubit York-town device obtained using QITE, which subsequently setsthe basis for the QLanczos algorithm. We benchmarkedthese results against exact calculations, and obtained goodagreement when error mitigation was employed.

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RESULTS

The Ising model Hamiltonian with periodic boundary con-ditions can be written as

H = −J∑i∈ZNs

XiXi+1 − hT∑i∈ZNs

Zi , (1)

where Xi, Yi, Zi are the Pauli matrices at the ith site, i =0, 1, . . . , Ns − 1, Ns is the number of spatial sites, J is thenearest-neighbor coupling strength, and hT is the transversemagnetic field. We impose periodic boundary conditions byidentifying Ns ≡ 0. At each site, we place a qubit on whichthe Pauli matrices act, and define the occupation numberof the ith site by ni = I−Zi

2 with corresponding eigenstates|ni〉 (|0〉 (|1〉) denotes an unoccupied (occupied) site). Avector in the computational basis |x〉 (x = 0, 1, . . . , 2Ns−1)is specified by the sites which are occupied corresponding tothe digits of x equal to 1 (e.g., for Ns = 4, the state |0000〉has no particles, whereas |0101〉 consists of two particles atsites 1 and 3).

To study the time evolution of the system, we prepareit in the initial state |initial〉, evolve it for time t with theevolution operator U(t) = e−iHt, and then measure it, thusprojecting it onto a state |final〉. This process leads to thequantum computation of the transition probability

Pfi(t) = |Afi(t)|2 , Afi(t) = 〈final | U(t)|initial〉 . (2)

In particular, in this work we study single-particle propaga-tion and two-particle scattering. In both cases, we preparethe system in the computational basis state |initial〉 = |x〉.For single-particle propagation, x contains a single digitequal to 1, whereas for two-particle scattering, it containstwo digits equal to 1. At the end of the quantum computa-tion, the measurement projects the system onto a differentcomputational basis state |final〉 = |x′〉. Being in the com-putational basis, both initial and final states are easy toconstruct. However, the unitary U(t) is difficult to imple-ment. We use the QLanczos algorithm to accomplish this,which is based on the quantum imaginary-time evolution(QITE) algorithm [11].

To calculate the transition probabilities (2), we employ ahybrid quantum-classical algorithm to solve the eigenvalueproblem of the Hamiltonian (1) (see “Methods” for details),

H|ψI〉 = EI |ψI〉 , I = 0, 1, . . . , 2Ns − 1 . (3)

Here, we present the readout error mitigated experimen-tal data (notated as ROEM exp. data in figure legends) ob-tained from data on the 5-qubit IBM Q Yorktown hardwarefor transition probability amplitudes (Fig. 1), occupationnumber at each spatial site (Fig. 2), and average transversemagnetization (Fig. 3) for number of spatial sites Ns = 3and Ns = 4. We chose the parameters of the system Hamil-tonian in (1) to be hT = 1 and J = 0.6. In ref. [1] it wasfound that errors arising from quantum hardware becomeworse as the coupling J increases. In this section we present

results which show small hardware errors even as one movesaway from the weak coupling regime.

In Figs. 1 (a)-(d), we show numerical values of the tran-sition amplitudes calculated from given exact |initial〉 and|final〉 states, and compare them with values obtained fromROEM experimental data produced by the QLanczos quan-tum algorithm that calculates energy eigenvalues and cor-responding eigenstates. Figs. 1 (a) and (c) show the one-particle propagation probability, and Figs. 1 (b) and (d)show the probability of two-particle scattering.

Similarly, in Figs. 2 (a)-(d), we show a comparison be-tween the numerical value of occupation numbers at variousspatial sites calculated from a given exact |initial〉 state andthe one calculated experimentally from the ROEM energyeigenvalues and corresponding eigenstates using the QLanc-zos algorithm. It should be noted that when the particlesare initially at sites 0, 2 (i.e., |initial〉 = |1010〉) the timeevolution of the occupation number at even (odd) sites isthe same, i.e., 〈n0(t)〉 = 〈n2(t)〉 (〈n1(t)〉 = 〈n3(t)〉).

Finally, in Figs. 3 (a)-(d), we present a comparison be-tween the numerical value of the average transverse mag-netization calculated from a given exact |initial〉 state andthe experimental average transverse magnetization obtainedfrom ROEM energy eigenvalues and corresponding eigen-states using the QLanczos algorithm.

Figs. 1, 2, and 3 demonstrate that for number of sitesNs = 3, the exact results and experimental data are in ex-cellent agreement. For a larger system (Ns = 4), the exactand experimental data are still in very good agreement. Inthe latter case, the quantum circuit used to calculate theenergy expectation values includes more single-qubit rota-tion and CNOT gates, which result in more error in themeasurements. This can be seen by comparing Fig. 10 withFig. 8 in the “Methods” section.

DISCUSSION

In this work, we discussed a hybrid quantum-classicalmethod to calculate physical properties of the Ising spinchain model as a function of time, such as transition ampli-tudes, occupation numbers at various sites, and transversemagnetization, using the QLanczos algorithm as a tool. Wetook advantage of the symmetry of the system to simplifythe quantum computation of the eigenvalues and eigen-states of the Hamiltonian of the system which were thenused for the computation of various physical quantities ofinterest. We ran experiments on IBM Q Yorktown hard-ware for Ns = 3 and Ns = 4 spatial sites. Although weused readout error mitigation only, the results show goodagreement with the exact values of the physical quantitiesof interest. It should be pointed out that although the useof the initialize function in the IBM Qiskit library gives en-ergy expectation value calculations at each QITE step whichare very close to the exact value in the noiseless simulatorcase, our results show how different the noisy simulator andthe hardware data can be from each other as well as exactcalculations. Our data constitute the first demonstration of

3

(a) (b)

(c) (d)

FIG. 1: Transition probabilities vs. time calculated using the energies obtained from exact diagonalization and compared to thosefrom ROEM energies using the QLanczos algorithm on IBM Q Yorktown hardware. The transitions are (a) |100〉 → |010〉, (b)|110〉 → |011〉, (c) |1000〉 → |0100〉, and (d) |0101〉 → |1010〉. The parameters are set to J = 0.6 and hT = 1. Nruns = 3, and the

shaded regions are showing one-standard-deviation error.

quantum imaginary-time evolution in a 4-qubit system onNISQ hardware, and can be useful for benchmarking pur-poses.

Notably, the use of the symmetry of the system in simpli-fying the QITE and QLanczos algorithms reduces the num-ber of steps in the quantum calculations, which leads to asignificant reduction in error due to NISQ hardware. TheQITE and QLanczos algorithms converge to the minima de-termined by the symmetry subgroup of the chosen initialstate. Further, higher excited states can be obtained by re-versing the sign of the Hamiltonian as needed. These twofeatures enabled us to find energy levels that otherwise weredifficult to compute due to the numerical difficulty associ-ated with increasing the number of vectors in the Krylovspace.

METHODS

A. Unitary Evolution Operators

The unitary evolution operator is expressed in terms ofthe eigenvalues and eigenstates of the Hamiltonian (1) as

U(t) =

2Ns−1∑I=0

e−iEIt|ψI〉〈ψI | . (4)

Let t be the unitary transformation from the eigenstates ofH to the computational basis. Its matrix elements are

tIx = 〈ψI |x〉 . (5)

All components of the eigenstates |ψI〉 are real, therefore,tIx ∈ R. This will simplify the computation of the compo-nents of the eigenstates.

Scattering data can be expressed in terms of transitionamplitudes between an initial and a final state, both mem-bers of the computational basis, |xin〉 and |xfin〉, respec-tively. A transition amplitude over time t,

Afi(t) ≡ 〈xfin|U(t)|xin〉 (6)

4

(a) (b)

(c) (d)

FIG. 2: Occupation numbers 〈ni(t) at the ith spatial site vs. time calculated using energies obtained from exact diagonalization andcompared to those calculated from ROEM energies using QLanczos algorithm on IBM Q Yorktown hardware. The initial states are(a) |100〉, (b) |110〉, (c) |1000〉, and (d) |1010〉. The parameters are set to J = 0.6 and hT = 1. Nruns = 3, and the shaded regionsshow one-standard-deviation error. In (a), (b), and (c) 〈n1(t)〉 and 〈n2(t)〉, and in (d) 〈n0(t)〉 and 〈n2(t)〉, as well as 〈n1(t)〉 and

〈n3(t)〉 overlap with each other.

(a) (b)

(c) (d)

FIG. 3: Exact magnetization vs. time calculated using energies obtained from exact diagonalization and compared to thosecalculated from ROEM energies using QLanczos algorithm on IBM Q Yorktown hardware. The initial states are (a) |100〉, (b)|110〉, (c) |1000〉, and (d) |1010〉. The parameters are set to J = 0.6 and hT = 1. Nruns = 3, and the shaded regions show

one-standard-deviation error.

can be calculated classically using the matrix t (eq. (5). We obtain

Afi(t) =

2Ns−1∑I=0

tIxintIxfine−iEIt . (7)

5

It should be noted that, while this calculation leads to moreaccurate results for NISQ devices, as we will demostrate, itis not scalable, therefore for a large number of qubits, it ismore efficient to use Trotterization on the evolution unitaryU(t).

The time evolution of the occupation number for the ithsite (i = 1, . . . , Ns) can be calculated using the expression(4) of the evolution operator. We obtain the average in thestate |x〉 at time t,

〈x|ni(t)|x〉 =

2Ns−1∑I,J,y=0

yitIxtJxtIytJyei(EJ−EI)t , (8)

where yi is the ith digit in the binary expansion of y. Wededuce the transverse magnetization as,

〈mz(t)〉 ≡1

Ns

Ns−1∑i=0

〈Zi(t)〉 = 1− 2

Ns

Ns−1∑i=0

〈ni(t)〉 . (9)

One can also simulate the thermal evolution of the system[10] by computing the ensemble average of any operator Oat finite temperature, T ,

〈O(β)〉 =1

Z

2Ns−1∑I=0

eβEI 〈ψI |O|ψI〉 , (10)

where β = 1kBT

, kB is the Boltzmann constant, and Z =∑I e−βEI is the partition function.

The phase transition can also be studied by using theprobability of the system being in the ferromagnetic state,PFM, as an order parameter, as studied in [4] using a trappedion quantum computer. We leave these calculations to afuture study.

Symmetry of the system

Here, we discuss the symmetry of the system and explainhow it can be utilized to reduce the number of the steps inquantum computations.

A conserved quantity of the system is parity, (−)F , where

F =

Ns∑i=1

ni . (11)

is the total occupation number. Indeed, it is easy to checkthat parity commutes with the Hamiltonian (1),

[(−)F , H] = 0 . (12)

Therefore all eigenstates of the Hamiltonian have definiteparity, starting with the ground state that has even parity((−)F = +1).

The Hamiltonian (1) is also symmetric under permuta-tions of the sites, P : i 7→ (i + 1)modNs, and reflection

around, say, i = 0, R : i 7→ (−i)modNs. If |ψI〉 is an eigen-state of the Hamiltonian (eq. (3)), then P|ψI〉 and R|ψI〉are also eigenstates of H belonging to the same eigenvalueEI . If the energy level EI is non-degenerate, then the cor-responding eigenstate must be invariant under permutationand reflection of the sites. Moreover, since R2 = I, eachenergy level consists of states which are either even or oddunder reflection of the spatial sites.

Let us first consider the case Ns = 3. The ground statemust be parity and reflection even. Since the ground state isnon-degenerate, it must also be invariant under permutationof the sites. It follows that it has to be of the form

|ψ0〉 = a|000〉+ b(|011〉+ |101〉+ |110〉) . (13)

The first excited state must be odd under parity and re-flection. These properties are incompatible with symmetryunder permutation of sites, indicating that the energy levelis degenerate. It is a double degeneracy with the spacespanned by {|ψ1〉,P|ψ1〉} (P2|ψ1〉 is a linear combinationof the other two states, since P3 = I, and so P2 = −P−I).We may choose

|ψ1〉 =1√2

(|001〉 − |010〉) (14)

so that P|ψ1〉 = 1√2(|100〉 − |001〉). Thus, we were able

to determine the first excited states solely from symmetryconsiderations.

For Ns = 4, the ground state is of the form

|ψ0〉 = a|0000〉+ b(|0011〉+ |0110〉+ |1001〉+ |1100〉)+c(|0101〉+ |1010〉) + d|1111〉 , (15)

easily checked to be parity and reflection even, as well asinvariant under permutation.

The first excited state is of the form

|ψ1〉 = a(|0001〉+ |0010〉+ |0100〉+ |1000〉)+b(|0111〉+ |1011〉+ |1101〉+ |1110〉) . (16)

It is parity odd, reflection even, and invariant under permu-tation.

The next level is degenerate and spanned by the states

1√2

(|0001〉 − |0100〉) , 1√2

(|0010〉 − |1000〉) (17)

which are parity and reflection odd.The next level is of the same form as the ground state

and orthogonal to it.The next level is of the form

a(|0001〉 − |0010〉+ |0100〉 − |1000〉)+ b(|0111〉 − |1011〉+ |1101〉 − |1110〉) , (18)

which is parity odd, reflection even, invariant under permu-tation and orthogonal to the first excited state.

The next level is degenerate and spanned by the states

1√2

(|0101〉 − |1010〉) , 1√2

(|0011〉 − |0110〉) ,

1√2

(|0110〉 − |1001〉) , 1√2

(|1001〉 − |1100〉) (19)

6

all of even parity.Another set of parity and reflection odd, degenerate

higher energy level states are

1√2

(|1110〉 − |1011〉) , 1√2

(|1101〉 − |0111〉) . (20)

To access higher energy levels, it is advantageous to flip thesign and use −H as the Hamiltonian and start by computingits ground state which corresponds to the highest energylevel of H. The same symmetry considerations apply tothe Hamiltonian with flipped sign, −H, and one obtainsexpressions for the higher-level states of H that are similarto the lower-level states obtained above.

Algorithms

As mentioned earlier, to calculate the energy levels andcorresponding eigenstates of our system we will use a hy-brid quantum-classical method based on the QLanczos al-gorithm which uses the QITE algorithm first proposed in[11]. Therefore, in this section we will give a brief overviewof these quantum algorithms.

We start by discussing the QITE algorithm whose clas-sical counterpart was introduced in order to simulate thedynamics of many-body systems. It is advantageous to sep-arate the Hamiltonian into local, but non-commuting, com-ponents, H =

∑m hm. The number of these local terms

in the Hamiltonian scales polynomially with the number ofparticles in the many-body system. Since we are only deal-ing with a small number of qubits, there is no need to splitthe Hamiltonian in our case.

QITE relies on evolution in imaginary time. To imple-ment it, we need to set t → −iβ in eq. (4) and define theimaginary-time evolution operator U = e−βH which is nolonger unitary. Starting with the state |Ψ0〉, the evolvedstate is found in n steps each evolving the system in imag-inary time ∆τ , where n = β

∆τ ,

|Ψ(β)〉 = cn(e−∆τH

)n |Ψ0〉 , (21)

with cn being a normalization constant (c−2n =

〈Ψ0|U2|Ψ0〉). In the zero-temperature limit (β →∞), thisstate converges to the ground state of the system.

The QITE algorithm simulates this non-unitaryimaginary-time evolution by approximate unitary up-dates. Thus, the sth step of the imaginary-time evolution,

|Ψs〉 =cscs−1

e−∆τH |Ψs−1〉 , (22)

with s = 1, 2, . . . , n and c0 = 1, can be approximated as

|Ψs〉 ≈ e−i∆τA[s]|Ψs−1〉 , (23)

where A[s] can be written in terms of Pauli operators (σ ∈{X,Y, Z}) involving Ns qubits as

A[s] =∑

i1,...,iNs

a[s]i1...iNsσi1 . . . σiNs

. (24)

Once the a[s] coefficients are calculated, these unitary up-dates can be implemented on a quantum computer. Thesecoefficients can be calculated up to order O(∆τ2) by solv-

ing a linear system of equations (S + ST ) · a = b, where

SI,I′ = 〈σi1 . . . σiNsσi′1 . . . σi′Ns

〉 , (25)

and

bI = −i√cs−1

cs〈σi1 . . . σiNs

H〉 , (26)

with I = {i1, . . . , iNs}, and the expectation values evalu-ated at the state computed in the previous step, |Ψs−1〉.These expectation values involve strings of Pauli matricesand can be evaluated with quantum algorithms recursively.By solving this linear system of equations classically, we ob-tain the minimum distance between |Ψs〉 and the unitaryupdate (23) to lowest order in ∆τ [11]. A solution of thelinear system of equations can also be found with a quan-tum algorithm, but we will not do this here as our focus isimplementation on NISQ hardware. This kind of quantumalgorithm would require implementation of unitary opera-tions with a circuit depth that NISQ hardware could nothandle.

In our previous work [12], we found out that these unitaryupdates for the systems we considered were in the form of aunitary coupled cluster (UCC) Ansatz. This is also the casefor the current Ising spin chain model.

The initial state |Ψ0〉 determines which eigenstate of thesystem the QITE algorithm will converge to. It will convergeto the ground state as long as |Ψ0〉 has a finite overlap withit. For convergence to an excited state, |Ψ0〉 must be or-thogonal to the ground state. As we discussed in the “Sym-metry of the System” section above, utilizing the symmetryof the system helps us make an educated choice of initialstate. In our Ising model, we can exploit the parity and re-flection symmetries to choose an initial state for QITE thatwill be orthogonal to low-level states and therefore convergeto the desired energy level. This minimizes the number ofrequired calculations.

The vector b has 3Ns elements and S is a 3Ns × 3Ns

matrix, therefore we need to perform 3Ns(3Ns + 1) mea-surements in order to calculate all elements in b and S.Since the Hamiltonian is real, so are these matrix elements.Therefore, in the calculation of b, only the elements thathave an odd number of Y Pauli matrices will contributewhile the rest will vanish. Similarly, the S +ST matrix ele-ments which have an even number of Y Pauli matrices willnot contribute. Additionally, the S +ST matrix is symmet-ric and its diagonal elements are all the same. Using thisinformation, we can reduce the number of measurementssignificantly.

As explained in more detail in the next section, althoughall of the above steps can be performed on quantum hard-ware, in view of limited resources, we only implemented thequantum circuit that produced |Ψn〉 from |Ψ0〉 on quantumhardware, aided by the initialize function in the IBM Qiskit

7

library, and performed the remaining steps using quantumsimulation. This also limited the error produced by quantumhardware. If all steps are implemented on NISQ hardware,then the error we are reporting on here will be larger anddepend on the NISQ device used.

Next, we apply the QLanczos algorithm which makesuse of the QITE algorithm to improve the calculationof the energy levels and corresponding eigenstates ofthe system. Although the classical Lanczos algorithmuses the Krylov space K spanned by a set of vectors{|Φ〉, H|Φ〉, H2|Φ〉, . . . }, in its quantum version (QLanc-zos), K is spanned by {|Φ0〉, |Φ2〉, . . . }, where |Φl〉 =cle−l∆τH |Ψs〉.The number of required QLanczos states |Φl〉 in the

Krylov space is determined by the number of eigenstates ofthe system that have non-zero overlap with the initial state,|Ψ0〉. Numerical calculations show that having a smallernumber of QLanczos states in the Krylov space than thenumber of eigenstates with non-zero overlap will result inconvergence at a higher number of QITE steps.

After filling the Krylov space with QLanczos states, weform the overlap (T ) and Hamiltonian (H) matrices whoseelements can be calculated in terms of the energy expecta-tion values, respectively, as

Tl,l′ = 〈Φl|Φl′〉 =clcl′

c2r, (27)

and

Hl,l′ = 〈Φl|H|Φl′〉 = Tl,l′〈Φr|H|Φr〉 , (28)

where r = l+l′

2 , and l, l′ are even. The normalization con-stants can be calculated recursively in terms of expectationvalues using

1

c2r+1

=〈Φr|e−2∆τH |Φr〉

c2r. (29)

Thus, all matrix elements of T andH can be computed witha quantum circuit as expectation values in the states gener-ated by the QITE algorithm. We then solve the generalizedeigenvalue equation

Hx = ET x , (30)

classically and find approximations to the eigenvalues andeigenstates of the system Hamiltonian, which depend onthe choice of initial state |Ψ0〉, in terms of the elements ofthe eigenstates of the generalized eigenvalue equation. Fora given eigenvalue E, denote the corresponding eigenvector

by x(E) = (x(E)0 , x

(E)1 , . . . )T . We obtain the approximation

to an eigenstate of the Hamiltonian (1),

|Ψ[E]〉 = cE

(x

(E)0 |Φ0〉+ x

(E)1 |Φ1〉+ . . .

), (31)

where c−1E = ‖

∑l=0,1,... x

(E)l |Φl〉‖, at energy level

E = 〈Ψ[E]|H|Ψ[E]〉 . (32)

These approximations are easily expressed in terms of quan-tities that were deduced from QITE. We obtain

E =

∑l,l′=0,2,... x

(E)l Hl,l′x

(E)l′∑

l,l′=0,2,... x(E)l Tl,l′x

(E)l′

(33)

By choosing different initial states |Ψ0〉 informed by symme-try considerations, we obtain approximations to all energylevels of the Hamiltonian (1).

To avoid spurious energy levels E, we compute the uncer-tainty in energy, ∆E ≡

√〈H2〉 − 〈H〉2 in the state |Ψ[E]〉.

As discussed above, 〈H〉 = E is computed classically interms of physical quantities obtained in QITE (eqs. (27)and (28)). 〈H2〉 can also be extracted from QITE usingthe matrix elements

Hl,l′ = 〈Φl|H2|Φl′〉 = Tl,l′〈Φr|H2|Φr〉 , (34)

We discard eigenvalues E, if their uncertainty exceeds acertain value, by demanding ∆E ≤ δ.

Quantum Program

To calculate the time evolution of various physical quanti-ties, we need the eigenvalues and eigenstates of the system.In our previous work, we demonstrated the practical calcu-lation of the energy spectrum of many-body chemical andnuclear systems by implementing the QITE/QLanczos al-gorithm on NISQ devices [12]. Here, we extend our workto the calculation of energy levels and corresponding eigen-states of the Ising model Hamiltonian (1).

Since the QLanczos algorithm makes use of output fromthe QITE algorithm, we start with the calculation of energyexpectation values of imaginary-time evolution with differ-ent initial states informed by symmetry considerations ofthe system. Using the QITE algorithm outlined above, wecalculate the unitary updates (eqs. (23) and (24)) at everyimaginary-time step using a small value of the imaginary-time parameter ∆τ and the Hamiltonian (1). Starting withthe state |Ψ0〉, after s unitary updates, we obtain the state

|Ψs〉 = e−i∆τA[s]e−i∆τA[s−1] · · · e−i∆τA[1]|Ψ0〉 (35)

which we implement with a quantum circuit. We simpli-fied these circuits following the methods discussed in [15],as implemented with the initialize function in the IBM QQiskit library. Examples of 3- and 4-qubit quantum circuitsfor the states (35) are depicted in Fig. 4 in terms of single-qubit rotation gates Ry(θ) and two-qubit CNOT gates. Atevery imaginary-time step, the angles change, as they de-pend on the state |Ψs〉, but the depth of the circuit remainsthe same. Therefore, in terms of economizing the numberof gates and operations in the quantum circuit, our resultsare similar to those in our earlier work [12]. It should benoted that, depending on the topology of the quantum hard-ware, interactions between physical qubits matching those

8

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|0i Y (✓s02) Y (�✓s02) • • •

|0i X X • Y (�✓s03) Y (✓s03) • X

|0i X Y (�2✓s01) • • •

(1)

|0i Ry(⇡/2) Ry(⇡/2)

|0i U3(✓2, 0, 0) U3(✓3, 0, 0) • •

|0i U3(✓1, 0, 0) • • • •

(2)

|0i Ry(⇡/2) Ry(⇡/2)

|0i Ry(✓2) Ry(✓3) • •

|0i Ry(✓1) • • • •

(3)

A = Ry(✓2) Ry(✓3)

Ry(✓1) • • •

(4)

|0i

A

Ry(⇡/2) Ry(⇡/2)

|0i • •

|0i •

(5)

|0i Ry(⇡/4) Ry(⇡/4) Ry(⇡/4) Ry(⇡/4)

|0i

A

Ry(✓4) Ry(✓5) Ry(✓5) Ry(✓4) • •

|0i • • • •

|0i • • •

1

|0i Y (✓s02) Y (�✓s02) • • •

|0i X X • Y (�✓s03) Y (✓s03) • X

|0i X Y (�2✓s01) • • •

(1)

|0i Ry(⇡/2) Ry(⇡/2)

|0i U3(✓2, 0, 0) U3(✓3, 0, 0) • •

|0i U3(✓1, 0, 0) • • • •

(2)

|0i Ry(⇡/2) Ry(⇡/2)

|0i Ry(✓2) Ry(✓3) • •

|0i Ry(✓1) • • • •

(3)

A = Ry(✓2) Ry(✓3)

Ry(✓1) • • •

(4)

|0i

A

Ry(⇡/2) Ry(⇡/2)

|0i • •

|0i •

(5)

|0i Ry(⇡/4) Ry(⇡/4) Ry(⇡/4) Ry(⇡/4)

|0i

A

Ry(✓4) Ry(✓5) Ry(✓5) Ry(✓4) • •

|0i • • • •

|0i • • •

1

|0i Y (✓s02) Y (�✓s02) • • •

|0i X X • Y (�✓s03) Y (✓s03) • X

|0i X Y (�2✓s01) • • •

(1)

|0i Ry(⇡/2) Ry(⇡/2)

|0i U3(✓2, 0, 0) U3(✓3, 0, 0) • •

|0i U3(✓1, 0, 0) • • • •

(2)

|0i Ry(⇡/2) Ry(⇡/2)

|0i Ry(✓2) Ry(✓3) • •

|0i Ry(✓1) • • • •

(3)

A = Ry(✓2) Ry(✓3)

Ry(✓1) • • •

(4)

|0i

A

Ry(⇡/2) Ry(⇡/2)

|0i • •

|0i •

(5)

|0i Ry(⇡/4) Ry(⇡/4) Ry(⇡/4) Ry(⇡/4)

|0i

A

Ry(✓4) Ry(✓5) Ry(✓5) Ry(✓4) • •

|0i • • • •

|0i • • •

FIG. 4: Typical quantum circuits for unitary updates |Ψs〉 obtained with the aid of the IBM Qiskit initialize function. The energyexpectation value at each QITE step is obtained from measurements on these quantum circuits. (a) A 3-qubit quantum circuit. (b)The 3-qubit gate used in the 3- and 4-qubit quantum circuits expressed in terms of Ry(θ) rotation and CNOT gates. (c) A 4-qubit

quantum circuit.

in the quantum circuit implementing (35) may not be read-ily available, necessitating the addition of SWAP gates tothe circuits in Fig. 4.

FIG. 5: The quantum circuits were run on 5-qubit IBM QYorktown (version v2.0.5) hardware because of its periodic

topology. The arrows in the figure indicate the direction of theCNOT gates.

The experiments were run on 5-qubit IBM Q Yorktownhardware. The number of shots for the each experimentwas 8192 and each experiment was run Nruns = 3 timesto calculate the statistical error in the measurements. Thereason for choosing this quantum computer out of otherIBM Q’s cloud accessible devices is its periodic topology asseen in Fig. 5. Using a quantum computer with periodictopology reduces the number of required SWAP gates forour periodic Ising spin chain Hamiltonian which reduces thenumber of required CNOT gates. This is important be-cause CNOT gates are the dominant source of the error ina quantum circuit. For comparison, Honeywell’s ion trapquantum computer offers connectivity between all physi-cal qubits. Therefore, the error in this type of quantumsystem might be smaller since it does not require the ad-dition of SWAP gates for the type of interaction Hamilto-nian considered here. The basis gates which can be directlyimplemented on IBM Q Yorktown quantum computer aresingle-qubit gates U and the two-qubit CNOT gate, where

U(θ, φ, λ) =

(cos θ2 −eiλ sin θ

2

eiφ sin θ2 ei(φ+λ) cos θ2

), (36)

is a general three-parameter single-qubit gate. In Fig. 4,we used the single-qubit rotation gate Ry(θ) which can beexpressed in terms of the basis gates as Ry(θ) = U(θ, 0, 0).

As mentioned in the ‘Algorithms’ section, simulating eachQITE step requires significant number of measurements onhardware. Even using the aforementioned properties of band S matrices there needs to be a large number of measure-ment done to apply the QITE algorithm on hardware. Forexample, for Ns = 3 we were able to reduce the number ofmeasurements from 756 to 187 at every QITE step. Due tolimitations in cloud access to the quantum hardware (suchas long queue and connection interruptions) we simulatedthe quantum circuits for the states |Ψs〉 and implementedthem on quantum hardware to obtain the energy expecta-tion values for various values of imaginary time. With fullimplementation on a NISQ device, additional errors will oc-cur. To estimate these additional errors, we considered ageneric case and fully implemented it on simulated quan-tum hardware. We obtained energy expectation values forvarious values of imaginary time for three sites, Ns = 3,using the initial state |Ψ0〉 = |100〉, and the Ising modelwith parameters J = 0.6 and hT = 1. We implementedthe QITE algorithm and obtained the operator A[s] frommeasurements on the noisy simulator of the same backend.We used Nshots = 8192 and the calibration parameters from04/24/2020. In Fig. 6 we compare the convergence of theenergy expectation values to the first excited state energy inthree different cases, (a) from exact calculation of the state|Ψs〉 as well as energy expectation values, (b) from a noisysimulation of the state |Ψs〉 and exact energy expectationvalues, and (c) from a noisy simulation of both the state|Ψs〉 and energy expectation values. The energy expectationvalues obtained using methods (a) and (b) are very close toeach other, showing that the main source of additional erroris due to measurements. It follows that the use of simulatedstates does not introduce significant errors. However, theenergy expectation values obtained from measurements on

9

quantum hardware differ from results from noiseless simula-tions. In what follows, we use simulated states, implementtheir quantum circuits on quantum hardware, and performmeasurements to obtain energy expectation values.

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

2.4

2.2

2.0

1.8

1.6

1.4

1.2

1.0

E()

3-qubit Ising Model PBC = 0.1, J = 0.6 for | 0 = |100Calculated energy with noisy sim.ROEM measured energy with noisy sim.exact1st ESE

FIG. 6: Energy vs. imaginary time calculated exactly andcompared to the one calculated using a noisy simulator, andROEM measured energy from the noisy hardware of IBM QYorktown. Initial state is |100〉. The parameters are set tohT = 1 and J = 0.6. Time step is ∆τ = 0.1. Energies

converge to energy level −2.4.

The results of measurements on these quantum circuitsproduced by the QITE algorithm as a function of imaginarytime for different initial states are depicted in Figs. 8, 9,and 10 for Ns = 3 and Ns = 4 spatial sites of our Isingmodel with parameters hT = 1 and J = 0.6. In the Ns =3 case, the initial states |Ψ0〉 are chosen as |100〉, |010〉,1√3(|110〉+ |101〉+ |011〉), and |111〉 shown in Fig. 8 (a)-

(d), respectively.Similarly, in the Ns = 4 case, the initial states are chosen

as |1000〉, |0100〉, 12 (|0001〉 + |0010〉 + |0100〉 + |1000〉),

12 (|0001〉 − |0010〉 + |0100〉 − |1000〉), and 1√

7(|0000〉 +

|1100〉+ |0110〉+ |0101〉+ |1010〉+ |1001〉+ |1111〉). Theyare shown in Fig. 10 (a)-(d), respectively. These initialstates are chosen by taking the symmetry of the systeminto consideration, as explained in the “Symmetry of theSystem” section.

The QITE algorithm converges to the minimum of thesymmetry group that the initial state belongs to. Therefore,it might be challenging to access higher-value energy levelsusing the QITE and QLanczos algorithms. To facilitate thealgorithm’s convergence to higher levels, we reversed thesign of the Hamiltonian (1) so that high energy levels turninto low levels whereas the corresponding eigenstates remainthe same. We applied this strategy to calculate some ofthe high energy levels and corresponding eigenstates of oursystem, e.g., for the 4th and 5th excited states in the 3-qubit (Ns = 3), and the 15th excited state in the 4-qubit(Ns = 4) case. The results of this strategy for the QITEalgorithm, including energy expectation values, can be seenin Figs. 10(e) and 9. In these examples, since we are looking

for the minimum of the reverse Hamiltonian −H, we chosethe initial states |Ψ0〉 to be reflection and parity symmetric,namely |110〉, |011〉, |101〉, and |0000〉, respectively.

As mentioned in the “Symmetry of the System” section,some of the eigenstates are completely constrained by thesymmetry of the system, therefore calculating them is re-dundant. For example, in the Ns = 4 case for parame-ters J = 0.6 and hT = 1 the zero eigenvalue is degener-ate and the corresponding exact eigenstates are given byeq. (19). Similarly, the eigenstates correponding to thedegenerate energy level −2 are given analytically by eq.(17). We took advantage of the exact expressions forthese eigenstates in our calculations. Although, we usedthe exact eigenstates obtained using symmetry constraints,we measured the energy expectation values for the eigen-states demonstrated as the first state in (19) and statesin (17) on hardware (IBM Q Yorktown) using the quan-tum circuits seen in Fig. 7 (a), (b), and (c), respectively.These circuits were run on hardware Nruns = 3 times witheach run having Nshots = 8192 on 08/12/2020 with qubitlayout [q0, q1, q2, q3] = [1, 0, 3, 2] and the ROEM averageenergy values obtained are 0.037 ± 0.006, −2.06 ± 0.02,and −2.01 ± 0.01 (where the ± error is the standard de-viation of the mean) compared to the exact eigenvalues of0 and -2, respectively. For the same coupling and magne-tization parameters the states expressed in (20) correspondto eigenvalue 2 and it is degenerate. Although these statescorrespond to 3 occupied sites and since we study singleparticle propagation and two-particle scattering only we didnot need them in our calculations we obtained an experi-mental ROEM mean value of 2.05± 0.03 for the first statein (20) using the circuit in Fig. 7(d). The experimentswere run on IBM Q Yorktown, Nruns = 3 times with eachrun having Nshots = 8192 on 08/13/2020 with qubit layout[q0, q1, q2, q3] = [0, 1, 2, 3].

In our current study, we used two-dimensional Krylovspaces. Although, depending on the choice of initial state,convergence might take longer for a low-dimensional Krylovspace, adding more dimensions causes numerical instabili-ties and does not guarantee convergence to eigenstates ofthe Hamiltonian (1). Interestingly, we were able to ob-serve convergence to the eigenvalues of the system by us-ing a three-dimensional Krylov space together with our un-certainty criterion to exclude spurious states (∆E ≤ δ).However, we did not obtain three distinct energy eigen-states. In general, results were numerically more accuratein two-dimensional Krylov spaces for the QLanczos algo-rithm. Adding more dimensions decreased the number ofcases where off-diagonal T matrix elements were < 1 result-ing in spurious eigenstates. Our numerical results indicatethat using two-dimensional Krylov space is the best choicefor the QLanczos algorithm implemented on noisy quantumdevices. Further application of the error mitigation strate-gies, such as Richardson extrapolation (an example of appli-cation of Richardson extrapolation to QITE algorithm canbe seen in [12]), might improve the numerical stability andcan provide faster convergence in higher-dimensional Krylovspaces.

10

2

The circuit for | 0i = |0010i � |1000i is as seen below.

|0i

|0i Ry(⇡)

|0i

|0i Ry(�⇡/2) •

The circuit for | 0i = |0001i � |0100i is as seen below.

|0i Ry(⇡)

|0i

|0i Ry(�⇡/2) •

|0i

The circuit for | 0i = |0101i � |1010i is as seen below.

|0i

|0i Ry(⇡)

|0i

|0i Ry(�⇡/2) •

The circuit for | 0i = |1110i � |1011i is as seen below.

|0i Ry(�⇡/2) •

|0i Ry(⇡)

|0i Ry(⇡)

|0i Ry(⇡)

2

The circuit for | 0i = |0010i � |1000i is as seen below.

|0i

|0i Ry(⇡)

|0i

|0i Ry(�⇡/2) •

The circuit for | 0i = |0101i � |1010i is as seen below.

|0i Ry(⇡) •

|0i Ry(⇡)

|0i Ry(⇡)

|0i Ry(�⇡/2) • •

2

The circuit for | 0i = |0010i � |1000i is as seen below.

|0i

|0i Ry(⇡)

|0i

|0i Ry(�⇡/2) •

The circuit for | 0i = |0001i � |0100i is as seen below.

|0i Ry(⇡)

|0i

|0i Ry(�⇡/2) •

|0i

The circuit for | 0i = |0101i � |1010i is as seen below.

|0i

|0i Ry(⇡)

|0i

|0i Ry(�⇡/2) •

2

The circuit for | 0i = |0010i � |1000i is as seen below.

|0i

|0i Ry(⇡)

|0i

|0i Ry(�⇡/2) •

The circuit for | 0i = |0101i � |1010i is as seen below.

|0i Ry(⇡) •

|0i Ry(⇡)

|0i Ry(⇡)

|0i Ry(�⇡/2) • •

(a) (b) (c) (d)

FIG. 7: The quantum circuits used to calculate energy levels with exact initial states |Ψ0〉: (a) 0, with 1√2(|0101〉 − |1010〉), (b) -2,

with 1√2(|0010〉 − |1000〉), (c) -2, with 1√

2(|0001〉 − |0100〉), and (d) 2, with 1√

2(|1110〉 − |1011〉). The parameters were set to

J = 0.6 and hT = 1.

(a) (b)

(c) (d)

FIG. 8: Energy vs. imaginary time calculated exactly using −H, and compared to IBM Q Aer QASM noiseless and noisy simulator,IBM Q Yorktown hardware raw and ROEM energy expectation values. The initial state is (a) |100〉, (b) |010〉, (c)

1√3(|110〉+ |101〉+ |011〉), (d) |111〉. Data on (a), (b) and (c), and (d) collected on 04/23-24/2020, 04/19-22/2020, 05/01/2020

and 04/22-24-25/2020, respectively. For the hardware data, Nrun = 3 and the error bars are ±σ. Runs for (a), (b), (d) were onqubits [q0, q1, q2] = [0, 1, 2], whereas for (c) on [2, 3, 4], because on the respective days of the runs, the backend properties were

better for those qubits. The parameters are set to hT = 1 and J = 0.6. Time step is ∆τ = 0.1. Energies converge to first-excitedenergy level −2.4 ((a), (b) and (d)) and ground-state energy level −3.4 ((c)).

As mentioned in the “Algorithms” section, we decide onthe convergence to the eigenstates and eigenvalues of thesystem and discard spurious states by using the uncertainty

criterion, ∆E ≤ δ. Two examples involving the ground andexcited states that are specific to a given initial state, |Ψ0〉,are shown in Fig. 12. Specifically, the uncertainty ∆E is

11

(a) (b) (c)

FIG. 9: Energy vs. imaginary time calculated exactly using −H, and compared to IBM Q Aer QASM noiseless and noisy simulator,IBM Q Yorktown hardware raw and ROEM energy expectation values. The initial state is (a) |110〉, (b) |011〉, (c) |101〉. Data werecollected on days 06/12/2020-06/13/2020. For the hardware data, Nrun = 3 and the error bars are ±σ. Runs to obtain these data

were done on qubits [q0, q1, q2] = [2, 3, 4]. The parameters are set to hT = 1 and J = 0.6. Time step is ∆τ = 0.1. Energiesconverge to energy level −1.6.

(a) (b) (c)

(d) (e) (f)

FIG. 10: Energy vs. imaginary time calculated exactly and compared to IBM Q Aer QASM noiseless and noisy simulator, IBM QYorktown hardware raw and ROEM energy expectation values. The initial state is (a) |1000〉, (b) |0100〉, (c)

12(|0001〉+ |0010〉+ |0100〉+ |1000〉), (d) 1

2(|0001〉 − |0010〉+ |0100〉 − |1000〉), (e)

1√7(|0000〉+ |1100〉+ |0110〉+ |0101〉+ |1010〉+ |1001〉+ |1111〉), (f) |0000〉 (with −H). Runs to obtain these data were done on

days (a) 04/26/2020-05/13/2020, (b) 05/15/2020, (c) 04/22, 28/2020, (d) 04/22/2020, (e) 05/06, 08/2020, (f) 06/01/2020,respectively using qubits [q0, q1, q2, q3] = [0, 1, 2, 3]. For the hardware data, Nrun = 3 and the error bars are ±σ. The parameters are

set to hT = 1 and J = 0.6. Time step is ∆τ = 0.1. Energies converge to first-excited energy level −3.4 ((a), (b) and (c)), theground-state energy level −4.4 ((e) and (f)), and the energy level −1.1 ((d)).

shown for Ns = 4 and various values of (l,m), where l,mare even integers and label the basis states of the Krylovspace, which is spanned by {|Φl〉, |Φm〉}. Results of the3 different runs on IBM Q Yorktown hardware are shown.We keep increasing l and m, which correspond to QLanc-zos states with higher QITE steps, until ∆E < 1, and wechoose the eigenvalues and eigenstates that give the mini-mum uncertainty.

After the application of this process we were able to ob-

tain energy eigenvalues of (1) experimentally for Ns = 3and Ns = 4 with parameters J = 0.6 and hT = 1 usingeither the exact states obtained from symmetry or usingQLanczos algorithm with a Krylov space of size 2. As seenin Fig. 11 (a) (Ns = 3) and (b) (Ns = 4) the experimen-tal eigenvalues are in very good agreement with the exacteigenvalues of the Hamiltonian.

Finally, after the quantum computation of the eigenvaluesand eigenstates of our Hamiltonian for Ns = 3 and Ns = 4,

12

(a) (b)

FIG. 11: Exact and experimental eigenvalues of the Hamiltonian for Ns = 3 ((a)) and Ns = 4 ((b)). The parameters are set toJ = 0.6 and hT = 1. For Ns = 4, the 2nd, 3rd, and 6th energy levels were obtained from the exact eigenstates (Eqs. (17) and

(19)). The remaining eigenvalues were obtained using the QLanczos algorithm implemented on Yorktown hardware. Theexperiments were run Nruns = 3 times and the error bars represent one standard deviation.

we calculate the coefficients 〈xin|ψI〉 and 〈xfin|ψI〉 whichare then used in Eqs. (7), (8), and (9).

We summarized our method to calculate the transitionamplitudes, occupation number, and transverse magnetiza-tion using QLanczos algorithm in the pseudocode below inFig. 13.

Error Mitigation

Running the quantum circuits on NISQ devices brings er-rors of various sources such as noise from the implementa-tion of the circuit gates and noise due to the measurementreadout errors. To mitigate these errors in the measure-ments error mitigation strategies are employed. In this work,we will only use a readout error mitigation technique sinceQLanczos algorithm gives results that are good agreementwith the exact values. One can use further error mitigationstrategies such as Richardson extrapolation as we did in ref.[12] or reduced density matrix purification ([16]) to improvethe results obtained using the QITE algorithm.

In this paper, we use local readout error mitigation strat-egy that we used in our previous work [12] in which thecorrected expectation values of the Pauli terms is calculated

using

〈Zi . . . Zj〉 =∑

x∈possible outcomes

p(x)

× (−1)xi − p−i1− p+

i

× · · · ×(−1)xj − p−j

1− p+j

,

(37)

where p(x) is the probability of each qubit outcome and ittakes 2N values. Here, we only consider the expectationvalues for Z terms since we do the measurements in Zbasis. The terms with X and Y Pauli operators are rotatedto be measured in Z basis. We define the symmetric andanti-symmetric combinations of the probability of i-th qubitflipping from 0 to 1 (pi(0|1)) or from 1 to 0 (pi(1|0)) as

p±i = pi(0|1)± pi(1|0) , (38)

with

p(1|0) =# of states expected in |1〉 measured in |0〉

# of shots(39)

or vice versa for p(0|1).

DATA AVAILABILITY

The data that support the findings of this study are avail-able from the authors upon reasonable request.

CODE AVAILABILITY

The code that is used to produce the data presented inthis study is available from the authors upon reasonablerequest.

13

FIG. 12: The uncertainty in energy ∆E for different Krylov (K) space parameters [l,m], where {|Φl〉, |Φm〉} span K, and initialstates (a) |1000〉 (cf. with Fig. 10(a)) and (b) |0000〉 (cf. with Fig. 10(f)). Experimental results from Nruns = 3 runs on IBM Q

Yorktown hardware.

12

(a) (b)

FIG. 11: Exact and experimental eigenvalues of the Hamiltonian for Ns = 3 ((a)) and Ns = 4 ((b)). The parameters are set toJ = 0.6 and hT = 1. For Ns = 4, the 2nd, 3rd, and 6th energy levels were obtained from the exact eigenstates (Eqs. (17) and

(19)). The remaining eigenvalues were obtained using the QLanczos algorithm implemented on Yorktown hardware. Theexperiments were run Nruns = 3 times and the error bars represent one standard deviation.

we calculate the coefficients 〈xin|ψI〉 and 〈xfin|ψI〉 whichare then used in Eqs. (7), (8), and (9).

We summarized our method to calculate the transitionamplitudes, occupation number, and transverse magnetiza-tion using QLanczos algorithm in the pseudocode below.

Algorithm 1 The process of calculating the transitionamplitudes, occupation number, and average transverse

magnetization using the QLanczos algorithm

Choose initial state |Ψ0〉.Calculate unitary updates, A[s].Find imaginary-time evolved state by using the unitary update.

Use initialize function to find the quantum circuit takes initialstate to that particular state.Do measurements using the quantum circuits obtained.Calculate the T and H matrices using the energy expectationvalues.Solve the generalized eigenvalue equation and find eigenvaluesand eigenstates.Use minimum uncertainty criterion to decide about the con-vergence.Calculate the transition amplitudes, occupation number, av-erage magnetization, etc.

Error Mitigation

Running the quantum circuits on NISQ devices brings er-rors of various sources such as noise from the implementa-tion of the circuit gates and noise due to the measurementreadout errors. To mitigate these errors in the measure-ments error mitigation strategies are employed. In this work,we will only use a readout error mitigation technique sinceQLanczos algorithm gives results that are good agreementwith the exact values. One can use further error mitigationstrategies such as Richardson extrapolation as we did in ref.[12] or reduced density matrix purification ([16]) to improvethe results obtained using the QITE algorithm.

In this paper, we use local readout error mitigation strat-egy that we used in our previous work [12] in which thecorrected expectation values of the Pauli terms is calculatedusing

〈Zi . . . Zj〉 =∑

x∈possible outcomes

p(x)

× (−1)xi − p−i1− p+

i

× · · · ×(−1)xj − p−j

1− p+j

,

(37)

where p(x) is the probability of each qubit outcome and ittakes 2N values. Here, we only consider the expectationvalues for Z terms since we do the measurements in Zbasis. The terms with X and Y Pauli operators are rotatedto be measured in Z basis. We define the symmetric andanti-symmetric combinations of the probability of i-th qubitflipping from 0 to 1 (pi(0|1)) or from 1 to 0 (pi(1|0)) as

p±i = pi(0|1)± pi(1|0) , (38)

with

p(1|0) =# of states expected in |1〉 measured in |0〉

# of shots(39)

or vice versa for p(0|1).

DATA AVAILABILITY

The data that support the findings of this study are avail-able from the authors upon reasonable request.

CODE AVAILABILITY

The code that is used to produce the data presented inthis study is available from the authors upon reasonablerequest.

FIG. 13: The algorithm where the process of calculating thetransition amplitudes, occupation number and average

transverse magnetization is summarized.

[1] E. Gustafson, Y. Meurice, and J. Unmuth-Yockey, “Quan-tum simulation of scattering in the quantum Ising model”,Phys. Rev. D 99, 094503 (2019).

[2] E. Gustafson, P. Dreher, Z. Hang, Y. Meurice, “Efficient er-ror mitigation for real-time evolution of a (1+1) field theoryusing a quantum computer”, arXiv:1910.09478v2 [hep-lat]

14

[3] H. Lamm, and S. Lawrence, “Simulation of NonequilibriumDynamics on a Quantum Computer”, Phys. Rev. Lett. 121,170501 (2018).

[4] K. Kim, S. Korenblit, R. Islam, E. E. Edwards, M.-S. Chang,C. Noh, H. Carmichael, G.-D. Lin, L.-M. Duan, C. C. JosephWang, J. K. Freericks and C. Monroe, “Quantum simula-tion of the transverse Ising model with trapped ions”, NewJournal of Physics 13 (2011) 105003.

[5] A. Smith, M. S. Kim, F. Pollman, and J. Knolle, “Sim-ulating quantum many-body dynamics on a current digi-tal quantum computer”, npj Quantum Information 5, 106(2019).

[6] J. Preskill, “Quantum Computing in the NISQ era and Be-yond,” Quantum 2, 79 (2018).

[7] N. Klco, E. F. Dumitrescu, A. J. McCaskey, T. D. Morris,R. C. Pooser, M. Sanz, E. Solano, P. Lougovski, and M.J. Savage, “Quantum-classical computation of Schwingermodel dynamics using quantum computers”, Phys. Rev. A98, 032331 (2018).

[8] A. Alexandru, P. F. Bedaque, and S. Lawrence, “Quan-tum Algorithms for Disordered Physics”, Phys. Rev. A 101,032325 (2020).

[9] K. Seki, T. Shirakawa, and S. Yunoki, “Symmetry-adaptedvariational quantum eigensolver”, Phys. Rev. A 101,052340 (2020).

[10] A. Cervera-Lierta, “Exact Ising model simulation on a quan-tum computer”,Quantum 2, 114 (2018).

[11] M. Motta, C. Sun, A. T. K. Tan, M. J. O’Rourke, E. Ye,A. J. Minnich, F. G. S. L. Brandao, and G. K.-L. Chan,“Determining eigenstates and thermal states on a quantumcomputer using quantum imaginary time evolution”, NaturePhysics (2019).

[12] K. Yeter-Aydeniz, R. C. Pooser, G. Siopsis, “PracticalQuantum Computation of Chemical and Nuclear EnergyLevels Using Quantum Imaginary Time Evolution and Lanc-zos Algorithms”, npj Quantum Information 6, 63 (2020).

[13] H. Nishi, T. Kosugi, and Y. Matsushita, “Implementationof quantum imaginary-time evolution method on NISQ de-vices: Nonlocal approximation”, arXiv:2005.12715 [quant-ph](2020).

[14] N. Gomes, F. Zhang, N. F. Berthusen, C.-Z. Wang, K.-M. Ho, P. P. Orth, and Y. Yao1, “Efficient step-mergedquantum imaginary time evolution algorithm for quantum

chemistry , arXiv:2006.15371 [physics.comp-ph] (2020).[15] V. V. Shende, S. S. Bullock and I. L. Markov, “Synthesis of

quantum-logic circuits,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems, 25, 6(2006).

[16] A. J. McCaskey, Z. P. Parks, J. Jakowski, S. V. Moore, T. D.Morris, T. S. Humble, and R. C. Pooser, “Quantum chem-istry as a benchmark for near-term quantum computers”,npj Quantum Information 5, 99 (2019).

[17] B. Eastin, and S. T. Flammia, “Q-circuit Tutorial”,arXiv:quant-ph/0406003v2 (2004).

ACKNOWLEDGMENTS

This manuscript has been authored by UT-Battelle, LLC,under Contract No. DE-AC0500OR22725 with the U.S. De-partment of Energy. The quantum circuits were drawn usingQ-circuit package [17]. This work was supported by theQuantum Information Science Enabled Discovery (Quan-tISED) for High Energy Physics program at ORNL underFWP number ERKAP61 and used resources of Oak RidgeLeadership Computing Facility located at ORNL, which issupported by the Office of Science of the Department ofEnergy under contract No. DE-AC05-00OR22725. The au-thors acknowledge use of the IBM Q for this work. Theviews expressed are those of the authors and do not reflectthe official policy or position of IBM or the IBM Q team.

AUTHOR CONTRIBUTIONS

K. Y. A. and G. S. designed the study, K. Y. A. collecteddata and produced figures. R. C. P. and G. S. supervised theresearch. All authors discussed the results and contributedto the final paper.

COMPETING INTERESTS

The authors declare that there are no competing interests.