Improving the accuracy of the energy estimation by combining quantum annealing with classical computation
Takashi Imoto, Yuya Seki, Yuichiro Matsuzaki, Shiro Kawabata
IImproving the accuracy of the energy estimation by combining quantum annealingwith classical computation
Takashi Imoto, Yuya Seki, Yuichiro Matsuzaki, ∗ and Shiro Kawabata † Research Center for Emerging Computing Technologies,National Institute of Advanced Industrial Science and Technology (AIST),1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan. (Dated: February 11, 2021)Quantum chemistry calculations are important applications of quantum annealing. For practicalapplications in quantum chemistry, it is essential to estimate a ground state energy of the Hamil-tonian with chemical accuracy. However, there are no known methods to guarantee the accuracyof the estimation of the energy calculated by quantum annealing. Here, we propose a way to im-prove the accuracy of the estimate of the ground state energy by combining quantum annealingwith classical computation. In our scheme, before running the QA, we need a pre-estimation of theenergies of the ground state and first excited state with some error bars (corresponding to possibleestimation error) by performing classical computation with some approximations. We show that, ifan expectation value and variance of the energy of the state after the QA are smaller than certainthreshold values (that we can calculate from the pre-estimation), the QA provides us with a betterestimate of the ground state energy than that of the pre-estimation. Since the expectation valueand variance of the energy can be experimentally measurable by the QA, our results pave the wayfor accurate estimation of the ground state energy with the QA.
I. INTRODUCTION
Recently, quantum chemistry calculations have at-tracted attention as new application for quantum devicesbecause of its potential use in medical areas. One of themain purposes of the quantum chemistry is to calculatethe energy of the molecule Hamiltonian written by thesecond quantized form. The high accuracy of the en-ergy of chemical materials is required at least 1 . × − hartree where 1hartee = e / π(cid:15) a = 27 . a = 1bohr = 0 . × − m. This accuracy is calledchemical accuracy. The energy with a chemical accuracyallows us to estimate the chemical reaction rate at a roomtemperature using the Eyring equation[1]There are sophisticated techniques to map themolecule Hamiltonian with the second quantized forminto a spin Hamiltonian. These techniques are impor-tant to implement the quantum chemistry calculationswith the quantum devices composed of qubits, becausethe Hamiltonian to describe the molecules in the quan-tum devices should be written by the Pauli matrices.We can map the second-quantized many-body Hamil-tonians onto those of qubits systems by Bravyi-Kitaevtransformation[2–6].There is an improvement over the Jordan-Wignertransformation in the requirement for the number of thequbit operators per a fermionic operator. The Jordan-Wigner transformation maps one of n fermionic operatorsto O ( n ) qubits operators. On the other hand, Bravyi-Kitaev transformation maps one of n fermionic operatorsto O ( log ( n )) qubits. The comparison of the gate numberfor the Bravyi-Kitaev tansformation and Jordan-Wigner ∗ [email protected] † [email protected] transformation to get the ground state and the lowest en-ergy with the Trotter decomposition is reported[7]. Also,Babbush et al . represent the Hamiltonian using only 2-local interaction between spins[8].In fault tolerant quantum computation, quantum al-gorithms have been proposed in quantum chemistrycalculations[9–11]. Molecular energies are obtained us-ing phase-estimation algorithms[12, 13]. However, thefault tolerant quantum computer require many qubitswith high fidelity gate operations beyond the capabilityof near-term quantum computer to use error-correction.So the algorithm for quantum chemistry is not experi-mentally implemented with a practically useful size yet.Recently, Noisy Intermediate-Scale Quantum(NISQ)computing is proposed [14–16]. One of the promisingalgorithms with NISQ is the variational quantum eigen-solver(VQE) with the variational method[17, 18]. TheVQE gives the lowest eigenvalue of a Hamiltonian suchas that of a chemical material. The VQE is a hybridquantum-classical algorithm. Variational algorithm isalso used to simulate quantum dynamics [19, 20]. Theenergy variance was used to know how close the quantumstate is to the energy eigenstate in the NISQ algorithm[21].Quantum annealing(QA) is also a promising way toimplement the quantum chemistry calculations. The QAwas traditionally used to solve the combinatorial opti-mization problem[22–24]. We map the combinatorialoptimization problem into the Ising Hamiltonian H P ,and we call this a problem Hamiltonian whose groundstate corresponds to the solution of the combinatorialoptimization problem. On the other hand, we use an-other Hamiltonian H D that represents transverse mag-netic fields, which we call a driver Hamiltonian. In theQA, we prepare a ground state of H D , and the totaltime-dependent Hamiltonian is changed from H D to H P a r X i v : . [ qu a n t - ph ] F e b within an annealing time T . As long as an adiabatic con-dition is satisfied, an adiabatic theorem guarantees thatwe can obtain the ground state of the problem Hamil-tonian by the QA. Importantly, by replacing the H P with the molecule Hamiltonian, the QA can be usedto estimate an energy of the ground state in quantumchemistry[6, 25–28]. In addition, the excited states searchin quantum chemistry is discussed [29, 30].D-wave systems, Inc. [31] have realized quantum an-nealing machines composed of thousands of the qubits.They use superconducting flux qubits to implementthe quantum annealing. There are many experimentaldemonstrations of the QA by the device of the D-wavesystems, Inc. [32–35]. Especially, quantum chemistry cal-culations were demonstrated with the QA to estimate theground state energy for a small size molecule [27].The potential problem to use the QA for practicalquantum chemistry calculations is an intrinsic error dur-ing the QA. Non-adiabatic transitions induce a transitionfrom the ground state to excited states. Also, decoher-ence due to the coupling with an environment causes un-wanted decay of the quantum states during the QA. Dueto these problems, it is not clear whether we can achievethe chemical accuracy in quantum chemistry calculationsby the QA. So it is essential to achieve a higher accuracyto estimate the ground state energy in the QA.In this paper, we propose a way to estimate an en-ergy of the target Hamiltonian with improved accuracyby combining quantum annealing with classical compu-tation. We show that, if the population of the groundstate is more than 1 / / / Pre-estimationQA
𝐻 < !" ! " $ − %& ! " $ If Otherwise 𝐸& ’ − δ𝑀 ’ < 𝐸 ’ < 𝐸& ’ + δ𝑀 ’ 𝐻 Δ𝐸 $ measure andNo guarantee of accuracy Δ𝐸 $ ≥ |𝐸 ’ − 𝐻 | $ 𝐸& ( − δ𝑀 ( < 𝐸 ( < 𝐸& ( + δ𝑀 ( Classic computerQuantum annealer
FIG. 1. A flow chart showing how to estimate the ground stateenergy of target Hamiltonian in our protocol. We need to pre-estimate the ground state energy by using a classical computerwith some approximation, and need to know ˜ E , ˜ E , δM ,and δM where ˜ E ( ˜ E ) is the approximated ground (firstexcited) state energy from pre-estimation and δM ( δM ) isthe bound of the error of the pre-estimation. In addtion, E ( E ) denotes the true energy of the ground (first excited)state and (cid:104) H (cid:105) ( (cid:104) ∆ E (cid:105) ) denotes the expectation (variance) ofthe Hamiltonian of the state after the QA. In our protocol,when (cid:104) H (cid:105) is smaller than ( ˜ E − ˜ E − δM + δM ), the energyvariance of the state after the QA can be an upper bound ofthe estimation error. II. QUANTUM ANNEALING
Let us review the QA for the ground state search. Wealso regard the driving Hamiltonian as the transversefield. The total Hamiltonian for the QA is described asfollows H ( t ) = tT H P + (cid:16) − tT (cid:17) H D (1)where T is the annealing time. First, we prepare theground state of the transverse field H D = − (cid:80) Ni =1 ˆ σ xi , | Ψ(0) (cid:105) = | + · · · + (cid:105) where the quantum state | + (cid:105) expressesthe eigenstate of σ x with the eigenvalue +1. Second,the driver Hamiltonian is adiabatically changed into theproblem Hamiltonian. Finally, we obtain the groundstate of the problem Hamiltonian if the dynamics is adi-abatic, and so the measurements of an observable H P provides the ground state energy.Various noise deteriorates the accuracy of the QA. Themain noise sources are environmental decoherence andnon-adiabatic transitions. There is a trade-off betweenthese two errors. We should implement the QA slowly toavoid the non-adiabatic transitions, while the slower dy-namics tend to increase the error due to the decoherence.There are many attempts to suppress non-adiabatictransitions and decoherence. The use of non-stochasticHamiltonians has been proposed to increase an energygap during the QA for a specific model, which could con-tribute the suppression of the non-adiabatic transitions.Inhomogeneous driving Hamiltonian for a p-spin modelis known to contribute to the speedup of the QA for spe-cific cases [36, 37]. Direct estimation of the energy gapbetween the ground state and the first excited state us-ing the quantum annealing was proposed, and this wasshown to be more robust against non-adiabatic transi-tions than the conventional scheme [23]. Both theoreti-cal and experimental studies have been made to suppressthe decoherence during the QA. We can use error correc-tion techniques [38], spin lock techniques [39–41], anddecoherence free subspace [41] for the suppresion of thedecoherence. A method using non-adiabatic transitionand quench for an efficient QA is also investigated [42–48]. Despite the efforts of previous research, there are nouniversal ways to suppress both environmental decoher-ence and non-adiabatic transitions during the QA, andthis fact makes it difficult to guarantee the accuracy ofthe results of the QA. III. METHOD
Here, we present our scheme to estimate a ground stateenergy with improved accuracy in a certain condition.
A. Bounds on the error of the energy
In practice, the quantum state become mixed states,because non-diagonal terms in the density matrix decayfrom the decoherence. After implementing the QA, wemeasure only Hamiltonian and energy variance. In thiscase, we can show that the non-diagonal terms in theenergy basis does not affect the expectation values. So wecan describe the quantum state after the QA either a purestate or a mixed state as long as the energy populationis the same between them. For simplicity, we would usea pure state for the description.Suppose that we obtain a state of | φ ( ann )0 (cid:105) after theQA. We rewrite this state as follows. | φ ( ann )0 (cid:105) = (cid:112) − (cid:15) | φ (cid:105) + (cid:88) m (cid:54) =0 (cid:15) m | φ m (cid:105) . (2)where | φ (cid:105) denotes the ground state, | φ m (cid:105) ( m > m -th excited state, (cid:15) m denotes the ampli-tude of the m -th excited state, and (cid:15) denotes the ampli-tude of all the states except the ground state. In otherwords, √ − (cid:15) denotes the amplitude of the groundstate. Due to the normalization, we have a condition of (cid:15) = (cid:80) m (cid:54) =0 (cid:15) m . Since we consider an expectation valueof the Hamiltonian and the energy variance, the relativephase between the energy eigenstate does not affect ourresults. So we can assume (cid:15) m to be real values withoutloss of generality throughout our paper.First, let us explain the estimation error and energyvariance. The estimation error of the energy eigenvalueof the problem Hamiltonian is given by (cid:104) φ ( ann )0 | H P | φ ( ann )0 (cid:105) − (cid:104) φ | H P | φ (cid:105) = (cid:88) m (cid:54) =0 (cid:15) m ( E m − E )(3) where H P is the problem Hamiltonian and E m is the m -th energy eigenvalue of the problem Hamiltonian. On theother hand, the energy variance ∆ E is given by∆ E = (cid:104) φ ( ann )0 | H P | φ ( ann )0 (cid:105) − (cid:104) φ ( ann )0 | H P | φ ( ann )0 (cid:105) = (cid:88) m (cid:54) =0 (cid:15) m ( E m − E ) − (cid:16) (cid:88) m (cid:54) =0 (cid:15) m ( E m − E ) (cid:17) . (4)We subtract the energy dispersion ∆ E from the errorsquared of the energy as follows.∆ E − (cid:16) (cid:104) φ ( ann )0 | H P | φ ( ann )0 (cid:105) − (cid:104) φ | H P | φ (cid:105) (cid:17) = (cid:88) m (cid:54) =0 (cid:15) m ( E m − E ) − (cid:16) (cid:88) m (cid:54) =0 (cid:15) m ( E m − E ) (cid:17) (5)We derive the next theorem. Theorem 1.
If the amplitude of all the states except theground state (cid:15) satisfies (cid:15) ≤ , then ∆ E − (cid:16) (cid:104) φ ( ann )0 | H P | φ ( ann )0 (cid:105) − (cid:104) φ | H P | φ (cid:105) (cid:17) ≥ . (6) Proof. we consider the relation between the energy vari-ance and the estiamtion error of the energy eigenvalue.We remark that the following inequality is hold from theCauchy–Schwarz inequality. (cid:16) (cid:88) m (cid:54) = n (cid:15) m ( E m − E n ) (cid:17) ≤ (cid:16) (cid:88) m (cid:54) = n (cid:15) m (cid:17)(cid:16) (cid:88) m (cid:54) = n (cid:15) m ( E m − E n ) (cid:17) . (7)The lower bound of difference between the energy vari-ance and the error of the square of energy (5) is given asfollows.∆ E − (cid:16) (cid:104) φ ( ann )0 | H P | φ ( ann )0 (cid:105) − (cid:104) φ | H P | φ (cid:105) (cid:17) = (cid:88) m (cid:54) =0 (cid:15) m ( E m − E ) − (cid:16) (cid:88) m (cid:54) =0 (cid:15) m ( E m − E ) (cid:17) ≥ (cid:88) m (cid:54) = n (cid:15) m ( E m − E n ) − (cid:15) (cid:88) m (cid:54) = n (cid:15) m ( E m − E n ) = (1 − (cid:15) ) (cid:88) m (cid:54) = n (cid:15) m ( E m − E n ) (8)where we have applied the inequality (7) to rewrite theinequality. Finally, from (cid:80) m (cid:54) = n (cid:15) m ( E m − E n ) ≥ (cid:15) ≤ . (9) B. Pre-estimation of the energy before the QA byperforming classical computation
In this subsection, we show a way to check ≥ (cid:15) , i.e.the population of the ground state to be more than 1 / ≥ (cid:15) is satisfied, and so wecan use the energy variance to obtain an upper boundof the error estimation, as we will explain later. Let ˜ E n denote the approximate value of obtained from the pre-estimation.There are many ways to calculate the ground energyand excited state energy of molcules in quantum chem-istry with a classical computer. For example, variationaltrial function gives us an upperbound of the ground stateenergy. There is a way to estimate the lower boud of theground state energy as shown [49]. Also,various ways toobtain an energy gap between the ground state energyand exited states is known[50–53]. The combination ofthese technique would provide the range of the groundstate energy and first excited state energy.The estimation error of the approximate eigenvalue ˜ E n from the true eigenvalue E n is denoted by δ ˜ E n . Thus,we remark the equality˜ E n = E n + δ ˜ E n . (10)We assume that the estimation errors are bounded asfollows. | δ ˜ E | < δM , | δ ˜ E | < δM . (11)where δM and δM denotes error bars representing theaccuracy of the pre-estimation. We can show that thesufficient condition of inequality ≥ (cid:15) is E ( ann )0 ≤
12 ( E + E ) . (12)Substituting the equality (10) into the inequality (12),we obtain( E < ) E ( ann )0 <
12 ( ˜ E + ˜ E ) −
12 ( δ ˜ E + δ ˜ E ) (13)We obtain a sufficient condition for the inequality (13)as follows.( E < ) E ( ann )0 <
12 ( ˜ E + ˜ E ) −
12 ( | δ ˜ E | + | δ ˜ E | ) (14)From | δ ˜ E | < δM and | δ ˜ E | < δM , a sufficient condi-tion for the inequality (14) is( E < ) E ( ann )0 <
12 ( ˜ E + ˜ E ) −
12 ( δM + δM ) . (15) From the approximate energy eigenvalues by the pre-estiamtion ( ˜ E and ˜ E ) and the upper bound of the esti-mation errors ( δM and δM ), the inequality (15) is thesufficient condition of (9). This means that, as long as(15) is satisfied, we can use the energy variance as the newerror bar (corresponding to the upper-bound of the esti-mation error) of the energy estimation. Especially whenthe new error bar given by the energy variance is smallerthan δM , the accuracy to estimate the ground state en-ergy is better than that from just the pre-estimation.The condition (15) is not always satisfied. If thereare significant effect of decoherence and/or non-adiabatictransitions, E ( ann )0 could be large so that the sufficientconditions would not be satisfied. Alternatively, if theestimation error ( δM + δM ) is large, again, it becomesharder to satisfy the sufficient condition. In these cases,we should try other approaches such as optimizing thequantum annealing schedule, fabricating new sampleswith lower decoherence, or more precise pre-estimationwith a longer calculation time using a classical computerto make the condition (15) satisfied. C. Measurement of the energy and variance of theHamiltonian
We describe how to measure the energy and varianceof the Hamiltonian in the QA. We assume that we canperform any single qubit measurements in the QA. TheHamiltonian is now written in the form of H = (cid:80) j ˆ P j where ˆ P j denotes the product of Pauri matrices (suchas ˆ σ z , ˆ σ z , . . . , and ˆ σ x ˆ σ x ˆ σ y ˆ σ y ). After the preparationof the ground state with the QA, we can implementsingle qubit measurements to obtain (cid:104) ˆ P (cid:105) . This meansthat, by repeating the experiments (that are composedof the ground state preparation and single qubit measure-ments), we can measure (cid:104) ˆ P j (cid:105) for every j , and we obtain (cid:104) H (cid:105) by summing up them. Similarly, we can measure (cid:104) H (cid:105) , and so we can also measure the variance of theenergy. These techniques are used in the algorithm inNISQ devices[54, 55]. IV. NUMERICAL RESULT
In this section, we perform the numerical simulationsto estimate the error of the energy using our method.We consider the hydrogen molecule. The Hamiltonianof the hydrogen molecule can be described by the Paulimatrices. To consider the decoherence, we simulate theQA with the Lindblad master equation, and discuss therelation between decoherence rate and the accuracy ofthe energy estimation. In addition, we plot the improvederror bars obtained from our methods.We introduce the Lindblad master equation. We con-sider the time dependent system Hamiltonian H ( t ) un-der a noisy environment. The Lindblad master equationwhich we use in this paper is given by dρ ( t ) dt = − i [ H ( t ) , ρ ( t )] + (cid:88) n γ [ σ ( k ) n ρ ( t ) σ ( k ) n − ρ ( t )] (16)where σ ( k ) j ( k = x, y, z ) denotes the Pauli matrix act-ing at site j , γ denotes a decoherence rate and ρ ( t ) isa density matrix of the quantum state at time t . Wesolve the Lindblad master equation numerically with theQuTiP [56, 57]. Throughout of this paper, we choose thedecoherence type σ zj as the Lindblad operator. This typeof noise has been studied in a previous work to considerthe effect of noise on the superconducting qubits [58].The Hamiltonian of hydrogen is given by H = h I + h ˆ σ z + h ˆ σ z + h ˆ σ z + h ˆ σ z + h ˆ σ z ˆ σ z + h ˆ σ z ˆ σ z + h ˆ σ z ˆ σ z + h ˆ σ z ˆ σ z + h ˆ σ z ˆ σ z + h ˆ σ z ˆ σ z + h ˆ σ y ˆ σ y ˆ σ x ˆ σ x + h ˆ σ x ˆ σ y ˆ σ y ˆ σ x + h ˆ σ y ˆ σ x ˆ σ x ˆ σ y + h ˆ σ x ˆ σ x ˆ σ y ˆ σ y (17)where we used STO-3G basis and Jordan-Wigner trans-formation. The coefficients of the Hamiltonian (17) h , . . . , h depend on the interatomic distance. We con-sider the interatomic distance is 0 . h . h . h . h − . h − . h . h . h − . h − . h . h . h . h . h . h . The most promissing device for the quantum anneal-ing is a superconducting qubit. We mainly consider theimplementation of the superconducting qubits. The typ-ical energy scale of the superconducting qubit is aroungGHz [60]. So we adopt thie energy scale to describe theHamiltonian.The relation between the measured energy and anneal-ing time is shown in FIG2. We can choose the annealingtime to minimize the energy of the problem Hamiltonianafter the QA. Throughout of our paper, we choose suchan optimized annealing time for the plots. Importantly,as the decoherence rate increases, the minimum energy after the optimization increases. This is because the de-coherence could induce a transition from a ground statto excited states.
FIG. 2. Plots to show the relation between the anneal-ing time and the ground state energy by quantum annealing.We consider the hydrogen molecule with interatomic distance0 . T for the each decoherence rate γ . By applying the method discussed in the subsectionIII B, we numerically determine the conditions that sat-isfies ≥ (cid:15) with the pre-estimation when we estimatethe ground state energy of the hydrogen molecules. Inother words, we show the region where we can use the en-ergy variance as the upper bound of the estimation error.Such a region is plotted in FIG3 (a). As the decoherencerate increases, pre-estimation should be done more pre-cisely to satisfy ≥ (cid:15) . On the other hand, even whenwe can use the energy variance obtained from the QA asthe new error bound (due to the satisfaction of the con-dition of ≥ (cid:15) ), the pre-estimation could be still betterif the variance is too large. We plot the condition whenthe energy variance can be smaller than δM while thecondition of ≥ (cid:15) is satisfied, as shown FIG3 (b).In the FIG4, we plot the estimation of the ground stateenergy and the error bar obtained by the energy variancewhen we use our scheme in the QA. As the decoherencerate becomes smaller, the error bar (corresponding to theenergy variance) becomes smaller. Also, we confirm thatthe variance is actually larger than the estimation errorwhen the decoherence rate is larger. V. CONCLUSION
In this paper, we propose a way to estimate an en-ergy of the target Hamiltonian with improved accuracyby combining quantum annealing with classical compu-tation. We show that, if the population of the groundstate is more than 1 / / δ𝑀 ! + δ𝑀 " (i) success(ii) fail 𝐸 " − 𝐸 ! (ii)(i) [GHz] [ G H z ] (a) Threshold decoherence rate for our scheme to beapplied δ𝑀 ! + δ𝑀 " (i) success(ii) fail (ii) (i) [GHz] [ G H z ] (b) The threhold decoherence rate for our scheme tobe more precise than pre-estimatationFIG. 3. (a)The threshold decoherence rate for our scheme tobe applied. The holizontal axis denotes the estimation errorwhen we perform pre-estimation by a classical computer. Aslong as the decoherence rate of the QA is below the thresh-old, we can apply our scheme, and so the energy variancecan be an upper-bound of the estimation error of the QA.Here, we define that our scheme succeeds (fails) when wecan (cannot) apply our scheme based on this prescription. If( δM + δM ) / E − E ) / pre-estimation of the energy of the ground state and firstexcited state. More precisely, we obtain the approximateenergy of the problem Hamiltonian with possible errorbars for the ground state and first excited state by per-forming classical computation with some approximation(such as mean field technique). From the values obtainedby the pre-estimation, we can calculate a threshold, and [ G H z ] [GHz] FIG. 4. The energy expectation value with the error barin our scheme. The dashed line is the exact ground stateenergy. The solid line is the energy expectation value ( E ( ann )0 )obtained from the QA. if the energy of the state after the QA is smaller thanthe threshold, the population of the ground state is morethan 1 / ACKNOWLEDGMENTS
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