Variational quantum solver employing the PDS energy functional

Variational quantum solver employing the PDS energy
 functional
 Bo Peng and Karol Kowalski

 Physical and Computational Science Division, Pacific Northwest National Laboratory, Richland, Washington 99354, United
 States of America

 Recently a new class of quantum algo- 1 Introduction
 rithms that are based on the quantum
 computation of the connected moment ex- Quantum computing (QC) techniques attract
 pansion has been reported to find the much attention in many mathematics, physics,
 ground and excited state energies. In and chemistry areas by providing means to
arXiv:2101.08526v6 [quant-ph] 9 Jun 2021

 particular, the Peeters-Devreese-Soldatov address insurmountable computational barriers
 (PDS) formulation is found variational and for simulating quantum systems on classical
 bearing the potential for further combin- computers.[2, 3, 43, 47, 53, 61] One of the fo-
 ing with the existing variational quantum cus areas for quantum computing is quantum
 infrastructure. Here we find that the PDS chemistry, where Hamiltonians can be effectively
 formulation can be considered as a new en- mapped into qubit registers. In this area, sev-
 ergy functional of which the PDS energy eral quantum computing algorithms, including
 gradient can be employed in a conventional quantum phase estimator (QPE) [5, 12, 15, 26,
 variational quantum solver. In compar- 38, 52, 58, 69] and variational quantum eigen-
 ison with the usual variational quantum solver (VQE), [16, 24, 29, 31, 32, 44, 50, 55, 60]
 eigensolver (VQE) and the original static have been extensively tested on benchmark sys-
 PDS approach, this new variational quan- tems corresponding to the description of chemical
 tum solver offers an effective approach to reactions involving bond-forming and breaking
 navigate the dynamics to be free from get- processes, excited states, and strongly correlated
 ting trapped in the local minima that refer molecular systems. In more recent applications,
 to different states, and achieve high accu- several groups reported quantum algorithms for
 racy at finding the ground state and its en- imaginary time evolution,[42, 46] quantum fil-
 ergy through the rotation of the trial wave ter diagonalization,[48] quantum inverse iteration
 function of modest quality, thus improves algorithms,[36] and quantum power/moments
 the accuracy and efficiency of the quantum methods. [59, 67] The main thrust that drives
 simulation. We demonstrate the perfor- this field is related to the efficient encoding of
 mance of the proposed variational quan- the electron correlation effects that are needed to
 tum solver for toy models, H2 molecule, describe molecular systems.
 and strongly correlated planar H4 sys- Basic methodological questions related to an
 tem in some challenging situations. In all efficient way of incorporating large degrees of free-
 the case studies, the proposed variational dom required to capture a subtle balance be-
 quantum approach outperforms the usual tween static and dynamical correlations effects
 VQE and static PDS calculations even at still need to be appropriately addressed. A typ-
 the lowest order. We also discuss the lim- ical way of addressing these challenges in VQE
 itations of the proposed approach and its approaches is by incorporating more and more
 preliminary execution for model Hamilto- parameters (usually corresponding to excitation
 nian on the NISQ device. amplitudes in a broad class of unitary coupled-
 cluster methods [4, 17, 27, 35, 37, 64]). Un-
 fortunately, this brute force approach is quickly
 Bo Peng: peng398@pnnl.gov
 stumbling into insurmountable problems associ-
 Karol Kowalski: karol.kowalski@pnnl.gov
 ated with the resulting quantum circuit com-

 Accepted in Quantum 2021-06-08, click title to verify. Published under CC-BY 4.0. 1

plexity and problems with numerical optimiza- 2 Method
tion procedures performed on classical machines
(the so-called barren plateau problem reported in 2.1 PDS formalism
Refs.[10, 11, 41, 45, 51, 66, 68]). In this section we will give a brief description
 In this paper, we propose a new solution to of the PDS formalism. The detailed discus-
these problems. Instead of adding more parame- sion of the PDS methodology and highly relevant
ters to the trial wave function, we choose to opti- connected moment expansion (CMX) formalisms
mize a new class of energy functionals (or quasi- have been given in the original work[49, 62] as
functionals, where the energy is calculated as a well as our recent work[14, 34] and many earlier
simple equation solution) that already encom- literatures (see for example Refs. [18, 19, 33, 39,
passes information about high-order static and 40, 54, 65]). The many-body techniques used in
dynamical correlation effects. An ideal choice for the derivation of PDS expansions originate in the
such high-level functional is based on the Peeters, effort to provide upper bounds for the free ener-
Devreese, and Soldatov (PDS) formalism,[49, 62] gies, and to provide alternative re-derivation of
where variational energy is obtained as a solu- the Bogolubov’s [8] and Feynman’s [20] inequal-
tion of simple equations expressed in terms of ities. Since the Gibbs-Bogolubov inequality re-
the Hamiltonian’s moments or expectations val- duces to the Rayleight-Ritz variational principle
ues of the powers of the Hamiltonians operator in zero temperature limit, these formulations can
defined for the trial wave function. In Ref. [34] be directly applied to quantum chemistry. Here
we demonstrated that in such calculations high- we only provide an overview of basic steps in-
level of accuracy can be achieved even with very volved in the derivations of the PDS formulation.
simple parametrization of the trial wave functions A starting point of the studies of upper bounds
(capturing only essential correlation effects) and for the exact ground-state energy E0 is the analy-
low-rank moments. We believe that merging the sis of function Γ(t) (defined for trial wave function
PDS formalism with the quantum gradient based |φi having non-zero overlap with the ground-state
variational approach would be considered as a wave function)
more interesting alternative for by-passing main
problems associated with the excessive number of Γ(t) = hφ|e−tH |φi , (1)
amplitudes that need to be included to reach the
 and its Laplace transform f (s)
so-called chemical accuracy.
 Z +∞
 In the following sections we will briefly intro- f (s) = e−st Γ(t)dt . (2)
duce the PDS formalism and describe how the 0

PDS energy functional can be incorporated with It can be proved that, for a complex scalar s,
the minimization procedures that are based on Eq. (2) exists if the real part of s −E0 .
the quantum gradient approach [25, 42, 45, 57, 71] Under this condition, for Hamiltonian H defined
to produce a new class of variational quantum by discrete energy levels Ei and corresponding
solver (which is called PDS(K)-VQS for short in eigenvectors |Ψi i (i = 0, 1, . . . , M )
the rest of the paper) to target the ground state
and its energy in a quantum-classical hybrid man- M
 X
ner. Furthermore, we will test its performance, H= Ei |Ψi ihΨi | , (3)
 i=0
in particular the performance of the more afford-
able lower order PDS(K)-VQS (K = 2, 3, 4) ap- f (s) takes the form
proaches combining with the trial wave function
expressed in low-depth quantum circuits, at find- M
 X ω(Ei )
ing the ground state and its energy for the Hamil- f (s) = (4)
 i=0
 s + Ei
tonians describing toy models and H2 molecular
system, as well as the strongly correlated planar where ω(Ei ) = |hΨi |φi|2 . The PDS formalism is
H4 system, in some challenging situations where based on introducing parameters into expansion
the barren plateau problem precludes the effec- (4) using a simple identity (with a real parameter
tive utilization of the standard VQE approach. a)

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1 1 En − a (En − a)2
 = − + . (5)
 s + En s + a (s + a)2 (s + En )(s + a)2

When the above identity is applied for the first a1 ) one gets the following expression for the f (s)
time to Eq. (4) (introducing the first parameter function

 M
 " #
 X 1 Ei − a1 (Ei − a1 )2
 f (s) = ω(Ei ) − + (6)
 i=0
 s + a1 (s + a1 )2 (s + Ei )(s + a1 )2

The transformation (6) can be repeated K times (with each time introducing a new parameter ai ,
 i = 1, . . . , K) to reformulate the f (s) function as

 f (s) = RK (s, a1 , . . . , aK ) + WK (s, a1 , . . . , aK ) , (7)

 where

  
 M K
 X ω(Ei ) Y (Ei − aj )2  ≥ −E0 (if −E0 ),
 RK (s, a1 , . . . , aK ) =  (8)
 i=0
 s + Ei j=1
 (s + aj )2
   
 M  K j−1 2
 X X  1 Ei − aj  Y (Ei − an ) 
 WK (s, a1 , . . . , aK ) = ω(Ei )  − . (9)
 i=0
 
 j=1
 s + aj (s + aj )2 n=1 (s + an )2 

The K-th order PDS formalism (PDS(K) for It can be shown that the optimal parameters in
 (K) (K)
short henceforth) is then associated with defining the PDS(K) formalism, (a1 , . . . , aK ), are the
the introduced K real parameters (a1 , . . . , aK ) roots of the polynomial PK (E),
that minimize the value of RK (s, a1 , . . . , aK ). In K
this minimization process the necessary extreme
 X
 PK (E) = E K + Xi E K−i , (12)
conditions are given by the system of equations i=1

 ∂RK (s, a1 , . . . , aK ) and these roots provide upper bounds for the ex-
 = 0, (i = 1, . . . , K), (10) act ground and excited state energies, e.g., for the
 ∂ai
 ground state energy we have
which can be alternatively represented by the ma- (K) (K)
 E0 ≤ min(a1 , . . . , aK ) ≤ hφ|H|φi . (13)
trix system of equations for an auxiliary vector
X = (X1 , · · · , XK )T Note that, as shown in Refs.[49, 62] the PDS for-
 malism also applies to the Hamiltonian charac-
 MX = −Y. (11) terized by discrete and continuous spectral reso-
 lutions together.
Here, the matrix elements of M and vector Y are
defined as the expectation values of Hamiltonian
 2.2 PDS(K)-VQS formalism
powers (i.e. moments), Mij = hφ|H 2K−i−j |φi,
Yi = hφ|H 2K−i |φi (i, j = 1, · · · , K) (for simplic- In the variational method, we approximate the
ity, we will use the notation hH n i ≡ hφ|H n |φi). quantum state using parametrized trial state

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|Ψi ≈ |φi. Using Variational
 a quantum circuit,quantum the solver
 trial employing
 quantum analog the PDS of thefree energy
 Fisher functional
 information ma-
 1, a) 1, b)
state can be prepared by Bo Peng
 applying and Karol Kowalski
 a sequence of trix in the classical natural gradient,[1] to mea-
parametrized unitary gatesPhysical
 on theSciences and Computational Division, Pacific Northwest National Laboratory, Richland, Washington 99354,
 initial state
 United States of America |0i, sure the distance in the space of pure quantum
 (Dated: 18 January 2021)
 ~
 Riemannian metric Rij (θ)
 ~ = · · · Uk (θk ) · · · U1 (θ1 )|0i
 |φi = |φ(θ)i (14)
 In our previous work (J. Chem. Phys. 2020, 153, 201102),
 GD ij
 with the resulting quantum
 δ
 circuit complexity and problems
(θ~ = {θ1 , · · · , θn }). weHere Uka(θ
 reported k )class
 new is the k-th algorithms
 of quantum uni- that are based with numerical optimization procedures performed on classi-
 on the quantum computation of the connected moment ex- cal machines
 ~ ∂|φ(θ)i~ (the so-called
 ~ barren plateau problem
 ~ reported
  
tary single- or two-qubit gate that is controlled NGDthe < in∂θ ∂hφ( θ)|
 − ∂hφ( θ)| ~ ~ ∂|φ(θ)i
 pansion to find the ground-state energy. In particular, Refs.
 i
 ? ∂θ
 ). j ∂θi |φ(θ)ihφ(θ)| ∂θj
by parameter θk . Peeters-Devreese-Soldatov
 The goal is to approach the is found vari-
 (PDS) formulation In this paper, we propose a new solution to these prob-
ground-state energy ational
 of and bearing the potential
 a many-body for further combining with
 Hamilto- lems. Instead  of∂hφ(
 adding more
 ~ ∂|φ(
 θ)| ~ parameters
 θ)i
  to the trial wave
 the existing variational quantum infrastructure. Following
 ITE the function, we < choose∂θto optimize
 ∂θ a new class of energy func-
nian, H, by findingdirection, the values of these
 here we propose param-quantum algorithm tionals (or quasi-functionals, where the energy is calculated
 an variational
 i j

 ~ that minimize
eters, θ, based on the expectation
 the energy gradient ofvalue
 this newofPDS energy expres- as a simple equation solution) that already encompasses infor-
the Hamiltonian sion. In comparison with the original PDS, the present varia-
 tional approach helps find the ground state energy at the low
 mation about high-order static and dynamical correlation ef-
 fects. An ideal choice for such high-level functional is based
 Table 1: Three Rimannian metric forms, ordinary gradi-
 order PDS expansion with the trial wave-function of modest on the Peeters, Devreese, and Soldatov (PDS) formalism,?
 ~ ~ ent descent (GD), natural gradient descent (NGD), and
 quality, thus
 Emin = minhφ(θ)|H|φ(θ)i. improves the accuracy and
 (15) efficiency. The im- where variational energy is obtained as a solution of simple
 provement can even be witnessed from some toy examplesimaginary time evolution
 equations expressed (ITE), in exploited
 terms of theinHamiltonian’s the presentmoments
 θ~
 and simple chemical systems. study. or expectations values of the powers of the Hamiltonians op-
 ?
 erator defined for the trial where wave thefunction.
 coefficient In Ref.
 X = we (Xdemon-
 1 , · · · , XK ) is
 T
 To do this, the conventional VQE starts by con- strated that in such calculations high-level of accuracy can be
 ~ and measuring the cor- solving the following linear equation
structing the ansatzI.|φ( INTRODUCTION
 θ)i states. Thewave
 achieved even with very simple parametrization of the trial
 quantum Fubini-Study metric de-
 functions (capturing only essential correlation
 MX =effects) Y.
responding expectation value of the Hamiltonian scribes the and low-rank moments.
 curvature of the ansatz We believeclass that therather proposed algo-
 Quantum computing (QC) techniques attract much at-
using a quantum computer,
 tention in many and then relies
 mathematics, physics,onand chemistry
 than the ar- learning rithm maylandscape,be an interesting withbut thealternative
 element
 often ofperformsfor by-passing
 matrix M andmain vector Y bei
 eas by providing means
a classical optimization routine to obtain new θ. to address ~
 insurmountable compu- problems associated with
 the the excessive
 expectation number
 values of of amplitudespowers (i
 Hamiltonian
 tational barriers for simulating quantum systemsas well as Hessian
 on classical that need tobased be included methods
 Mijto = reachhH 2K the(e.g. i, YBFGS
 so-called
 i j chemical
 i = hH
 2K accu-
 i
 i (i, j = 1, · ·
During the parameter computers.optimization
 ? (or areas
 One of the focus dynam- for quantumoptimizer
 computing that racy. approximatesInthe the variational
 Hessian method, of the we approximate
 is quantumthat
ics), the set of parameters chemistry, where Hamiltonians
 is updated at the can becost effectively In this paper we discuss state the using minimization
 parametrized procedures
 trial state based
 | i ⇡ | i. U
 mapped into qubit registers. In this area, several functionon using
 quantum the natural first-order
 gradients gradient,
 approach for PDS see Ref.
 formulation toprepared
 cal-
k-th step (k > 1) can be written as tum circuit, the trial state can ?
 be by a
 computing algorithms, including quantum phase [70]estimator
 fortheacoefficient culate ground-
 recent detailed and excited-state
 discussion).
 quence of electronic
 parametrizedThere energies.are gates
 unitary Using on the init
 where X = (X , · · · , X ) T
 is obtained by VQE framework relies on new PD
 (QPE)? , and variational quantum eigensolver (VQE) ?
 , have
 solving the followingthis combined
 linear
 1
 equationapproach, we demonstrate ahHrapid
 K
 convergence
 ~
 and their ✓-derivatives being
 also some other options for the
 of the minimization algorithm in| situations
 Riemannian k
 i met-
 θ~k = θ~k−1been− ηR extensively
 −1 ~ testedθ).
 (θ)∇E( ~on benchmark (16) systems correspond- i(2) ~ where
 = | (circuits.
 ✓)i = ·In· the
 the barren
 · Ukfollowing,
 (✓k ) · · · from
 U1 (✓sev 1)
 ing to the description of chemical reactions involving ric including
 bond- imaginary-time
 plateau problem
 MX = Y.precludesevolution the effective (ITE)
 utilization or
 of the how
 ther demonstrate stan-the correspo
 constructed and howWe the PDS-VQE
 forming and breaking processes, excited states, even andwith the elementdard
 strongly
 classical variational
 Fisher metric quantum thateigensolver (VQE) approach.
 (Y✓~ being 1 , ·have been Heredis-
 of matrix M and vector = {✓defined · · ,as✓n }). Uk (✓k ) is the k-th uni
 ~ correlated
 ~ molecular systems. In more recent applications,
 the expectation also demonstrate
 values of Hamiltonian that
 powers combining
 (i.e. a new class of energy
 moments), func-
where ∇E(θ) = ∂E/∂ θ is the energy gradient vec- cussed in two-qubit gate that isthe
 Incontrolled by parameter ✓
 several groups reported quantum algorithms’ applications Mij = hH to2Ksome i j
 i, Yrecent
 tional i = hH 2Konreports.[42,
 based i
 j = 1, · · · , K).63,
 ia(i,straightforward form 71] ofIII. Tab.
 trial wavefunc-
 NUMERICAL EXAMPLES
tor, and η is the step size (or learning rate). R( ~
 θ) In the variational
 tion method,
 expressed we approximate
 in to approach
 terms of the quantum
 low-depth the ground-state
 quantum energy
 circuit and of a many
 describe the imaginary time evolution of quantum systems. commonly used flavors of the Rieman-
 1, state
 three using parametrized trial state | i ⇡ tonian, | i. Using H, by
 a quan- finding the values of these param
 The mainmatrix
is the Riemannian metric thrust that θ~ that
 atdrives this is
 field is related to tum
 flex- effi- the low-rank
 the circuit, trial
 PDS
 state can R(
 expansions
 ~ are
 be prepared
 (PDS(n),
 by applying
 n=2,3),
 A. Toy weHamoltonians
 are able to
 cient encoding electron correlation effects needed toquence
 nian metric
 describe
 matrix
 achieve a chemical θ) minimize
 level listed
 of thea se-
 accuracy and
 expectation
 even will
 for be
 value
 strongly of the
 cor- Hamilton
 of parametrized unitary gates on the initial state |0i,
ible to characterize molecular
 the singular systems. point in the pa- used in the related following molecular case problems.
 studies. We also discuss
 Remarkably, We algorithms
 first consider two for slightly lar
 ~ =encoding Emin = minh 0(✓)|H| ~ ~1
 (✓)i.
rameter space and isBasic methodological
 essentially questionstorelated
 related theto an efficient way | i =simple
 | (✓)i · · · Uk (✓k )trial
 · · · Uwave
 1 (✓1 )|0i function(3)and its derivatives. 1 0 a
 As
 as pointed out in Refs.
 benchmark systems
 [42,we73], employ
 thea difference
 set of model
 be-
 ~
 ✓
 Hamiltonians
 0 0
indistinguishability of ofincorporating
 ~
 E(θ).[71] large degrees of freedom required to~capture
 It is worth men-
 a subtle balance between static dynamical correlations
 (✓ = {✓1 , · · · , ✓n }). Here Uk (✓k )?is the k-th unitary single- or
 tween natural
 effects testedgradient
 in Refs. , the descent H2 ,To and (NGD)
 H4 systems.and ITE
 HA = @
 B 0 2 0 0 C
 0 A byB
 0 0 3 starts
 , H
 two-qubit gate that is controlled by parameter ✓k .do Thethis,
 goal isthe conventional VQE 0 0 0 0
tioning that Eq. (5)stilloriginates from natural
 need to be appropriately gra- A typical
 addressed. to way
 accounts of for
 approach the ground-state
 the global energyphase,of a the ansatz
 many-body andHamil- ~introducing
 | if(✓)i and measures the correspondin
dient learning method addressing in thethese general
 challenges in VQE approaches is by
 nonlinear incor-
 tonian, H, by finding the values of these value parameters, of the~ that
 ✓, Hamiltonian with ansatze
 using a quantum comp
 porating more and more parameters (usually corresponding a time-dependent II. METHOD
 minimize the expectation phase gate
 value of the Hamiltonian to the trial state,
optimization framework especially targeting ma-of unitary relies on a classical optimization
 | A (✓1 , ✓2 )i routine
 = R̃Y0,1 (✓to
 2 )R obtai
 0
 X (✓1
 to excitation amplitudes in a broad class thecoupled-
 Riemannian Emin = metric ~
 minh (✓)|H| employing
 ~ing the parameter
 (✓)i. 1,2
 NGD will
 (4) optimization, 0 be the ✓set of para
 cos
 1
 cluster methods).Here,Unfortunately, thisgra-
 brute force approach is In their✓~ original work, Peetters, Devreese, Band Soldatov
 1
chine learning problems.[1] the natural equivalent tohave theshownmetric updated at the
 employing ITE.k-th step (kB> 1) can0 be written
 2
 C a
 quickly stumbling into insurmountable problems associated To do this, the conventional that
 VQE thestarts
 K-th by order upper bound=of
 constructing @ thei sin
 ground-
 ✓1 ✓2 A .
 C
dient is the optimizer that accounts for the geo- ~ state energy is 2 cos 2
 the ansatz | (✓)i and measures thethe root of theexpectation
 corresponding polynomial ~ PK ~
 ✓k =Classical
 (E), i sin ✓11 ~sin ✓2 ~
 ✓k 1 ⌘R 2 (✓)rE( 2 ✓).
metric structure of the parameter space. For the Quantum
 value of the Hamiltonian using !a quantum computer, PDS and then
 ⃑a !classical optimization X K | B (✓1 , ✓2 )i = R̃Y0,1 (✓2 )RX 0
 ✓1 )
 | (
 relies on ) routine to obtain new ~
 ✓. Dur-
curved (or nonorthonormal)
 a) Electronic mail:parameter
 peng398@pnnl.govmanifold ! ⁄ ⃑ K (E)
 ing the parameter optimization, thePset ofwhere= E n, rE(
 parameters + that~ X=i E @E/@
 ✓) is
 n i ✓ 0
 , ~ is the energy (1) gradien
 i
 sin(✓
 2
 b) Electronic mail: karol.kowalski@pnnl.gov B 1
 ~ +iscos(
that exhibits the Riemannian character (e.g. in updated at the k-th step (k > 1) can be written ⌘ is as the step i=1size (or learning
 Converge? =B rate). R( 2 (1
 ✓) the
 @ sin2 (( ✓1 )) cos ✓2 +
large neural networks), natural gradient learning ✓~k = ✓~k 1 ⌘R 1 (✓)rE( ~ metric
 ~
 ✓). matrix(5) at ✓~ ℰthat are , ⃑ flexible
 sin 2 to
 (( ✓1 characteriz
 2
 )) sin ✓2
 2
 !"# 2 2
 point No in the Yesparamter space and are essentially
method is often employed to avoid the plateaus where rE(✓) ~ = @E/@ ✓~ is the energy gradient vector, and
 indistinguishability etofal.E(to✓). Note that ~ same toy models have b
 In Tab. I, three co
 ~ is the Riemannian demonstrate the difference
in the parameter space.[42, 63, 71] ⌘ is the step size (or learning rate). R(✓) ~
 Figure 1: The ~ workflow of flavors of the
 variational Riemannian
 quantum timemetric
 inarysolver evolution matrix
 (ITE) andR(grad ✓)
 metric matrix at ✓ that are flexible to characterize the singular ing the ground-state energy of Ham
 Note that when the parameter space is a Eu- employing
 point in the the paramter PDS spaceenergy will
 functional.
 and are essentially be used
 related to thein the following numerical example
 Here, R̃Yp,q (✓) is a controlled Y
 indistinguishability of E(✓). ~ In Tab. I, three To get theused
 commonly energyqubit gradient in the PDS framew
clidean space with orthonormal coordinate sys- p and target qubit q, and R
 flavors of the Riemannian metric matrix R( derivative
 ~ are listed
 ✓) w.r.t.
 and ✓i on both
 qubit sides of
 p around theEq. (1),The
 x-axis. we rt
tem the Riemannian metric tensor will reduce To get
 will be used the energynumerical
 in the following gradient in the PDSdefined
 examples. frame-
 as R (✓) = e i✓ /20with j
 j

 To get the energy gradient in the PDS framework, take the spin matrices. Based on the ansat E
to the unity matrix (see Tab. 1). In VQE set- work, take
 derivative w.r.t.the derivative
 ✓i on both sides of Eq. (1), we then θ
 w.r.t. i on both
 have sides
 the Hamiltonian can be expressed
 @E 1 B a
 = @
ting, one can define the Riemannian metric as of Eq. (12), and after reorganizing @✓0 K 1 1T the terms
 iE
 K
 XEA1(✓1 , ✓we
 2 ) = h A (✓1 , ✓2 )|HA | A

the quantum Fubini-Study metric, which is the @E
 can@✓express
 = the 1
 energy B .. KE
 derivative
 @ . A
 C K@X1 +
 as (K i)Xi E2 K ⇣ ✓ i⌘ 1
 1
 i
 KX 1 @✓ i
 = cos
 i=1
 + 3 sin2
 1 2
 KE K 1 + (K i)Xi E K i 1
 EB (✓1 , ✓2 ) = h B (✓1 , ✓2 )|HB | B
 i=1
 ⇣✓ ⌘
 (6) 2 1
 = 1 + sin cos(
 where @✓ @X
 i
 can be obtained by solving the
 2 fol
 where can be obtained by solving @X
 the following
 equation linear
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 equation
 @✓i
 4
 Further, for k 0, k 2 Z we will h
 @X @Y @M @X @Y @M
 M = X (7) M = ⇣ ⌘X
 2 ✓1
 @✓i @✓i @✓i @✓ k
 hHi A i✓1 ,✓2
 @✓
 = cos
 i @✓i + 3k si
 2 ⇣
 with @Yi /@✓k = @hH 2K i i/@✓k and @Mij /@✓k = 1
 with @Yiof/@✓ k = @hH @hHk
 2K i
 and1 ) @M
 ~ i/@✓2k sin(✓ 1
 @hH 2K i j i/@✓k . Fig. ?? summarizes the workflow
 2K i jPDS- A i/@ ✓ =
 i/@✓k . Fig. ?? summarizes the work
 k
 3 2
 @hH 2 sin

 T
 E K−1
 ∂E −1 
  ∂X
 .. 
 = .  ∂θ , (17)
 
 ∂θi K−1 
 X i
 K−i−1
 KE K−1 + (K − i)Xi E 1
 i=1

where ∂X
 ∂θi is associated with the θi -derivative of example, as we will show later for the H4 system
Eq. (11), that comprises 184 Pauli strings in the Hamil-
 tonian, the effective number of Pauli strings re-
 ∂X ∂Y ∂M quired for arbitrary hH n i (n = 2, 3, 4) measure-
 M =− − X, (18)
 ∂θi ∂θi ∂θi ments can be dropped from 1842 , 1843 , and 1844
 to 1774, 3702, and 4223, respectively, after the
and can be obtained by solving Eq. (18) as a lin- Pauli reduction, and the 4223 strings will not
ear equation with ∂Yi /∂θk = ∂hH 2K−i i/∂θk and be changed for more complex hH n i’s (n > 4).
∂Mij /∂θk = ∂hH 2K−i−j i/∂θk . Fig. 1 summa- Similar findings have also been reported in Ref.
rizes the workflow of PDS(K)-VQS, where on the [67], where by grouping the Pauli strings into
classical side the PDS(K) module includes two tensor-product basis sets the authors examined
steps, (i) solving two consecutive linear problems the operator counts for hH 4 i of Heisenberg model
to get X and ∂X/∂θi , and (ii) solving for roots defined on different lattice geometries for the
of polynomial (12) and computing Eq. (17). On number of qubits ranging from 2 up to 36, and
the quantum side, in comparison with the con- found that the effective number of Pauli strings
ventional VQE, the present PDS(K)-VQS infras- to be measured drops by several orders of magni-
tructure relies on quantum circuits to measure tude with sub-linear scaling in a given number of
 ~
hH n i and their θ-derivatives. qubits. For larger systems, the number of mea-
 In the present work, due to the relatively small surements can be further reduced by introduc-
system size, we directly exploit the Hadamard ing active space and local approximation. Alter-
test to compute the real part of hH n i for the natively, one can approximate hH n i by a linear
Hamiltonians that are represented as a sum of combination of the time-evolution operators as
Pauli strings. It is worth mentioning that for typ- introduced in some recent reports.[7, 59] For the
ical molecular systems that can be represented estimation of ∂hH n i/∂θk , in the present work we
by N qubits, the number of hH n i measurement limit Uk (θk ) exploited in the state preparation to
scales as O(N 4n ), which nevertheless can be re- be only one-qubit rotations. Then, following Ref.
duced once the Pauli strings are multiplied and [57], ∂hH n i/∂θk can be obtained by measuring
their expectation values are re-used as the con- hH n i twice using the same circuit but shifting θk
tributions to the higher order moments.[34] For by ± π2 separately, i.e.

 ∂hH n i(··· ,θk ,··· ) 1 n 
 = hH i(··· ,θk + π2 ,··· ) − hH n i(··· ,θk − π2 ,··· ) . (19)
 ∂θk 2

If θk parametrizes more than one one-qubit ro- 3 Numerical examples
tations in the circuit, then based on the prod-
uct rule ∂hH n i/∂θk will have contributions from In this section, with several examples, we will
all one-qubit θk rotations, each of which will be demonstrate how the PDS(K)-VQS performs in
obtained by applying (19) on the corresponding some challenging situations, and its difference in
rotation. comparison to the conventional VQE and static
 PDS(K) expansions.

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3.1 Toy Hamoltonians PDS(2)-VQS VQE

We first test the PDS(K)-VQS on two toy Hamil-
tonians GD
 HA = 1.5I4×4 + 0.5(I2×2 ⊗ σz − 2σz ⊗ σz )
  
 1 0 0 0
  
  0 2 0 0 
 ,
 =
 
  0 0 3 0 
 
 NGD
 0 0 0 0 (ITE)
 HB = I4×4 + 0.5(I2×2 ⊗ σz − σz ⊗ σz )
  
 1 0 0 0
  
  0 1 0 0 
 = , Figure 2: Variational trajectories on the PDS(2) energy
 
  0 0 2 0 
  surface (left panels) and original potential energy sur-
 0 0 0 0 face (right panels) discovering the ground state energy
 of Hamiltonian, HA , explored by gradient descent (top
with ansatze panels) and natural gradient descent/imaginary time
 |φA (θ1 , θ2 )i = R̃Y0,1 (θ2 )RX
 0
 (θ1 )|00i, evolution (bottom panels). On the background energy
 surfaces, the dark blue and white colors correspond to
 |φB (θ1 , θ2 )i = R̃Y0,1 (θ2 )RX
 0 1
 (θ1 )RX (θ1 )|01i, the global maximum and minimum energies, respec-
 tively. The arrows indicate the trajectories of the dy-
that have been exploited by McArdle et al. [42] manics, and are colored green if the trajectory converges
to demonstrate the performance of different Rie- to the ground state energy, and red otherwise. The step
mannian metrics in the conventional VQE ap- size η = 0.05 in all the calculations.
proach for finding the ground-state energy of the
same Hamiltoinans. Here, R̃Yp,q (θ) is a controlled
Y rotation of θ with control qubit p and target local minima as singular points, the VQE would
 p
qubit q, and RX (θ) is a rotation of θ on qubit p still get trapped. This can be observed from
around the x-axis. The rotation about the j-axis the VQE performance for system B, where both
is defined as Rσj (θ) = e−iθσj /2 with σj being one NGD and ITE fail to escape the local minima,
of the Pauli spin matrices. (θ1 , θ2 ) ∼ (± 3π
 8 , 0), in the dynamics due to the
 Figs. 2 and 3 show the performances of the pro- fact that the local minima are not the singular
posed PDS(K)-VQS (K = 2, i.e. PDS(2)-VQS) points of R in either NGD or ITE.
and the conventional VQE approaches for finding In contrast, the PDS(2)-VQS robustly converge
the ground state energy of the toy Hamiltonians. to the true ground state for both systems regard-
As can be seen, the ability of VQE navigation to less of the employed Riemannian metric. The
avoid the local minima on the conventional PES success of PDS(K)-VQS in these toy examples
depends on the Riemannian metric exploited. For can be essentially attributed to the fact that, in
system A, in comparison to GD, the NGD (or comparison to the original PES where the local
equivalently ITE in this case) is able to avoid the minima corresponding to a non-ground state, the
local minimum at (θ1 , θ2 ) = (0, 0). This is be- entire PDS(K) energy surface, except the singu-
cause the Riemannian metric, lar areas (see the infinitesimal white strips on the
 sin2 ( θ21 ) + 14 cos2 ( θ21 ) 0
 !
 left panels of Fig. 2 at θ1 = 0) where the fi-
 R= 1 2 ( θ1 )
 , delity of the trial wave function w.r.t the target
 0 4 sin 2 state is strictly zero, provides an upper bound
used in the NGD/ITE correctly characterizes any energy surface for the same (ground) state. This
rotation pair with θ1 = 0 as a singular point (i.e. state-specific nature makes the PDS(K)-VQS es-
det |R| = 0) such that R−1 will numerically navi- sentially explore a lower upper bound of the
gate the dynamics (e.g. via singular value decom- ground state at a given PDS order, and there-
position) to avoid collapsing in this local mini- fore the dynamics will not be trapped at a lo-
mum once the trajectory is getting close. There- cation that is associated with a different state.
fore, if the metric is unable to characterize the It worth mentioning that, a lower bound of the

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PDS(2)-VQS VQE 3.2 H2 and H4 systems
 We further employ the proposed PDS(K)-VQS
GD
 approach to find the ground state energy of H2
 and H4 molecular systems. For H2 molecule, we
 exploit an effective Hamiltonian and an ansatz
 exploited by Yamamoto [71] and Bravyi et al.[9],

 H = 0.4(σz ⊗ I + I ⊗ σz ) + 0.2σx ⊗ σx
NGD
 ~ = R0 (2θ3 )R1 (2θ4 )Ũ 0,1 R0 (2θ1 )R1 (2θ2 )|00i
 |φ(θ)i Y Y N Y Y
 p,q
 where ŨN denotes the CNOT gate with control
 qubit p and target qubit q.

 GD NGD/ITE

ITE

Figure 3: Variational trajectories on the PDS(2) en-
ergy surface (left panels) and original potential energy
surfaces (right panels) discovering the ground state en-
ergy of Hamiltonian, HB , explored by gradient descent
(top panels), natural gradient descent (middle panels),
and variational imaginary time (bottom panels). On the
background energy surfaces, the dark blue and white col-
ors correspond to the global maximum and minimum en-
ergies, respectively. The arrows indicate the trajectories
of the methods, and are colored green if the trajectory
converges to the true ground state energy, and red oth-
erwise. The step size η = 0.05 in all the calculationss.

 Figure 4: The computed ground state energy (top pan-
 els), energy deviation w.r.t. exact energy (middle pan-
 els), and fidelity of the trial state (bottom panels) of
 the H2 molecule iterate in the conventional VQE and
 PDS(K)-VQS (K = 2, 3, 4) infrastructures employing
ground state energy can also be obtained from gradient descent (left panels) and natural gradient de-
a static, and more costly, higher order PDS(K) scent/imaginary time evolution (right panels). The ini-
standalone calculation as demonstrated in our tial rotation is given by θ~ = (7π/32, π/2, 0, 0). The step
previous work.[34] From this perspective, the size η = 0.05 in all the calculations.
PDS(K)-VQS approach provides an effective way
to explore the possibility of pushing the low or- Fig. 4 compares the VQE and PDS(K)-VQS
der PDS(K) results towards high accuracy that performances exploiting the above-mentioned
would otherwise require higher-order and more ansatz to find the ground state energy of the H2
expensive PDS(K) calculations. Besides, since Hamiltonian. As can be seen, starting from the
generalized variational principle applies in the given initial rotation, the VQE is unable to con-
PDS framework,[49, 62] if other roots of Eq. (12) verge to the ground state energy within 100 itera-
are concerned, the PDS(K)-VQS will also be able tions, but rather drops to an excited state energy
to navigate the dynamics to give lower upper (−0.2 a.u. in this case). Actually, it has been
bounds for excited states as long as the fidelity shown that,[71] starting from the same initial ro-
of the trial wave function with respect to the tar- tation, the VQE needs to go through a “plateau”
get state is non-zero. that resides at this energy value and spreads over

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∼400 iterations before hitting the ground state proximate the target state, while is still able to
energy (∼−0.8 a.u. in this case) regardless of the provide systematically improvable upper bounds
employed Riemannian metric. of the expectation value of the target state by
 To achieve a higher level of accuracy (e.g. exploring the Krylov subspace. The benefit is
chemical accuracy kE(θ) ~ − Eexact )k < 1.5 × 10−3 the great simplification of the state preparation.
a.u.), low order PDS(K)-VQS typically needs The limitation is also obvious in that it some-
more iterations than high order PDS(K)-VQS. times would be challenging to further improve the
As shown in the middle left panel of Fig. 4, quality of the trial state within the PDS(K)-VQS
by employing GD in the dynamics, it takes the framework if the energies were well converged al-
PDS(4)-VQS

state of the linear hydrogen chain systems.[2] For in which simultaneous
 

 measurement can be per-
the planar H4 system whose ground state is a formed. cov Pi , Pj is the covariance between two
triplet, the circuit with close-to-zero initial rota- Pauli strings bounded by
tions would generate a trial state that is almost 
 q
singlet, which makes the conventional VQE and cov Pi , Pj ≤ | var(Pi ) · var(Pj )| (21)
the static PDS(K) (K = 2, 3, 4) simply fail. On
the other hand, as shown at the bottom of Fig. with the variance being computed from var(Pi ) =
5, the PDS(K)-VQS (K = 2, 3, 4) are capable 1 − hPi i2 . Here, we assume the covariances be-
of dealing with such a tough situation and again tween different Pauli strings to be zero for the
outperform. As can be seen, within 200 itera- brevity of the discussion. We can apply the above
tions, PDS(K)-VQS (K = 2, 3, 4) are able to con- metric to, for example, estimate the number of
verge to the ground state energy well below chem- measurements of H n (n = 1, 2, 3) required by the
ical accuracy and improving the fidelity of the PDS(2)-VQS calculation for the complete active
trial wave function to be >0.96. It is worth not- space (4 electrons, 4 spin-orbitals) of the pla-
ing that even though the converged rotations ob- nar H4 system. Given  ∼ 0.5mHartree, since
tained from the PDS(K)-VQS calculations gen- H n (n = 1, 2, 3) can be generated from at most
erate a high fidelity state, the expectation value ∼ 3700 Pauli strings, the estimated number of
of the generated state is still ∼ 0.02 a.u. above measurements needs to be done is ∼ 4.8 × 109 ,
the exact energy, and it then becomes challeng- which is one order of magnitude higher than that
ing to further improve the fidelity employing the for hHi (∼ 1.2 × 108 ). Thus, given the same trial
same circuit infrastructure through varying the state in this H4 case, if the number of conven-
rotations. Therefore, the circuit used here might tional VQE iterations is no more than one order
not be sufficient for preparing true ground state of magnitude larger than that of the PDS(2)-VQS
in practice if higher fidelity is desired. We here iterations, VQE would outperform PDS(2)-VQS
intentionally employ the circuit to artificially gen- in terms of total number of measurements, and
erate an extreme challenging case to show the per- PDS(2)-VQS outperforms otherwise. It is worth
formance difference between conventional VQE mentioning that, during the PDS(K)-VQS pro-
and PDS(K)-VQS approaches. cess for the ground state and energy, the excited
 state energies can also be estimated directly from
 the higher roots of the polynomial (12) without
4 Discussion any additional measurement (although accurate
 excited state energies would require higher order
From Section III, it has been seen that the
 PDS(K)-VQS calculations). In contrast, the con-
PDS(K)-VQS approach bears the potential of
 ventional VQE would need distinct trial states,
speeding up the iterations in comparison with the
 and thus different measurements, for targeting
conventional VQE approach. However, it is worth
 different states.
noting that the measurement effort of evaluating
 Generally speaking, as long as the relatively
hH n i’s (n > 1) and their derivatives are usually
 large number of measurements of the Pauli
more expensive than that of hHi and its deriva-
 strings becomes manageable, the PDS(K)-VQS
tive, and the actual cost saving will therefore be
 approach can be potentially applied for target-
compromised.
 ing the exact solutions for the system sizes that
 To have a close look at the measurement of
 are not classically tractable, in particular for the
the hH n i (and its impact on the total cost), we
 systems whose true ground and excited states we
employ the following metric to give an estimate
 have little knowledge of, or are challenging to ob-
for the number of measurements, M ,[23, 56, 69]
 tain classically. To reduce the measurement de-
 P qP 
 !2
 G i,j,∈G hi hj cov Pi , Pj mand, typical strategy is to partition the Pauli
 M= , (20) strings (that contribute to the moments) into
 
 commuting subsets that follow a certain rule, e.g.
where  is the desired precision and hi ’s and Pi ’s qubit-wise commutativity (QWC)[31, 44], general
are the coefficients and Pauli strings represent- commutativity[22, 72], unitary partitioning[30],
ing a moment (i.e. H n =
 P
 i hi Pi ) and hav- and/or Fermionic basis rotation grouping[28] to
ing been partitioned into certain groups, G’s, name a few. The applications of these commuting

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rules to the single Hamiltonian have shown that, measurements and ill-conditioning arising from
at a cost of introducing additional one-/multi- high Hamiltonian powers, lower oder PDS(K)-
qubit unitary transformation before the measure- VQS approaches are usually more feasible.
ment, the total number of required measure-
ments can be significantly reduced from O(N 4 )
to O(N 2∼3 ), or for simpler cases even O(N ). For
 |Ψ!"#$% ⟩ =
higher order moments, as we mentioned in the
method section, early study of applying QWC
bases to Heisenberg models represented by up to
36 qubits exhibits a sub-linear scaling of the num-
ber of measurements in the number of the qubits
(Ref. [67]), which then leads us to expect similar
scaling behaviors of the number of required mea-
surements for evaluating moments for molecu-
lar systems. Beside exploring the commutativity
of Pauli strings, other approaches including the
linear combinations of unitary operators (LCU)
technique[13], direct block-encoding[6, 21], and
quantum power methods[59] might also be worth
studying for reducing the number of measure-
ments at the cost of circuit depth. In the light of Figure 6: The performance of VQE and PDS(2)-VQS
that, we plan to perform a comprehensive bench- employing ordinary gradient descent (GD) to compute
mark as a follow-up work. the ground state energy of the four-site 2D Heisenberg
 model. (Top right) The circuit employed to generate the
 Since the PDS(K)-VQS formalism involves trial vector, where only the first rotation in RY gate is
solving linear system of equations and polyno- treated as a variational parameter θ, and other three
mial, there is a concern of numerical instability rotations are fixed to (0, 3, 3). Two initial rotations
 θ0 = −2.0 and θ0 = −3.0 are chosen for performance
when applying the PDS(K)-VQS approach in op- comparison. The exact ground state energy of the 2D
timization. Theoretically, the numerical instabil- Heisenberg model is Eexact = −3.6 a.u. (Center) The
ity of the PDS(K)-VQS approach might come VQE and PDS(2)-VQS energies and (bottom) the corre-
from two sources, (a) the singularity and ill- sponding fidelity changes of the trial vectors w.r.t. true
conditioning of the matrix M in Eq. (11) that ground state in the first ten iterations in the conventional
might consist of high order moments, and (b) the VQE and PDS(2)-VQS noise-free calculations. The step
singularity of the Riemannian metric (R) used size η = 1.0/Iteration in all the calculations.
in the dynamics (16). In particular, the singu-
larity of matrix M can be easily observed if the Ultimately, one would be concerned about
trial vector becomes very close to the exact wave how the PDS(K)-VQS applies to general mod-
function (det|M| = 0 if we replace the trial vec- els and how it performs on the real quantum
tor with exact vector). Numerically, the singu- hardware subject to the device noise. To ad-
larity problem can be avoided by adding a small dress these concerns and explore the potential of
positive number (e.g. 10−6 ) to the eigenvalue of the PDS(K)-VQS approach, we have started to
the matrix M or R via singular value decomposi- launch the PDS(K)-VQS calculations for more
tion (SVD). However, it is worthing noting that general Hamiltonians on both simulator and the
adding small perturbation to M might violate the real quantum hardware. Figs. 6 and 7 ex-
variationality of the PDS approach, and would hibit some preliminary results for a four-site 2D
not be recommended to use if the strict upper- Heisenberg model with external
 magnetic field,
 P P
bounds to the true energy are concerned. The ill- H = J hiji Xi Xj + Yi Yj + Zi Zj + B i Zi with
conditioning of matrix M could occur in the high J/B = 0.1. The simple circuit employed for the
order PDS calculations, where high order mo- state preparation in both VQE and PDS(K)-VQS
ments could make the condition number of ma- simulations is shown in Fig. 6, where, for the
trix M very large. Thus, from the practical point brevity of our discussion, we only treat one rota-
of view, due to the potentially larger number of tion in the state preparation as the variational pa-

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calculations, the accuracy of the results systemat-
 ically improves. For example, in the PDS(4)-VQS
 approach both the computed ground state energy
 and the trial state (and thus the magnetization)
 converge within 10 iterations being very close to
 the exact solutions.
 5 Conclusion

 In summary, we propose a new variational quan-
 tum solver that employs the PDS energy gradient.
 In comparison with the usual VQE, the PDS(K)-
 VQS helps identify an upper bound energy sur-
Figure 7: The computed ground state energy (top left) face for the ground state, and thus frees the dy-
and magnetization (top right) of the four-site 2D Heisen- namics from being trapped at local minima that
berg model and the corresponding changes of the fidelity refer to non-ground states. In comparison with
(bottom left) and variational parameter θ (bottom right) the static PDS(K) expansions, the PDS(K)-VQS
in the first ten PDS(K)-VQS (K = 2, 3, 4) iterations
 guides the rotation of the trial wave function of
running on IBM Toronto quantum hardware. The phys-
ical setup, error sources, and computed expectation val- modest quality, and is able to achieve high accu-
ues of Hamiltonian moments (up to hH 7 i) and the asso- racy at the expense of low order PDS(K) expan-
ciated standard deviations are shown in Fig. 8. In all the sions. We have demonstrated the capability of
calculations ordinary gradient descent (GD) is employed. the PDS(K)-VQS approach at finding the ground
The initial rotation θ~0 = −3.0. The exact ground state state and its energy for toy models, H2 molecule,
energy and magnetization of P the 2D Heisenberg model and strongly correlated planar H4 system in some
are Eexact = −3.6 a.u. and i hσzi i = −4.0 a.u., re- challenging situations. In all the case studies, the
spectively. The step size η = 1.0/Iteration in all the
 PDS(K)-VQS outperforms the standalone VQE
calculations.
 and static PDS(K) calculations in terms of effi-
 ciency even at the lowest order. We also discussed
rameter, and fix all other three rotations. As can the limitations of the PDS(K)-VQS approach at
be seen from the noise-free simulations in Fig. 6, the current stage. In particular, the PDS(K)-
the PDS(2)-VQS results quickly converge within VQS approach may suffer from large amount of
five iterations achieving ∼ 0.99 fidelity, while the measurements for large systems, which can nev-
performance of VQE exhibits strong dependence ertheless be reduced at the cost of circuit depth
on the initial rotation (for θ~0 = −2.0, the con- by working together with some measurement re-
ventional VQE is able to converge in 10 itera- duction methods. Finally, we have started to
tions with ∆E < 0.05 a.u. and Fidelity ∼ 0.97). launch PDS(K)-VQS simulations for more gen-
When running the PDS(K)-VQS simulations for eral Hamiltonians on IBM quantum hardware.
the same model on the IBM Toronto quantum Preliminary results for Heisenberg model indi-
hardware, as shown in Fig. 7, in comparison to cate that higher order PDS(K)-VQS approach
the ideal curves, the PDS(2/3)-VQS optimization exhibits better noise-resistance than the lower or-
curves on the real hardware significantly slows der ones. The discussed approach can be ex-
down, and deviate from the exact solutions due tended to any variational formulation based on
to the error from the real machine. However, if we the utilization of hH n i moments (e.g. Krylov sub-
increase the PDS order to perform PDS(4)-VQS space algorithms).

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(a) (b)
 T1(μs) T2(μs)
 !"#$
 ̅ = 1.074×10%&
 0 1
 '( = 3.410×10%& Q0 88.17 50.09
 2 ') = 3.990×10%& Q1 118.95 140.21
 3 '& = 8.270×10%& Q2 114.08 130.99
 '* = 1.400×10%& Q3 75.79 109.07
 ibmq_toronto
 (c)

Figure 8: (a) Quantum processor device map for ibmq_toronto showing the four qubits (Qn , n = 0 − 3) used
in the present computation. (b) Average CNOT error, 1-qubit readout assignment error, and thermal relaxation
time constant (T1) and dephasing time constant (T2) in the four qubits used in the present computation. (c)
The expectation values of the Hamiltonian moments, hH n i (n = 1 − 7), assembled
 P from the measurements
 
 of the
 P
expectation values of 21 QWC bases for four-site 2D Heisenberg model H = J hiji Xi Xj +Yi Yj +Zi Zj +B i Zi
with J/B = 0.1. The data points correspond to mean value from the calculations on IBM Quantum processor
ibmq_toronto with statistical error bars corresponding to 5 × 8192 shots (per point). The trial state is constructed
using the circuit given in Fig. 6 with initial rotation θ0 = −3.0.

6 Acknowledgement putation, 10(2):251–276, 1998. DOI:
 https://doi.org/10.1162/089976698300017746.
B. P. and K. K. were supported by the “Em-
bedding QC into Many-body Frameworks for [2] Frank Arute, Kunal Arya, Ryan Bab-
Strongly Correlated Molecular and Materials Sys- bush, Dave Bacon, Joseph C. Bardin,
tems” project, which is funded by the U.S. De- Rami Barends, Sergio Boixo, Michael
partment of Energy, Office of Science, Office of Broughton, Bob B. Buckley, David A.
Basic Energy Sciences (BES), the Division of Buell, Brian Burkett, Nicholas Bushnell,
Chemical Sciences, Geosciences, and Biosciences. Yu Chen, Zijun Chen, Benjamin Chiaro,
B. P. and K. K. acknowledge the use of the IBMQ Roberto Collins, William Courtney, Sean
for this work. The views expressed are those of Demura, Andrew Dunsworth, Edward Farhi,
the authors and do not reflect the official policy Austin Fowler, Brooks Foxen, Craig Gidney,
or position of IBM or the IBMQ team. Marissa Giustina, Rob Graff, Steve Habeg-
 ger, Matthew P. Harrigan, Alan Ho, Sabrina
 Hong, Trent Huang, William J. Huggins, Lev
7 Data Availability Ioffe, Sergei V. Isakov, Evan Jeffrey, Zhang
 Jiang, Cody Jones, Dvir Kafri, Kostyan-
The data that support the findings of this study tyn Kechedzhi, Julian Kelly, Seon Kim,
are available from the corresponding author upon Paul V. Klimov, Alexander Korotkov, Fedor
reasonable request. Kostritsa, David Landhuis, Pavel Laptev,
 Mike Lindmark, Erik Lucero, Orion Mar-
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