PRX QUANTUM 2, 020310 (2021)

Qubit-ADAPT-VQE: An Adaptive Algorithm for ConstructingHardware-Efficient Ansätze on a Quantum Processor

Ho Lun Tang ,1,* V.O. Shkolnikov,1 George S. Barron,1 Harper R. Grimsley,2 Nicholas J. Mayhall,2Edwin Barnes,1 and Sophia E. Economou1,†

1Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA

2Department of Chemistry, Virginia Tech, Blacksburg, Virginia 24061, USA

(Received 5 February 2020; revised 6 June 2020; accepted 30 March 2021; published 28 April 2021)

Quantum simulation, one of the most promising applications of a quantum computer, is currently beingexplored intensely using the variational quantum eigensolver. The feasibility and performance of thisalgorithm depend critically on the form of the wave-function ansatz. Recently in Ref. [Nat. Commun. 10,3007 (2019)], an algorithm termed ADAPT-VQE was introduced to build system-adapted ansätze withsubstantially fewer variational parameters compared to other approaches. This algorithm relies heavily ona predefined operator pool with which it builds the ansatz. However, Ref. [Nat. Commun. 10, 3007 (2019)]did not provide a prescription for how to select the pool, how many operators it must contain, or whetherthe resulting ansatz will succeed in converging to the ground state. In addition, the pool used in that workleads to state-preparation circuits that are too deep for a practical application on near-term devices. Here,we address all these key outstanding issues of the algorithm. We present a hardware-efficient variant ofADAPT-VQE that drastically reduces circuit depths using an operator pool that is guaranteed to containthe operators necessary to construct exact ansätze. Moreover, we show that the minimal pool size thatachieves this scales linearly with the number of qubits. Through numerical simulations on H4, LiH andH6, we show that our algorithm (“qubit-ADAPT”) reduces the circuit depth by an order of magnitude whilemaintaining the same accuracy as the original ADAPT-VQE. A central result of our approach is that theadditional measurement overhead of qubit-ADAPT compared to fixed-ansatz variational algorithms scalesonly linearly with the number of qubits. Our work provides a crucial step forward in running algorithmson near-term quantum devices.

DOI: 10.1103/PRXQuantum.2.020310

I. INTRODUCTION

Finding the ground state of a many-body interactingelectronic Hamiltonian is one of the most important prob-lems in modern quantum chemistry and physics. As thedimension of the Hamiltonian scales exponentially withthe number of particles, accurate classical simulationscan only be performed for systems with few electrons.Although many classical computational techniques havebeen developed to approximate the ground electronicstate, no classical method is available that can performaccurately for arbitrary systems with polynomial effort.While density-functional theory (DFT) [1,2] has been

*holuntang@vt.edu†economou@vt.edu

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license. Fur-ther distribution of this work must maintain attribution to theauthor(s) and the published article’s title, journal citation, andDOI.

tremendously useful for unraveling the microscopic detailsof weakly correlated molecules and materials, the mostaccurate form (Kohn-Sham DFT [1]) relies on a singleSlater determinant wave function, which fails to describestrong correlation with current functionals. Unlike DFT,whose accuracy relies on a density functional, which isnot able to be systematically improved, wave-function-based methods such as configuration interaction (CI) orcoupled cluster (CC) can be used to describe many-bodysystems with clear paths to arbitrary accuracy. However,in the presence of strong correlation such methods incuran exponential computational cost due to the large numberof Slater determinants involved. Alternatively, methodsbased on tensor network states [3–7] (most notably den-sity matrix renormalization group [8–10]) can exploit ranksparsity in low-energy states of one-dimensional systemsto simulate strongly correlated systems, with polynomialcost. This polynomial scaling is lost in higher dimensions,and the computational cost again grows exponentially. Aradically different approach is Feynman’s proposal to studyquantum systems using quantum computers [11]. Recent

2691-3399/21/2(2)/020310(16) 020310-1 Published by the American Physical Society

HO LUN TANG et al. PRX QUANTUM 2, 020310 (2021)

reviews provide a comprehensive background and discussnew developments in quantum simulation with quantumcomputers [12–14].

The long-term method for simulating chemical systemswith quantum computers is the quantum phase estima-tion algorithm (PEA) [15,16], which shows an exponen-tial speedup over classical algorithms [17,18]. Since thenumber of gates, i.e., unitary operations, involved in thisalgorithm is very large, it requires a long coherent evolu-tion, which can only be realized in fault-tolerant quantumcomputers [19]. These scalable, error-correcting devicesmay take decades to realize experimentally. In the mean-time, the community is exploring algorithms that can beapplied to existing and near-term processors, namely noisyintermediate-scale quantum (NISQ) devices [19].

A promising algorithm for NISQ hardware is the vari-ational quantum eigensolver (VQE) [20,21]. VQE isa hybrid method that combines classical computationalpower with a quantum processor. Based on the variationalprinciple in quantum physics, the VQE algorithm con-structs a trial wave function by applying gates on a quan-tum device and estimates the average energy by measuringthe Hamiltonian on that device. This measured energy isthen minimized by tuning the quantum circuit. The cir-cuit parameter optimization is performed by a classicalcomputer, and quantum resources are only used for theclassically intractable parts (state preparation and energyevaluation) of the calculation. Compared to PEA, VQE hasmuch more modest requirements on the coherence timesof the quantum processor, and it has already been realizedon NISQ devices, such as superconducting qubits [22,23],photons [20], and trapped ions [24,25]. The accuracy ofVQE is highly dependent on the explicit form of the wave-function ansatz and can only obtain the exact ground-stateenergy if the ansatz is capable of representing any state inthe subspace that contains the ground state. For example,if symmetries of the ground state are known, then one canconstruct an ansatz that represents any state in the corre-sponding symmetry subspace [26], guaranteeing that theexact ground state can be expressed in terms of the ansatz.

One of the commonly used ansätze for VQE is uni-tary coupled-cluster singles and doubles (UCCSD) [27,28]. Stemming from the coupled cluster theory used inchemistry, this unitary version is more suitable for quan-tum circuit implementation. The “singles and doubles” inUCCSD means only single and double excitation operatorsare included in the ansatz, each carrying its own variationalparameter. One drawback of UCCSD is that includingall the singles and doubles operators leads to a (poten-tially unnecessarily) deep circuit and a large number ofparameters to optimize. To reduce this complexity, severalalternatives have been proposed, which attempt to keeponly the most important operators [29–36]. A second draw-back is that UCCSD is also generally not exact and suffersfrom ambiguities in operator ordering upon factorization

into a product of exponentiated operators (Trotterization),which is a necessary step in converting the ansatz to a state-preparation circuit [37,38]. Another approach to buildingthe ansatz is to use the most accessible gates in the quan-tum device, alternating single-qubit gates and two-qubitgates layer by layer. This is referred to as a hardware-efficient ansatz [23]. Although this approach lessens thedemands on the quantum processor, the resulting wave-function ansatz can lead to difficulties in parameter opti-mization [39]. This problem was addressed by constructingparticle-conserving entangling gates instead of ordinarytwo-qubit gates [40,41] and symmetry-preserving circuits[26]. While these fixed-ansatz approaches can be appliedto any problem and can reduce the number of varia-tional parameters and circuit depths, further improvementmay still be possible by tailoring the ansatz to a givensimulation problem.

Recently, a new algorithm that provides a systematicmethod to build an ansatz dynamically was introduced.This algorithm, termed adaptive derivative assembledpseudo-Trotter (ADAPT) VQE [29], employs a predeter-mined pool of operators from which the ansatz is dynami-cally constructed. ADAPT-VQE differs substantially fromapproaches based on compiling fixed circuits. Instead, theansatz is grown iteratively, such that at each step, theoperator that affects the energy the most is added to theansatz. Using fermionic operators as a pool, Ref. [29]demonstrated that ADAPT-VQE substantially outperformsUCCSD, in terms of both number of variational parame-ters and accuracy. This result demonstrates the promise ofthe ADAPT algorithm. However, due to the gate overheadof the fermion-to-spin mapping, the operators consideredin Ref. [29] translate to a fairly large number of quantumgates, and, therefore, while the number of parameters isvery low, the circuit depth (which is significantly reducedcompared to UCCSD) may still be impractically large, lim-iting the applicability of ADAPT-VQE to NISQ devices.Even more importantly, it is not clear (i) how the operatorpool should be chosen in general, (ii) how many operatorsit should contain, and (iii) what guarantees that the poolis complete, i.e., that it enables convergence to the groundstate.

In this paper, we address these issues by introduc-ing qubit-ADAPT-VQE, an algorithm that substantiallyreduces both the number of measurements and the cir-cuit depths needed to achieve convergence. We term thisalgorithm qubit-adaptive derivative assembled problem-tailored (qubit-ADAPT) VQE, in contrast to the imple-mentation in Ref. [29], which we refer to as fermionic-ADAPT in this paper. Through classical simulations ofseveral different molecules, we demonstrate that comparedto fermionic-ADAPT, qubit-ADAPT reduces the circuitdepth by an order of magnitude while maintaining thesame accuracy. This compactness of qubit-ADAPT andits iteratively grown ansatz are expected to reduce the

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risk of barren plataeus. Moreover, we introduce a poolcompleteness criterion that determines whether a givenpool will generate an exact ADAPT ansatz. We prove thatthe minimal number of pool operators that satisfy this con-dition grows linearly with the number of qubits. This ismuch smaller than the quartic scaling originally assumedin fermionic-ADAPT, and it demonstrates that the addi-tional measurement overhead of ADAPT-VQE remainsmodest for larger systems (increasing only linearly overconventional, fixed-ansatz VQEs). Our results pave theway toward both practical and accurate VQE algorithmson NISQ devices.

This paper is organized as follows. First, we brieflyreview fermionic-ADAPT and estimate the correspondingcircuit depth in Sec. II. In Sec. III, we introduce qubit-ADAPT and provide a detailed description of the opera-tor pool. In Sec. III A, we compare the performance ofqubit-ADAPT and fermionic-ADAPT through numericalsimulations of H4, LiH, and H6 molecules. We show thatthe minimal size of the operator pool for qubit-ADAPTscales linearly in the number of qubits in Sec. III B. Weconclude in Sec. IV. Details of the estimate for the circuitdepth of fermionic-ADAPT and a constructive proof of thelinear scaling of minimal complete pools are included inappendices.

II. CIRCUIT DEPTH ESTIMATE FORFERMIONIC-ADAPT

The ADAPT ansatz is grown by one operator τi = −τ †i

at each iteration, and after the nth iteration is given by∣∣∣ψ

ADAPT(�θ)⟩

= eθk τk . . . eθ2 τ2eθ1 τ1∣∣ψHF⟩ , (1)

where∣∣ψHF

⟩

is the HF state. These operators are selectedfrom an operator pool defined upfront. At each iteration,the operator, which induces the maximum change to theenergy, is selected. This energy response is represented bythe gradient of the energy with respect to the correspondingparameter, i.e.,

∂

∂θi〈E〉 = 〈ψ | [H , τi] |ψ〉 , (2)

which can be measured on the quantum device. The ansatzkeeps growing until the norm of the gradient vector,

||�g|| =√√√√

∑

i

(∂

∂θi〈E〉

)2

, (3)

is zero or, in practice, smaller than a chosen threshold,ε. Compared to ordinary VQE, ADAPT-VQE requiresadditional measurements to obtain the gradient at each iter-ation; the number of these measurements is roughly equal

to the size of the pool times the number of terms in theHamiltonian [42], although the number of measurementsneeded to compute the mean energy can be decreased usingrecently developed techniques [43].

In order to attain a compact circuit for the ADAPTansatz, we want to minimize both the number of param-eters and the circuit depth. While the first requirementcan be satisfied by the general structure of the ADAPT-VQE algorithm, the second one is not guaranteed anddepends on the chosen pool. Using fermionic operators, aparameter-efficient operator pool can be constructed fromspin-adapted single excitation operators:

τ1 ∝ |↑〉a 〈↑|b + |↓〉a 〈↓|b − h.c., (4)

and double excitation operators:

τ2,T ∝ |T, 1〉pq 〈T, 1|rs + |T, −1〉pq 〈T, −1|rs

+ |T, 0〉pq 〈T, 0|rs − h.c.,

τ2,S ∝ |S, 0〉pq 〈S, 0|rs − h.c.,

(5)

where a, b, p , q, r, s are spatial orbitals and T, S refer totriplets and singlets formed by p , q or r, s. To implementADAPT-VQE on qubits, we have to map these fermionicoperators to Pauli operators, which has the consequencethat a parameter-efficient pool is most likely not gate-efficient. In this paper, we employ the Jordan-Wigner (JW)mapping, i.e., ai → ∏i−1

j =0 Zj (Xi − iYi), although othermappings [44–46] could be used instead.

In the JW mapping, a double excitation operator maycontain more than one fermionic operator a†

pa†qaras − h.c.,

which is transformed into at most eight Pauli strings. (Aproduct of four fermionic operators gives 16 terms in total,but each symmetric term is cancelled by its Hermitian con-jugate [35].) These excitation operators conserve S2, Szand particle number by summing a number of Pauli stringstogether, which results in a high gate count per excitationoperator.

In order to make a rough estimate of the number ofCNOT gates involved in one operator from the fermionicpool, we consider the generalized singles-doubles exci-tation fermionic pool. “Generalized” indicates that theexcitations are not restricted to occur only from occupiedto virtual orbitals. Rather, all combinations of excitationsare included. To obtain the gate count, we perform first-order Trotterization of each unitary, i.e., eθi τi ≈ ∏

j eθj Pj ,where Pj are the Pauli strings appearing in τi after theJW mapping. For simplicity, we assume only doubleexcitation operators are picked by the algorithm, whichmakes this a conservative estimate as any single excita-tion included would result in a smaller CNOT-to-parametercount ratio. The number of CNOTs needed for a singlePauli string Pi with length q is 2(q − 1) [48]. Here, thelength q is the number of nonidentity Pauli operators in

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HO LUN TANG et al. PRX QUANTUM 2, 020310 (2021)

the string. The average number of CNOTs involved in aspin-adapted double excitation operator is approximatelyNPauli(6 + 2NZ)Nspin, where NPauli is the average numberof Pauli strings in a doubles operator a†

pa†qaras − h.c., NZ

is the average number of Pauli Z’s in a Pauli string due tothe anticommutation relation of fermionic operators, andNspin is the average number of doubles operators summedin a spin-adapted operator [Eq. (5)]. By Appendix A, thesequantities are given by

NPauli = 8m(5m − 7)m(5m − 1)− 8

, (6)

NZ = 12m3 − 30m2 + 34m − 203(5m2 − m − 8)

, (7)

Nspin = 5m2 − m − 8m2 + m

, (8)

where m is the number of spatial orbitals. For large m,the number of CNOTs in a spin-adapted doubles operatoris approximately 64m.

III. QUBIT-ADAPT

Because the spin-adapted fermionic operators eachintroduce a large number of CNOTs into the state-preparation circuit, we are motivated to construct a newpool consisting of operators that involve fewer CNOTs.One way is to break down the spin-adapted fermionicoperators after the JW mapping and choose the individ-ual Pauli strings as the operator pool τ = P = i

∏

i pi, pi ∈{X , Y, Z}. This more hardware-efficient choice effectivelyreduces Nspin and NPauli to 1, while it contains the samebasic elements as the spin-adapted fermionic pool. Animportant property of this qubit pool is that it containsonly Pauli strings with odd numbers of Y’s because thefermionic operators are real, hence P = i

∏

i pi has to bereal. We refer to these as “odd” Pauli strings. The remain-ing “even” Pauli strings have no effect on the energy. Thisis because H is symmetric (time-reversal symmetry is pre-served), and so the expectation value of the commutatorin Eq. (2) will vanish for real |ψ〉 if τi is symmetric (i.e.,even). We can therefore restrict τi to the odd Pauli strings,which are antisymmetric. Using such operators in the poolalso ensures that the ansatz remains real throughout thequbit-ADAPT algorithm, which should be the case whenH is time-reversal symmetric. The length of these stringsranges from 2 to n due to the Pauli-Z chain responsi-ble for the fermionic anticommutation relation. To furtherreduce gate depth, we remove these Pauli-Z chains fromthe operators. Numerically, we find that these two pools(with and without Z chains) perform similarly. The factthat qubit-ADAPT does not require the Z chains in orderto achieve convergence suggests that this algorithm avoidsissues related to the long-range correlations inherent in the

JW mapping. The pool without Z chains gives Pauli stringswith maximum length 4. The size of this pool is muchsmaller than the full set of Pauli strings with maximumlength 4, as we only pick operators that already appearin the fermionic pool, which are capable of transforming∣∣ψHF

⟩

to the ground state. We refer to this reduced poolas the “qubit pool”. Below, we demonstrate that this poolproduces significantly shallower state preparation circuitscompared to fermionic-ADAPT. We also show that the sizeof the qubit pool can be reduced dramatically down to asize that scales only linearly in the number of qubits, whichsubstantially cuts down on the number of measurementsneeded to run qubit-ADAPT.

A. Numerical simulations

We compare the performance of the qubit pool to thefermionic pool in terms of the number of parameters andthe number of CNOTs for different molecules, H4, LiH, andH6 (Fig. 1). In each case, we choose a bond distance suchthat correlation effects are significant: r = 1.5 Å for H4and H6, and r = 2 Å for LiH. All the calculations are per-formed using an STO-3G basis and start from restrictedHartree-Fock orbitals without using a frozen-core approx-imation such that we have eight spin orbitals for H4 and12 spin orbitals for each of the other two molecules. AllCNOT gate counts are obtained using the qiskit commandcount_ops, where we consider the case of all-to-all qubitconnectivity for concreteness. Although these counts willincrease as the connectivity is reduced, we do not expectthe relative performance of the two algorithms to changesignificantly. In the case of fermionic-ADAPT, we showgate counts before and after transpilation is used. Thecounts before transpilation agree well with the estimatesobtained in Appendix A.

It is evident from Fig. 1 that in the case of the qubitpool, more parameters are used compared to the fermionicpool. On the other hand, the number of CNOTs is reducedsignificantly, by about an order of magnitude in the caseof H6. Switching from the fermionic pool to the qubitpool increases the number of parameters in the ansatz,which is the price for compressing the circuit depth. How-ever, the increase in parameter number increases only therequired classical computational power during the clas-sical optimization, while the decrease in circuit depthreduces the demands on the quantum processor. For NISQdevices, the number of CNOTs that can be implementedwithin the coherence time is very limited, so the abilityof qubit-ADAPT to divert more of the computational costaway from the quantum processor and onto the classicaloptimizer should be advantageous overall.

To evaluate the effectiveness of using the gradient asa means to grow the ADAPT ansatz, we compare theperformance of qubit-ADAPT with random operatororderings drawn from the same qubit pool. In Fig. 2, we

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QUBIT-ADAPT-VQE... PRX QUANTUM 2, 020310 (2021)

0 10 20 3010 14

10 12

10 10

10 8

10 6

10 4

10 2

100

E (

itera

tions

)-E

(F

CI)

Har

tree

(a)

0 50 100 150 200

ADAPT iteration number

(b)

0 200 400

(c)

0 1 210 14

10 12

10 10

10 8

10 6

10 4

10 2

100

E (

itera

tions

)-E

(F

CI)

Har

tree

(d)

0 3 6

Number of CNOTs (103)

(e)

qubit-ADAPT

ADAPT

ADAPT transpiled

0 10 20 30

(f)

H H H H HLi H H H HH H

FIG. 1. Ground-state energy difference between ADAPT and the exact (FCI) result for (a),(d) H4 at bond distance 1.5 Å, (b),(e)LiH at bond distance 2 Å, and (c),(f) H6 at bond distance 1.5 Å. The bond distances are chosen to ensure that correlation effects aresubstantial. All the calculations are performed using an STO-3G basis and start from restricted Hartree-Fock orbitals without using afrozen-core approximation such that we have eight spin orbitals for H4, 12 spin orbitals for LiH, and 12 spin orbitals for H6. Resultsare shown for qubit-ADAPT (blue), fermionic-ADAPT (orange), and transpiled fermionic-ADAPT (green). (a)–(c) Energy differenceas a function of ADAPT iterations (which is equal to the number of variational parameters). (d)–(f) Energy difference as a functionof the number of CNOTs in the corresponding state-preparation circuit. All CNOT gate counts are obtained using the qiskit commandcount_ops. The transpiled counts are obtained using qiskit.transpile [47] with heavy optimization, which includes canceling back-to-back CNOTs and “commutative cancellation” (see Supplementary Material [56] for the code). Although qubit-ADAPT requires morevariational parameters, it entails significantly fewer CNOTs compared to fermionic-ADAPT.

see that qubit-ADAPT always converges much faster thanthe random orderings for both H4 and LiH. When randomorderings are used for H4, convergence to the ground staterequires 46–78 parameters, compared to only 30 param-eters for qubit-ADAPT. In the case of LiH, the randomorderings require more than 3 times as many parametersas qubit-ADAPT to converge. We can provide only a lowerbound on the number of parameters needed to converge therandom-ordering results in Fig. 2(b) due to the long com-putational times needed in this case. These findings suggestthat the role of the gradient selection in qubit-ADAPT iscrucial for larger problems.

These findings suggest that qubit-ADAPT will likelyavoid issues with barren plateaus. Studies of barrenplateaus have revealed that circuit depth plays a key role[39,49], indicating that qubit-ADAPT-VQE may avoid

this issue due to the compactness of the ansätze it pro-duces. In fact, there are two recent papers that followsimilar strategies as ADAPT-VQE, and which demon-strate that variable ansätze are much more resilient tobarren plateaus. One of these works, the so-called lay-erwise learning, grows the ansatz by adding layers withfixed operator structure, thus optimizing the parametersgradually [50]. Similar conclusions were also drawn inRef. [51]. The features that protect against barren plateausin these works are also naturally present in the qubit-ADAPT-VQE ansatz-growth strategy. Additionally, it hasbeen demonstrated that correlating random parameters inthe ansatz can also prevent barren plateaus [52]. Again,we expect that qubit-ADAPT-VQE may also have thisfeature since the parameters of the ansatz at each layer arerelated to the parameters in the previous iteration. More

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HO LUN TANG et al. PRX QUANTUM 2, 020310 (2021)

E (

itera

tions

)-E

(F

CI)

Har

tree

E (

itera

tions

)-E

(F

CI)

Har

tree

qubit-ADAPT

LiH

H4

Random ordering

(a)

(b)

FIG. 2. Energy error of qubit-ADAPT versus ansätze with ran-dom operator orderings as a function of the number of iterationsfor (a) H4 and (b) LiH. The qubit pool is used in all cases, andthe orbital bases and bond distances are chosen as in Fig. 1.

generally, barren plateaus are believed to arise becausethe parameterized ansatz has too much expressivity in thespace of unitaries. Since qubit-ADAPT-VQE is problemtailored, each ansatz is only intended to be used for a sin-gle Hamiltonian. This is unlike the “hardware-efficient”approach where one ansatz is intended to solve multi-ple Hamiltonians. In this way, the ansätze resulting fromqubit-ADAPT-VQE have lower expressivity and thus lesssusceptibility to barren plateaus.

Figure 3 shows that the performance of qubit-ADAPTremains essentially the same across different bond dis-tances. Results are shown for LiH, where the bond distanceis varied from 1 to 3 Å. The fact that the curves cor-responding to different bond lengths largely overlap oneanother shows that the rate of convergence does not changesignificantly. Because the amount of correlation in theground state depends on the bond distance (more cor-relations tend to arise as the bonds are stretched), thissuggests that the convergence of qubit-ADAPT is not sen-sitive to the strength of correlations. This finding indicatesthat qubit-ADAPT is a promising approach to studyingstrongly correlated systems.

B. Operator pool reduction

So far, a drawback of the qubit pool is its large sizecompared to the fermionic pool, which in turn leads to

(a)

(b)

(b)

ADAPT iteration number

Bond distancec (Å)

Ene

rgy

(Har

tree

)E

nerg

y er

ror

(Har

tree

)E

nerg

y er

ror

(Har

tree

)

Number of CNOTs

qubit-ADAPT

ADAPTADAPT transpiled

FIG. 3. qubit-ADAPT and fermionic-ADAPT energy error ofLiH for various bond distances. The bond distance varies from1 to 3 Å with darker curves in (a), (b) corresponding to largerbond distance. The energy error is plotted against (a) the numberof parameters and (b) the number of CNOTs. The performanceremains similar across different bond lengths and hence acrossdifferent amounts of correlation in the ground state. The orbitalbasis used is the same as in Fig. 1. (c) The final energies obtainedby qubit-ADAPT (points marked by ×) for each bond distanceconsidered are plotted along with the exact ground-state energies(solid curve).

proportionally more measurements at each iteration. Eventhough the qubit pool is defined from the fermionic pool,which is a small subset of the full Pauli group, the pool sizegrows quickly with the number of orbitals. However, manyof these operators are redundant in the sense that eliminat-ing them has no effect on the convergence of the algorithm.For example, there are pairs of operators that are related bya global rotation, e.g., X0Y1Y2Y3 and Y0X1X2X3, so we needonly to keep one of them in the pool; discarding the otherdoes not have any significant effect on the performance ofqubit-ADAPT.

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The redundancy in the pool can be illustrated by remov-ing randomly selected operators from the pool and mon-itoring the impact on convergence. First we randomlyremove 3/4 of the operators in the pool. As shown inFig. 4, despite this large reduction of the pool size, theperformance of the algorithm is similar to that with theoriginal pool. However, as is also evident in Fig. 4, ifwe further remove more operators, the pool is some-times incomplete, and the energy may not converge tothe ground-state energy. We can understand the toleranceof the algorithm to the drastic reduction of the pool bystudying the Hilbert space spanned by the pool operators.Starting from the expression for

∣∣∣ψADAPT(�θ)

⟩

in Eq. 1 andusing the Baker-Campbell-Hausdorff formula repeatedly,we obtain

∣∣∣ψ

ADAPT(�θ)⟩

= e∑

i φiAi∣∣ψHF⟩ , (9)

where the Ai include all the pool operators and their com-mutators, and the φi are functions of �θ . Therefore, theHilbert space spanned by the pool is determined by theset {Ai}. Note that if the original pool is comprised of oddPauli strings, then so is {Ai}. If the operators in {Ai} cantransform the reference state to any real state in the n-qubitHilbert space, then the qubit-ADAPT ansatz is guaranteedto be exact, and it is capable of converging to the groundstate. (Here, the only symmetry we impose is time rever-sal.) Note that if {Ai} includes all the odd Pauli strings[of which there are 2n−1(2n − 1), which scales exponen-tially with the number of qubits], then we could create anarbitrary orthogonal transformation in Eq. (9). However,spanning the Hilbert space requires only a subset of theodd Pauli strings, because only 2n − 1 real parameters are

qubit-ADAPT

H4

qubit-ADAPT 1/4 pool

qubit-ADAPT 1/16 poolqubit-ADAPT 1/32 poolE

(ite

ratio

ns)-

E (

FC

I) H

artr

ee

FIG. 4. qubit-ADAPT energy error of H4 using the full qubitpool, and randomly chosen 1/4, 1/16, and 1/32 pools are plottedagainst the number of iterations. For the 1/4 and 1/16 pools, thealgorithm can converge for almost every run. For the 1/32 pool,most of the runs can reach only an energy error of 10−3. Theorbital basis and bond distance are chosen as in Fig. 1.

required to create an arbitrary real state. In particular, weneed {Ai} to be such that for an arbitrary state |ψ〉, the statesAi |ψ〉 form a complete basis. In this case, we refer to {Ai}as a complete basis of operators.

The problem then is to determine the minimal pools thatproduce a complete basis of operators. For a given pool,we define the overlap matrix as Mij = 〈ψ | A†

i Aj |ψ〉 where|ψ〉 is an arbitrary real state. If the rank of M satisfiesr(M ) ≥ 2n − 1, this pool is called complete. To determinethe smallest complete pools, we randomly generate manydifferent pools of increasing size and compute r(M ) in eachcase to test for completeness. We did this for up to sevenqubits. Our numerical investigations reveal that, surpris-ingly, the minimal pool size required for the overlap matrixto have the required rank of 2n − 1 is only 2n − 2. This isevident in Fig. 5, which shows the fraction of pools thatare complete for various pool sizes and numbers of qubits.In each case, complete pools are found for pool sizes thatcontain at least 2n − 2 operators. Note that not only is thismuch smaller than the Hilbert space, it is also much smallerthan the size of the fermionic pool, which scales like n4.

Randomly selecting pools of size 2n − 2 and testingfor completeness by computing r(M ) is a numericallyintensive process that quickly becomes infeasible as thenumber of qubits increases. This is further exacerbatedby the fact that the fraction of complete pools of size2n − 2 becomes smaller as n increases, as is evident inFig. 5, which raises the question of whether completepools of size 2n − 2 even exist for large values of n.In Appendix B, we prove analytically using inductionthat complete pools of size 2n − 2 exist for any n. Infact, the proof is constructive: We present two families ofminimal pools that are provably complete for any num-ber of qubits. One of these pools, which we call {Vj }nwhere j = 1, . . . , 2n − 2, is defined recursively as {Vj }n ={Zn{Vk}n−1, iYn, iYn−1}, starting from the pool for n = 2qubits: {Vj }2 = {iZ2Y1, iY2}. This pool is comprised ofgenerators for single-qubit Y rotations and conditional Y

0 2 4 6 8 10 120.0

0.2

0.4

0.6

0.8

1.0

Number of operators in pool

Com

plet

epo

ols

(fra

ctio

n)

FIG. 5. The fraction of pools that are complete as a functionof pool size for two (blue circles), three (yellow squares), four(green diamonds), and five (red triangles) qubits.

020310-7

HO LUN TANG et al. PRX QUANTUM 2, 020310 (2021)

rotations, and it contains operators that act on up to alln qubits. In Appendix B, we prove that these operatorsare sufficient to rotate any real state to any other realstate in the Hilbert space. As shown in Appendix C, {Vj }ncan be mapped to a second family of minimal completepools, which we call {Gj }n, that has a very local struc-ture. This pool contains all two-qubit operators of the formiZk+1Yk that act on two neighboring qubits labeled by kand k + 1. There are n − 1 such operators. {Gj }n also con-tains all single-qubit Pauli iY operators except on the firstqubit. Therefore, this pool contains 2n − 2 operators. InAppendix C, we show that the Gj can be obtained fromcommutators of the Vj for any n, and so the completenessof the former follows from that of the latter.

Ener

gy (

arbi

trar

y un

its)

Ener

gy (

arbi

trar

y un

its)

Ener

gy (

arbi

trar

y un

its)

(a)

(b)

(c)

FIG. 6. Energy error curves for three different kinds of mini-mal complete pools versus incomplete pools. Results for three,four, and five-qubit random Hamiltonians are shown in (a), (b),and (c), respectively.

To investigate the importance of the pool being com-plete, we run qubit-ADAPT for random real Hamiltoniansof three, four, and five qubits with random initial states andfor pools consisting of 2n − 2 operators randomly chosenfrom the set of odd Pauli strings. We also run simulationsusing the minimal complete pools {Vj }n and {Gj }n. Theoperator coefficients in each Hamiltonian (which is takento be real and symmetric) are obtained by sampling uni-formly in the range [−2, 2] with ten samples for each ofthese three cases. The corresponding energy error curvesare illustrated in Fig. 6. Each curve is the result for a dif-ferent random Hamiltonian. All of the curves that fail toconverge correspond to incomplete pools. For these cases,even though the gradient goes to zero the ground state isnot reached because important operators are never gen-erated. On the other hand, the runs with complete poolsalways converge, highlighting the importance of this cri-terion. For the cases considered, we find that 20%–40%of pools containing 2n − 2 operators are complete. Thesefindings shed light on the question of what constitutes agood operator pool for ADAPT-VQE. These points werenot appreciated in the original paper [29], and moreover,they suggest that the fermionic pool used in that work isovercomplete. Furthermore, the fact that the minimal poolsize is linearly proportional to n means that the numberof additional measurements needed for each step of qubit-ADAPT is also linear in n. Thus, the extra measurementoverhead of qubit-ADAPT remains modest as the problemsize increases.

IV. CONCLUSIONS

In conclusion, we introduce a more efficient and NISQ-compatible version of the ADAPT-VQE algorithm, calledqubit-ADAPT. The basic idea of qubit-ADAPT is to use apool consisting of Pauli strings (rather than fermionic oper-ators) so that the number of CNOT gates associated witheach pool operator is reduced. We establish a completenesscondition that guarantees that a pool will generate an exactADAPT ansatz, and we prove that the smallest possiblepool that obeys this criterion scales linearly with the num-ber of qubits. We also provide a constructive approach togenerating minimal complete pools for an arbitrary num-ber of qubits. These results remove the ad hoc elements ofthe ADAPT algorithm and lead to a substantial reductionin the depths of state-preparation circuits and in the numberof measurements needed to run ADAPT-VQE on realistichardware.

ACKNOWLEDGMENTS

This research is supported by the National ScienceFoundation (Award No. 1839136) and the US Departmentof Energy (Award No. DE-SC0019199).

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QUBIT-ADAPT-VQE... PRX QUANTUM 2, 020310 (2021)

APPENDIX A: CNOT ESTIMATION

From the structure of the pool operators, we can tryestimating the number of CNOT gates as a function ofthe number of variational parameters. These estimates aredescribed below.

1. Qubit-ADAPT

In order to get a conservative estimate on the CNOTcount, we assume that all operators picked are four-qubitPauli strings, since exponentiating an n-qubit Pauli stringrequires 2(n − 1) CNOT gates, the number of CNOT gates is

NCNOT = 6. (A1)

2. Fermionic-ADAPT

In the case of fermionic-ADAPT, the number of CNOTsused can be estimated as

NCNOT = (6 + 2 × NZ)× NPauli × Nspin, (A2)

where NZ is the average number of Pauli Z used for fix-ing the antimsymmetry of fermionic operators, Nspin isthe average number of fermionic operators in each spin-adapted group and NPauli is the average number of Paulistrings from each fermionic operator.

Nspin = Totalno.offermionicterms(a†pa†

qaras)

Total spin adapted double ops in pool= no. of fermionic terms in a spin adapted op,

NPauli = Total number of Pauli strings after JWTotal fermionic terms

= no. of Pauli strings from each fermionic term,

NZ = TotallengthofPauliZfromallfermionicTotal no. of fermionic terms

= increases the length of Pauli strings.

(A3)

To compute these total counts, we can first classify thecombination of spatial orbitals into five groups:

1. p �= q �= r �= s.2. One of p , q = one of r, s.3. p = q �= r �= s.4. p = q = r �= s.5. p = q �= r = s.

Since some of them can have both triplet and singletwhile the others can only have singlet. And the numberof fermionic terms in spin-adapted grouping also dependson the combination of spatial orbitals.

For group 1, the triplet operator reads as

τ2,T ∝ |T, 1〉pq 〈T, 1|rs + |T, −1〉pq 〈T, −1|rs

+ |T, 0〉pq 〈T, 0|rs − h.c.

= |pq〉 〈rs| + 12(|pq〉 + |pq〉) (〈rs| + 〈rs|)

+ |p q〉 〈rs| − h.c.

→ a†pa†

qaras + 12

(

a†pa†

qaras + a†pa†

qaras

+ a†p a†

qaras + a†p a†

qaras

)

+ a†p a†

qaras − h.c.,

so there are six terms in total (12 terms including Hermitianconjugate). For the singlet,

τ2,S ∝ |S, 0〉pq 〈S, 0|rs − h.c.

= 12(|pq〉 − |pq〉) (〈rs| − 〈rs|)− h.c.

→ 12

(

a†pa†

qaras − a†pa†

qaras − a†p a†

qaras + a†p a†

qaras

)

− h.c.,

so there are four terms in total (eight terms including Her-mitian conjugate). For group 2, the triplet operator reads as(for q = r)

τ2,T → a†pa†

qaqas + 12

(

a†pa†

qaqas + a†pa†

qaqas

+ a†p a†

qaqas + a†p a†

qaqas

)

+ a†p a†

qaqas − h.c.,

while the singlet operator is

τ2,S → 12

(

a†pa†

qaqas − a†pa†

qaqas − a†p a†

qaqas

+ a†p a†

qaqas

)

− h.c.

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HO LUN TANG et al. PRX QUANTUM 2, 020310 (2021)

For group 3, there is only singlet

τ2,S → 12

(

a†pa†

p aras − a†pa†

p aras − a†p a†

paras

+ a†p a†

paras

)

− h.c.

= a†pa†

p aras + a†pa†

p aras − h.c.

For group 4, there is only singlet

τ2,S → 12

(

a†pa†

p apas − a†pa†

p apas − a†p a†

papas

+ a†p a†

papas

)

− h.c.

= a†p a†

papas + a†pa†

p apas − h.c.

For group 5, there is only singlet

τ2,S → 12

(

a†pa†

p arar − a†pa†

p arar − a†p a†

parar

+ a†p a†

parar

)

− h.c.

= 2a†pa†

p arar − h.c..

Then we can perform Jordan-Wigner transformation toeach of these terms. Each anti-Hermitian pair of fermionicoperators having four different indices, i.e., a†

i a†j akal −

h.c., would result in eight Pauli strings, i.e.,

a†i a†

j akal − h.c.

= 2i∏

n

Zn

(

YiXj XkXl + XiYj XkXl − XiXj YkXl − XiXj XkYl

− XiYj YkYl − YiXj YkYl + YiYj XkYl + YiYj YkXl

)

.

The other case is two of the four indices are the same,

a†i a†

j aj al − h.c. =∏

n

Zn(Xi + iYi)Zj (Xl − iYl)− h.c.

= 4i∏

n

Zn

(

YiZj Xl − XiZj Yl

)

,

only two Pauli strings left. From the expression of thefive cases’ fermionic operators, we can then determine theaverage number of Pauli strings per anti-Hermitian pair foreach case.

For the length of Pauli Z, it is equivalent to count theinterval between i and j and between k and l where i ≤ j ≤k ≤ l, we calculate them for separate cases, for m spatialorbitals,

1. p �= q �= r �= s:The total number of Pauli Z is given by the sum

12

(

42

)

× (6 + 4)

×∑

m1<m2<m3<m4

[

(2m4 − 2m3 − 1)(2m2 − 2m1 − 1)]

= m2(2m4 − 15m3 + 40m2 − 45m + 18).

The prefactor 12

(

42

)

is counting the ways we choose two

of {p , q, r, s} to be in the dagger side, while the prefactor(6 + 4) is from the number of fermionic operators for eachcombination. If we have {i, j , k, l} = {1, 2, 3, 4}, which isin group 5, the Jordan-Wigner transformation would bringa Pauli-Z chain Z1Z2Z3 for the creation and annihilationoperator on index 4 and a Pauli-Z chain Z1Z2 for oper-ator on index 3, so only Z3 can survive, but it wouldcombine with the X3 ± iY3 from the operator on index 3,so there is no Pauli Zs effectively. Therefore, we shouldcount the number of integers can be fitted in the intervals2m4 − 2m3 and 2m2 − 2m1, that explains the terms −1 inthe calculation.

In this counting, we treat all the spin orbitals to bespin up (even number indices) even though each term in|T, 0〉 〈T, 0| and |S, 0〉 〈S, 0| has two spin-down and twospin-up operators. It is because whenever there are termswith both spin-up and spin-down spin orbitals, there wouldbe another term with the spin totally flipped, so the count-ing of these two terms would cancel each other. For exam-ple, consider p < q < r < s, the |T, 0〉 〈T, 0| terms readas 1

2

(

a†pa†

qaras + a†pa†

qaras + a†p a†

qaras + a†p a†

qaras

)

− h.c.,the first term and the last term are different by a totalspin flip, the number of Pauli Z would be 2s + 1 −2r − 1 + 2q + 1 − 2p − 1 and 2s − (2r + 1)− 1 + 2q −(2p + 1)− 1, so the +1 due to the odd index switches signby the spin flip. Therefore, the average number of Pauli Zof this pair is the 2s − 2r − 1 + 2q − 2p − 1, which is thenumber of Pauli Z in the all spin-up term a†

pa†qaras (or the

all spin-down term).2. One of p , q = one of r, s.The total number of Pauli Z is given by the sum

10 ×[ ∑

m1=m2<m3<m4

(2m4 − 2m3 − 1)

+∑

m1<m2=m3<m4

(2m4 − 2m3 − 1 + 2m3 − 2m1 − 1)

+∑

m1<m2<m3=m4

(2m2 − 2m1 − 1)]

= 10m3(m3 − 4m2 + 5m − 2).

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QUBIT-ADAPT-VQE... PRX QUANTUM 2, 020310 (2021)

Here, we do not have the counting prefactor in group 1 aswe already decide that one of the two identical indices isin the dagger side.

3. p = q �= r �= s.The total number of Pauli Z is given by the sum

2 ×[ ∑

m1=m2<m3<m4

(2m4 − 2m3 − 1)

+∑

m1<m2=m3<m4

(2m4 − 2m3 − 1 + 2m3 − 2m1 − 1)

+∑

m1<m2<m3=m4

(2m2 − 2m1 − 1)]

= 2m3(m3 − 4m2 + 5m − 2).

It is only different from case 2 by the prefactor 2 instead of10 as it has two fermionic operators in total.

4. p = q = r �= s.The total number of Pauli Z is given by the sum

2 ×[ ∑

m1=m2=m3<m4

(2m4 − 2m1 − 1)

+∑

m1<m2=m3=m4

(2m4 − 2m1 − 1)]

= 2m3(2m2 − 3m + 1).

5. p = q �= r = s.Since the operators have a form a†

pa†p arar, they do not

have Pauli Z.

Using Eq. (A3) and Table I, the average number can thenbe calculated as follows:

Nspin = 5m2 − m − 8m2 + m

, (A4)

TABLE I. Top: Average number of Pauli strings per anti-Hermitian pair of spin-orbital operators. Middle and bottom: For each group,the number of combinations are calculated. For each combination, it has two spin group if it has both triplet and singlet operators, onlyone spin group if it only has singlet. For each singlet and triplet operator, the number of fermionic terms are calculated. And then wecalculate the number of Paulis in each term. Finally, we count the length of Pauli Z.

Group 1 2 3 4 5

No. of Fermi ops (T) 6 6 NA NA NANo. of Fermi ops (S) 4 4 2 2 1Average no. of Paulis (T) 8 (4 × 2 + 2 × 8)/6 = 4 NA NA NAAverage no. of Paulis (S) 8 (2 × 2 + 2 × 8)/4 = 5 8 2 8

Group 1 2 3

No. of combinations(

m2

) (

m − 22

)

/2 m(

m − 12

)

m(

m − 12

)

No. of spin groups(

m2

) (

m − 22

)

/2 × 2 m(

m − 12

)

× 2 m(

m − 12

)

No. of Fermi ops(

m2

) (

m − 22

)

/2 × (6 + 4) m(

m − 12

)

× (6 + 4) m(

m − 12

)

× 2

No. of Pauli strings(

m2

)(

m − 22

)

/2 × (6 + 4)× 8 m(

m − 12

)

× (6 × 4 + 4 × 5) m(

m − 12

)

× 2 × 8

No. of Pauli Zm2(2m4 − 15m3 + 40m2 − 45m + 18)

10m3(m3 − 4m2 + 5m − 2)

2m3(m3 − 4m2 + 5m − 2)

Group 4 5 Total

No. of combinations 2(

m2

) (

m2

)

m(m3 + 2m2 − m − 2)/8

No. of spin groups 2(

m2

) (

m2

)

m2(m2 − 1)/4

No. of Fermi ops 2(

m2

)

× 2(

m2

)

m(5m3 − 6m2 − 7m + 8)/4

No. of Pauli strings 2(

m2

)

× 2 × 2(

m2

)

× 8 2m(5m3 − 15m2 + 14m − 4)

No. of Pauli Z2m3(2m2 − 3m + 1) 0 m(6m4 − 21m3 + 32m2 − 27m + 10)/6

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HO LUN TANG et al. PRX QUANTUM 2, 020310 (2021)

NPauli = 8(5m3 − 15m2 + 14m − 4)5m3 − 6m2 − 7m + 8

, (A5)

NZ = 12m3 − 30m2 + 34m − 203(8 + m − 5m2)

. (A6)

We can estimate the CNOTs required for the H4 molecule(m = 4) is

NCNOT ≈ Npars × NPauli × (6 + 2 × NZ)× Nspin

= 11 × NPauli × (6 + 2 × NZ)× Nspin

= 1928.41,

which is very close to the number we obtain.

H4 LiH H6

Estimated no. of CNOTS per parameters 175 303 303No. of parameters 11 30 91Total estimated no. of CNOTS 1928 9078 27536No. of CNOTS 2208 6824 28632

This estimation is accurate when all the spin orbitals areevenly explored (H6 and H4), i.e., no “core” orbitals.

APPENDIX B: CONSTRUCTIVE PROOF OFMINIMAL COMPLETE POOLS

In this appendix, we present a constructive proof thatthere exist complete pools for qubit-ADAPT containingonly 2n − 2 operators, which we refer to as minimalcomplete pools. We first prove that the set of operators{Vi}n=3 = {iZ3Z2Y1, iZ3Y2, iY3, iY2} forms a complete poolfor n = 3 qubits, i.e., using operators of the form

∏

k eθkVk ,we can rotate any real state into any other real state.We then prove that the recursively defined pool, {Vi}n ={Zn{Vi}n−1, iYn, iYn−1}, is a complete pool for n qubits, suchthat

∏

k eθkVk |ψ〉 = |ϕ〉 for any two real states |ψ〉 and |ϕ〉in the Hilbert space.

1. Complete pool for three qubits

Here, we prove that {Vi}n=3={iZ3Z2Y1, iZ3Y2, iY3, iY2} isa minimal complete pool for three qubits. We do this byshowing that we can map any state |ψ〉 to |000〉 using onlyoperators of the form

∏

k eθkVk .We begin by decomposing an arbitrary real state |ψ〉 as

follows:

|ψ〉 = |0〉3 |ψ0〉21 + |1〉3 |ψ1〉21 . (B1)

The subscripts outside the kets indicate which qubit(s)the ket belongs to. In general, the two-qubit states |ψ0〉

and |ψ1〉 do not have the same norm: 〈ψ0|ψ0〉 �= 〈ψ1|ψ1〉.However, we can always perform a rotation eiθY3 on qubit3 (the leftmost qubit) to transform |ψ〉 to the form

|ψ ′〉 = eiθY3 |ψ〉 = |0〉3 |ψ ′0〉21 + |1〉3 |ψ ′

1〉21, (B2)

where

|ψ ′0〉 = cos θ |ψ0〉 + sin θ |ψ1〉 , (B3)

|ψ ′1〉 = cos θ |ψ1〉 − sin θ |ψ0〉 . (B4)

The norms of these states are

〈ψ ′0|ψ ′

0〉 = cos2 θ 〈ψ0|ψ0〉 + sin2 θ 〈ψ1|ψ1〉+ sin(2θ) 〈ψ0|ψ1〉 , (B5)

〈ψ ′1|ψ ′

1〉 = cos2 θ 〈ψ1|ψ1〉 + sin2 θ 〈ψ0|ψ0〉− sin(2θ) 〈ψ0|ψ1〉 . (B6)

We can make these two norms equal by choosing θ suchthat

tan(2θ) = 〈ψ1|ψ1〉 − 〈ψ0|ψ0〉2 〈ψ0|ψ1〉 . (B7)

Note that this procedure works for arbitrary states |ψ0〉 and|ψ1〉, not just for two-qubit states.

The heart of the proof amounts to showing that a stateof the form in Eq. (B2), where 〈ψ ′

0|ψ ′0〉 = 〈ψ ′

1|ψ ′1〉 = 1/2,

can be transformed into a form where qubit 3 has beenfactored out:

|ψ ′〉 → (|0〉 + |1〉)3 |χ〉21 , (B8)

where |χ〉 is some two-qubit state. We do this usingthe (slightly reduced) set of pool operators {Vred

i }3 ={iZ3Z2Y1, iZ3Y2, iY2}. These are the same as {Vi}3, exceptthat iY3 is not included. In the next subsection, we general-ize Eq. (B8) to n qubits and then use it to prove that {Vi}nis a complete pool.

To prove Eq. (B8), let us begin by expanding |ψ ′〉 asfollows:

|ψ ′〉 = |00〉 |ψ ′0a〉 + |01〉 |ψ ′

0b〉 + |10〉 |ψ ′1a〉 + |11〉 |ψ ′

1b〉.(B9)

From this point onward, we suppress the subscripts onthe kets for notational simplicity. Similarly to how wemade the states |ψ ′

0〉 and |ψ ′1〉 have the same norm, here

we can apply the conditional operations eiθ0[(1+Z3)/2]Y2 andeiθ1[(1−Z3)/2]Y2 to make

〈ψ ′0a|ψ ′

0a〉 = 〈ψ ′0b|ψ ′

0b〉 = 〈ψ ′1a|ψ ′

1a〉 = 〈ψ ′1b|ψ ′

1b〉 = 1/4.(B10)

We assume that this has already been done in Eq. (B9), andthat |ψ ′〉 has been redefined accordingly.

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QUBIT-ADAPT-VQE... PRX QUANTUM 2, 020310 (2021)

Next, we apply the conditional rotation eiφ1Z3Z2Y1 onqubit 1 to bring the state to

eiφ1Z3Z2Y1 |ψ ′〉 = (|00〉 + |01〉) |χ0〉 + |10〉 |ψ ′′1a〉

+ |11〉 |ψ ′′1b〉. (B11)

Here, we use the fact that |ψ ′0a〉 and |ψ ′

0b〉 are real single-qubit states, and thus can be viewed as two-dimensionalvectors lying in the same plane. The conditional rotationeiφ1Z3Z2Y1 rotates these two vectors in opposite directionsbecause one is conditioned on the state |00〉 and the otheron |01〉, and eventually the two vectors coincide at |χ0〉for a particular value of the rotation angle φ1. This opera-tion also affects the other terms in Eq. (B11) (as indicatedwith the extra primes), but their form remains the same.This conditional operation will be used repeatedly in whatfollows.

The next step is to apply a conditional rotation on qubit2 to bring it to the state |0〉 in the first term:

eiφ2[(1+Z3)/2]Y2eiφ1Z3Z2Y1 |ψ ′〉=

√2 |00〉 |χ0〉 + |10〉 |ψ ′′

1a〉 + |11〉 |ψ ′′1b〉. (B12)

We can then apply another conditional operation on qubit1 to rotate the states |ψ ′′

1a〉 and |ψ ′′1b〉 into each other:

eiφ3Z3Z2Y1eiφ2[(1+Z3)/2]Y2eiφ1Z3Z2Y1 |ψ ′〉=

√2 |00〉 ∣

∣χ ′0

⟩ + (|10〉 + |11〉) |χ1〉 . (B13)

This operation brings the two states to some state |χ1〉. Italso changes |χ0〉 to

∣∣χ ′

0

⟩

. Another conditional rotation onqubit 2 simplifies the second term:

eiφ4[(1−Z3)/2]Y2eiφ3Z3Z2Y1eiφ2[(1+Z3)/2]Y2eiφ1Z3Z2Y1 |ψ ′〉=

√2 |00〉 ∣

∣χ ′0

⟩ +√

2 |10〉 |χ1〉 . (B14)

We need one more conditional operation on qubit 1 to bringthis to the desired form:

eiφ5Z3Z2Y1 eiφ4[(1−Z3)/2]Y2 eiφ3Z3Z2Y1 eiφ2[(1+Z3)/2]Y2 eiφ1Z3Z2Y1 |ψ ′〉=

√2 (|00〉 + |10〉) |χ〉 = (|0〉 + |1〉) |0〉 |χ〉

√2. (B15)

We thus prove Eq. (B8). For reasons that will become clearin the next subsection, it is important to note that we didthis using only the reduced pool {Vred

i }3.All that remains is to show that Eq. (B15) can be mapped

to the state |000〉. This can be done using a Y3 rotationfollowed by a conditional operation on qubit 1:

eiφ6Z3Z2Y1ei(π/4)Y3eiφ5Z3Z2Y1eiφ4[(1−Z3)/2]Y2

× eiφ3Z3Z2Y1eiφ2[(1+Z3)/2]Y2 × eiφ1Z3Z2Y1 |ψ ′〉 = |000〉 .(B16)

Since we can map any state to |000〉, it follows that wecan map any three-qubit state to any other three-qubit state

using only the pool operators. We thus show that {Vi}3 ={iZ3Z2Y1, iZ3Y2, iY3, iY2} is a minimal complete pool forthree qubits.

2. Proof of the complete pool for n qubits

In the previous subsection, we prove that the pool {Vi}3is a complete pool for three qubits. A key piece of thisproof is the statement that we can use the reduced pool{Vred

i }3 = {iZ3Z2Y1, iZ3Y2, iY2} to factor out the third qubitEq. (B8):

|0〉 |ψ ′0〉 + |1〉 |ψ ′

1〉 → (|0〉 + |1〉) |χ〉 , (B17)

where we assume 〈ψ ′0|ψ ′

0〉 = 〈ψ ′1|ψ ′

1〉, and |χ〉 is sometwo-qubit state. We prove that a similar statement holdsfor n qubits.

Theorem: We can factor out the nth qubit from a genericn-qubit state:

|0〉 |ψ ′0〉 + |1〉 |ψ ′

1〉 → (|0〉 + |1〉) |χ〉 , (B18)

where 〈ψ ′0|ψ ′

0〉 = 〈ψ ′1|ψ ′

1〉, and |χ〉 is some (n − 1)-qubitstate, using only the reduced n-qubit pool:

{Vredi }n = {Zn{Vi}n−1, iYn−1}

= {Zn{Vredi }n−1, iZnYn−1, iYn−1}. (B19)

This pool is defined recursively starting from {Vredi }3, and

it differs from the full pool by only one operator (iYn):

{Vi}n = {Zn{Vi}n−1, iYn, iYn−1}. (B20)

We use induction to prove this theorem. We assume itholds for n qubits and then show that it also holds for n + 1qubits. This proof will closely follow what we did for threequbits above. Thus, we start with the (n + 1)-qubit state:

|0〉n+1 |ψ ′0〉 + |1〉n+1 |ψ ′

1〉, (B21)

where |ψ ′0〉 and |ψ ′

1〉 are n-qubit states that havethe same norm. We want to factor out the leftmost(n + 1)th qubit using only the operators {Vred

i }n+1 ={Zn+1{Vred

i }n, iZn+1Yn, iYn}. We decompose the state fur-ther:

|00〉 |ψ ′0a〉 + |01〉 |ψ ′

0b〉 + |10〉 |ψ ′1a〉 + |11〉 |ψ ′

1b〉. (B22)

As before, we can assume that all four states |ψ ′0a〉,|ψ ′

0b〉, |ψ ′1a〉, |ψ ′

1b〉, have the same norm of 1/4, since thiscan be achieved by applying two conditional operationseiθ0[(1+Zn+1)/2]Yn and eiθ1[(1−Zn+1)/2]Yn .

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We next apply a conditional operation built from theoperators eφ1Zn+1{Vred

i }n to map the state to

(|00〉 + |01〉) |χ0〉 + |10〉 |ψ ′′1a〉 + |11〉 |ψ ′′

1b〉. (B23)

We use the induction hypothesis for n qubits to obtain thisresult. Since the leftmost qubit is in the state |0〉 in the firsttwo terms, the effect of eφ1Zn+1{Vred

i }n is the same as eφ1{Vredi }n

on these terms, and the hypothesis for n qubits then allowsus to bring these two terms into a factorized form as above.The last two terms also evolve under this operation, buttheir general form remains the same since the n + 1 qubitremains in the state |1〉 under this operation. We followthis with a conditional operation eiφ2[(1+Zn+1)/2]Yn on the nthqubit to rotate it to |0〉 in the first term:

√2 |00〉 |χ0〉 + |10〉 |ψ ′′

1a〉 + |11〉 |ψ ′′1b〉. (B24)

Another application of eφ3Zn+1{Vredi }n can be used to simi-

larly factorize the last two terms:

√2 |00〉 ∣

∣χ ′0

⟩ + (|10〉 + |11〉) |χ1〉 . (B25)

Note that the form of the first term remains the same sincethis conditional operation does not rotate the (n + 1)thor nth qubits. This is because all the operators in {Vred

i }nhave either a Z or a I on the nth qubit. Next applyeiφ4[(1−Zn+1)/2]Yn to rotate the nth qubit in the second termto |0〉:

√2 |00〉 ∣

∣χ ′0

⟩ +√

2 |10〉 |χ1〉 . (B26)

Additional applications of eφZn+1{Vredi }n bring this to

√2 (|0〉 + |1〉) |0〉 |χ〉 , (B27)

as desired. In order to make this final step, we need to usea slight extension of the theorem in which we factor outthe (n + 1)th qubit instead of the nth qubit. Do we have allthe necessary operators to make this extension work? Yes.To see this, recall that in order to factor out the nth qubit,we need {V red

i }n = {Zn{V redi }n−1, iZnYn−1, iYn−1}. However,

here we have

Zn+1{Vredi }n = {Zn+1Zn{Vred

i }n−1, iZn+1ZnYn−1, iZn+1Yn−1}.(B28)

The operators Zn+1Zn{V redi }n−1 and iZn+1ZnYn−1 pro-

duce exactly the same conditional rotations on thestate in Eq. (B26) as Zn{V red

i }n−1 and iZnYn−1 pro-duce on

√2 |0〉 ∣

∣χ ′0

⟩ + √2 |1〉 |χ1〉. However, the theorem

also requires unconditional Yn−1 rotations on∣∣χ ′

0

⟩

and|χ1〉. These can be implemented using the compositeoperator

ei(π/4)[(1−Zn+1)/2]YneiφZn+1ZnYn−1ei(π/4)[(1−Zn+1)/2]Yn , (B29)

which first rotates Eq. (B26) into a state in which the(n + 1)th and nth qubit states in the two terms have thesame parity before performing the Zn+1ZnYn−1 rotation, sothat

∣∣χ ′

0

⟩

and |χ1〉 undergo the same rotation. The (n + 1)thand nth qubits are then restored to their original statesby the final rotation in Eq. (B29). Given that we alreadyshow above that this procedure works for n = 3 qubits, thiscompletes the proof of the theorem.

With the theorem proven, it is easy to prove that {Vi}n+1is a complete pool. We already know that a rotation gener-ated by Yn+1 is enough to bring an arbitrary (n + 1)-qubitstate to the form needed to apply the theorem. As we justsaw, we can then use Zn+1{V red

i }n, iZn+1Yn, and iYn to fac-tor out the (n + 1)th qubit and arrive at Eq. (B27). If wethen apply another Yn+1 rotation to bring the (n + 1)thqubit to |0〉, we can perform the conditional operationeiφ6Zn+1{V red

i }n to rotate√

2 |χ〉 to |0〉⊗n−1. Thus, we canrotate an arbitrary (n + 1)-qubit state to |0〉⊗n+1 using onlythe generators {Vi}n+1 = {Zn+1{V red

i }n, iZn+1Yn, iYn+1, iYn}.This in turn implies that we can rotate any (n + 1)-qubitstate to any other (n + 1)-qubit state using the same setof operators. Therefore, {Vi}n+1 is a complete pool, and itcontains 2n − 2 operators.

APPENDIX C: MAPPING BETWEEN MINIMALCOMPLETE POOLS

In this section, we show that the two examples of min-imal complete pools discussed in the main text can bemapped into one another. The first example is the pool {Vi}that is constructed iteratively in Appendix B. The secondexample is the pool comprised of operators that act only onadjacent qubits in a linear array, {Gi}. To facilitate the fol-lowing analysis, we order the elements in these two poolsas follows (where we ignore the factors of i in the definitionof the pool operators for simplicity):

V1 = ZZ . . . ZY, V2 = ZZ . . . ZYI ,

V3 = ZZ . . . ZYII , . . . , Vn−1 = ZYII . . . I , Vn = YII . . . I ,

Vn+1 = ZZ . . . ZIYI , Vn+2 = ZZ . . . ZIYII ,

. . . , V2n−3 = ZIYII . . . I , V2n−2 = IYII . . . I .

(C1)

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QUBIT-ADAPT-VQE... PRX QUANTUM 2, 020310 (2021)

G1 = ZYII . . . I , G2 = IZYII . . . I ,

G3 = IIZYII . . . I , . . . , Gn−2 = II . . . IZYI , Gn−1 = II . . . IZY,

Gn = YII . . . I , Gn+1 = IYII . . . I ,

Gn+2 = IIYII . . . I , . . . , G2n−3 = II . . . IYII , G2n−2 = II . . . IYI .

(C2)

We show that the Vi can be obtained from commuta-tors of the Gi. This means that the two pools share thesame operator basis, and so completeness of the one poolimplies completeness of the other. Since we already proveabove that the Vi are complete in Appendix B, this demon-strates that the Gi also form a minimal complete pool forany number n of qubits.

Recall that two Pauli strings do not commute if theydiffer by an odd number of Pauli operators, ignoring anyqubits with identity operators in at least one of the twostrings. In this case, the result of the commutator is propor-tional to the product of the two Pauli strings. Now noticethat we can write the Vi as products of the Gi:

Vn−k ∼ G1

k∏

j =2

Gj Gj +n−1, k = 2, . . . , n − 1,

Vn−1 = G1,

Vn = Gn,

V2n−k ∼ Gk−1GkVn−k, k = 3, . . . , n − 1,

V2n−2 = Gn+1.

(C3)

The operators in the product in the first line should beordered such that the operators with the smallest valuesof j are on the left and the ones with the largest valuesare on the right. Notice that no two adjacent operators inthese products commute. Therefore, we can rewrite themas nested commutators of the Gi. This proves that the Viand Gi produce the same operator basis when all possiblecommutators of each pool are computed. We already showin Appendix B that the Vi are complete. Therefore, the Gialso constitute a minimal complete pool. This pool is par-ticularly useful for quantum processors containing a lineararray of qubits with nearest-neighbor coupling only. Inter-estingly, the entangling operators in the Gi pool are sim-ilar to the cross-resonance interaction in fixed-frequencysuperconducting qubits, such as those in the IBM quantumprocessors [53–55]. This similarity could be exploited toimplement qubit-ADAPT with native hardware operations.

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