Non-Hermitian Quantum Annealing in the Antiferromagnetic Ising Chain
Alexander I. Nesterov, Gennady P. Berman, Juan C. Beas Zepeda, Alan R. Bishop
aa r X i v : . [ qu a n t - ph ] F e b Non-Hermitian Quantum Annealing in theAntiferromagnetic Ising Chain
Alexander I. Nesterov
Departamento de F´ısica, CUCEI, Universidad de Guadalajara, Av. Revoluci´on 1500,Guadalajara, CP 44420, Jalisco, M´exico
Gennady P. Berman
Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87544, USAE-mail: [email protected]
Juan C. Beas Zepeda
Departamento de F´ısica, CUCEI, Universidad de Guadalajara, Av. Revoluci´on 1500,Guadalajara, CP 44420, Jalisco, M´exicoE-mail: [email protected]
Alan R. Bishop
Los Alamos National Laboratory, STE, MS A127, Los Alamos, NM, 87544, USAE-mail: [email protected]
Abstract.
A non-Hermitian quantum optimization algorithm is created and used to find theground state of an antiferromagnetic Ising chain. We demonstrate analytically andnumerically (for up to N = 1024 spins) that our approach leads to a significantreduction of the annealing time that is proportional to ln N , which is much lessthan the time (proportional to N ) required for the quantum annealing based on thecorresponding Hermitian algorithm. We propose to use this approach to achieve similarspeed-up for NP-complete problems by using classical computers in combination withquantum algorithms. LA-UR: 13-21324
Submitted to:
New J. Phys.
PACS numbers: 03.67.Ac, 64.70.Tg, 03.65.Yz, 75.10.Jm
It is generally recognized that quantum annealing (QA) algorithms can find applicationsfor solving many practically important problems including optimization of complexnetworks, finding the global minimum of multi-valued functions, and cost minimization.Instead of classical annealing algorithm in which the temperature effects are utilized,the QA can operate, in principle, at zero temperature. In this case, one first formulatesthe optimization problem in terms of finding the ground state of the effective quantumHermitian Hamiltonian, H , which describes a quantum time-evolution of a quantumregister with N qubits. During the slow and directed time-evolution, the quantumsystem tunnels to the global ground state of this effective Hamiltonian. One of themain challenges is to accelerate the speed of QA algorithms, so that the annealingtime grows not exponentially, but polynomially with the size of the quantum register[1, 2, 3, 4, 5, 6, 7, 8].Many approaches dealing with finding the ground state of the Hamiltonian, H ,using QA algorithms, can be found in [1, 2, 3, 4, 5] (see also references therein).To illustrate the idea of QA algorithms, consider the time-dependent Hamiltonian, H ( t ) = H + Γ( t ) H , where H is the Hamiltonian to be optimized and H is anauxiliary (“initial”) Hamiltonian. In order to initiate a non-trivial quantum dynamicsof a register, the condition, [ H , H ] = 0, is imposed. The external time-dependentfield, Γ( t ), decreases from a large enough value to zero during the quantum annealing.The ground state of H is the initial state, which is assumed to be known. Then,the adiabatic theorem guarantees that if Γ( t ) decreases sufficiently slow, the systemapproaches the ground state of H , at the end of the quantum annealing.One of the requirements of the adiabatic theorem is the existence of a finite gapbetween instantaneous ground state and first excited state of the total Hamiltonian, H ( t ). However, usually in quantum optimization of N -qubit models the minimal energygap becomes exponentially small, producing an exponential increase of the annealingtime [2, 4, 9, 10, 11]. Then, the main problem is on how to speed up the performanceof the QA algorithms and, at the same time, find the ground state of H .In [12], we proposed an adiabatic quantum optimization algorithm based on non-Hermitian quantum mechanics. Our approach demonstrated that when using the non-Hermitian term, H , an effective level repulsion takes place for the total Hamiltonian,greatly increasing the minimum energy gap.In [13, 14], we applied this non-Hermitian quantum annealing (NQA) algorithmto Grover’s problem of finding a marked item in an unsorted database, and to studythe transition to the ground state in a 1-dimensional ferromagnetic Ising chain. Wedemonstrated that, for a suitable choice of relaxation and other parameters, the searchtime is proportional to the logarithm of the number of qubits (spins).In this paper, we apply our previously developed NQA to study the transitionaldynamics to the ground state of H in a 1-dimensional antiferromagnetic Ising spin chainin a time-dependent transverse magnetic field. We also assume that the dissipationvanishes at the end of evolution. When the annealing is completed, the system isgoverned by the Hermitian Hamiltonian, H .The main reason for choosing the antiferromagnetic Ising spin chain is that theground state in this model is much more complicated than in the correspondingferromagnetic chain. Namely, it includes rapid spatial oscillations (correlations) of spins.This allows us to demonstrate that our (NQA) algorithm works well for this case, andsignificantly decreases the annealing time. Another reason is to consider the behavior ofquantum correlation functions (which are more complicated than in the ferromagneticchain) and formation of defects generated in the process of annealing.We show that the NQA significantly increases the probability for the system toremain in the ground state. In particular, a comparison with the results of the HermitianQA reveals that the NQA reaches the ground state of H with much larger probability,when we use the same annealing scheme for both. We show that the NQA has acomplexity of order ln N , where N is the number of spins. This is much better than thequantum Hermitian adiabatic algorithm which has the complexity for this problem oforder N .This paper is organized as follows. In Sec. II, we introduce a dissipative one-dimensional antiferromagnetic Ising system in a transverse magnetic field governed bya non-Hermitian Hamiltonian. In Sec. III, we study the quench dynamics of our systemusing both analytic and numerical methods. In Sec. IV, we study the correlationproperties of our system and defects formation at the end of evolution. We concludein Sec. V with a discussion of our results. In the Appendix we present some technicaldetails.
2. Description of the model
We consider the 1-dimensional antiferromagnetic Ising chain in a transverse magneticfield with dissipation governed by the following non-Hermitian Hamiltonian: H = J N X n =1 ( gσ xn + σ zn σ zn +1 − i δσ + n σ − n ) , (1)with the periodic boundary condition, σ N +1 = σ . The external magnetic field isassociated with the parameter, g , the spontaneous decay is described by the parameter, δ , and σ ± n = ( σ zn ± iσ yn ) / σ xn = 1 − c † n c n , (2) σ yn = i ( c † n − c n ) Y m 2. Thus, for S x = 0and N F = N/ periodic ( antiperiodic ) boundary conditions if N/ odd ( even ). Note that since the parity of the fermions is conserved, the imposed boundaryconditions are valid for all values of the parameters, g and δ .Next, by applying the Fourier transformations, c n = e − iπ/ √ N X k c k e i πkn/N , (7)we find that the Hamiltonian (5) can be recast in Fourier space as follows: H = − X k J (cid:16) g − cos ϕ k ) c † k c k − g + sin ϕ k ( c † k c †− k + c − k c k ) (cid:17) , (8)where ˜ g = g + iδ and ϕ k = 2 πk/N . For periodic boundary conditions, c N +1 = c , thewave number, k , takes the following discrete values: k = − N , . . . , , , . . . , N − , (9)and for antiperiodic boundary conditions, c N +1 = − c , we obtain k = ± , ± , . . . , ± N − , (10)In what follows, we impose the antiperiodic boundary conditions for the fermionicoperators.The Hamiltonian, H , can be diagonalized using the generalized Bogoliubovtransformation [18]. Its spectrum is given by ε ± ( k ) = ε ± ε k , in which ε = J cos ϕ k − iJ δ , and ε k = J p ˜ g − g cos ϕ k + 1. There are two (right) eigenstates foreach k , | u + ( k ) i = cos θ k sin θ k ! , (11) | u − ( k ) i = − sin θ k cos θ k ! , (12)in which cos θ k = cos ϕ k − ˜ g p ˜ g − g cos ϕ k + 1 , (13)sin θ k = − sin ϕ k p ˜ g − g cos ϕ k + 1 , (14)with θ k being a complex angle.Since for each k , the ground state lies into the two-dimensional Hilbert spacespanned by | i k | i − k and | i k | i − k , it is sufficient to project H k onto this subspace.For a given value of k , both of these states can be represented as a point on the complextwo-dimensional sphere, S c . In this subspace, the Hamiltonian, H k , takes the form H k = ε − J ˜ g − cos ϕ k sin ϕ k sin ϕ k − ˜ g + cos ϕ k ! . (15)Its ground state can be written as a product of qubit-like states, | ψ g i = N k | u − ( k ) i , sothat: | ψ g i = O k (cid:16) cos θ k | i k | i − k − sin θ k | i k | i − k (cid:17) , (16)where, | i k , is the vacuum state of the mode c k , and | i k is the excited state: | i k = c † k | i k .For | ˜ g | ≫ 1, the ground state is paramagnetic with all spins oriented along the x axis. From Eq. (13) we obtain cos θ k → − | ˜ g | → ∞ . From here it follows that, | u − ( k ) i → (cid:16) (cid:17) . Thus, the north pole of the complex Bloch sphere corresponds to theparamagnetic ground state. On the other hand, when | g | ≪ z -axis. The real part of the complex energy reaches its minimum at the pointdefined by cos θ k = 1, and, hence, the south pole of the complex sphere is related to thepure antiferromagnetic ground state. 3. Quantum annealing In this section, we consider the NQA for the time-dependent Hamiltonian of Eq. (1)written as ˜ H τ ( t ) = H + H ( t ) , (17)where H = J N X n =1 σ zn σ zn +1 , (18) H ( t ) = J N X n =1 ( g ( t ) σ xn − i δ ( t ) σ + n σ − n ) . (19)We use with the ground state of the auxiliary Hamiltonian, H (0), as the initialstate. This state is “paramagnetic” with all spins oriented along the x axis. For g ≫ H τ (0), is dominated by H (0), and the ground state of the totalHamiltonian, ˜ H τ , is determined by the ground state of H (0). The term, H , causesquantum tunneling between the eigenstates of the Hamiltonian, H . At the end ofthe NQA, we obtain, ˜ H τ ( τ ) = H . If the quench is sufficiently slow, the adiabatictheorem guarantees reaching the ground state of the main Hamiltonian, H , at the endof computation.As shown in Sec. II, the total Hamiltonian, ˜ H τ ( t ), in the momentum representationsplits into a sum of independent terms, ˜ H τ ( t ) = P k H k ( t ). Each H k acts in the two-dimensional Hilbert space spanned by | k i = | i k | i − k and | k i = | i k | i − k . Thewavefunction can be written as, | ψ ( t ) i = Q k | ψ k ( t ) i , where | ψ k ( t ) i = c ( k, t ) | k i + c ( k, t ) | k i . (20)Choosing the basis as, k = (cid:16) (cid:17) and k = (cid:16) (cid:17) , we find that the Hamiltonian, H k ( t ), projected on this two-dimensional subspace takes the form H k ( t ) = ε ( t )11 − J ˜ g ( t ) − cos ϕ k sin ϕ k sin ϕ k − ˜ g ( t ) + cos ϕ k ! , (21)where ε ( t ) = J cos ϕ k − iJ δ ( t ) and ˜ g ( t ) = g ( t ) + iδ ( t ). Further we assume lineardependence of ˜ g ( t ) on time:˜ g ( t ) = ( γ ( τ − t ) , ≤ t ≤ τ , t > τ (22)where γ = ( g + iδ ) /τ , and g , δ are real parameters.The general wave functions, | ψ k i and h ˜ ψ k | , satisfy the Schr¨odinger equation and itsadjoint equation i ∂∂t | ψ k i = H k ( t ) | ψ k i , (23) − i ∂∂t h ˜ ψ k | = h ˜ ψ k |H k ( t ) . (24)Presenting | ψ k ( t ) i as a linear superposition, | ψ k ( t ) i = ( u k ( t ) | k i + v k ( t ) | k i ) e i R ε ( t ) dt , (25)and inserting expression (25) into Eq. (23), we obtain i ˙ u k = J ((˜ g − cos ϕ k ) u k − sin ϕ k v k ) , (26) i ˙ v k = − J ( sin ϕ k u k + (˜ g − cos ϕ k ) v k ) . (27)The solution can be written in terms of the parabolic cylinder functions, D ν ( z ): U k ( z k ) = B k D iν k ( iz k ) + √ iν k A k D − iν k − ( z k ) , (28) V k ( z k ) = A k D − iν k ( z k ) − i √ iν k B k D iν k − ( iz k )) . (29)Here we introduced the new functions: u k ( t ) = U ( z k ), v k ( t ) = V ( z k ) and z k ( t ) = e iπ/ s Jγ (cid:16) γ ( τ − t ) − cos ϕ k (cid:17) , (30) ν k = J sin ϕ k γ . (31)The constants, A k and B k , in Eqs. (28) and (29) are determined by the initial conditions.In the adiabatic basis the wavefunction, | ψ k ( t ) i , can be written as follows: | ψ k ( t ) i = ( α k ( t ) | u − ( k, t ) i + β k ( t ) | u + ( k, t ) i ) e i R ε ( t ) dt . (32)From Eqs. (25) and (32) it follows that α k ( t ) = u k ( t ) cos θ k ( t )2 − v k ( t ) sin θ k ( t )2 , (33)and β k ( t ) = v k ( t ) cos θ k ( t )2 + u k ( t ) sin θ k ( t )2 . (34)We assume that the evolution begins from the ground state. Then, for any k thisimplies, | α k (0) | = 1 and β k (0) = 0. Since cos θ k (0) = − 1, one has, u k (0) = 0 and v k (0) = − 1. Using these results, we obtain B k = 0, and the solution of the Schr¨odingerequation can be written as follows: U k = A k √ iν k D − iν k − ( z k ) , (35) V k = A k D − iν k ( z k ) , (36)in which A k = ( D − iν k ( z k (0))) − .Since for non-Hermitian systems the norm of the wavefunction is not conserved, foreach k we define the “intrinsic” probability to stay in the ground state as P gsk ( t ) = | α k ( t ) | | α k ( t ) | + | β k ( t ) | . (37)For each k the evolution of the system is independent, and the probability for the wholesystem to stay in the ground state is given by P gs ( t ) = Q k P gsk ( t ). In what follows,we use the asymptotic expansion for Weber functions to calculate the probability, P gs = Q k P gsk ( τ ), to remain in the ground state at the end of evolution,For long wavelength modes with | ϕ k | ≪ π/ 4, using Eqs. (35), (36), we obtain P gsk ( τ ) ≈ 11 + | D − iν k ( z k ( τ )) | |√ iν k D − iν k − ( z k ( τ )) | . (38)Using the asymptotic formulas for Weber functions with the large argument values, andtaking into account that δ ≪ g , we obtain, as in [14], P k ( τ ) = 1 − e − π ℜ ν k − e − π ℜ ν k + e − π ℜ ν k −ℜ z k ( τ ) , (39)where ℜ ν k = J τ π k / (2 gN ) , (40) ℜ z k ( τ ) = 2 δJ τ /g . (41)For the Hermitian QA, δ = 0, and Eq. (39) leads to the Landau-Zener formula [19, 20], P k ( τ ) = 1 − exp (cid:16) − J τ π k gN (cid:17) . (42)Similar considerations of the short wavelength modes with 3 π/ < | ϕ k | ≤ π yields, P gsk ( τ ) = 1 + O ( p | ν k | ). For short wavelength modes, with π/ < | ϕ k | ≤ π/ 2, employingthe large-order asymptotics for Weber functions, we obtain, P gsk ( τ ) = 1 + O (1 / p | ν k | ).Our theoretical predictions are confirmed by numerical calculations performed forup to N = 1024 qubits. (See Fig. 1.) One can observe that while short wavelengthexcitations are essential at the critical point, at the end of evolution their contributionto the transition probability from the ground state to the first excited state is negligible.Since for each k , the evolution is an independent, the probability of the wholesystem to remain in the ground state at the end of evolution is the product P gs = Y k P gsk ( τ ) . (43)In the long wavelength approximation one can take into account only ϕ k = π/N , andestimate P gs as P gs ≈ − e − πτ/τ − e − πτ/τ + e − πτ/τ −ℜ z ( τ ) , (44) Figure 1. (Color online) Left panel. The probability, P gsk , to remain in the groundstate as a function of the scaled time, s = t/τ , for the Hermitian QA ( δ = 0, J = 0 . g = 10, τ = 10 , N = 1024). From bottom to top: k = 1 , , , , 64. Right panel.The probability, P gsk , to remain in the ground state as a function of the scaled time, s = t/τ , for the NQA ( δ = 0 . J = 0 . g = 10, τ = 10 , N = 1024). Blue line( k = 1), red line ( k = 4), green line ( k = 16), orange line ( k = 32), black line ( k = 64). where τ = 2 gN / ( π J ) and ℜ z ( τ ) = 2 δJ τ /g .For the Hermitian QA ( δ = 0), Eq. (44) yields the Landau-Zener formula [19, 20] P gs = 1 − e − πτ/τ . (45)From here it follows that P gs ≈ 1, if τ ≥ τ . Thus, the computational time for theHermitian QA should be of order N .For the NQA, assuming τ ≪ τ , we obtain P gs ≈ 11 + τ πτ e − Jδτ/g . (46)From here, in the limit of δ → 0, we obtain P gs → 11 + τ πτ ≪ . (47)The obtained result is expected, as in this case the time of Hermitian annealing, τ , issmall with respect to the characteristic time, τ : τ ≪ τ .Next, assuming2 J δτg − ln τ πτ ≫ . (48)we obtain P gs ≈ − τ πτ e − Jδτ/g . (49)As one can see, P gs ≈ 1, if the conditions of Eq. (48) are satisfied. From (48) we obtainthe following rough estimate of the computational time for NQA: τ & ( g / J δ ) ln N .0 Figure 2. (Color online) Annealing time, τ , as function of N . Left panel. HermitianQA, δ = 0 ( J = 0 . g = 10). Right panel. Non-Hermitian QA, from top to bottom: δ = 0 . , . , In Fig. 2, the annealing time, τ , as a function of the number of spins is shown forthe Hermitian QA (left panel) and for the NQA (right panel). The comparison showsthat for large number of spins ( N ∼ δ & . 25, the annealing time of theNQA is ≈ times smaller than the time for the Hermitian QA.These results demonstrate that even moderate dissipation increases the transitionprobability. The characteristic time of non-Hermitian annealing, even for small butfinite , is determined not only by the number of spins, (as in the Hermitian annealing),but mainly by the dissipation rate, . The non-Hermitian quantum annealing has acomplexity of order , which is significantly better than the corresponding complexity forthe quantum Hermitian adiabatic algorithm ( ∼ N ). 4. Correlation functions and defects formation During QA the system does not remain in the ground state at all times. At the criticalpoint, the quantum system becomes excited, and its final state is determined by thenumber of defects. In the case of the antiferromagnetic Ising chain, the system ends inthe state such as | . . . ↑↓↑↓↓↑↓↑↓↑↓↑↓↑↑↓↑↓↑↓↑↓↑↓↓↑↓↑↓↑↓↑↓↑↓ , . . . i (50)with neighboring spins polarized on opposite directions along the z -axis and separatedby walls (defects) in which the polarization of spins has the same orientation.The antiferromagnetic spin-spin correlation functions provide information aboutlong-range order and defects formation. We restrict ourselves consider of theantiferromagnetic correlation function only in the final state, χ ( p ) = 1 N N X n =1 ( h σ zn σ zn + p i − h σ zn ih σ zn + p i ) , (51)1defined at ˜ g ( τ ) = 0.Using the Jordan-Wigner transformation, the correlation function can be recast asfollows: χ ( p ) = 1 N N X n =1 h B n A n +1 B n +2 A n +3 . . . B n + p − A n + p i , (52)where A m = c † m + c m and B m = c † m − c m .Next, using Wick’s theorem, we can express the average in Eq. (52) in terms ofcontractions of pairs: h B m A n i , h B m B j i and h A m A n i [21]. When the pairing h B m B j i = 0and h A m A n i = 0 for m = n , the correlation function can be written as a determinantof the Toeplitz matrix, χ τ ( p ) = det G , where G = G G − . . . G − p +1 G G . . . G − p +2 ... ... ... ... G p − G p − . . . G (53)Its components are defined as follows: G p ( τ ) = 1 N X k | v k | − | u k | − i ( u k v ∗ k + u ∗ k v k ) | v k | + | u k | e i ( m − n ) ϕ k . (54)In the thermodynamic limit, N → ∞ , we obtain G k = 12 π Z π − π ( | v | − | u | − i ( uv ∗ + u ∗ v )) e ikϕ dϕ | v | + | u | . (55)Computation of the matrix elements of the Toeplitz matrix for the ground state ofthe Hamiltonian, H , yields the following result [21, 22, 23, 24]: G k = ( − , k = 00 , k = ± , ± , . . . (56)Using this result, we obtain the correlation function of the antiferromagnetic chain as χ ( p ) = ( − p .For the time-dependent problem, the correlation function is not defined by adeterminant, since in the general case h A m A n i = δ mn + 2 i ℑ β mn , (57) h B m B n i = − δ mn + 2 i ℑ β mn , (58)where β mn = 12 πi Z π − π uv ∗ e iϕ ( m − n ) dϕ | u | + | v | . (59)In Fig. 3, the results of our numerical calculation for N = 512 are shown. As can beobserved (see inset) for the correlation distance, p ≫ 1, we can neglect the contributionof all ℑ β mn . This is valid for both Hermitian QA and for NQA, as well. Thus, thelong-range correlation function can be calculated using the Toeplitz determinant with G k = 12 π Z π − π ( | v | − | u | ) e ikϕ dϕ | v | + | u | . (60)2 Figure 3. (Color online) Dependence of ℑ β on p = m − n , δ = 0 (blue), δ = 0 . δ = 0 . δ = 1 (black) ( τ = 1000, g = 10, J = 0 . N = 512). The asymptotics of the Toeplitz determinant can be obtained by applying the Szeg¨olimit theorem. We use the results of Ref. [25] to obtain for the Hermitian QA ( δ = 0)the following asymptotic formula for the correlation function: χ ( p ) ∼ ( − p exp (cid:16) − . p ˆ ξ (cid:17) cos (cid:16)r ln 22 π p ˆ ξ + ϕ (cid:17) , (61)where the phase, ϕ , depends on the parameter, ˆ ξ = p J τ / g , (for details see Ref. [25]).Asymptotically, the correlation function exhibits decaying oscillations. This agreeswith the results obtained in Ref. [26] for the ferromagnetic Ising chain, where theasymptotic behavior of the correlation function is defined by Eq. (61), without thefactor, ( − p . For the ferromagnetic Ising chain the oscillatory behavior, correspondsto alternating magnetization signs in neighboring ordered domains, separated by kinks[25, 26]. For the antiferromagnetic Ising chain the neighboring spins, being polarizedon opposite directions, are separated by walls (defects) where the polarization of spinshas the same orientation. In both cases, the Kibble-Zurek (KZ) correlation length[26], ˆ ξ , determines the characteristic domain size as L = π ˆ ξ p π/ ln 2. In Fig. 4,the results of our numerical simulations are depicted for N = 512 spins and τ = 25.The theoretical prediction of the domain size, L ≈ . 5, agrees with the results of thenumerical calculation.The decaying oscillatory behavior of the correlation functions is confirmed byour numerical calculations presented in Figs. 4 and 5. One can observe that for3 Figure 4. (Color online) Spin antiferromagnetic correlation function and schematicposition of spins. Spins, being polarized on opposite directions inside of domain,are separated by walls (red arrows) where the polarization of spins has the sameorientation. The correlation length and domains size are determined by the KZparameter, ˆ ξ . Red diamonds: correlation function calculated for N = 512 spins( τ = 25, δ = 0). Solid line: envelope in Eq. (61). Figure 5. (Color online) Long range behavior of the correlation function at the endof annealing. From bottom to top: δ = 1 , . , . , τ = 1000, g = 10, J = 0 . N = 512). Solid lines present the asymptotics given by Eq. (61). NQA the long-range behavior of the correlation functions is much better then for thecorresponding Hermitian QA, and the characteristic domain size is much longer for NQAthan L , even for modest values of δ .4To evaluate the efficiency of the QA one should calculate the number of defects. Thecomputational time of the QA is the time required to achieve the number of defects belowsome acceptable value. The number of defects is equal to the number of quasiparticlesexcited at ˜ g = 0 (final state) [16]. It is given by N = h ψ τ | ˆ N | ψ τ i , whereˆ N = X k b † k b k = 12 N X n =1 (1 + σ zn σ zn +1 ) . (62)In the adiabatic regime we can approximate the average number of defects as,¯ N = 1 − P k ( τ ) , (63)with the lowest ϕ k = π/N [26]. Using Eq. (44), we obtain¯ N = e − πτ/τ −ℜ z ( τ ) − e − πτ/τ + e − πτ/τ −ℜ z ( τ ) . (64)When τ ≫ τ we find¯ N ≈ exp (cid:16) − πττ (cid:17) . (65)So, for large annealing times, the NQA does not have any advantage in comparison withthe Hermitian QA.In the opposite limit, τ ≪ τ , and under the condition2 J δτg − ln τ πτ ≫ , (66)the average number of defects is approximated by¯ N ≈ τ πτ e − Jδτ/g . (67)For the parameters chosen in our numerical simulations: N = 512, J = 0 . g = 10, andfor τ = 10 we obtain ¯ N ≈ e − δ . Thus, while for the Hermitian QA, the numberof defects, separating the domains with antiferromagnetic order, is ¯ N ≈ 40, for theNQA with the same protocol, the number of defects decreases as e − δ . For instance,for δ = 0 . 25 we obtain ¯ N ≈ n = lim N →∞ N N = 12 (1 + χ (1)) . (68)As shown in the previous section, during the slow evolution only long wavelength modescan be excited. So, one can use the Gaussian distribution by replacing sin ϕ ≈ ϕ andcos ϕ k ≈ 1. In the limit p J τ /g ≫ 1, we can employ Eqs. (39) - (41) to calculate thedensity of defects, n = 1 π Z π e − π ℜ ν −ℜ z dϕ − e − π ℜ ν + e − π ℜ ν −ℜ z , (69)where ϕ k → ϕ as N → ∞ . Performing the integration with ℜ ν = J τ ϕ / g and ℜ z = 2 δJ τ /g , we obtain n = n e − δτJ/g Φ (cid:16) − e − δτJ/g , , (cid:17) , (70)5 Figure 6. (Color online) Density of defects for NQA as a function of the dissipationparameter, δ ( τ = 10 , J = 0 . g = 10). where n = 12 π r gJ τ (71)denotes the density of defects for the Hermitian LZ problem [16], and Φ( x, a, c ) is theLerch transcendent [27].In Fig. 6 the density of defects as a function of the decay parameter, δ , is shown.As one can see, even moderate dissipation greatly decreases the number of defects inthe system. 5. Conclusion Previously, research on quantum annealing could be conventionally divided by two parts:(i) combining classical computers with quantum algorithms and by (ii) building realquantum computers. Many schemes and approaches of quantum annealing algorithmshave been proposed [1, 2, 3, 4, 5, 6, 7, 8, 28, 29, 30] (see also references therein). Themain objective of these publications is to significantly decrease the time of annealing.The very popular test models are the one-dimensional Ising spin chains, ferromagneticand antiferromagnetic, which are also useful for practical purposes. In this case, thequantum annealing algorithms are used to find the ground state of the chain.The approach presented in this paper is related to item (i) above, and was appliedfor the one-dimensional antiferromagnetic spin chain. We have chosen an auxiliaryHamiltonian in such a way that the total Hamiltonian is non-Hermitian. This allows usto shift the minimal gap in the energy spectrum in the complex plane, and to significantlyreduce the time required to find the ground state. Our approach leads to an annealingtime proportional to: ln N , where N is the number of spins. This is much less thanthe time of Hermitian annealing ( ∼ N ) for the same problem. We also demonstrated6the behavior of quantum correlation functions and their relation to the spin defectsgenerated during quantum annealing.It is interesting to mention here, that similar to the ferromagnetic chain, the maincontribution to the destruction of the final ground state in the process of quantumannealing is produced by the long-wavelength modes, as expected. This is in spite of thefact that in the antiferromagnetic chain the ground state represents a rapidly-oscillatingspatial structure with short wavelengths (opposite to the ferromagnetic chain).The dissipative term which we use corresponds to tunneling of the system to itsown continuum, as usually happens when one applies the Feshbach projection methodto intrinsic states in nuclear physics and quantum optics. In our case, the intrinsicstates are the states of the quantum computer (register). So, the probability of ourquantum computer to survive during the NQA can be small. That is why we use a ratioof two probabilities – the ratio of the probability for the system to remain in the groundstate to the probability of survival of the quantum computer. This relative probability(we call it “intrinsic” probability) is well-defined, and remains finite during the NQA.So, our approach cannot be directly used in the experiments on QA, but rather as acombination of classical computer and protocols for NQA to significantly decrease thetime of annealing. Also, the dissipative term which we use in this paper is rather artificialin the sense that it has no direct relation to physical dissipative mechanisms. At thesame time, we note that our dissipation term corresponds, in principal, to tunnelingeffects in the superconducting phase qubits if tunneling is realized mostly from thelowest energy levels.Our hope is that the observed reduction of the annealing time can serve as a guideto finding solutions of the NP-complete problems by using a combination of classicalcomputers and NQA algorithms. We also would like to mention that many importantproblems still remain to be considered. One of them is the application of this dissipativeapproach to more complicated Ising models with frustrated interactions, in which theground state can be strongly inhomogeneous. Another approach (“Ising machine”) wasproposed recently in [31, 32] in which both energy dissipation and pumping effects wereutilized for mapping of Ising models onto laser systems. In this case, the system canwork in the regime of a limit cycle which could be advantageous for performing quantumalgorithms. Both of these approaches to applying NQA algorithms to Ising models arenow in progress. Acknowledgments The work by G.P.B. and A.R.B. was carried out under the auspices of the NationalNuclear Security Administration of the U.S. Department of Energy at Los AlamosNational Laboratory under Contract No. DE-AC52-06NA25396. A.I.N. acknowledgesthe support from the CONACyT, Grant No. 15439. J.C.B.Z. acknowledges the supportfrom the CONACyT, Grant No. 171014.7 Appendix A.Exact solution of the Non-Hermitian Landau-Zener problem The non-Hermitian Hamiltonian, H k ( t ), projected on the two-dimensional subspacespanned by | k i = (cid:16) (cid:17) and | k i = (cid:16) (cid:17) , takes the form H k ( t ) = − ε ( t )11 − J ˜ g ( t ) − cos ϕ k sin ϕ k sin ϕ k − ˜ g ( t ) + cos ϕ k ! , (A.1)where ε ( t ) = J cos ϕ k + iJ δ ( t ) and ˜ g ( t ) = g ( t ) − iδ ( t ). We assume a linear dependenceof the function, ˜ g ( t ), on time:˜ g ( t ) = ( γ ( τ − t ) , ≤ t ≤ τ , t > τ , (A.2)where, γ = ( g − iδ ) /τ , and g , δ are real parameters.The general wave functions, | ψ k i and h ˜ ψ k | , satisfy the Schr¨odinger equation and itsadjoint equation i ∂∂t | ψ k i = H k ( t ) | ψ k i , (A.3) − i ∂∂t h ˜ ψ k | = h ˜ ψ k |H k ( t ) . (A.4)Presenting | ψ k ( t ) i as a linear superposition | ψ k ( t ) i = ( u k ( t ) | k i + v k ( t ) | k i ) e i R ε ( t ) dt , (A.5)and inserting (A.5) into Eq. (A.3), we obtain i ˙ u k = J ((˜ g − cos ϕ k ) u k − sin ϕ k v k ) , (A.6) i ˙ v k = − J ( sin ϕ k u k + (˜ g − cos ϕ k ) v k ) . (A.7)Let z k ( t ) = e iπ/ p J/γ ( γ ( τ − t ) − cos ϕ k ) be a new variable. Then, for newfunctions, u k ( t ) = U k ( z k ) and v k ( t ) = V k ( z k ), we obtain ddz k U k = z k U k − √ iν k V k , (A.8) ddz k V k = − z k V k − √ iν k U k , (A.9)where ν k = ( J/ γ ) sin ϕ k , and the complex ‘time’ z k runs from z k (0) = e iπ/ p J/γ ( γτ − cos ϕ k ) to z k ( τ ) = − e iπ/ p J/γ cos ϕ k .From Eqs. (A.8), (A.9), we obtain the second order Weber equations d dz k U k − (cid:16) 12 + z k iν k (cid:17) U k = 0 , (A.10) d dz k V k + (cid:16) − z k − iν k (cid:17) V k = 0 . (A.11)Solution of the Weber’s equations is given by the parabolic cylinder functions, D − iν k ( ± z )and D iν k − ( ± iz ).8We obtain the solutions of Eqs. (A.8, A.9) in the form U k ( z k ) = B k D iν k ( iz k ) + √ iν k A k D − iν k − ( z k ) , (A.12) V k ( z k ) = A k D − iν k ( z k ) − i √ iν k B k D iν k − ( iz k )) , (A.13)where the constants, A k and B k , are determined from the initial conditions. References [1] Tadashi Kadowaki and Hidetoshi Nishimori. 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