Dynamical Mean Field Theory Algorithm and Experiment on Quantum Computers I

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Dynamical mean field theory algorithm and experiment on quantum computers I. Rungger, N Fitzpatrick, H. Chen, C Alderete, H. Apel, A Cowtan, A Patterson, D. Muñoz, Y. Zhu, N Nguyen, et al.

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I. Rungger, N Fitzpatrick, H. Chen, C Alderete, H. Apel, et al.. Dynamical mean field theory algorithm and experiment on quantum computers. 2020. hal-02515827

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I. Rungger,1, ∗ N. Fitzpatrick,2 H. Chen,3, 4 C. H. Alderete,5, 6 H. Apel,2, 7 A. Cowtan,2 A. Patterson,7 D. Muñoz Ramo,2 Y. Zhu,5 N. H. Nguyen,5 E. Grant,3, 4 S. Chretien,1 L. Wossnig,3, 4 N. M. Linke,5 and R. Duncan2, 8 1National Physical Laboratory, Teddington, TW11 0LW, United Kingdom 2Cambridge Quantum Computing Ltd, 9a Bridge Street, Cambridge, United Kingdom 3Rahko Ltd., Finsbury Park, N4 3JP, United Kingdom 4Dept. of Computer Science, University College London, Gower Street, London, WC1E 6BT, United Kingdom 5Joint Quantum Institute, Department of Physics, University of Maryland, College Park, MD 20742, USA 6Instituto Nacional de Astrof´ısica, Optica´ y Electrónica, Calle Luis Enrique Erro No. 1, Sta. Ma. Tonantzintla, Pue. CP 72840, Mexico 7Dept. of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT, United Kingdom 8Department of Computer and Information Sciences, University of Strathclyde, 26 Richmond Street, Glasgow, United Kingdom The developments of quantum computing algorithms and experiments for atomic scale simulations have largely focused on quantum chemistry for molecules, while their application in condensed matter systems is scarcely explored. Here we present a quantum algorithm to perform dynamical mean field theory (DMFT) calculations for condensed matter systems on currently available quantum computers, and demonstrate it on two quantum hardware platforms. DMFT is required to properly describe the large class of materials with strongly correlated electrons. The computationally challenging part arises from solving the effective problem of an interacting impurity coupled to a bath, which scales exponentially with system size on conventional computers. An exponential speedup is expected on quantum computers, but the algorithms proposed so far are based on real time evolution of the wavefunction, which requires high-depth circuits and hence very low noise levels in the quantum hardware. Here we propose an alternative approach, which uses the variational quantum eigensolver (VQE) method for ground and excited states to obtain the needed quantities as part of an exact diagonalization impurity solver. We present the algorithm for a two site DMFT system, which we benchmark using simulations on conventional computers as well as experiments on superconducting and trapped ion qubits, demonstrating that this method is suitable for running DMFT calculations on currently available quantum hardware.

I. INTRODUCTION electron-electron correlations are strong, the dynamical mean field theory (DMFT) is the state-of-the-art cor- Computers have an integral role in the design process rection to DFT [13–15]. In DMFT one separates out the of new materials and chemicals, predicting and explain- strongly interacting local orbitals as an effective impurity ing the behavior of the systems in question [1–4]. Such from the remaining part of the system, which is treated as simulations are mostly based on density functional theory an effective bath together with the orbitals of neighbor- (DFT) due to its low computational cost and simplicity ing atoms. Example cases where DMFT overcomes the of use [1]. DFT allows for the computation of important failures of DFT include Mott insulators [6] and super- properties, such as atomic structures, chemical reaction conducting systems [16], molecules on surfaces exhibit- rates and the electronic structure of solids. However, in ing Kondo behavior [17], and even extend to biochemical many instances DFT fails to correctly predict the ob- systems [18] and metalloproteins such as haemoglobin [8]. arXiv:1910.04735v2 [quant-ph] 8 Jan 2020 served behavior [5], for example Mott insulator transi- Metalloproteins play an important role within the phar- tions [6] or the binding of oxygen to haemoglobin [7, 8]. maceutical sector, such as the therapeutic application of The reason is that DFT treats the quantum mechan- hemocyanins [19], a very promising class of anti-cancer ical interactions between electrons within an effective therapeutics. mean field approximation, which becomes invalid when However, the high computational cost required for ac- the electrons are strongly correlated with each other. curate solutions limits DMFT to small systems on the A number of corrections have been developed to over- currently available conventional computers. Quantum come the limitations of DFT [9], such as hybrid func- computers can in principle solve correlated electronic tionals [10], self-interaction corrections [11], or the GW structure problems in polynomial time [20–22] using al- approximation [12]. For solid state systems, where local gorithms such as phase estimation [23]. While large-scale quantum computers are still out of reach, small, so called noisy intermediate scale quantum computers [24] have re- cently become available, and have sparked a growing in- ∗ [email protected] terest in demonstrating applications on existing devices. 2

In particular, quantum-classical hybrid algorithms, such index is a collective index that includes both orbital and as the variational quantum eigensolver (VQE) [25], have spin degrees of freedom, so that for example α = 1 rep- been shown to enable simulations for small molecules and resents the spin orbital (1, ↑), where the first integer in- simplified model Hamiltonians [26, 27], perform learning dicates the site index and the arrow the spin direction, tasks [28–31] or even algebraic operations [32]. However, α = 2 represents (1, ↓), α = 3 represents (2, ↑), and so due to the noise levels and limited size of current quan- on, until the index α = 2Nimp represents (Nimp, ↓). The tum computers simulations of large molecules or mate- spin orbitals associated with the Nb bath orbitals are la- rials systems are still out of reach. Embedding methods beled with i, j subscripts, and the ones associated with such as DMFT may overcome this problem, since here the any of the Nimp + Nb orbitals with n and m subscripts. computationally demanding part of the calculation is per- Within ED the impurity Hamiltonian is usually written formed only on a smaller subsystem. A recent proposal as [33, 39, 43] demonstrates that such methods can also be formulated using quantum algorithms [33, 34]. In these approaches Hˆ = Hîmp + Hˆbath + Hˆmix, (1) the computationally intensive part of the DMFT calcula- X X Hˆ = ( − µ)σ ˆ−σˆ+ + U σˆ−σˆ−σˆ+σˆ+, tion is executed on a quantum device, while the remain- imp α α α αβγδ α β γ δ ing part is done classically. This hybrid approach has α αβγδ (2) been simulated on a conventional computer for a small ˆ X − + ∗ − + prototype system of 2 sites [35] within the framework of Hmix = Vαiσˆα σî + Vαiσî σˆα , (3) 2-site DMFT [36]. αi To date no hybrid quantum-classical algorithm that X Hˆ = σˆ−σˆ+, (4) solves the DMFT formalism has been implemented on bath i i i i an actual quantum device, because prior methods have been unable to deal with the high levels of noise in these where µ is the chemical potential, α are the internal on- machines. Here we expand upon prior work by intro- site energies of the impurity, Uαβδγ are the electron in- ducing a VQE-based quantum-classical hybrid algorithm, teraction energies, Vαi are the hopping matrix elements which allows us to compute the 2-site DMFT system on between the impurity and the bath, and the i denote the a quantum device. The quantum computer solves the ef- onsite energies of the bath orbitals. As part of the stan- fective quantum impurity problem, which in the DMFT dard Jordan-Wigner transform here we have introduced loop is self-consistently determined via a feedback be- a modified form of the Pauli ladder operators that takes tween the quantum and classical computation. We use into account the fermionic nature of the electrons VQE to implement an exact diagonalization solver [37– α−1  39], and demonstrate that our algorithm can tackle the Y 1 σˆ± = σˆz (ˆσx ± iσˆy ) , (5) electronic structure problem of correlated materials us- α  β 2 α α ing quantum devices available today. This method can β=1     be scaled to larger system sizes as the available number Nimp i−1 Y Y 1 of qubits increases, where its efficiency relies on the scal- σˆ± = σˆz σˆz (ˆσx ± iσˆy) . (6) i  β  j  2 i i ability of the underlying VQE, which is an active area of β=1 j=1 research [40, 41]. With that, we believe our method can allow many open questions in quantum materials to be − + With this definitionσ ˆα(i) [ˆσα(i)] creates [destroys] an elec- resolved once an intermediate scale quantum computer tron on spin orbital α(i). The electron number operator with around 100 qubits becomes available. − + on spin orbital α(i) is then given byn ˆα(i) =σ ˆα(i)σˆα(i), D z E D z E so that for σˆα(i) = 1 [ σˆα(i) = −1] spin orbital α(i) II. QUANTUM ALGORITHM is empty [filled]. The total number of qubits to represent Hˆ is 2(Nimp + Nb). A. Dynamical mean field theory We can calculate all the energy eigenvalues of Hˆ , EN,n, and the corresponding eigenvectors, |ψN,ni. Here the Within DMFT the extended lattice model is mapped integer N denotes the number of electrons of the state, to an effective Anderson impurity problem with Hamilto- and the integer n goes from 0 to the number of eigenstates nian operator Hˆ , where the interacting region is coupled with N electrons minus one, MN . We order the states by to an infinitely extended bath [13–15]. Here we use the increasing energy, so that |ψN,0i is the ground state for exact diagonalization (ED) approximation as the impu- N electrons, |ψN,1i is the first excited state and so on, rity solver, where a finite number of effective sites is used until the highest excited state for N electrons |ψN,MN i. to represent the bath [37–39]. To map this ED Hamil- We denote as N0 the number of electrons of the overall tonian to a quantum computer we perform a Jordan- ground state, and label the ground state (GS) as |ψ0i =

Wigner transform [42]. As a matter of notation, we la- |ψN0,0i, with E0 = EN0,0. bel quantities associated with the Nimp impurity spin or- Within DMFT the central quantities are the local re- bitals with Greek subscripts α, β, γ, δ. The spin orbital tarded Green’s functions (GFs) of the original lattice 3 model, Glat(ω), and of the impurity problem, G(ω). Here ω is the real or imaginary energy. The central condition Classical computa�on εα of DMFT is that locally on the interacting site the impu- Σα(ω) Σlat,α(ω)=Σα(ω) εi rity problem is equivalent to the lattice problem, which G (ω) G (ω) α lat,α Vαi leads to the condition that Glat(ω) = G(ω). This condition can only be exactly satisfied if an infinite number of bath sites is used. For a finite number of bath sites in ED one typically aims at effectively minimizing the difference Quantum-classical hybrid computa�on between Glat(ω) and G(ω). In practice one starts from some initial guess for impurity Hamiltonian parameters VQE loop α, i and Vαi, and uses them to calculate G(ω). From E0 ,N0 these quantities and the original lattice Hamiltonian one θoptimal ωp/h,n obtains Glat(ω). At this stage then one updates α, i and λp/h,α,n Vαi to effectively aim to minimize the difference between G(ω) and Glat(ω). One then iterates this procedure until α, i and Vαi reach self-consistency. This process is denoted as the DMFT loop (Fig. 1)[39]. We now assume that the local GF’s off-diagonal terms FIG. 1. Schematic of the ED DMFT loop: we start by are small enough that they can be neglected. Note that choosing initial values of the impurity Hamiltonian parame- we use this assumption only to simplify the notation, ters {α, i,Vαi}, which we then use as input to the quantum and one can equivalently formulate the algorithm also computing part of the loop; here we use a VQE quantum cir- for dense GFs. We use the Lehman representation of the cuit to obtain as output all the {ωp/h,n, λp/h,α,n}. These are diagonal elements of the zero temperature impurity GF, then passed as input to the classical computing part of the loop, where the GFs and self-energies are computed to ob- Gα(ω), which are generally given by tain updated parameters {α, i,Vαi}. These are then again passed to the quantum computing part of the loop, and the MN0−1 MN0+1 X λh,α,n X λp,α,n process is iterated until the {α, i,Vαi} are equal within a Gα(ω) = + , ω + iδ − ωh,n ω + iδ − ωp,n specified tolerance between subsequent iterations. Once this n=0 n=0 DMFT self-consistency is achieved, one can use the obtained (7) self-consistent impurity Hamiltonian parameters to calculate where the first summation goes over all states with one the electronic structure of the system. electron removed from the ground state (hole states), while the second summation goes over all states with one electron added to it (particle states)[39]. Here δ is an where the so-called hybridization function is given by infinitesimally small positive number, and 2 X |Vαi| ωh,n = E0 − EN −1,n, (8) ∆ (ω) = , (13) 0 α ω + iδ −   2 i i * α−1 + Y z x λh,α,n = ψN0−1,n  σˆβ σˆα ψ0 , (9) and we have introduced the many-body self-energy,

β=1 Σα(ω), which includes all the modifications to the non-interacting Green’s function, G (ω) = ωp,n = EN +1,n − E0, (10) 0,α 0 −1   2 [ω + iδ − α + µ − ∆α(ω)] , induced by the interaction * α−1 + Y term proportional to U in the Hamiltonian. It can be z x −1 λp,α,n = ψN0+1,n  σˆβ σˆα ψ0 . (11) −1 written in general form as Σα(ω) = G0,α(ω) − Gα (ω). β=1 In our quantum algorithm we determine Gα(ω) with calculations on a quantum computer, which means that we Note that in Eqs. (9) and (11) we use the fact that the evaluate the quantities in Eqs. (8-11) on a quantum de- number of electrons of the states in the brackets differs vice. The remaining computations of the DMFT calcu- by one (see Appendix A). The matrix elements λ p/h,α,n lation are performed on a classical computer (Fig. 1). have values between zero and one, and satisfy the sum P rule n(λp,α,n + λh,α,n) = 1. For increasing Nimp + N the total number of excited states M becomes b N0±1 B. Quantum Circuits exponentially large. However, in practical calculations one typically needs only the low energy excitations up to a specified energy cutoff, which are needed to represent In principle ωp/h,n and λp/h,α,n can be obtained by performing a Fourier transform of the real-time GF [33, 35]. Gα(ω) in the desired energy range around 0. However, the error induced by the finite time steps used We can equivalently write Gα(ω) as to evaluate the real-time GF grows for larger interaction 1 strengths, and therefore the method requires very small Gα(ω) = , (12) ω + iδ − α + µ − ∆α(ω) − Σα(ω) time steps. On the currently available quantum comput- 4 ers this leads to noise levels in the results that do not C. Regularization allow for accurate computations of these quantities. A recent proposal to mitigate the errors in the Trotter evo- With the quantities in Eqs. (8-11) calculated on a lution is to use effectively averaged integrated quantities quantum computer one can in principle perform the full [44]. Our proposed method is to calculate ωp/h,n and DMFT loop in Fig. 1. However, the small deviations of λp/h,α,n via total energy calculations based on a vari- the calculated quantities from the exact values lead to ational quantum eigensolver (VQE), which is generally the presence of unphysical poles in Σ(ω), and hence usu- more resilient to noise [25, 26, 45, 46]. This approach has ally the DMFT loop does not converge. These unphysi- also been suggested in a recent article for general Green’s cal divergences are found at the energies where G0,α(ω) function based calculations [47]. In the following we only −1 −1 is 0, since Σα(ω) = G (ω) − G (ω). The exact in- present the method to calculate ω and λ , since 0,α α p,n p,α,n teracting G−1(ω) has divergences at the same energies, the procedure to obtain ω and λ is analogous. α h,n h,α,n which exactly cancel out the divergences due to G−1 (ω). We represent a general eigenstate of Hˆ on the quantum 0,α For the approximate Gα(ω) calculated with the quan- computer by tum circuit the cancellation is only partial. We therefore need to perform a regularization of the calculated |ψN,ni = UˆN,n |0i , (14) quantities, in which these are modified in order to re- where UˆN,n is a unitary operator, and the |0i state repre- store the exact cancellation of the divergences of G0,α(ω). z The set of energies, at which G (ω) = 0, corresponds sents the state with hσˆmi = 1 for all m. We typically use 0,α to the one where ∆α(ω) in Eq. (13) has a pole, and a hardware efficient ansatz to apply UˆN,n to the state |0i in our quantum circuit [48]. Note that for the common hence is equal to the set {i}. Cancellation of the divergences in Σ (ω) therefore requires that G ( ) = 0 case where the parameters in Hˆ are real, all the expan- α α i and dGα(ω)/dω = dG0,α(ω)/dω| for all i. Together sion coefficients of the |ψN,ni can be made real as well, so ω=i with Eq. (7) we then obtain the sum rules that UˆN,n can be restricted to real operators. We therefore typically do not need to include any Rx rotations MN0−1 MN0+1 X λh,α,n X λp,α,n in our ansatz, since these would only be required if the + = 0. (17) coefficients have a complex component, and instead use − ω − ω n=0 i h,n n=0 i p,n only Ry rotations. MN0−1 MN0+1 We first prepare the state |ψ0i by varying the param- X λh,α,n X λp,α,n 1 + = . (18) eters of the ansatz quantum circuit to minimize the total 2 2 V 2 n=0 (i − ωh,n) n=0 (i − ωp,n) i energy and obtain E0 and with it N0. To then calculate the spectrum of the N0 + 1 and N0 − 1 electron states we In principle one can therefore evaluate all quantities on use a modified Hamiltonian, a quantum computer, and then perform a constrained minimization procedure, where all λ and ω ˆ 2 p/h,α,n p/h,n H˜ = Hˆ + β Nˆ − Ntarget . (15) are modified by as little as possible to satisfy the sum rules above. This can be a challenging task for increas- ˆ P P ing system size. We therefore propose to perform the Here N = α nˆα + i nî is the total number operator. constrained minimization only on the λ , while fix- In H˜ˆ a penalty term proportional to a real parameter p/h,α,n ing the ω . The reason is that in general we ex- β is added to enforce the target electron number N p/h,n target pect that the ω are calculated more accurately than [49, 50], in our case N = N ± 1. To calculate the p/h,n target 0 the λ , since the quantum circuit for the λ ˆ p/h,α,n p/h,α,n ground state and excited states of H˜ one can then use a is longer. Since Eqs. (17) and (18) are linear in the number of methods [51–53]. Here we use the algorithm λp/h,α,n, the constrained minimization is straight forward proposed in Ref. [52], which finds the excited states by even for larger system sizes [54]. Overall the regulariza- penalizing the overlap of the higher energy eigenstates tion is therefore scalable to large system sizes. with the previously found lower energy eigenstates. With this method Eqs. (8-11) can be evaluated. To calculate λp,α,n without increasing the number of qubits III. 2-SITE DMFT SIMULATIONS AND we combine Eqs. (11) and (14) to EXPERIMENTS   2 * α−1 + A. 2-site DMFT model ˆ † Y z x ˆ λp,α,n = 0 U  σˆβ σˆαUN0,0 0 . (16) N0+1,n β=1 As a proof of concept we consider the 2-site DMFT system presented in Refs. [36] and [35], which solves the We therefore obtain λ on a quantum device p,α,n single-band Hubbard model on the Bethe lattice with in- by evaluating the quantum circuit corresponding to finite connectivity using exact diagonalization of a two Uˆ † Qα−1 σˆz σˆxUˆ and measuring the proba- N0+1,α β=1 β α N0,0 site impurity problem with one interacting and one bath bility of finding the |0i state at the end. site. Here we only present the parts required to perform 5 the quantum computing DMFT calculations, for a detailed description of the model we refer to Ref. [36]. For this system Hˆ from Eq. (1) becomes

U µ U Hˆ = σˆzσˆz + − (ˆσz +σ ˆz) − 2 (ˆσz +σ ˆz) 4 1 3 2 4 1 3 2 2 4 V + (ˆσxσˆx +σ ˆyσˆy +σ ˆxσˆx +σ ˆyσˆy) . (19) 2 1 2 1 2 3 4 3 4 FIG. 2. a) Quantum circuit for the 4-qubit DMFT calculation based on the Penalty Term (PT) approach; b) and c) 2-qubit Note that here we use a modified mapping of indices be- circuits used in the Circuit Reduction (CR) approach: b) cir- tween spin orbitals and qubits from that used in Eq. (1) cuit for the 2 electron Sz = 0 ground state calculation; c) in order to slightly reduce the number of resulting Pauli circuit for the 1 and 3 electron spectrum calculations. Since operators: qubit 1 (2) represents ↑ electrons on site 1 the expansion coefficients of our states are real, we only use R rotations. (2), while qubit 3 (4) represents a ↓ electron on site 1 y (2). We can then introduce the operator Sˆz for the total z component of the spin as (in units of ~/2) generally applicable also away from ph symmetry, and we present an example for that in Appendix D. At each Sˆz =n ˆ1 +n ˆ2 − nˆ3 − nˆ4, (20) step of the iterative procedure we have a fixed pair of U and V as input to the quantum computation, which then with expectation value Sz = hSˆzi. Since Hˆ is equivalent provides as output the corresponding values for ω for up- and down-spin electrons we have G3 = G1 and p/h,n and λ . At ph symmetry we have ω = −ω Σ3 = Σ1, so that we need to evaluate the Green’s function p/h,α,n h,n p,n and λ = λ , so that we need to perform the com- and self-energy only for one spin (we evaluate G = G1 h,α,n p,α,n putations only for the particle contributions. To demon- and Σ = Σ1). In 2-site DMFT there are only two bath parame- strate the applicability of our DMFT method on current ters that can be optimized in the DMFT self-consistent quantum hardware we therefore need to show that these quantities computed in an experiment deviate only lit- loop, 2 and V = V12. The conditions for DMFT self- consistency derived in Ref. [36] are tle from the exact numerical results, which we do in the remaining part of the manuscript.

nimp = nlat, (21) We independently use both the approach of circuit re- V 2 = z, (22) duction (CR) and of the additional penalty term (PT) in the Hamiltonian (Eq. (15)) to obtain energies at the re- where we have implicitly set the unit of energy equal quired number of electrons (see Sec. II B). Within the PT to the unscaled hopping of the Bethe lattice. Here approach we use the full 4-qubit Hamiltonian in Eq. (19) 0 ˆ R to calculate E0, N0 and UN0,0, and then the states for nimp = −∞ DOSimp(ω)dω is the occupation of the impurity, with the impurity density of states (DOS) N0 + 1 electrons to obtain all quantities in Eqs. (8-11). 2 To represent the operators UˆN,n on the quantum com- given by DOSimp(ω) = − π Im [G(ω + iδ)], and nlat = 0 puter we use a circuit based on the hardware efficient R DOS (ω)dω is the occupation of one Bethe lat- −∞ lat approach for real Hamiltonians, with two layers of ro- tice site in the periodic system, with the DOS of tation gates and one layer of entangling gates, giving a the Bethe lattice DOS (ω) = 2ρ [ω + µ − Σ(ω)], and √ lat 0 total of 8 angles to be optimized (Fig. 2a). ρ (x) = (1/2π) 4 − x2. In Eq. (22) We have in- 0 For the CR approach we design a separate reduced troduced the quasi-particle weight z, also known as circuit for each different N and S , with a corresponding the wave-function re-normalization factor, defined as z mapping to a reduced Hamiltonian (see Appendix B). In dRe[Σ(ω)] Im[Σ(iδ)] z = 1/ 1 − dω = 1/ 1 − δ . 2-site ω=0 this case no penalty term is needed, and one can just DMFT self-consistency therefore requires updating 2 run normal VQE on the individual circuits to determine and V until Eqs. (21) and (22) are satisfied within a the ground state, and subsequently calculate the excited desired tolerance. states for the N0 ± 1 circuits. For 2-site DMFT the circuits can be reduced to 2 qubits for all N, which then also allows the calculation of the overlap terms with Eq. (16) B. Quantum computing implementation (see Appendix. B). For one and three electrons the resulting effective 2-qubit Hamiltonian has identical eigenval- For our simulations in this section we consider the ues due to particle-hole symmetry (E3,n = E1,n), and is ˆ U z x ˆ U z x particle-hole (ph) symmetric case, where µ = U/2 and H = 4 σˆ2 + V σˆ2 for one electron, and H = − 4 σˆ2 + V σˆ2 2 = 0, so that the condition in Eq. (21) is auto- for three electrons. The state of the first qubit does not matically√ satisfied, and we only have to update V until affect the Hamiltonian, reflecting the fact that there are |V − z| < η to satisfy the condition in Eq. (22), with always two degenerate states, one with Sz = 1 and one η a small finite tolerance [35]. Note that our scheme is with Sz = −1. The 2 qubit circuit required to obtain all 6

a 1.1 b 1 TABLE I. Results of simulations on a classical computer with- Experiment (IBM) Simulation out noise, comparing exact numerical results with those on a 1 0.8 No regularization simulator for 4 qubits using the PT approach and for 2 qubits Exact using the CR approach, for U = 4 and V = 0.745356. The 0.9 0.6

“∞ Shots” VQE runs use the information directly from the V DMFT wave-function state vector rather than simulations of individ- 0.8 z 0.4 ual measurements; the obtained “Optimal θ” angles are then used for a single step calculation with 5000 shots to indicate 0.7 0.2 the statistical noise induced by a finite number of measure- 0.6 0 ments. These optimal angles are then used also for the fixed- 0 1 2 3 4 5 6 7 0 2 4 6 8 10 parameters runs on quantum hardware. DMFT loop iteration U 4 qubits 2 qubits VQE Optimal θ VQE Optimal θ FIG. 3. a) Simulation and experiment on the IBM quantum Exact (∞ Shots) (5000 Shots) (∞ Shots) (5000 Shots) computing hardware of the DMFT algorithm for the change of the bath parameter V as function of DMFT loop iteration E0 -1.795 -1.795 -1.804 -1.795 -1.811 for a fixed value of U = 4. Using the regularization procedure E3,0 -1.247 -1.247 -1.232 -1.247 -1.279 to calculate λ (Eq. (23)) the bath parameter V converges E3,2 1.247 1.247 1.260 1.247 1.244 λ 0.262 0.262 0.288 0.262 0.273 quickly and smoothly to the correct value, and the experimental data follow rather closely the simulated results; without regularization no convergence is possible. b) Value of the quasi-particle weight z at DMFT self-consistency obtained in the possible eigenvalues of these Hamiltonians is shown in simulations, where z ≈ V 2. The exact numerical value agrees Fig. 2c, and consists of only one rotation; the degenerate well with the numerical result obtained using the quantum eigenstate with opposite Sz is then obtained by adding simulation. a Ry(π) rotation on the first qubit. For two electrons there is one state with Sz = 2, one with Sz = −2, and four with Sz = 0, for which the Hamiltonian mapped to 2 ˆ U z z x x qubits is H = 4 σˆ1 σˆ2 +V (ˆσ1 +σ ˆ2 ). The general 2-qubit ansatz for an eigenstate of this Hamiltonian is shown in q U 2 √ Fig. 2b. Note that the CR approach is generally prefer- 1 − 6 for U < 6, and V = z = 0 for U ≥ 6, with able over the PT approach, since it requires fewer angles N0 = 2 and Sz = 0 as GS. As first benchmarking test of as optimization parameters, and hence fewer gates and the quantum algorithm we use a value of U = 4 and the fewer qubits. The drawback is that it requires to find the corresponding analytic DMFT solution V = 0.745356, projection of the Hamiltonian to a reduced set of qubits and compute E0, E3,0, E3,2, and λ on the quantum com- for the specified number of electrons and other conserved puter. We first compute the GS using VQE with simu- quantities, which can be a challenging task for a general lations of the algorithm on a classical computer, and for Hamiltonian. both the PT and the CR approaches we find that the GS Since the Hamiltonian is spin-symmetric, for N = 3 is the N0 = 2 and Sz = 0, in agreement with the exact electrons we have 2 doubly degenerate independent en- numerical result. The values of E0 for the quantum sim- ergy eigenvalues (E3,0 = E3,1 and E3,2 = E3,3), where ulation with effectively infinite statistical measurements one has Sz = 1 and one Sz = −1. For convenience we are identical to the exact numerical results, and the same order the states in such a way that when they are de- is true for E3,0,E3,2 and λ (see Tab. I), showing that our generate, for N = 3 the smaller n refers to the Sz = −1 ansatz circuits can fully represent the eigenstates. For state (Sz = 1 for N = 1), and the larger n to the Sz = 1 this simulation we directly use the actual wave function state (Sz = −1 for N = 1). The poles in the GF (Eq. of the system to evaluate the expectation values, which 7) are then found at ωp,0 = ωp,1 = E3,0 − E0 and ωp,2 = corresponds to the information obtained after an infinite ωp,3 = E3,2 − E0, as well as at −ωp,0 and −ωp,2 due to amount of measurements. Any deviation from the exact ph symmetry. Furthermore, we have λp,1,1 = λp,1,3 = 0, results on a quantum device can therefore be attributed since these involve matrix elements between states with to the finite number of measurements (“shots”) and noise P in the quantum computer. We can then fix the angles θ in different spin. Since n(λp,α,n + λh,α,n) = 1, we then obtain λp,1,2 = 1/2 − λp,1,0, so that we have only one the circuit to the optimal values obtained with the VQE independent parameter, λ = λp,1,0. To solve the DMFT calculations for an effectively infinite number of shots, problem we therefore need to calculate the three energies and perform a single step calculation with a finite num- E0, E3,0 and E3,2, as well as λ on the quantum computer. ber of shots at these “Optimal θ” values. For 5000 shots One important advantage of the chosen system is that the discrepancies in the energy are within about 2%, and it has exact analytical solutions for the ph-symmetric sys- for λ they are also within about 10%. tem (away from ph-symmetry we can still obtain essentially exact numerical solutions), which we can therefore Next we perform the full DMFT loop simulation using use to benchmark the quality of the quantum computa-√ the quantum circuit. Importantly, we need to regular- tions. The analytic DMFT solution for V is V = z = ize the calculated λ to satisfy the sum rule constraints 7 derived in Sec. II C, which in this case simplify to TABLE II. Experimental data obtained on the IBM superconducting qubit quantum computer: the results are anal- ω2 V 2 − ω2 λ = p,0 p,2 . (23) ogous to those obtained by classical simulation in Tab. I. 2 2 2 2V ωp,0 − ωp,2 The “Optimal θ” results have been measured using the angles optimized with the simulator, while for the data in the When applying regularization in the DMFT loop we “VQE” columns the VQE optimization loop has been per- therefore do not need to calculate λ on the quantum com- formed on the IBM device, and for the “DMFT+VQE” col- puter, since it is obtained with this relation. In this case umn the whole DMFT loop has been performed on a quantum computer (converged value for V is V = 0.755). The value of we only need to obtain ωp,0 and ωp,2 via total energy calculations on the quantum computer. We find that in this λ given for “DMFT+VQE” is the one used in the DMFT self- consistency loop based on the regularization sum rules (Eq. way the DMFT loop with the quantum algorithm con- (23)). verges well for all U, and is only limited by the statistical 4 qubits 2 qubits noise of the finite number of shots, while without regu- Exact Optimal θ Optimal θ VQE DMFT+VQE larization convergence cannot usually be achieved (Fig. E -1.795 -1.500 -1.700 -1.823 -1.809 3). This is due to the appearance of unphysical diver- 0 E3,0 -1.247 -1.111 -1.259 -1.248 -1.245 gences in the self-energy when Eq. (23) is not exactly E3,2 1.247 1.025 1.253 1.248 1.244 fulfilled, which the regularization removes, as discussed λ 0.262 0.113 0.210 0.275 0.271 in Sec. II C. Here we use 10000 shots to obtain the expectation values. The calculated values of z at DMFT self-consistency, zDMFT, agree well with the exact ana- TABLE III. Experimental data obtained on the ion trap lytical ones for all values of U (Fig. 3b). The only minor quantum computer at the University of Maryland: the re- difference is that the transition from finite to zero z is sults are analogous to those obtained by classical simulation somewhat smoothed out in the simulated results due to in Tab. I. The “Optimal θ” results have been measured using the fact that we only use a finite number of shots for the the angles optimized with the simulator. evaluation of expectation values. 4 qubits 2 qubits Exact Optimal θ Optimal θ E0 -1.795 -1.691 -1.742 E3,0 -1.247 -1.178 -1.208 C. Experiments on quantum computers E3,2 1.247 1.144 1.230 λ 0.262 0.224 0.258 To demonstrate that the algorithm runs successfully on current quantum computers we first perform the calculation for the pair U = 4 and V = 0.745356 on two on a chain of seven ions, of which five are used as qubits. independent types of quantum computing hardware, the For the 4(2) qubit runs we use 4000(5000) shots per mea- IBM superconducting qubit architecture and the ion trap surement. SPAM errors on the output distribution are quantum computer at the University of Maryland. We corrected via the inverse of an independently measured then perform a full DMFT loop on the IBM quantum state-to-state error matrix. Importantly, this error char- computer for U = 4 to demonstrate that it can also ob- acterization scales linearly with system size. In Appendix tain the self-consistent value of V to a good accuracy. We C we compare 4-qubit results without and with SPAM use the quantum chemistry package EUMEN by Cam- correction for superconducting and trapped ion qubits. bridge Quantum Computing (CQC) in combination with We first perform a measurement for fixed circuit pa- the t|keti compiler for circuit optimization, which is freely rameter values set equal to the ones optimized on the available to researchers [55]. To partly compensate for classical simulator with VQE (Tab. I), using both the 4 noise in the quantum hardware we apply standard cor- qubit PT approach and the 2 qubit CR approach. The rections to the outputs for state preparation and mea- results for the IBM quantum computer are given in Tab. surement (SPAM) errors [56]. II, and the ones for the quantum computer at the Univer- We run the experiments on superconducting qubits on sity of Maryland are presented in Tab.III (more detailed the IBM quantum computers, and use 4096(8192) shots results are given in Appendix C). The 4 qubit exper- for a measurement on 4(2) qubits. We use the SPAM iments on the IBM device give energies that are within library in CQC’s t|keti compiler for the 2 qubit measure- about 10-20% of the exact values, and for the 2 qubit CR ments [57], and the equivalent Qiskit’s Ignis library for experiments we obtain energies within about 5% of the the 4 qubit measurements[58]. exact ones. For λ the deviation is larger, about 20%(60%) The ion trap quantum computer at the University of from the exact values when using the CR(PT) method. Maryland is based on a chain of individual 171Yb+ ions This larger error is expected, since the circuit to evaluate confined in a Paul trap [59, 60]. The native operations λ is longer than the one for the energies. Importantly, of the system are single qubit rotations, or R gates, and for a DMFT calculation λ is obtained from the exact sum two-qubit entangling interactions, or XX gates, which are rule (Eq. (23)), so that the accuracy of the final result created by coupling any pair of qubits via the motional is entirely determined by the accuracy of the energies. modes in the trap [61]. This experiment is performed On the ion trap device, the measurement outcomes are 8

15 closer to the exact values. The results on 4(2) qubits a Non-interacting b for the energy are within about 6-8%(1-3%) of the exact Interacting (DMFT Experiment) ones, and λ for the CR method essentially matches the Interacting (Exact) 10 exact value (Tab. III). Note that here we consider the lat 0.4 ph-symmetric case, since it allows to reduce the number imp DOS

DOS 0.3 of computations. To show that the approach is valid also 5 for the general case, in Appendix D we present an ex- 0.2 ample for a simulation and experiment for the general 0.1 non-particle-hole symmetric case. 0 0 -4 -2 0 2 4 -4 -2 0 2 4 These results show that the required energies can be ω ω calculated rather accurately with experiments on the con- sidered quantum devices. They also confirm that the FIG. 4. (a) DOS on the impurity site and (b) corresponding DMFT DOS on the Bethe lattice for U = 4, computed with CR approach generally gives results that are more noise- many-electron interactions as a DMFT+VQE experiment us- resilient. We therefore apply the CR method to perform ing the 2 qubit reduced circuit approach on the IBM quantum a full VQE loop on the IBM quantum computer. For computer (green curve). We use the output of the DMFT self- this experiment we perform the VQE optimization with consistent loop on the quantum computer for V (V = 0.755) the Rotosolve optimization algorithm.[62] The VQE re- The results of the experiment compare well with the exact sults in Tab. II show that it converges to within 2% of values at the exact V = 0.745356 (dashed curve). For com- the exact results. The value of λ is accurate to about parison we also present the DOS without many-electron in- 5%. These results are better than those obtained at the teractions (orange curve). theoretical optimal angles on the same device, and are therefore indicative that the VQE optimization on the quantum device partly compensates the calibration er- small differences in the energies of the VQE experiment rors in the hardware. and the exact values. For comparison we also present So far in order to reduce the runtime of the experiments the DOS obtained for the non-interacting system, which we have only presented VQE results for a fixed V at ex- for the Bethe lattice has only a single broad peak, and act DMFT self-consistency. The experimental time for a therefore lacks the correct three peak structure obtained full DMFT loop is the product of the single VQE loop with DMFT. time, and the number of iterations in the outer DMFT loop to update V to self-consistency (Fig. 1). Given the high accuracy of the VQE results we expect that a full IV. CONCLUSIONS DMFT loop will exhibit similar convergence as the simulated one, and therefore reach self-consistency at about We have presented an algorithm that performs DMFT ten iterations (Fig. 3a). We verify this by running a full calculations on currently available quantum computers. DMFT loop with the 2-qubit CR method on the IBM Our benchmarks on superconducting and trapped ion quantum computer. The results are shown in Fig. 3a, qubits for 2-site DMFT show that such practical calcu- and indeed the convergence of the DMFT loop on the lations are possible with low levels of error. The reason is that the method is based on VQE total energy calcu- quantum hardware is very similar to the simulated√ results. Convergence to a set tolerance of |V − z| < 0.01 lations, which are generally more resilient to noise than is reached after 7 iterations, and the final self-consistent the real time evolution of states. The method will extend value for V is V = 0.755, and therefore very close to the to growing system sizes compatible with near term quan- exact one. In the last column of Tab. II we also show tum devices, and will benefit from the development for that the resulting energies and λ are close to the exact VQE calculations for quantum chemistry calculations of values (in the DMFT loop λ is calculated with the regu- molecules, since the required quantum circuits are simi- larization Eq. (23)). Note that due to the lower accuracy lar. We expect that our proof of concept demonstration of the 4-qubit PT approach on the IBM quantum com- that DMFT can be run on current quantum hardware puter we expect worse convergence of the DMFT loop will spark additional research into quantum algorithms on 4 qubits when compared to the 2-qubit CR approach, for condensed matter physics systems. while on the trapped ion quantum computer a similar DMFT convergence can be expected also on 4 qubits due to the higher accuracy. V. ACKNOWLEDGMENTS Having ran the DMFT self-consistency on the quantum computer to obtain V , one can then determine the IR acknowledges financial support from the UK De- electronic structure of the system. Using the energies cal- partment of Business, Energy and Industrial Strategy culated with VQE, and λ regularized using Eq. (23), we (BEIS). IR/RD/DR/NF acknowledge financial support plot the DOS on the impurity site and on the correspond- from Innovate UK through the Analysis for Innovators ing Bethe lattice (Fig. 4). The agreement with the exact Scheme A4I R3 Mini Project 104936. HC acknowl- results is good, the small deviations are caused by the edges the support though a Teaching Fellowship from 9

UCL, LW acknowledges kindly the support through the and equivalently Google PhD Fellowship in Quantum Computing. CHA   * α−1 + acknowledges ﬁnancial support from CONACYT (doc- + Y z y ψN−1,m σˆα ψN,n = i ψN−1,m  σˆβ σˆα ψN,n , toral grant no. 455378). NML acknowledges ﬁnan- cial support from the NSF Physics Frontier Center at β=1 JQI (grant no. PHY-1430094). EG is supported by (A10)   the UK EPSRC [EP/P510270/1]. AP is supported by * α−1 + the InQUBATE Training and Skills Hub grant EPSRC − Y z y ψN+1,m σˆα ψN,n = −i ψN+1,m  σˆβ σˆα ψN,n .

EP/P510270/1. AP and IR thank the members of the β=1 EPSRC (grant No. EP/S005021/1) Prosperity Partner- (A11) ship in Quantum Software for Modeling and Simulation for useful discussions. By inserting Eqs. (A8-A9) into Eqs. (A2-A3) we obtain the relations for the matrix elements λp/h,α,n given in Eqs. (9) and Eq. (11). Appendix A: Matrix elements relations

In Sec. II A we have used a standard Jordan-Wigner Appendix B: Circuit reduction transform and as part of it we have introduced a modiﬁed form of the Pauli ladder operators that takes into account 1. General relations the fermionic nature of the electrons as

α−1  Here we perform the transformations to reduce the Y 1 σˆ± = σˆz (ˆσx ± iσˆy ) . (A1) number of qubits within 2-site DMFT in order to restrict α  β 2 α α the wavefunction to a fixed number of electrons and total β=1 z component of the spin. Such a restriction is possible, − + With this definitionσ ˆα (ˆσα ) creates (destroys) an elec- since the Hamiltonian in Eq. (19) commutes with both tron on spin orbital α. The matrix elements λp/h,α,n are the total number operator then generally given by Nˆ =n ˆ1 +n ˆ2 +n ˆ3 +n ˆ4, (B1) + 2 λh,α,n = ψN0−1,n σˆα ψ0 , (A2) and the operator for the total z component of the spin − 2 (in units of ~/2) λp,α,n = ψN0+1,n σˆα ψ0 . (A3) ˆ Using the definition in Eq. (A1) we obtain Sz =n ˆ1 +n ˆ2 − nˆ3 − nˆ4, (B2) − + α−1  wheren ˆα =σ ˆα σˆα is the number operator on spin orbital Y z x + − α. As outlined in the main text, qubits 1 and 2 corre-  σˆβ σˆα =σ ˆα +σ ˆα , (A4) spond to ↑ spin orbitals, while qubits 3 and 4 correspond β=1 to ↓ spin orbitals. Note that in this section we use Greek   α−1 subscripts for both the bath and interacting sites. As Y z y + − i  σˆβ σˆα =σ ˆα − σˆα . (A5) outlined in Sec. III A, the qubit indices are mapped to β=1 site and spin indices as follows: qubit 1 (2) represents ↑ electrons on site 1 (2), while qubit 3 (4) represents a ↓ Furthermore, sinceσ ˆ− (ˆσ+) creates (destroys) one elec- α α electron on site 1 (2). tron on spin orbital α, we also have The Pauli operators satisfy the commutator relation + + ψN 0,m σˆα ψN,n = δN 0,N−1 ψN−1,m σˆα ψN,n , a b c σˆ , σˆ = 2iδαβabcσˆ , (B3) (A6) α β α − − where abc is the Levi-Civita symbol, and the anti- ψ 0 σˆ ψ = δ 0 ψ σˆ ψ . N ,m α N,n N ,N+1 N+1,m α N,n commutator relation (A7) a b {σˆ , σˆ } = 2δab. (B4) Combining Eqs. (A4-A7) we obtain α α   Using these relations one obtains * α−1 + Y a b c + z x σˆ σˆ = δab + iabcσˆ . (B5) ψN−1,m σˆα ψN,n = ψN−1,m  σˆβ σˆα ψN,n , α α α

β=1 With this equation we simplify the number operator for (A8) a spin orbital to * α−1  + − + 1 x y x y − Y z x nˆα =σ ˆ σˆ = (ˆσ − iσˆ ) (ˆσ + iσˆ ) ψN+1,m σˆα ψN,n = ψN+1,m  σˆβ σˆα ψN,n , α α α α α α 4 β=1 1 = (1 − σˆz ) . (B6) (A9) 2 α 10

We can therefore rewrite Nˆ and Sˆz as spin-preserving electron hops between sites for the second operator. This corresponds to the ladder operators ˆ 1 z z z z N = 2 − (ˆσ1 +σ ˆ2 +σ ˆ3 +σ ˆ4 ) , (B7) + − + − + 2 σ¯ˆ1 =σ ˆ3 σˆ1 +σ ˆ4 σˆ2 , (B12) 1 z z z z ˆ+ − + − + Sˆ = (−σˆ − σˆ +σ ˆ +σ ˆ ) . (B8) σ¯2 =σ ˆ2 σˆ1 +σ ˆ4 σˆ3 , (B13) z 2 1 2 3 4 and resulting x and y operators One can then verify that the commutators between Hˆ ˆ ˆ 1 (Eq. (19)), N, and Sz vanish σ¯ˆx = (ˆσxσˆx +σ ˆyσˆy +σ ˆxσˆx +σ ˆyσˆy) , (B14) 1 2 1 3 1 3 2 4 2 4 h i h i h i x 1 x x y y x x y y H,ˆ Nˆ = H,ˆ Sˆz = N,ˆ Sˆz = 0. (B9) σ¯ˆ = (ˆσ σˆ +σ ˆ σˆ +σ ˆ σˆ +σ ˆ σˆ ) , (B15) 2 2 1 2 1 2 3 4 3 4 1 We can therefore evaluate the eigenstates of the Hamil- σ¯ˆy = (−σˆxσˆy +σ ˆyσˆx − σˆxσˆy +σ ˆyσˆx) , (B16) 1 2 1 3 1 3 2 4 2 4 tonian separately for each given number of electrons, N, y 1 x y y x x y y x and total z component of the spin, Sz. In the following σ¯ˆ2 = (−σˆ1 σˆ2 +σ ˆ1 σˆ2 − σˆ3 σˆ4 +σ ˆ3 σˆ4 ) . (B17) subsections we derive eﬀective Hamiltonians with a re- 2 duced number of qubits for each pair of N and Sz. As It can be veriﬁed that for all states with N = 1 these new a matter of notation, we denote the expectation value of operators satisfy the commutator relations Eqs. (B3-B4), an operator by removing the hat from the symbol of the and therefore can be represented as system of two qubits. z z operator, so that for example σ1 = hσˆ1 i. We then de- For a system with N = 1, in order to have the single note a basis vector of the system by the list of σz values z α electron on site 1 with spin up we need to haveσ ¯1 = within the ket: |σz, σz, σz, σzi. z 1 2 3 4 −Sz = −1 andσ ¯2 = −1, while for all other eigenstates of the 2-qubit system there is no electron on site one with spin up. This allows us to construct the inverse 2. N =0 projection as

z 1 z z For N = 0 from Eq. (B7) we obtain the condition σˆ1 = − 1 − σ¯ˆ1 1 − σ¯ˆ2 + 1. (B18) z z z z z 2 σ1 +σ2 +σ3 +σ4 = 4. Since |σα| = 1 this condition implies z z z z that σ1 = σ2 = σ3 = σ4 = 1, resulting also to Sz = 0. For the other sites and spins the inverse transformation Therefore we have only one possible state, with energy can be constructed in an analogous way: E = h1, 1, 1, 1| Hˆ |1, 1, 1, 1i = µ − U/4 − . N=0,Sz =0 2 1 σˆz = − 1 − σ¯ˆz 1 + σ¯ˆz + 1, (B19) 2 2 1 2 1 3. N =1 σˆz = − 1 + σ¯ˆz 1 − σ¯ˆz + 1, (B20) 3 2 1 2 1 z z z ˆz ˆz For N = 1 Eq. (B7) gives the condition σ3 + σ4 = σˆ4 = − 1 + σ¯1 1 + σ¯2 + 1. (B21) z z z z 2 2−σ1 −σ2 . With Eq. (B8) we then have Sz = 1−σ1 −σ2 . z z Since in general σ3 + σ4 ≤ 2 we also have the additional It is straight forward to verify that these relations do z z z z z relation 0 ≤ σ1 + σ2 ≤ 2. Since |σα| = 1, with these con- indeed give Eqs. (B10-B11) when solved for σ¯ˆ1 and σ¯ˆ2 . ditions the possible values for Sz are {−1, 1}. This is gen- With Eqs. (B14-B21) we can transform the 4-qubit erally the case for an arbitrary number of qubits, where Hamiltonian (Eq. 19) for N = 1 to the 2-qubit system one adds or removes a single electron from a Sz = 0 state. as The operator Sˆ therefore determines the spin-imbalance z of such systems, and to fully characterize the state one ˆ µ 2 z x µ U 2 H = + σ¯ˆ2 + V σ¯ˆ2 + − − . (B22) only needs to add the information about the electron dis- 2 2 2 4 2 tribution across the various sites. We therefore introduce the new operators to uniquely determine the state: 4. N =2 1 σ¯ˆz = (ˆσz +σ ˆz − σˆz − σˆz) , (B10) 1 2 1 2 3 4 For N = 2 from Eq. (B7) we obtain the condi- z z z z z 1 z z z z tion σ3 + σ4 = −σ1 − σ2 , so that with Eq. (B8) σ¯ˆ2 = (ˆσ1 − σˆ2 +σ ˆ3 − σˆ4 ) , (B11) z z z 2 Sz = −σ1 − σ2 . Since |σα| = 1, the possible values for Sz are {−2, 0, 2}. For Sz = 2 there is only z ˆ z z z where σ¯ˆ1 = −Sz determines the spin-imbalance, and σ¯ˆ2 one possible state with σ1 = σ2 = −1, with energy ˆ determines the imbalance of electron number across the EN=2,Sz =2 = h−1, −1, 1, 1| H | − 1, −1, 1, 1i = −U/4. two sites. Raising or lowering the expectation values of Analogously, for Sz = −2 we only have the state with z z these states involves spin-ﬂips for the ﬁrst operator, and σ1 = σ2 = 1, with energy EN=2,Sz =−2 = −U/4. 11

z z For Sz = 0 we have the additional condition σ2 = −σ1 , up. We can impose this condition by constructing the z z which since N = 2 also implies σ4 = −σ3 . We introduce inverse projection for this site and spin as a new set of Pauli operators, which guarantee that these 1 conditions are automatically satisfied: σˆz = 1 + σ¯ˆz 1 + σ¯ˆz − 1. (B32) 1 2 1 2 σ¯ˆz =σ ˆz, (B23) 1 1 For the other sites and spins the inverse transformation z z σ¯ˆ2 =σ ˆ3 , (B24) can be constructed in an analogous way: x x x σ¯ˆ1 =σ ˆ1 σˆ2 , (B25) z 1 z z x x x σˆ2 = 1 + σ¯ˆ1 1 − σ¯ˆ2 − 1, (B33) σ¯ˆ2 =σ ˆ3 σˆ4 , (B26) 2 y y x 1 σ¯ˆ =σ ˆ σˆ2 , (B27) z z z 1 1 σˆ3 = 1 − σ¯ˆ1 1 + σ¯ˆ2 − 1, (B34) y y x 2 σ¯ˆ2 =σ ˆ3 σˆ4 . (B28) z 1 z z σˆ4 = 1 − σ¯ˆ1 1 − σ¯ˆ2 − 1. (B35) The z components are equal to the ones of the original 2 z-components for qubit 1 and 3, while the x and y com- It can be verified that these relations give Eqs. (B10- ponents also make sure that when the values of qubits 1 z z B11) when solved for σ¯ˆ1 and σ¯ˆ2 . and 3 are flipped, the values of qubits 2 and 4 are flipped With Eqs. (B18-B35) we can transform the 4-qubit at the same time to satisfy the conditions above. One Hamiltonian (Eq. 19) for N = 3 to the 2-qubit system can verify that the new operators satisfy the commuta- as tor relations for Pauli operators (Eqs. (B3-B4)), which shows that they can be represented as new system of two ˆ µ 2 U z x µ U 2 H = + − σ¯ˆ2 + V σ¯ˆ2 − − − . qubits. Within this subspace with N = 2 and Sz = 0 2 2 2 2 4 2 the inverse operations can be directly obtained from the (B36) equations above, with the additional relations for qubits 2 and 4 given by 6. N =4 z z σˆ2 = −σ¯ˆ1 , (B29) z ˆz z σˆ4 = −σ¯2 . (B30) For N = 4 from Eq. (B7) we obtain the condition σ1 + z z z z σ2 + σ3 + σ4 = −4. Since |σα| = 1 this condition implies With these relations, and the commutator relations for z z z z that σ1 = σ2 = σ3 = σ4 = −1, resulting also to Sz = 0. Pauli operators (Eqs. (B3-B4)), we can represent the Therefore we have only one possible state, with energy Hamiltonian from Eq. (19) with the newly introduced EN=4,S =0 = h−1, −1, −1, −1| Hˆ | − 1, −1, −1, −1i = transformed Pauli operators as z −µ + 3U/4 + 2. ˆ U z z µ U 2 z z H = σ¯ˆ1 σ¯ˆ2 + − + σ¯ˆ1 + σ¯ˆ2 4 2 4 2 7. Matrix elements with circuit reduction x x + V σ¯ˆ1 + σ¯ˆ2 . (B31) On 4 qubits the hole matrix elements λh,1,n of the We can therefore evaluate the states for N = 2 and Sz = Green’s function (Eq. (7)) are given by 0 with this reduced Hamiltonian on 2 qubits. D E 2 ˆ † − ˆ λh,1,n = 0 U2,0σˆ1 U1,n 0 . (B37) 5. N =3 This would generally require a back-mapping from the z z 2-qubit states to 4-qubits. However, if the ordering of For N = 3 Eq. (B7) gives the condition σ3 + σ4 = z z the states in the reduced circuits is chosen appropriately, −2 − σ1 − σ2 . With Eq. (B8) we then have Sz = −1 − z z z z then one can perform this operation entirely on 2 qubits. σ1 − σ2 . Since in general −2 ≤ σ3 + σ4 , this results z z This is indeed the case for our projections for 2 and 1 to the additional relation 0 ≥ σ1 + σ2 ≥ −2. Since − z electrons, where the σ¯ˆ operator applied to the 1-electron |σα| = 1, with these conditions the possible values for Sz are {−1, 1}. For N = 3 we can therefore use the same states, and when using the 2-electron inverse mapping mapping to 2-qubits as for the 1-electron system, given for Sz = 0, gives the correct 2-electron states. We can in Eqs. (B10-B11) and Eqs. (B14-B17). The inverse therefore evaluate λh,1,n as mapping however needs to be modified to the appropriate D E 2 ¯ ¯ˆ † ˆ− ¯ˆ ¯ states with N = 3. λh,1,n = 0 U2,0σ¯1 U1,n 0 , (B38) For a system with N = 3, in order to have no electron z on site one with spin up we need to haveσ ¯1 = −Sz = where the bars over the quantities indicate that these z 1 andσ ¯2 = 1, while for all other eigenstates of the 2- are evaluated on 2 qubits. Since the 2-electron ground qubit system there is an electron on site one with spin state has Sz = 0, for λh,1,n to be non-zero the 1-electron 12 state needs to have S = −1, which impliesσ ¯z = 1 for z 1 TABLE IV. Experimental data obtained on the IBM super- ¯ˆ ¯ the 1-electron state. The 1-electron states U1,n |0i with conducting qubit quantum computer (IBM QC) and on the z σ¯1 = −1 therefore have λh,1,n = 0. For the 1-electron trapped ion quantum computer at the University of Maryland − ¯ˆ ¯ x ¯ˆ ¯ (UMD QC), extending the 4-qubit results presented in Tabs. states with Sz = −1 we have σ¯ˆ1 U1,n |0i = σ¯ˆ1 U1,n |0i, so that II and III for “Optimal θ”. We evaluate E3,n and λp,1,n for all the n ∈ {0, 1, 2, 3} on the quantum hardware. Further- D E 2 more, we compare the raw result as output by the quantum ¯ ¯ˆ † ˆx ¯ˆ ¯ λh,1,n = 0 U2,0σ¯1 U1,n 0 . (B39) computer with the SPAM corrected results. IBM QC UMD QC z Note that with the circuit of Fig. 2c one always hasσ ¯1 = Optimal θ Optimal θ Optimal θ Optimal θ Exact 1, and therefore the states with finite λh,1,n. Therefore (raw result) (SPAM-cor.) (raw result) (SPAM-cor.) λh,1,n can be evaluated entirely on the 2-qubit circuit E -1.795 -1.196 -1.500 -1.603 -1.691 with this equation. 0 E3,0 -1.247 -0.810 -1.111 -1.111 -1.178 The particle matrix elements λp,1,n of the Green’s func- E3,1 -1.247 -0.799 -0.981 -1.055 -1.112 tion on 2 qubits can be obtained in an analogous way, and E3,2 1.247 1.001 1.025 1.129 1.144 are given by E3,3 1.247 0.934 0.972 1.048 1.098 λ3,0 0.262 0.145 0.113 0.228 0.224 D E 2 ¯ ¯ˆ † + ¯ˆ ¯ λ3,1 0 0.010 0.002 0.002 0.002 λp,1,n = 0 U2,0σ¯ˆ1 U3,n 0 . (B40) λ3,2 0.236 0.189 0.159 0.211 0.221 λ3,3 0 0.011 0.004 0.001 0.000 Since the 2-electron ground state has Sz = 0, for λ to be non-zero the 3-electron state needs to have Sz = 1, which impliesσ ¯z = −1 for the 3-electron state. The 3-electron 1 Appendix D: Non-particle-hole symmetric system ¯ˆ ¯ z states U3,n |0i withσ ¯1 = 1 therefore have λp,1,n = 0. For the 3-electron states with S = 1 we have σ¯ˆ+U¯ˆ |0¯i = z 1 3,n The relations E3,n = E1,n and λp,1,n = λh,1,n are only x ¯ˆ ¯ σ¯ˆ1 U3,n |0i, so that valid at particle-hole (ph) symmetry. For a general set of parameters in the Hamiltonian of Eq. (19) this is not D E 2 λ = 0¯ U¯ˆ † σ¯ˆxU¯ˆ 0¯ . (B41) the case, and we need to compute the quantities for one p,1,n 2,0 1 3,n and three electrons separately. These can be computed using the same methods as for the ph-symmetric case, Note that with the circuit of Fig. 2c one always has so that moving away from ph-symmetry mainly leads to σ¯z = 1, and therefore the states with λ = 0. To 1 p,1,n double the amount of computations that need to be per- obtain the state at the same energy with σz = −1, which 1 formed on a quantum computer. Furthermore depending has finite λ , one simply needs to add a R rotation h,1,n y on the system it might be necessary to use a richer hard- by π to the first qubit. In this way we can perform the ware efficient ansatz with more parameters in order to be entire calculation of the λ on the 2-qubit system. p,1,n able to describe all the states of the more general Hamil- tonian. For our 2-qubit CR approach the circuits used Appendix C: Additional 4-qubit experimental data for the ph-symmetric case (Figs. 2bc) also fully describe all states of the system away from ph-symmetry. For 4- qubits however the circuit shown in Fig. 2a can only ap- As outlined in Sec. III B the particle-hole symmetric proximately reproduce the exact numerical results. We 2-site DMFT system is fully characterized by E0, E3,0, therefore needed to extend the circuit, and we verified E3,2, and λ = λp,1,0, and the results for these quantities measured on the quantum computers are given in Tabs. II and III. The remaining energies and matrix elements 푅푦 휃0 푅푦 휃5 are then obtained by the exact relations E3,1 = E3,0, E = E , λ = 1/2 − λ, and λ = λ = 0. 3,3 3,2 p,1,2 p,1,1 p,1,2 푅 휃 푅 휃 To evaluate to what extent these exact relations hold 푦 1 푦 6 when measured on quantum computers, in Tab. IV we 푅 휃 푅 휃 푅 휃 present the values of all these quantities measured on the 푦 2 푦 4 푦 7 quantum computers within the 4-qubit PT approach. It 푅 휃 푅 휃 can be seen that all relations are fulfilled to a good ap- 푦 3 푦 8 proximation. Furthermore we also present the results without and with SPAM correction, to evaluate the ef- FIG. 5. Quantum circuit for the 4-qubit DMFT calcula- fect of SPAM correction on the data. Overall it can be tion based on the perturbed Hamiltonian (PT) approach used seen that SPAM correction tends to improve the data, al- away from particle-hole symmetry. With this ansatz circuit though it is not always systematic, in particular the ma- we can reproduce the exact numerical results for all parameters used in the general 2-site DMFT Hamiltonian in Eq. trix elements λ3,1,n are not improved by adding SPAM corrections. (19). 13 that the circuit shown in Fig. 5 represents the smallest As an example demonstration we use the case of U = possible extension to the one of Fig. 2 that can fully re- 4 and µ = −0.16016. We perform the DMFT self- produce the exact results. Since the required extension consistency on a simulator to satisfy the conditions in is only one Ry rotation, the expected quality of results is Eqs. (21-22), and find that this gives the self-consistent similar to that obtained for ph-symmetry. impurity problem parameters 2 = −0.29764 and V = 0.93709. The resulting self-consistent DMFT impurity occupation is nimp = 0.5 (at ph-symmetry we have TABLE V. Experimental data obtained on the ion trap quan- nimp = 1). In the same way as for the ph-symmetric tum computer at the University of Maryland, and compared case we use the “Optimal θ” parameters computed on to the numerically exact values. The “Optimal θ” results have been measured using the angles for the circuit ansatz the simulator to set the parameters of the quantum cir- optimized with the simulator. For the 2-qubit results we com- cuit in the experiment, and evaluate the energies and pute the states for 1 and 3 electrons only for those cases that matrix elements for these. We perform the experiments give a finite λp/h,1,n, the remaining states could be obtained on the trapped ion quantum computer at the University by simply adding an additional Ry(π) rotation on the first of Maryland, using both the PT and CR approaches, and qubit. the results are given in Tab. V. The overall agreement 4 qubits 2 qubits between the experimental results and the exact data is Optimal θ Optimal θ Optimal θ Optimal θ analogous to the ph-symmetric case (Tab. IV), show- Exact (raw result) (SPAM-cor.) (raw result) (SPAM-cor.) ing that the method can equally be applied on existing

E0 -1.837 -1.600 -1.730 -1.752 -1.792 quantum hardware also away from ph-symmetry. E1,0 -1.033 -0.843 -0.927 -1.010 -1.031 E1,1 -1.033 -0.896 -0.987 E1,2 0.896 0.833 0.848 0.889 0.897 E1,3 0.896 0.926 0.940 λ1,0 0 0.003 0.001 λ1,1 0.217 0.195 0.193 0.221 0.215 λ1,2 0.033 0.044 0.041 0.038 0.035 λ1,3 0 0.003 0.002 E3,0 -0.624 -0.265 -0.507 -0.516 -0.581 E3,1 -0.624 -0.477 -0.550 E3,2 4.212 3.959 4.030 4.204 4.211 E3,3 4.212 3.952 4.042 λ3,0 0.644 0.584 0.589 0.621 0.621 λ3,1 0 0.006 0.002 λ3,2 0.106 0.118 0.113 0.109 0.101 λ3,3 0 0.002 0.001

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