Noise reduction using past causal conesin variational quantum algorithms

Omar Shehab∗1, Isaac H. Kim†2, Nhung H. Nguyen3, Kevin Landsman3,Cinthia H. Alderete3,4, Daiwei Zhu3, C. Monroe1,3, and Norbert M. Linke3

1IonQ, Inc., College Park, MD 207402Stanford Institute for Theoretical Physics, Stanford University, Stanford CA 94305

3Joint Quantum Institute, Department of Physics and Joint Center for Quantum Information and ComputerScience, University of Maryland, College Park, MD 20742

4Instituto Nacional de Astrofísica, Óptica y Electrónica, Calle Luis Enrique Erro No. 1, Sta. Ma. Tonantzintla,Pue. CP 72840, Mexico

June 14, 2019

Abstract

We introduce an approach to improve the accuracy and reduce the sample complexity of near termquantum-classical algorithms. We construct a simpler initial parameterized quantum state, or ansatz,based on the past causal cone of each observable, generally yielding fewer qubits and gates. We implementthis protocol on a trapped ion quantum computer and demonstrate improvement in accuracy and time-to-solution at an arbitrary point in the variational search space. We report a ∼ 27% improvement inthe accuracy of the calculation of the deuteron binding energy and ∼ 40% improvement in the accuracyof the quantum approximate optimization of the MAXCUT problem applied to the dragon graph T3,2.When the time-to-solution is prioritized over accuracy, the former requires ∼ 71% fewer measurementsand the latter requires ∼ 78% fewer measurements.

1 IntroductionThe variational quantum eigensolver algorithm (VQE) [1,2] has been proposed and demonstrated for eigen-value approximation problems on noisy intermediate-scale quantum (NISQ) computers [3]. The VQE al-gorithm off-loads part of the task onto a classical computer in a hybrid quantum-classical approach withshort-depth quantum circuits, as opposed to the more stringent gate fidelity requirements in the phase es-timation algorithm approach [4, 5]. Researchers have successfully implemented the algorithm on variousquantum hardware [1, 6–14], also showing that the VQE algorithm is robust to certain types of error [2, 8].

The quantum approximate optimization algorithm (QAOA) [15] has been proposed to solve combinatorialoptimization problems on a NISQ computer. While the domain of application and the details of QAOAimplementation differ significantly from VQE algorithms, from a high level point of view, these approachesare similar in nature. In this work, we focus on noise reduction techniques in VQE algorithms, but thediscussions, experiments, and results are equally pertinent to both VQE and QAOA algorithms.

VQE uses the Rayleigh-Ritz variational principle [16,17] to compute the eigenvalue of a Hamiltonian H.For a parameterized wavefunction Ψ(~θ), the energy expectation 〈Ψ(~θ)|H|Ψ(~θ)〉 is bounded from below bythe lowest eigenvalue E0 of the Hamiltonian, where ~θ is a vector of independent parameters. VQE relieson the efficient creation of candidate states |Ψ(~θ)〉 and the measurement of 〈Ψ(~θ)|H|Ψ(~θ)〉 using a quantumcomputer. By classically optimizing the parameters ~θ, the local minimum of the Hamiltonian cost function is∗Electronic address: shehab@ionq.co†Electronic address: isaac.kim.quantum@gmail.com

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taken as an approximate ground state energy E0 of the system. QAOAs arrive at a target state by applyingp layers of evolution. While increasing the total number of gates and variational parameters, each successivelayer refines the candidate state and improves the accuracy of the approximation.

To generate the parameterized wavefunction for VQE/QAOA, both a hardware efficient ansatz [8] and aphysically inspired ansatz [12,18] have been implemented with respective advantages and disadvantages. Thehardware efficient ansatz [18] suffers from the potential for getting stuck in the barren plateaus of the energylandscape [19]. The physically inspired ansatz can quickly lead to deep circuits as the complexity of thephysical system increases. For example, for the unitary coupled cluster (UCC) ansatz relevant to molecularsimulations, the numbers of gates and circuit depth scale as O

(M3N

)and O

(M2N

)respectively where

M is the number of spin-orbitals, and N is the number of electrons, assuming a single Trotter step [20, 21].Finally, the success of VQE also depends on a large number of measurements for statistical certainty.

The coefficient of the Pauli term with the largest absolute value in a qubit Hamiltonian determines theupper bound on the variance of the expectation value [8] and hence the hardware performance and number ofmeasurements needed to achieve a desired accuracy. It can be limited by a careful model choice, as done e.g.in [22] when computing the binding energy of the deuteron nucleus. Once an appropriate fermionic model isconstructed for the VQE algorithm, the accuracy of the result is determined by the number of measurementsand experimental details such as the gate fidelity and qubit connectivity. Clearly, any reduction in circuitcomplexity, the size of the parameter space, or the number of measurements is desirable for a successful VQEapplication.

Several approaches to optimize VQE circuits have been proposed, such as removing qubits stabilized bythe Hamiltonian [6], making use of block-diagonality [23] or symmetry [14,24], grouping Hamiltonian termsbased on their norms [25], Pauli grouping [8], resetting qubits in a tensor network representation [26,27], orsubspace expansion [28]. The effect of optimization on VQE accuracy has also been rigorously studied [8].However, choosing the appropriate values for the QAOA circuit parameters to reach global optima has beenshown to be a hard problem [19]. In this work, we improve the VQE/QAOA process fidelity by using reducedansatz circuits based on past causal cones of each observable, and experimentally demonstrate the advantageon a trapped ion quantum computer.

2 Past causal cones as a reduced variational ansatzThe reduced-ansatz variational quantum eigensolver (RA-VQE) algorithm (and similarly the reduced-ansatzquantum approximate optimization algorithm, RA-QAOA) leverages the construction of a reduced ansatzwith respect to the terms of the Hamiltonian. Our construction shares similarities to the deep multi-scaleentanglement renormalization ansatz (DMERA) proposed in [29,30], but can be applied more generally. Thealgorithm replaces the original ansatz with a set of reduced circuits computed from the past causal cone(PCC) [31] of each term in the Hamiltonian. The PCC of a term is the set of gates that can influence itsexpectation value, and can be computed, for instance, using the depth-first search [32] on the directed acyclicgraph representation of the original ansatz.

Consider the QAOA ansatz to compute the MAXCUT of the dragon graph T3,2 shown in Figure 1.One can easily show that the exact MAXCUT for this graph is 4. The negated QAOA Hamiltonian is− 1

2 (5− Z1Z2 − Z2Z3 − Z3Z4 − Z4Z5 − Z3Z5), reflecting the connectivity of the graph. The QAOA ansatzat p = 1 is shown in Figure 2. Figure 3 shows the reduced ansatz set for the five observables.

Figure 1: The dragon graph, T3,2.

2

|0〉 H • • Xβ

|0〉 H Z γ2

• • Xβ

|0〉 H Z γ2

• • • • Xβ

|0〉 H Z γ2

• • Xβ

|0〉 H Z γ2

Z γ2

Xβ

Figure 2: QAOA ansatz to compute MAXCUT of the dragon T3,2 graph at p = 1. The ZZ interactions aredecomposed using two CNOT and one rotation gates.

|0〉 H • • Xβ

|0〉 H Z γ2

• • Xβ

|0〉 H Z γ2

(a) Z1Z2

|0〉 H • •

|0〉 H Z γ2

• • Xβ

|0〉 H Z γ2

• • • • Xβ

|0〉 H Z γ2

|0〉 H Z γ2

(b) Z2Z3

|0〉 H • •

|0〉 H Z γ2

• • • • Xβ

|0〉 H Z γ2

• • Xβ

|0〉 H Z γ2

Z γ2

(c) Z3Z4

|0〉 H • • • •

|0〉 H Z γ2

• • Xβ

|0〉 H Z γ2

Z γ2

Xβ

(d) Z4Z5

|0〉 H • •

|0〉 H Z γ2

• • • • Xβ

|0〉 H Z γ2

• •

|0〉 H Z γ2

Z γ2

Xβ

(e) Z3Z5

Figure 3: Reduced ansatz for each term of the dragon graph Hamiltonian.

One can compute the expectation value of each term from its reduced ansatz and combine the results todetermine the expectation value of the full Hamiltonian. This has a number of advantages over the originalVQE/QAOA algorithm. Each reduced ansatz has lower or equal complexity and qubit number compared tothe original ansatz. This reduction in both depth and width generally leads to lower noise. We note thatfor certain problems, there exist special ansatz constructions using a similar approach that possess inherentrobustness to noise [29], while the current work is a more general method to minimize the effect of noise.Additionally, some of the sub-Hamiltonians may have lower maximum absolute coefficients which give atighter upper bound on the variance of the expectation value of the Hamiltonian, or conversely a tighterupper bound on the number of measurements needed to maintain a given variance. We experimentallydemonstrate this advantage by comparing the energy obtained from the RA-VQE to VQE, and that ofRA-QAOA to QAOA.

Given a qubit Hamiltonian of a VQE problem expressed as H =∑tα hαPα, has t Pauli observables, one

may create t reduced circuits and compute the expectation value for each individually. While giving the mostaccurate result, this strategy also increases the total number of measurements. Instead, once the past causalcone circuits are generated, one can consider the minimum number of reduced ansatz circuits that support allterms in the Hamiltonian, reducing the total number of measurements and hence the time-to-solution. For

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instance, the reduced ansatz for two Hamiltonian terms may share the same circuit when the correspondingHamiltonian terms are supported by the same qubits but measured in different bases. Similarly, one reducedansatz may be the subcircuit of another.

We compute the expectation value for all sub-Hamiltonians independently but minimize them together(or maximize in the case of QAOA) with the prescribed number of measurements based on the chosenstrategy. The RA-VQE algorithm can be outlined very coarsely as follows.

1. Construct the reduced ansatz set.

2. Group sub-Hamiltonians based on chosen strategy.

3. Execute the circuits and measure the expectation value of each observable.

4. Calculate the expectation value for the overall Hamiltonian.

5. Use a classical non-linear optimizer to minimize/maximize this expectation value.

3 Experimental demonstrationWe use the VQE algorithm to compute the binding energy of the deuteron using a pion-less effective fieldtheory. This problem has attracted attention as a benchmark algorithm, and was implemented on bothsuperconducting and a trapped-ion platforms [12, 22]. For a four qubit ansatz, the qubit Hamiltonian is28.657− 2.143X0X1− 3.913X1X2− 5.671X2X3− 2.143Y0Y1− 3.913Y1Y2− 5.671Y2Y3 + 0.218Z0− 6.125Z1−9.625Z2 − 13.125Z3 and the circuit is given in Figure 4.

|0〉 Xπ

|0〉 Yφ • •

|0〉 Yλ1 • •

|0〉 Yλ2 •

Figure 4: The canonical four qubit UCC ansatz for deuteron [12]. Here the canonical gate set means therotation gates, their controlled versions, Hadamard and CNOT gates.

We use a trapped ion quantum computer to find the deuteron binding energy using both the original andreduced ansatz circuits. Since, the main advantage of our approach is higher process fidelity, we focus on thein-silico global minimum of the energy landscape of −2.14 MeV at the parameter values 0.858, 0.958, and0.758 radians. With 5000 measurements per circuit, we determine the experimental binding energy as −1.5(2)MeV for the original VQE ansatz, and −2.0(2) MeV for the reduced ansatz. This is a ∼ 80% improvementin accuracy, making the energy consistent with the theoretical value. Grouping of the sub-Hamiltoniansgives a similar result. When we prioritize time-to-solution, 14600 measurements are sufficient to determinethe binding energy as −1.5(3) MeV. This is a ∼ 70% reduction compared to the 50000 measurements theoriginal VQE ansatz needed.

The experimentally determined binding energies and the absolute standard errors are plotted againstincreasing number of measurements per circuit in Figure 5. Appendix A contains similar results for eachindividual Hamiltonian term.

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(a) Binding energy vs. # of measurements (b) Absolute standard error vs. # of measurements

Figure 5: Theoretical and experimentally determined binding energies (a) and the absolute standard errors(b) for the original and the reduced VQE ansatz. Data for all individual Hamiltonian terms is reported inAppendix A.

We also demonstrate the advantage of the RA-QAOA ansatz on the trapped ion quantum computer bysolving the MAXCUT problem for the dragon graph introduced in Section 2. We again use the set ofparameters corresponding to the in-silico global minimum of −3.45, which are γ = 1.358, and β = 2.462.With 5000 measurements per circuit, the original QAOA ansatz gives −3.26(2) while the reduced versiongives −3.33(2), a ∼ 36(9)% improvement in accuracy.

5410 measurements (5500 measurements conducted in the experiment) are sufficient to determine theMAXCUT as −3.34(7) . This is a ∼ 78% reduction in the number of measurements compared to 25000measurements needed in the original QAOA ansatz.

The experimentally determined MAXCUT and the absolute standard errors are plotted against increasingnumber of measurements per circuit in Figure 6. Appendix A contains similar results for each individualHamiltonian terms. For both examples, our method achieves more accurate results with fewer measurementsthan the standard VQE/QAOA circuits.

(a) MAXCUT vs. # of measurements (b) Absolute standard error vs. # of measurements

Figure 6: Theoretical and experimentally determined MAXCUT (a) and the absolute standard errors (b) forthe original QAOA ansatz and the reduced ansatz at the parameters (γ = 1.358, β = 2.462, p = 1). Similarexperimental data for individual Hamiltonian term is reported in Appendix A.

5

4 DiscussionThe reduced ansatz methods developed and demonstrated here show how targeted circuit optimization cangive significant performance increases which are crucial for NISQ devices. Depending on the problem, thedesign of the original ansatz can even be informed by its potential to take advantage of subsequent reducedansatz formulations. In the future, we hope to investigate how the structural complexity of a problem mayadversely affect the advantages expected to be achieved from the reduced ansatz approach, and how thismethod can be adapted to other types of algorithms.

5 Methods

5.1 Trapped ion hardwareThe trapped-ion quantum computer uses the 2S1/2|F = 0,mF = 0〉 and |F = 1,mF = 1〉 states of individualtrapped 171Yb+ ions as qubits. The ions are initialized by optical pumping to |F = 0,mF = 0〉 and detectedby state-dependent fluorescence on the 2S 1

2to 2P 1

2transition. We use a pair of counter-propagating Raman

beams, one of which is split into an array of individual addressing beams, to drive gate operations. The twonative gates in the system are single-qubit R-gates which are rotations around an axis in the X/Y plane,single qubit Z-rotations by phase advances in the classical controllers, and two qubit entangling XX-gateswhich use the motional modes to create entanglement between any two qubits. Both the R as well as theXX angle can be varied continuously. For the details of the single and two qubit gate implementations werefer the reader to Appendix A of [33] and to [34–37]. Typical gate times are 10µs for single-qubit and210µs for XX-gates. The errors in state initialization and detection are corrected by applying the inverse ofan independently measured state-to-state error matrix. Typical gate fidelities are ≈ 99.5% for single qubitgates and ≈ 98.5% for XX-gates.

For the four-qubit deuteron ansatz, seven ions are loaded into the trap, where the inner five are usedas qubits, with the outermost pair being used to evenly space the middle five ions. The algorithmic qubits1, 2, 3, 4 are mapped onto physical qubits 1, 2, 3, 5. The average four-qubit readout fidelity is 97.1%. To beconsistent, the same physical qubits are used for the reduced ansatz.

For the dragon graph ansatz, the algorithmic qubits 1, 2, 3, 4, 5 are mapped onto physical qubits 1, 5, 3, 2, 4. The average read-out fidelity for five qubits is 94.3%. Similarly, the same mapping is used for the reducedansatz.

Error bars for the correlators are the one-sigma intervals of the asymmetric binomial distribution ofstate populations. Since the error bars for the correlators tend to a symmetric limit for large number ofshots, the error bar for the Hamiltonian can be approximated by a Gaussian distribution, which follows frompropagation of the errors of individual correlators.

5.2 Construction of the reduced ansatz for the deuteronIn our approach, the original VQE ansatz is divided into smaller ansatz circuits, one for each term in theHamiltonian as shown in Figure 7.

6

|0〉 Xπ

|0〉 Yφ • •

|0〉 Yλ1 •

(a) X1X2 or Y1Y2

|0〉 Yφ •

|0〉 Yλ1 • •

|0〉 Yλ2 •

(b) X2X3 or Y2Y3

|0〉 Yφ •

|0〉 Yλ1 •

|0〉 Yλ2 •

(c) X3X4, Y3Y4 or Z3

|0〉 Xπ

|0〉 Yφ •

(d) Z1

|0〉 Yφ •

|0〉 Yλ1 •

(e) Z2

|0〉 Yφ •

|0〉 Yλ1 •

|0〉 Yλ2

(f) Z4

Figure 7: Reduced ansatz for each term of the deuteron Hamiltonian.

5.3 Construction of the deuteron sub-HamiltoniansTable 1 shows the Hamiltonian terms supported by every reduced ansatz of the deuteron problem.

Term Sub-HamiltonianX1X2 or Y1Y2 28.657I − 2.143X1X2 − 2.143Y1Y2 + 0.218Z1 − 6.125Z2

X2X3 or Y2Y3 28.657I−3.913X2X3−5.671X3X4−3.913Y2Y3−5.671Y3Y4−6.125Z2−9.625Z3 − 13.125Z4

X3X4 or Y3Y4 or Z3 28.657I − 5.671X3X4 − 5.671Y3Y4 − 9.625Z3 − 13.125Z4

Z1 28.657I + 0.218Z1

Z2 28.657I − 6.125Z2

Z4 28.657I − 13.125Z4

Table 1: New sub-Hamiltonian simulation problems generated by the reduced approach

One can either run all the ansatz circuits in Table 1 to prioritize accuracy, or a minimal subset whichcovers every term to reduce time-to-solution. We experimentally demonstrate that both strategies withthe same number of measurements as in the original VQE algorithm, and determine the binding energymore accurately than the the original VQE ansatz. For the second strategy, we consider the first two sub-Hamiltonians. 〈Z2〉 is considered as a term of only the first sub-Hamiltonian to avoid repeated calculationsince the corresponding ansatz is shallower.

5.4 Estimating the number of measurements for shorter time-to-solutionIf accuracy is the priority, one should run the reduced VQE or QAOA ansatz with as many measurementsas possible. We run 5000 measurements per circuit, which would require 50000 measurements in total forten reduced ansatz circuits. If the accuracy of the original VQE or QAOA ansatz is sufficient, it may beachieved with fewer measurements with the reduced ansatz.

Since, both the original and reduced ansatz would not be used together in practice, the target accuracycan only be estimated from the previous experiments of the same scale. We run the original VQE ansatzand use a standard 1-σ error to estimate the number of measurements needed for the reduced approach to

7

maintain the same error. The target error rate, the absolute value of the coefficient of the target observable(hγ), and the coefficient of largest absolute value (hγ,max) are used to estimate the number of measurements

Sβ according to ε =

√T |h2

max|S which is equation 12 of the supplemental material of [8]. The results are

given in table Table 2. In this table, hγ is the coefficient of the corresponding Hamiltonian term, hγ,maxis the coefficient with the largest absolute value in the sub-Hamiltonian, and Tβ is the number of terms inthe corresponding sub-Hamiltonian. In the experiment we use the closest multiple of fifty as the prescribednumber of measurements. When the prescribed number is too small we replace it with 500 to avoid theinitial fluctuation.

Term |hγ | |hγ,max| Tβ SβX1X2 or Y1Y2 2.143 6.125 4 ∼ 436X2X3 or Y2Y3 3.913 13.125 7 ∼ 3500

X3X4 or Y3Y4 or Z3 5.671 or 9.625 13.125 4 ∼ 2000Z1 0.218 0.218 1 ∼ 1Z2 6.125 6.125 1 ∼ 109Z4 13.125 13.125 1 ∼ 500

Table 2: Estimated number of measurements for the deuteron ansatz to maintain the original VQE accuracy.Sβ is the number of estimated measurements, hγ is the coefficient of the corresponding Hamiltonian term,hγ,max is the coefficient with the largest absolute value in the sub-Hamiltonian, and Tβ is the number ofterms in the corresponding sub-Hamiltonian.

In a similar manner, the number of measurements per Hamiltonian term needed for the reduced-ansatzQAOA approach to maintain the similar error level as in the original QAOA algorithm is determined as1082. In the experiment, we use 1100 measurements with the goal to maintain the accuracy 0.034 found forthe original QAOA ansatz.

5.5 Circuit optimization for trapped ion hardwareThe following circuit identities are used to translate the canonical gates into the physical gates native to ourtrapped ion architecture [38,39].

•=

Yπ2

XXπ2

X−π2

Y−π2

X−π2

•

Yθ=

• •Y θ

2Y− θ

2

• •Yθ

=X−π

2Z−π

2

XXθ

Zπ2

Xπ2

Z−π2

Zπ2

(1)

These circuits are then optimized using known rules (refer to [39] for details). The goal is to reduce thenumber of XX and RX gates. The optimized physical version of the original VQE ansatz to compute thebinding energy of deuteron is given in Figure 8.

|0〉XX−π

2

Xπ2

|0〉 Xφ−π2

Zπ2

Z−πXX−λ1

2

X−π Z−π2

XX−π2

|0〉 Xλ12 −

π2

Z−π2

XX−λ22

Z−π2

XXπ2

|0〉 Xλ22 −

3π2

Z−π2

X−π Zπ2

X−π2

Figure 8: Optimized four qubit deuteron ansatz circuit , written over a native gate set for our trapped-ionquantum computer.

8

The optimized physical version of the reduced VQE ansatz is given in Figure 9.

|0〉XX−π

2

Xπ2

|0〉 Xφ−π2

Zπ2

Z−πXX−λ1

2

X−π Z−π2

XX−π2

|0〉 Xλ12 −

π2

Z−π2

X−π2

Z−π2

X−π2

(a) X1X2 or Y1Y2

|0〉 X−π2

Z−φ−π2

XX−λ12

X−π Z−π2

XX−π2

|0〉 Xλ12 −

3π2

Zπ2

XX−λ22

X−π Z−π2

XXπ2

|0〉 Xλ22 −

π2

Z− 3π2

Zπ2

X−π2

(b) X2X3 or Y2Y3

|0〉 X−π2

Z−φ−π2

XX−λ12

X−π Z−π2

X−π2

|0〉 Xλ12 −

π2

Z−π2

XX−λ22

X−π2

Z−π2

XXπ2

|0〉 Xλ22 −

π2

Z− 3π2

Zπ2

X−π2

(c) X3X4, Y3Y4 or Z3

|0〉 Xπ

XXπ2

Xπ2

|0〉 Xπ2−φ Z−π

2Zπ

2X−π

2

(d) Z1

|0〉 X−π2

Z−φ−π2

XX−λ12

X−π Z−π2

XXπ2

X−π

|0〉 Xλ12 +π

2Zπ

2Z−π

2X−π

2

(e) Z2

|0〉 X−π2

Z−φ−π2

XX−λ12

X−π Z−π2

X−π2

|0〉 Xλ12 −

π2

Z−π2

XX−λ22

X−π2

Z−π2

X−π2

|0〉 Xλ22 −π

Z−π2

X−π

(f) Z4

Figure 9: Optimized reduced deuteron VQE ansatz written over a native gate set for trapped-ion quantumcomputers.

The optimized physical version of the original QAOA ansatz to compute MAXCUT of the dragon graphis given in Figure 10.

9

|0〉XX γ1

2

X−π2

Z−π2

Xβ−π2

|0〉XX γ1

2

X−π2

Z−π2

Xβ−π2

|0〉XX γ1

2

XX γ12

X−π2

Z−π2

Xβ−π2

|0〉XX γ1

2

X−π2

Z−π2

Xβ−π2

|0〉 X−π2

Z−π2

Xβ−π2

Figure 10: Optimized QAOA ansatz to compute MAXCUT of the dragon graph when p = 1.

The optimized physical version of the reduced QAOA ansatz is given in Figure 11.

|0〉XX γ1

2

X−π2

Z−π2

Xβ−π2

|0〉XX γ1

2

X−π2

Z−π2

Xβ−π2

|0〉 X−π2

Z−π2

X−π2

(a) Z1Z2

|0〉XX γ1

2

X−π2

Z−π2

X−π2

|0〉XX γ1

2

X−π2

Z−π2

Xβ−π2

|0〉XX γ1

2

XX γ12

X−π2

Z−π2

Xβ−π2

|0〉 X−π2

Z−π2

X−π2

|0〉 X−π2

Z−π2

X−π2

(b) Z2Z3

|0〉XX γ1

2

X−π2

Z−π2

X−π2

|0〉XX γ1

2

XX γ12

X−π2

Z−π2

Xβ−π2

|0〉XX γ1

2

X−π2

Z−π2

Xβ−π2

|0〉 X−π2

Z−π2

X−π2

(c) Z3Z4

|0〉XX γ1

2

XX γ12

X−π2

Z−π2

X−π2

|0〉XX γ1

2

X−π2

Z−π2

Xβ−π2

|0〉 X−π2

Z−π2

Xβ−π2

(d) Z4Z5

|0〉XX γ1

2

X−π2

Z−π2

X−π2

|0〉XX γ1

2

XX γ12

X−π2

Z−π2

Xβ−π2

|0〉XX γ1

2

X−π2

Z−π2

X−π2

|0〉 X−π2

Z−π2

Xβ−π2

(e) Z3Z5

Figure 11: Optimized reduced QAOA ansatz written over a native gate set for trapped-ion quantum com-puters.

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AcknowledgementsThis project was conceived during the Aspen Winter Conference, Advances in Quantum Algorithms andComputation 2018 at the Aspen Center for Physics which is supported by National Science Foundationgrant PHY-1607611. OS and IK thank the organizers for the hospitality. NML gratefully acknowledgesfunding by the PFC@JQI, NSF grant number PHY-1430094. C.H.A. acknowledges financial support fromCONACYT doctoral grant No. 455378. The authors thank Matthew Keesan, Shantanu Debnath, VandiverChaplin, Yunseong Nam, Jungsang Kim, Kenneth Brown, Eugene Dumitrescu, Greg Quiroz, Bill Huggins,and Raphael Pooser for their insightful comments.

Author ContributionsTheory was developed by O.S. and I.K; experimental data collected and analyzed by H.N.N. with help fromK.L., C.H.A., D.Z. and N.M.L.; O.S. and I.K. performed the circuit design and in-silico simulation; and O.S.and I.K. prepared the manuscript with input from all authors.

Competing InterestsProvisional patent applications for this work were filed by IonQ, Inc.

Data availabilityAll data needed to evaluate the conclusions are available from the corresponding author upon request.

CorrespondenceCorrespondence and requests for materials should be addressed to Omar Shehab (email: shehab@ionq.co)or Isaac Kim (email: isaac.kim.quantum@gmail.com).

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A Experimental data for individual terms in the HamiltonianIn this section, we present the experimental data to show how the expectation values and absolute errorconverge as the number of measurements increase for individual Hamiltonian term for the reduced ansatz.The result is reported for both the direct and sub-Hamiltonian grouping approach for the RA-VQE ansatzand the direct approach for the RA-QAOA ansatz. For any observable O, ∆〈O〉 is the difference betweenthe experimental and in-silico expectation values.

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Figure 12: Experimental data for the expectation value of individual terms in the deuteron Hamiltonian.14

Figure 13: Experimental data for the absolute noise in the expectation value of individual terms in thedeuteron Hamiltonian. 15

Figure 14: Experimental data for the expectation value of individual terms in the deuteron Hamiltonianusing sub-Hamiltonian grouping approach.

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Figure 15: Experimental data for the absolute noise in the expectation value of individual terms in thedeuteron Hamiltonian using sub-Hamiltonian grouping approach.

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Figure 16: Experimental data for the expectation value of individual terms in the dragon graph Hamiltonian.

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Figure 17: Experimental data for the absolute noise in the expectation value of individual terms in thedragon graph Hamiltonian.

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