A proposal of noise suppression for quantum annealing
AA proposal of noise suppression for quantum annealing
Takayuki Suzuki
Department of Physics, Waseda University, Tokyo 169-8555, Japan
Hiromichi Nakazato
Department of Physics, Waseda University, Tokyo 169-8555, Japan andInstitute for Advanced Theoretical and Experimental Physics, Waseda University, Tokyo 169-8555, Japan
A method to suppress noise, which is one of the major obstacles to obtain an optimal solutionin quantum annealers, is proposed. We generalize the conventionally used Hamiltonian, i.e., thetransverse field Hamiltonian, by introducing an ancillary system, which leads to cancellation of theeffect of noise on the system under consideration for some typical cases. We also confirm numericallythat the method is effective for a kind of noise usually encountered in the case of flux qubit.
I. INTRODUCTION
Recently, quantum information technology has beenactively researched and developed. Quantum annealingis one of the algorithms to solve the combinatorial opti-mization problem [1, 2], although whether this algorithmis faster than the classical algorithms for combinatorialoptimization problem still remains open. In quantumannealing, the initial state is set in the ground state ofthe Hamiltonian H which is easily prepared. The finalHamiltonian H p describes the combinatorial optimiza-tion problem, which we want to solve, and the groundstate of this Hamiltonian can be reached if the Hamilto-nian is changed adiabatically from H to H p thanks tothe adiabatic theorem [3, 4].A device for quantum annealing composed of super-conductivity qubits is produced by D-Wave Systems Inc.[5, 6]. The main weakness of the device is due to de-coherence. It is known that the final state of quantumannealing in the device is not the ground state becausethe coherent time is shorter than the annealing time al-though some studies proposed to take advantage of thisphenomenon for quantum annealing [7–9].Even if we consider an ideal, i.e. noiseless, quantumannealer, the energy gap becomes smaller exponentiallywith the system size. This means that the calculationtime required becomes exponentially large, which is thesame as brute-force search. A previous research, however,shows that a Hamiltonian which has the term σ x ⊗ σ x resolves the difficulty in some models, i.e. the energygap of the Hamiltonian becomes small not exponentially,but polynomially with the system size [10]. The D-Wavedevice imitates transverse field Ising Hamiltonian and im-plementation of the term σ x ⊗ σ x is highly demanded byresearchers in the field of quantum annealing [11].In this paper, we point out that a term σ x ⊗ σ x + σ y ⊗ σ y is useful for noise suppression. Several methods to sup-press the noise have already been proposed, focusing ondynamical decoupling [12], error correction [13], energygap [14, 15], or spin-boson architecture [16]. Our methodis different from the previous ones. It is similar to themethod proposed in [17, 18], in which a different driverHamiltonian is introduced so that the dynamics under discussion is realized in a subspace where constraintsof the optimization problem are met. This method isextended to Quantum Approximate Optimization Algo-rithm as Quantum Alternating Operator Ansatz [19]. Weuse a similar idea for noise suppression, i.e. we confinethe dynamics to a subspace which is almost noise free.Remarkably, this method is successfully applicable to su-perconducting flux qubits, which the device of D-Waveis composed of, although the number of qubits has to bedoubled. II. SETUP
In this section, we consider a closed system. In theconventional quantum annealing, the Hamiltonian is de-signed as a time-dependent transverse field Ising model: H C ( t ) = − A ( t ) N (cid:88) i =1 σ xi + B ( t ) H zC,p , (1) H zC,p = N (cid:88) i =1 h i σ zi + N (cid:88) i =1 N (cid:88) j = i +1 J ij σ zi σ zj , (2)where A ( t ) and B ( t ) are positive functions of time t . These coefficients should satisfy A (0) (cid:29) B (0) and A ( T ) (cid:28) B ( T ), where T is an annealing time. We call H zC,p “problem Hamiltonian”, because this term usu-ally corresponds to the objective function of optimiza-tion problem. The initial state is set as ⊗ Ni =1 | + (cid:105) i wherethe states are defined as σ xi |±(cid:105) i = ±|±(cid:105) i . Because thisinitial state is the ground state of H C (0), the final statebecomes the ground state of the problem Hamiltonian,which solves the optimization problem, if the Hamilto-nian is changed adiabatically.We propose another Hamiltonian of quantum anneal-ing as follows: H ( t ) = A ( t ) N (cid:88) i =1 ( cσ x i − σ x i − σ y i − σ y i ) + B ( t ) H zp , (3) H zp = N (cid:88) i =1 h i σ z i + N (cid:88) i =1 N (cid:88) j = i +1 J ij σ z i σ z j , (4) a r X i v : . [ qu a n t - ph ] J un where c ∈ R is a constant parameter. We call odd-numbered qubits “ancilla qubits” and even-numberedqubits “physical qubits”. Note that H zp is equivalentto H zC,p and we also call H zp “problem Hamiltonian”.This Hamiltonian (3) does not change under the follow-ing transformation: (cid:40) σ x i − → − σ x i − , σ y i − → − σ y i − , σ z i − → σ z i − ,σ x i → − σ x i , σ y i → − σ y i , σ z i → σ z i (5)for each i in { , , . . . , N } . This is due to the fact that H ( t ) has N constants of motion σ z i − σ z i (1 ≤ i ≤ N )[20]. Moreover, there exists a unitary operator C i − , i transforming σ z i − σ z i into σ z i − because the spectrumof σ z i − σ z i is the same as that of σ z i − . The unitaryoperator is expressed as follows: C i − , i = (6)in the standard ordered basis B = {| (cid:105) i − , i , | (cid:105) i − , i , | (cid:105) i − , i , | (cid:105) i − , i } , (7)where the states are defined by σ zn | (cid:105) n = | (cid:105) n and σ zn | (cid:105) n = −| (cid:105) n . Notice that the operator C i − , i isnothing but a C-NOT gate C i − , i = σ x i − ⊗ ( I i − σ z i ) / I i − ⊗ ( I i + σ z i ) /
2. Let us introduce a unitaryoperator W := N (cid:79) i =1 C i − , i , (8)abbreviations |•(cid:105) A := |•(cid:105) , , ··· , N − , (9) |•(cid:105) P := |•(cid:105) , , ··· , N , (10)and a set A := { N (cid:122) (cid:125)(cid:124) (cid:123) · · · , · · · , · · · , · · · } , (11)where the number of elements is 2 N . Transforming H ( t )into ˜ H ( t ) = W † H ( t ) W , we get˜ H ( t ) = (cid:88) λ ∈A | λ (cid:105) A (cid:104) λ | ⊗ ˜ H λ ( t ) , (12)˜ H λ ( t ) = A ( t ) N (cid:88) i =1 ( c + f ( λ i − )) σ x i + B ( t ) N (cid:88) i =1 h i σ z i + N (cid:88) i =1 N (cid:88) j = i +1 J ij σ z i σ z j , (13) where we have introduced a function f ( x ) := 1 − x and λ i − is the i -th element of λ , i.e., λ =( λ , λ , · · · , λ N − ). We can see that the Hamiltonian˜ H λ ( t ) is the same as H C ( t ) in (1) up to the coefficient oftransverse field.Let us consider the dynamics of conventional quantumannealing using ˜ H λ ( t ) in (13). We define the time evolu-tion operator corresponding to ˜ H λ ( t ) as ˜ U λ ( t ). We alsoexpress the basis for subspace of physical qubits as fol-lows: (cid:40) N (cid:79) i =1 | s n,i (cid:105) i (cid:41) N n =1 = {| · · · (cid:105) P , | · · · (cid:105) P , · · · , | · · · (cid:105) P } . (14)If c + f ( λ i − ) in (13) is negative for all i , the ground statewhich is chosen as initial state is (cid:78) Ni =1 | + (cid:105) i . The stateat t is expressed as ˜ U (cid:126) ( t ) (cid:78) Ni =1 | + (cid:105) i , where λ = (cid:126) λ = (1 , , · · · , a n ( t ) as the coefficient ofthe state when the state is expanded with the basis in(11): ˜ U (cid:126) ( t ) N (cid:79) i =1 | + (cid:105) i = N (cid:88) n =1 (cid:32) a n ( t ) N (cid:79) i =1 | s n,i (cid:105) i (cid:33) . (15)Next, we calculate the time evolution in the originalspace. The Schr¨odinger equation i ddt U ( t ) = H ( t ) U ( t ) (16)is transformed to i ddt W † U ( t ) W = ˜ H ( t ) W † U ( t ) W = (cid:88) λ ∈A | λ (cid:105) A (cid:104) λ | ⊗ ˜ H λ ( t ) W † U ( t ) W . (17)The Hamiltonian in (17) indicates that the subdynamicsof the physical qubits connected to each ancilla qubitdoes not interfere each other. In other words, the timeevolution operator can be expressed as follows: W † U ( t ) W = (cid:88) λ ∈A | λ (cid:105) A (cid:104) λ | ⊗ ˜ U λ ( t ) . (18)We can easily confirm this equality by substituting (18)for (17). Therefore, the time evolution operator corre-sponding to H ( t ) in (3) is represented as U ( t ) = W (cid:32) (cid:88) λ ∈A | λ (cid:105) A (cid:104) λ | ⊗ ˜ U λ ( t ) (cid:33) W † . (19)If the initial state is | ψ I (cid:105) = N (cid:79) i =1 √ | (cid:105) i − , i + | (cid:105) i − , i ) , (20)then the state at t is calculated as follows: | ψ ( t ) (cid:105) = U ( t ) N (cid:79) i =1 √ | (cid:105) i − , i + | (cid:105) i − , i )= W (cid:32) (cid:88) λ ∈A | λ (cid:105) A (cid:104) λ | ⊗ ˜ U λ ( t ) (cid:33) N (cid:79) i =1 | (cid:105) i − | + (cid:105) i = W (cid:32) | (cid:126) (cid:105) A ⊗ ˜ U (cid:126) ( t ) N (cid:79) i =1 | + (cid:105) i (cid:33) = W | (cid:126) (cid:105) A ⊗ N (cid:88) n =1 (cid:32) a n ( t ) N (cid:79) i =1 | s n,i (cid:105) i (cid:33) = N (cid:88) n =1 (cid:32) a n ( t ) N (cid:79) i =1 | ¯ s n,i s n,i (cid:105) i − , i (cid:33) , (21)where we define | ¯0 (cid:105) = | (cid:105) and | ¯1 (cid:105) = | (cid:105) . If only thephysical qubits are measured, the probability that we geta state ⊗ Ni =1 | s m,i (cid:105) i is | a m ( t ) | . This is the same as theusual quantum annealing, as can be seen from (15). Weshould set c < | ψ I (cid:105) is the ground state of theinitial Hamiltonian H (0). III. NOISE SUPPRESSION
In this section, we discuss how and when noise canbe suppressed using the proposed Hamiltonian (3) whenthe system is open and exposed to the noise. In theconventional quantum annealing, such an open dynamicsis described by the following total Hamiltonian [21]: H tot,C ( t ) = H C ( t ) + ∞ (cid:88) k =1 ω k b † k b k + N (cid:88) i =1 ∞ (cid:88) k =1 (cid:0) g zi,k σ zi + g xi,k σ xi (cid:1) ⊗ (cid:16) b † k + b k (cid:17) , (22)where b k and b † k are bosonic annihilation and creationoperators satisfying the standard commutation relations[ b k , b † k (cid:48) ] = δ kk (cid:48) etc., and consist of a bosonic reservoir.The interaction between the i -th spin of the system andthe mode k of the reservoir is assumed to take placethrough the longitudinal and transversal couplings withthe form factors g zi,k and g xi,k , respectively. It has beenargued [21] that the effect of noise introduced by suchcouplings brings about considerable degradations in thequantum annealing process. In this paper, the conven-tional system Hamiltonian H C ( t ) is replaced with ourproposal H ( t ) in (3) and we discuss the open dynamics described by the total Hamiltonian H tot ( t ) = H ( t ) + ∞ (cid:88) k =1 ω k b † k b k + N (cid:88) i =1 ∞ (cid:88) k =1 (cid:0) g zi,k σ zi + g xi,k σ xi (cid:1) ⊗ (cid:16) b † k + b k (cid:17) . (23)Observe that our system, composed of physical (even-numbered) and ancillary (odd-numbered) spins, is inter-acting with the common reservoir. A. Effect of longitudinal ( σ z ) coupling In this subsection, we consider the case where thetransversal couplings are negligible | g zi,k | (cid:29) | g xi,k | ∼ ∀ i, k because it is recognized that this inequality holds insuperconducting flux qubits [5, 6, 9, 22]. In this case, thetotal Hamiltonian can be approximated as follows: H tot ( t ) = H ( t ) + ∞ (cid:88) k =1 ω k b † k b k + N (cid:88) i =1 ∞ (cid:88) k =1 g zi,k σ zi ⊗ (cid:16) b † k + b k (cid:17) . (24)This total Hamiltonian does not change under the trans-formation (5) and can be transformed in the same way.Transforming H tot ( t ) into ˜ H tot ( t ) = W † H tot ( t ) W , we get˜ H tot ( t ) = (cid:88) λ ∈A | λ (cid:105) A (cid:104) λ | ⊗ ˜ H tot, λ ( t ) , (25)˜ H tot, λ ( t ) = ˜ H λ ( t ) + ∞ (cid:88) k =1 ω k b † k b k + N (cid:88) i =1 ∞ (cid:88) k =1 (cid:0) f ( λ i − ) g z i − ,k + g z i,k (cid:1) σ z i ⊗ (cid:16) b † k + b k (cid:17) . (26)This indicates that the noise does not affect the systemdynamics, i.e., the system decouples from the noisy en-vironment, if λ i − = 1 and g z i − ,k = g z i,k hold for ar-bitrary i and k . In this case, the time evolution of thespin system is the same as that of a closed system. Evenif such exact equalities do not hold, when the differences | g z i − ,k − g z i,k | are much smaller than A ( t ) , B ( t ), and ω k for arbitrary i and k , the spin system and boson bath canbe regarded as approximately decoupled. It is remarkablethat the effect of noise caused through the longitudinalcouplings can be completely (or at least almost) canceledin the present protocol proposed here. B. Effect of both the longitudinal and transversalcouplings: Numerical simulations
In this subsection, we numerically examine the dynam-ics in (23) when the transversal couplings are not negli-gible. In the numerical calculation, we simulate the dy-namics in terms of (nearly) adiabatic quantum masterequation [21, 23]. According to the microscopic deriva-tion in [21], the master equation derived reads as follows(we assume an Ohmic bath and ignore the Lamb shift forsimplicity):˙ ρ ( t ) = − i [ H ( t ) , ρ ( t )]+ (cid:88) α = x,z (cid:88) ω N (cid:88) i,j =1 g αi, | ω | g αj, | ω | γ ( ω ) × (cid:20) L α,j,ω ( t ) ρ ( t ) L † α,i,ω ( t ) − (cid:26) L † α,i,ω ( t ) L α,j,ω ( t ) , ρ ( t ) (cid:27)(cid:21) , (27) L α,i,ω ( t ) = (cid:88) ω ba ( t )= ω | E a ( t ) (cid:105)(cid:104) E a ( t ) | σ αi | E b ( t ) (cid:105)(cid:104) E b ( t ) | , (28) γ ( ω ) = η | ω | e − | ω | ωc − e − β | ω | (Θ( ω ) + Θ( − ω ) e − β | ω | ) , (29)where β is the inverse temperature of the bosonic bath, ω c is a high-frequency cutoff and η is a positive constant.We defined the instantaneous eigenstates and eignevaluesof H ( t ) as | E a ( t ) (cid:105) and E a ( t ), the energy gaps as ω ba ( t ) = E b ( t ) − E a ( t ), and the Heaviside step function as Θ( ω ).In (29), only one of the step functions is supposed to benon-zero at ω = 0. We note that this master equationis derived under the assumption that the Hamiltonian ofthe system is adiabatic [21] or nearly adiabatic [23].We investigate the effect of g xi,k in the case of N =2, i.e., the number of qubits is 4. We set A ( t ) = at/T, B ( t ) = a − A ( t ), and a = 10GHz (in units suchthat (cid:126) = 1). Note that this energy scale is the same asthe energy scale of D-Wave 2000Q quantum annealer [9].For concreteness, we consider the problem Hamiltonian(4) with the following parameters: h = 1 , h = 14 , J = 18 . (30)The ground state at t = T is | E ( T ) (cid:105) = | (cid:105) =: | g (cid:105) . Inthe following numerical calculation, we set T = 1000nsand consider a uniform noise, i.e. g xi,k = g x and g zi,k = g z .We also set 1 /β = 1 . (cid:39) η = 0 . − ,and ω c = 8 π GHz in accordance with [21]. Time depen-dence of the energy spectrum is shown in FIG. 1.First, consider the case g x = 0. Figure 2 shows theprobability that the ground state is obtained when mea-sured at t = T . As we explained before, noises are can-celled completely in the proposed Hamiltonian (23). Wenote that the quantum adiabatic master equation (27)is valid in this case because the dynamics is adiabatic.This is why the probability to measure the ground stateis almost 1 in the present scheme.Next, we consider the case g x (cid:54) = 0. We consider twopatterns: 1) the strength of the noise g = (cid:112) ( g x ) + ( g z ) is fixed and the relative weight measured by tan θ = E n e r g y ( G H z ) E n e r g y ( G H z ) FIG. 1. Time dependence of energy eigenvalues of H ( t ) (left)and ˜ H (cid:126) ( t ) (right) with c = − /
2. The problem Hamilto-nian is given in (4) with parameters (30) and we set A ( t ) = at/T, B ( t ) = a − A ( t ), and a = 10GHz. c g z Probability of |g> c g z Probability of |g> c g z Difference of Probability
FIG. 2. Probabilities to measure the ground state at t = T , according to the present scheme H tot ( t ) (left) and to thestandard one H tot,C ( t ) (right) with g xi,k = 0 and g zi,k = g z .Their difference is shown at (bottom). We set 1 /β = 1 . (cid:39) η = 0 . − , and ω c = 8 π GHz. g x /g z is varied and 2) only g x is varied with g z keptfixed. The results of g = 0 . g z = 0 . g x (cid:46) . g z . We note that proposed method wouldlose its superiority over the conventional one only when θ (cid:39) π/ c Probability of |g> c Probability of |g> c Difference of Probability
FIG. 3. Probabilities to measure the ground state at t = T ,according to the present scheme H tot ( t ) (left) and to the stan-dard one H tot,C ( t ) (right) with g xi,k = g sin θ , g zi,k = g cos θ ,and g = 0 . /β = 1 . (cid:39) η = 0 . − , and ω c = 8 π GHz. c g x / g z Probability of |g> c g x / g z Probability of |g> c g x / g z Difference of Probability
FIG. 4. Probabilities to measure the ground state at t = T , according to the present scheme H tot ( t ) (left) and to thestandard one H tot,C ( t ) (right) with g xi,k = g x and g zi,k = g z =0 . /β =1 . (cid:39) η = 0 . − , and ω c = 8 π GHz.
IV. SUMMARY
We have reported a method to suppress noise in quan-tum annealing. In this proposal, we introduce a driverHamiltonian different from conventional one to make asubspace where there is almost no noise. In the exam-ple, we showed the superiority of the proposed methodin the problem where the number of variables is 2. Inthis case, the probability to measure the ground state at t = T is not very low even in the conventional methodbecause the energy gaps are large enough. If the num-ber of qubits becomes larger, however, the superiority ofthe proposed method will become more dramatic becausethe energy gaps become smaller in general, which causesno problem in the current scheme that is immune to de-coherence caused by longitudinal noise. This idea is alsovalid for adiabatic quantum computer if the type of noiseis the same.We emphasize that the proposed method is completelynew and is different from the idea of decoherence-freesubspace [24–27]. There are two main differences. First,in the idea of decoherence-free subspace, the state evolvesaccording to the Schr¨odinger equation with an effectiveHamiltonian which is different from the original systemHamiltonian. In contrast, the state evolves under thesystem Hamiltonian itself in this proposal. Second, inthe idea of decoherence-free subspace, the state shouldbe in a space that is spanned by the eigenstates of Lind-blad operators. If Lindblad operators depend on time,it is usually difficult to hold the condition [28]. In ourproposal, however, such a condition is not required be-cause the coupling cancels if it is almost longitudinal.This method could be utilized to suppress noise in futurequantum devices. ACKNOWLEDGMENTS
H. N. is partly supported by Waseda University Grantfor Special Research Projects 2020-C272. T. S. is partlysupported by Top Global University Project, WasedaUniversity.
Appendix A: Eigenvalues/ Eigenstates
We prove that the eigenvalues of ˜ H λ ( t ) are included inthose of H ( t ). The eigenequation of ˜ H λ ( t ) is as follows:˜ H λ ( t ) | ˜ ε n, λ ( t ) (cid:105) P =˜ ε n, λ ( t ) | ˜ ε n, λ ( t ) (cid:105) P . (A1)This equation can be transformed as follows: (cid:88) λ ∈A (cid:16) | λ (cid:105) A (cid:104) λ | ⊗ ˜ H λ ( t ) (cid:17) | λ (cid:48) (cid:105) A ⊗ | ˜ ε n, λ (cid:48) ( t ) (cid:105) P =˜ ε n, λ (cid:48) ( t ) | λ (cid:48) (cid:105) A ⊗ | ˜ ε n, λ (cid:48) ( t ) (cid:105) P ,H ( t ) W | λ (cid:48) (cid:105) A ⊗ | ˜ ε n, λ (cid:48) ( t ) (cid:105) P =˜ ε n, λ (cid:48) ( t ) W | λ (cid:48) (cid:105) A ⊗ | ˜ ε n, λ (cid:48) ( t ) (cid:105) P . (A2)This indicates that the eigenstates of H ( t ) are W | ˜ ε n, λ ( t ) (cid:105) P ⊗ | λ (cid:105) A and eigenvalues are ˜ ε n, λ ( t ), becausethe number of ˜ ε n, λ (cid:48) ( t ) is the same as that of eigenvaluesof H ( t ). Appendix B: Symmetry Preserving in LindbladEquation
We consider the case where the transversal couplingsare negligible | g zi,k | (cid:29) | g xi,k | ∼ ∀ i, k . We transform (27)and (28) using W :˙˜ ρ ( t ) = − i [ ˜ H ( t ) , ˜ ρ ( t )] + (cid:88) ω N (cid:88) i,j =1 g zi, | ω | g zj, | ω | γ ( ω ) × (cid:20) ˜ L j,ω ( t )˜ ρ ( t ) ˜ L † i,ω ( t ) − (cid:26) ˜ L † i,ω ( t ) ˜ L j,ω ( t ) , ˜ ρ ( t ) (cid:27)(cid:21) , (B1)˜ L i,ω ( t ) = (cid:88) ω ba ( t )= ω | ˜ E a ( t ) (cid:105)(cid:104) ˜ E a ( t ) | W † σ zi W | ˜ E b ( t ) (cid:105)(cid:104) ˜ E b ( t ) | , (B2)where we defined ˜ ρ ( t ) = W † ρ ( t ) W and the eigenvaluesand eigenstates of ˜ H ( t ) as | ˜ E a ( t ) (cid:105) and ˜ E a ( t ). If the ini-tial state is | ˜ ψ (0) (cid:105) = (cid:78) Ni =1 | (cid:105) i − | + (cid:105) i , the dynamics is confined to a subspace because the eigenstates of ˜ H ( t )are decomposed: | ˜ E a ( t ) (cid:105) = | λ (cid:105) A ⊗ | ˜ ε n, λ ( t ) (cid:105) P , (B3)where a corresponds to ( λ , n ). This allows us to trans-form (B1) into˙˜ ρ (cid:126) ( t ) = − i [ ˜ H (cid:126) ( t ) , ˜ ρ (cid:126) ( t )]+ (cid:88) ω N (cid:88) i,j =1 ( g z i, | ω | − g z i − , | ω | )( g z j, | ω | − g z j − , | ω | ) × γ ( ω ) (cid:20) ˜ L j,ω,(cid:126) ( t )˜ ρ (cid:126) ( t ) ˜ L † i,ω,(cid:126) ( t ) − (cid:26) ˜ L † i,ω,(cid:126) ( t ) ˜ L j,ω,(cid:126) ( t ) , ˜ ρ (cid:126) ( t ) (cid:27)(cid:21) , (B4)˜ L i,ω,(cid:126) ( t ) = (cid:88) ω ba ( t )= ω | ˜ ε a,(cid:126) ( t ) (cid:105)(cid:104) ˜ ε a,(cid:126) ( t ) | σ zi | ˜ ε b,(cid:126) ( t ) (cid:105)(cid:104) ˜ ε b,(cid:126) ( t ) | , (B5)where λ = (cid:126) λ = (1 , , · · · ,
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