THE JOURNAL OF CHEMICAL PHYSICS 141, 224108 (2014)
Transitionless driving on adiabatic search algorithm Sangchul Oh1,a) and Sabre Kais1,2,b) 1
Qatar Environment and Energy Research Institute, Qatar Foundation, Doha, Qatar Department of Chemistry, Department of Physics and Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA 2
(Received 7 September 2014; accepted 24 November 2014; published online 9 December 2014) We study quantum dynamics of the adiabatic search algorithm with the equivalent two-level system. Its adiabatic and non-adiabatic evolution is studied and visualized as trajectories of Bloch vectors on a Bloch sphere. We find the change in the non-adiabatic transition probability from exponential decay for the short running time to inverse-square decay in asymptotic running time. The scaling of the critical running time is expressed in terms of the Lambert W function. We derive the transitionless driving Hamiltonian for the adiabatic search algorithm, which makes a quantum state follow the adiabatic path. We demonstrate that a uniform transitionless driving Hamiltonian, approximate to the exact time-dependent driving Hamiltonian, can alter the non-adiabatic transition probability from the inverse square decay to the inverse fourth power decay with the running time. This may open up a new but simple way of speeding up adiabatic quantum dynamics. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4903451] I. INTRODUCTION
Quantum mechanics is the core of quantum information and computation in which information is stored in quantum states and processed quantum mechanically. The synergy between quantum mechanics and quantum information science has been growing rapidly over the past decade.1 For example, the concept of quantum entanglement was born when developing quantum theory, has drawn much attention in quantum information, and in turn becomes one of the key elements to understand quantum many-body systems. Quantum algorithms are being developed to solve quantum mechanical problems in physics and chemistry1–3 which are believed to be intractable with a classical digital computer. The quantum adiabatic theorem is one of the fundamental theorem in quantum mechanics and has a lot of applications such as Landau-Zener-Majorana-Stückelberg problems,4–7 geometric phases,8 adiabatic passages,9–11 etc. In quantum information science, it is central to the adiabatic quantum computation model12 and the quantum annealing method.13, 14 So the adiabatic search algorithm considered here would be a simple but good example to see the adiabatic and non-adiabatic quantum dynamics. is known to find a Grover’s quantum search algorithm15 √ marked one out of N entries with the O( N) queries on a quantum computer, otherwise the O(N) queries are needed on a classical computer. While it was initially designed to be implemented on a quantum circuit model, its adiabatic quantum computation version, called adiabatic search algorithm, was also proposed and the equivalence between them was proved.16–18 Much attention has been paid to solving an instantaneous eigenvalue problem of a time-dependent Hamiltonian of an adiabatic quantum algorithm because a) Email: [email protected] b) Email: [email protected]
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the minimum gap of a system determines the validity of an adiabatic quantum evolution and thus its computational complexity.12 The non-adiabatic transition to other states, i.e., the deviation from the adiabatic evolution, is the main concern in adiabatic quantum computation. To know in detail how the non-adiabatic transition decreases asymptotically with running time, the minimum gap of the instantaneous eigenvalues is not enough, so a time-dependent Schrödinger equation has to be solved. In this paper, we study quantum dynamics of adiabatic search algorithm with an equivalent two-level system to calculate its non-adiabatic transition probability. The adiabatic and non-adiabatic evolutions of a quantum state are represented by trajectories on a Bloch sphere. We show that the non-adiabatic transition probability changes from exponential decay for short running time to inverse square decay for long running time. The dependence of the critical running time on the problem size is written in terms of the Lambert W function. Finally, we show that a constant driving Hamiltonian could reduce significantly the non-adiabatic transition probability, which may speed up adiabatic quantum computation.
II. QUANTUM DYNAMICS OF ADIABATIC SEARCH ALGORITHM A. Adiabatic search algorithm as a two-level system
Let us start with introducing the time-dependent Hamiltonian for Grover’s adiabatic search algorithm.18, 19 The adiabatic quantum computation is based on the adiabatic theorem which states that if a time-dependent Hamiltonian changes slowly enough, then an eigenstate of an initial Hamiltonian, an input state, evolves to an eigenstate of a final Hamiltonian, an output state.9, 12 Grover’s search algorithm takes the input state as a superposition of all possible states |ϕin = √1N N−1 i=0 |i with N entries. It is the ground state of the
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initial Hamiltonian H0 = I − |ϕin ϕin | = I − N1 i,j |ij | where I is an N × N identity matrix. Note that i, j |ij| is a matrix with all entries 1 whose eigenvalues are 0 (N − 1 multiples) and N.20 The output or target state |w to find is the ground state of the final (or problem) Hamiltonian Hp = I − |ww|. The slow change from the initial to final Hamiltonians can be done as H(t) = f(s)H0 + g(s)Hp where s ≡ t/T is the dimensionless (or macroscopic) time,9, 21 T is the running time acting as the adiabatic parameter, and a turn-off function f(s) and turn-on function g(s) satisfy f(0) = g(1) = 1 and f(1) = g(0) = 0. The simplest choice of f and g is to interpolate H0 and Hp linearly, i.e., f(s) = 1 − s and g(s) = s. Grover’s search algorithm is understood as a rotation from the input state |ϕ in to the target state |w. This implies it is essentially a two-dimensional problem formed by two linearly independent vectors |ϕ in and |w. While √ in quantum circuit model the full rotation is done by O( N ) successive finite rotations, it is done by a continuous rotation in adiabatic quantum computation. The two vectors |w and |ϕ in are linearly independent but not orthogonal. An orthonormal basis is easily constructed from the matrix representation of I − H (s) = f (s)|ϕin ϕin | + g(s)|ww| where only the wth diagonal element is different. Thus, the timedependent Hamiltonian for Grover’s adiabatic search algorithm is represented with the orthonormal basis {|w, |w⊥ } as ⎤ ⎡ √ g N −1 f ⎣1+Nf ⎦, (1) H (s) = I − N √N − 1 N − 1 1 where |w⊥ = √N−1 convenient form as
N
i=w
|i . Hamiltonian (1) is written in ⎡
Z(s)
1 ⎣ (f + g) I− 2 2N X(s)
X(s)
⎤
⎦, (2) −Z(s) √ where Z(s) ≡ 2f + N(g − f) and X(s) ≡ 2f N − 1. Since the first term in Eq. (2) is not relevant to dynamics, it will be dropped. The Hamiltonian is written as ⎤ ⎡ sin θ (s) ¯ω(s) ⎣ cos θ (s) ⎦, (3) H (s) = − 2 sin θ (s) − cos θ (s) H (s) =
where the gap between the ground and excited states is given √ 1 by ¯ω(s) ≡ N Z 2 + X2 = (f − g)2 + N4 f g . Here, mixing angle θ is defined by tan θ (s) ≡ X(s)/Z(s). While a different choice of f and g gives rise to a different energy gap, hereafter we consider only a linear interpolation case. Hereafter, we set ¯ = 1. It is straightforward to write down the instantaneous eigenstates of H(s)|e± (s) = e± (s)|e± (s), ⎤ ⎤ ⎡ ⎡ cos θ2 − sin θ2 ⎦ , |e+ (s) = ⎣ ⎦ . (4) |e− (s) = ⎣ sin θ2 cos θ2 As represented by a Bloch vector in Fig. 1, the input state |ϕ in = |e− (0) is a vector with azimuthal angle tan θ √ = (2 − N )/2 N − 1. The target state |w = |e− (1) points
FIG. 1. Upper panel (a): Trajectories of a Bloch vector r(t) on a Bloch sphere for various running times (1) T = 10, (2) T = 100, (3) T = 300. (4) The blue longitudinal line represents the adiabatic path. Here, N = 4 is set. If N is large, the initial Bloch vector becomes closer to |w⊥ . Lower panel (b): Stereographic projection of the Bloch sphere to the plane tangent to the north pole. It maps the Bloch vector r = (x, y, z) on the Bloch sphere to the point 2y 2x P = (X, Y) on the plane where X = 1+z and Y = 1+z . Circles π /4 and π /2 correspond to latitude of 45◦ N and the equator, respectively.
to the north pole. Thus, like the Landau-Zener-MajoranaStückelberg problem,4–7 Grover’s adiabatic search algorithm is just a rotation of a single qubit driven by time-dependent Hamiltonian (3). B. Non-adiabatic quantum dynamics
To understand non-adiabatic effects, we solve numerically the time-dependent Schrödinger equation d |ψ(t) = HT (t)|ψ(t) , (5) dt where the time-dependent Hamiltonian HT (t) is given by Eq. (3). As illustrated in Fig. 1, a quantum state |ψ(t) = α(t)|w + β(t)|w⊥ is visualized by a Bloch vector r(t) ≡ ψ(t)|σ |ψ(t) with Pauli √ matrices σ k for k = x, y, z. In the adiabatic limit of T N , an evolved quantum state remains in the instantaneous ground state, that is, |ψ(t)
|e− (t) up to the dynamical and geometric phase factors. So, the Bloch vector rad (s) = e− (s)|σ |e− (s) for the adiabatic state travels to the north pole along the longitude line on a Bloch sphere, the blue line in Fig. 1. The adiabatic path is a good approximation to the exact evolution if the running time T is large enough, that is, the Hamiltonian changes slowly enough. For finite running time T, however, a real path deviates from the adiabatic path as illustrated in Fig. 1. To see this in detail, we examine how a quantum state |ψ(t) deviates from the instantaneous eigenstate |e− (t) as the adiabatic parameter T is varied. The evolved state |ψ(t) is written in terms of instantaneous eigenstates as |ψ(s) = a(s)|e− (s) + b(s)|e+ (s) . Fig. 2 plots the i¯
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C. Change in transition probability from exponential to power-law decays
FIG. 2. Transition probability P(s) of being in an instantaneous eigenstate |e+ (s) as a function of s for (a) T = 20, (b) T = 200, (c) T = 500, (d) T = 1000. Here, the system size N = 10 is taken. P(s) is magnified by 10, 300, 2000, 10 000 times, respectively.
transition probability P(s) = 1 − |a(s)|2 of being in an instantaneous ground state |e+ (s) for various running time T. For the short running time T, as shown in Fig. 2(a), the maximum of P(s) does not coincide with the location of the minimum energy gap. As depicted in Figs. 2(b)–2(d), P(s) becomes smaller and more symmetric and reaches at its peak at s = 1/2 as the running time T is increased. It should be noted that the oscillations in Figs. 2(c) and 2(d) are not numerical errors which are in the order of O(10−13 ). The oscillations are consistent with the oscillations of the non-adiabatic transition probability in Fig. 3.
The non-adiabatic transition probability P(1) at s = 1 indicates the error of adiabatic quantum computation. The asymptotic form of P(1) for the Landau-Zener-MajoranaStückelberg problem is known to decrease exponentially.4–7 The non-adiabatic transition probability, however, changes in the asymptotic limit. Santoro et al. reported the 1/T2 decay in the simulated annealing system.22 Suzuki and Okada23 calculated numerically the residual energy, the difference between the energy expectation E(s) = ψ(s)|H(s)|ψ(s) and the instantaneous ground energy e− (s), for a modified LandauZener-Majorana-Stükelberg problem. They showed the transition of the residual energy from exponential decay only for the short running time to the inverse-square decay for the long running time. A similar result for the adiabatic quantum algorithm was obtained by Rezakhani et al.24 The power decay for the adiabatic quantum teleportation was obtained by Oh et al.25 This phenomenon is ubiquitous. The change in the transition probability from the exponential decay at very short time to the power law decay was analyzed in connection with the wave packet localization or Anderson localization.26, 27 While it would be interesting to explore where these could be explained on the equal footing. As illustrated in Fig. 3, we calculate numerically the nonadiabatic transition probability P(1) as a function of the running time T and find P (1) ∼
exp (−A T )
for T < Tc
B/T 2
for T > Tc
.
(6)
The coefficients A, B, and the transition time Tc depend on the system size N, as shown in Fig. 4. The numerical data show A ∼ π /4N and B ∼ 4/N. The critical running time Tc can be defined by a solution of the transcendental equation e−A T = B/T2 in Eq. (6). It is given by √
π 2 A B 8N W − √ , Tc = − W−1 − ∼ A 2 π −1 4 N3
(7)
where W−1 is the lower branch of the Lambert W function.28, 29
FIG. 3. Log-log plots of the non-adiabatic transition probability P as a function of the running time T for (a) N = 2, (b) N = 10, and (c) N = 20. The green dashed line is exp ( − AT) and the black dotted line is B/T2 . The arrows indicate the critical running time Tc .
FIG. 4. Coefficients A and B in Eq. (6), and 1/Tc as a function of the system size N. The cyan solid line is a plot with Eq. (7).
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D. Transitionless driving
As shown before, the adiabatic quantum evolution would be successful only when the time-dependent Hamiltonian changes slowly enough. This means the running time should be long enough. To overcome the slowness of the adiabatic evolution, some techniques, called shortcuts to adiabaticity, have been developed. Demirplak and Rice,30 and Berry31 showed that a time-dependent Hamiltonian HD (t), called the counter or transitionless driving term, in addition to the original time-dependent Hamiltonian makes a quantum state follow the original adiabatic state exactly. The other methods include the dynamic invariants of a time-dependent system32–34 and fast-forward technique.35 The similarities and differences among the three methods are discussed in Ref. 34. Here, we use the transitionless driving technique to accelerate the adiabatic quantum algorithm. We would like to point out that while there are many ways (for example, quantum Brachistochone,36 quantum optimal control,37 and quantum spline38 ) for getting a target state, adiabatic quantum computation is different. If a target state were known, it could be obtained by rotating an initial state around the axis, perpendicular to the plane formed by the initial and target states, by angle between them. However, in adiabatic quantum computation the target state, the solution of the problem, is assumed to be unknown, so the non-adiabatic rotation based on the knowledge of the target state must be ruled out. The main idea of the transitionless driving method is to make the driving Hamiltonian cancel the non-adiabatic term seen in the adiabatic frame. The driving Hamiltonian HD (t) for Hamiltonian (3) reads θ˙ ∂U † (t) U (t) = −¯ σy , (8) ∂t 2 where the unitary operator U(t) is composed of instantaneous eigenstates |e± (t), ⎤ ⎡ cos θ2 − sin θ2 ⎦, (9) U (t) = ⎣ sin θ2 cos θ2 HD (t) = i¯
ds = T1 dθ . Note Pauli operator σ y is repreand θ˙ = dθ ds dt ds sented by σy = −i|ww⊥ | + i|w⊥ w|. For linear interpo√ ˙ = 2 N−1 [(1 − 2s)2 + 4 (1 − s)s]−1 . As lation, one has θ(t) NT N expected, the driving Hamiltonian goes to zero in the adiabatic limit, T 1. While the driving Hamiltonian HD (t) makes a quantum state evolve exactly along the longitudinal line (adiabatic path) regardless of T, it seems to be difficult to control the strength θ˙ even in linear interpolation case. So, we investigate whether an approximate but constant driving Hamiltonian, instead of the exact time-dependent driving Hamiltonian (8), could reduce some errors. We consider two constant driving Hamiltonians which are the minimum and maximum values of HD , respectively, √ √ ¯ N −1 ¯ N −1 min max σy , HD = − σy . (10) HD = − NT T Fig. 5 shows how the instantaneous eigenvalues change when the driving Hamiltonian HD (s) is added to H(s). The role of
FIG. 5. Instantaneous eigenvalues E0, 1 (s) of H(s) + HD (s) as a function of s for T = 10, 20, 30, 50, HDmin at T = 10, and HD = 0. The size of the system is N = 100.
HD is to make the gap at the avoided crossing wider. While the approximate driving Hamiltonian HDmin seems to make a very little change in adiabatic energy levels and the trajectory as shown in Fig. 6, it produces drastic change in the non-adiabatic transition probability for the long running time, from O(1/T2 ) to O(1/T4 ) as depicted in Fig. 7. Let us take a close look at it in connection with the adiabatic condition T
maxs |e+ (s)| dH |e− (s)| ds , mins E(s)2
(11)
where E is the energy gap. For two Hamiltonians H(s) and D with T = 40, while the numerators in Eq. (11) H (s) + Hmin are same, the denominators change slightly, to be more specific, from 0.01 to 0.010396. Although the right-hand side of the inequality (11) changes very little, P(1) for the long running time changes from the inverse square to fourth power decays. Note that HDmin also reduces P(1) for the short running time. From Figs. 1(b) and 6(b), one might guess why
FIG. 6. Upper panel (a): Trajectories of Bloch vectors on a Bloch sphere when the quantum evolution is driven (1) by adiabatically or exactly HD (t), (2) by HDmin , (3) by HDmax , and (4) without driving. Here, N = 4 and T = 40 are taken. Lower panel (b): Stereographic project of the trajectories on the plane as in Fig. 1.
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FIG. 7. Non-adiabatic transition probability P(1) as a function of the running time T with HDmin (blue) and without driving Hamiltonian (red). Here, N = 10 is taken.
the approximate driving Hamiltonian HDmin speeds up the adiabatic evolution. The non-adiabatic effect pulls the trajectories to west. The corresponding non-adiabatic Hamiltonian is proportional to σ y . The approximate driving Hamiltonian proportional to −σ y reduces its strength so that it makes the trajectories close to the adiabatic path. III. CONCLUSION
We studied quantum dynamics of Grover’s adiabatic search algorithm as a time-dependent two-level system. The transition from the non-adiabatic and adiabatic quantum evolutions were visualized by changes in trajectories of Bloch vectors on a Bloch sphere. We found a drastic change in the non-adiabatic transition probability from well-known exponential decay for the short running time to the inverse-square decay for the longer running time. The dependence of the critical running time on the problem size is obtained with Lambert W function. We showed an approximate but constant driving Hamiltonian could reduce the non-adiabatic transition probability significantly which becomes the inverse fourth power decay for the long running time. It would be interesting to see whether the results obtained in this paper could be applied to other quantum system, for example, a quantum Ising model,39 quantum optimization problems,40 or PT-symmetric quantum systems.41 While our results were obtained by numerical calculations, it would be interesting to seek an exact analytic solution. Another interesting question is whether the speeding up by the transitionless driving would be applicable for systems with many minima, for example, Ising spin glasses or potential energy surfaces for the analysis of chemical reaction dynamics. According to the numerical studies in Ref. 42, systems with many minima show an inverse logarithmic decay with T except for asymptotically large T, in contrast with the inverse square decay behavior of the systems without local minima.23 Also, Mukherjee and Chakrabarti,43 following result of Ray et al.,44 presented the simple argument on the same inverse logarithmic decay of residual energy with the annealing time T. Clearly, it would be interesting to investigate how significantly the approximate transitionless driving, obtained here, could modify the logarithmic decay behavior of a system with many minima.
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Transitionless driving on adiabatic search algorithm Sangchul Oh1,a) and Sabre Kais1,2,b) 1
Qatar Environment and Energy Research Institute, Qatar Foundation, Doha, Qatar Department of Chemistry, Department of Physics and Birck Nanotechnology Center, Purdue University, West Lafayette, Indiana 47907, USA 2
(Received 7 September 2014; accepted 24 November 2014; published online 9 December 2014) We study quantum dynamics of the adiabatic search algorithm with the equivalent two-level system. Its adiabatic and non-adiabatic evolution is studied and visualized as trajectories of Bloch vectors on a Bloch sphere. We find the change in the non-adiabatic transition probability from exponential decay for the short running time to inverse-square decay in asymptotic running time. The scaling of the critical running time is expressed in terms of the Lambert W function. We derive the transitionless driving Hamiltonian for the adiabatic search algorithm, which makes a quantum state follow the adiabatic path. We demonstrate that a uniform transitionless driving Hamiltonian, approximate to the exact time-dependent driving Hamiltonian, can alter the non-adiabatic transition probability from the inverse square decay to the inverse fourth power decay with the running time. This may open up a new but simple way of speeding up adiabatic quantum dynamics. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4903451] I. INTRODUCTION
Quantum mechanics is the core of quantum information and computation in which information is stored in quantum states and processed quantum mechanically. The synergy between quantum mechanics and quantum information science has been growing rapidly over the past decade.1 For example, the concept of quantum entanglement was born when developing quantum theory, has drawn much attention in quantum information, and in turn becomes one of the key elements to understand quantum many-body systems. Quantum algorithms are being developed to solve quantum mechanical problems in physics and chemistry1–3 which are believed to be intractable with a classical digital computer. The quantum adiabatic theorem is one of the fundamental theorem in quantum mechanics and has a lot of applications such as Landau-Zener-Majorana-Stückelberg problems,4–7 geometric phases,8 adiabatic passages,9–11 etc. In quantum information science, it is central to the adiabatic quantum computation model12 and the quantum annealing method.13, 14 So the adiabatic search algorithm considered here would be a simple but good example to see the adiabatic and non-adiabatic quantum dynamics. is known to find a Grover’s quantum search algorithm15 √ marked one out of N entries with the O( N) queries on a quantum computer, otherwise the O(N) queries are needed on a classical computer. While it was initially designed to be implemented on a quantum circuit model, its adiabatic quantum computation version, called adiabatic search algorithm, was also proposed and the equivalence between them was proved.16–18 Much attention has been paid to solving an instantaneous eigenvalue problem of a time-dependent Hamiltonian of an adiabatic quantum algorithm because a) Email: [email protected] b) Email: [email protected]
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the minimum gap of a system determines the validity of an adiabatic quantum evolution and thus its computational complexity.12 The non-adiabatic transition to other states, i.e., the deviation from the adiabatic evolution, is the main concern in adiabatic quantum computation. To know in detail how the non-adiabatic transition decreases asymptotically with running time, the minimum gap of the instantaneous eigenvalues is not enough, so a time-dependent Schrödinger equation has to be solved. In this paper, we study quantum dynamics of adiabatic search algorithm with an equivalent two-level system to calculate its non-adiabatic transition probability. The adiabatic and non-adiabatic evolutions of a quantum state are represented by trajectories on a Bloch sphere. We show that the non-adiabatic transition probability changes from exponential decay for short running time to inverse square decay for long running time. The dependence of the critical running time on the problem size is written in terms of the Lambert W function. Finally, we show that a constant driving Hamiltonian could reduce significantly the non-adiabatic transition probability, which may speed up adiabatic quantum computation.
II. QUANTUM DYNAMICS OF ADIABATIC SEARCH ALGORITHM A. Adiabatic search algorithm as a two-level system
Let us start with introducing the time-dependent Hamiltonian for Grover’s adiabatic search algorithm.18, 19 The adiabatic quantum computation is based on the adiabatic theorem which states that if a time-dependent Hamiltonian changes slowly enough, then an eigenstate of an initial Hamiltonian, an input state, evolves to an eigenstate of a final Hamiltonian, an output state.9, 12 Grover’s search algorithm takes the input state as a superposition of all possible states |ϕin = √1N N−1 i=0 |i with N entries. It is the ground state of the
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initial Hamiltonian H0 = I − |ϕin ϕin | = I − N1 i,j |ij | where I is an N × N identity matrix. Note that i, j |ij| is a matrix with all entries 1 whose eigenvalues are 0 (N − 1 multiples) and N.20 The output or target state |w to find is the ground state of the final (or problem) Hamiltonian Hp = I − |ww|. The slow change from the initial to final Hamiltonians can be done as H(t) = f(s)H0 + g(s)Hp where s ≡ t/T is the dimensionless (or macroscopic) time,9, 21 T is the running time acting as the adiabatic parameter, and a turn-off function f(s) and turn-on function g(s) satisfy f(0) = g(1) = 1 and f(1) = g(0) = 0. The simplest choice of f and g is to interpolate H0 and Hp linearly, i.e., f(s) = 1 − s and g(s) = s. Grover’s search algorithm is understood as a rotation from the input state |ϕ in to the target state |w. This implies it is essentially a two-dimensional problem formed by two linearly independent vectors |ϕ in and |w. While √ in quantum circuit model the full rotation is done by O( N ) successive finite rotations, it is done by a continuous rotation in adiabatic quantum computation. The two vectors |w and |ϕ in are linearly independent but not orthogonal. An orthonormal basis is easily constructed from the matrix representation of I − H (s) = f (s)|ϕin ϕin | + g(s)|ww| where only the wth diagonal element is different. Thus, the timedependent Hamiltonian for Grover’s adiabatic search algorithm is represented with the orthonormal basis {|w, |w⊥ } as ⎤ ⎡ √ g N −1 f ⎣1+Nf ⎦, (1) H (s) = I − N √N − 1 N − 1 1 where |w⊥ = √N−1 convenient form as
N
i=w
|i . Hamiltonian (1) is written in ⎡
Z(s)
1 ⎣ (f + g) I− 2 2N X(s)
X(s)
⎤
⎦, (2) −Z(s) √ where Z(s) ≡ 2f + N(g − f) and X(s) ≡ 2f N − 1. Since the first term in Eq. (2) is not relevant to dynamics, it will be dropped. The Hamiltonian is written as ⎤ ⎡ sin θ (s) ¯ω(s) ⎣ cos θ (s) ⎦, (3) H (s) = − 2 sin θ (s) − cos θ (s) H (s) =
where the gap between the ground and excited states is given √ 1 by ¯ω(s) ≡ N Z 2 + X2 = (f − g)2 + N4 f g . Here, mixing angle θ is defined by tan θ (s) ≡ X(s)/Z(s). While a different choice of f and g gives rise to a different energy gap, hereafter we consider only a linear interpolation case. Hereafter, we set ¯ = 1. It is straightforward to write down the instantaneous eigenstates of H(s)|e± (s) = e± (s)|e± (s), ⎤ ⎤ ⎡ ⎡ cos θ2 − sin θ2 ⎦ , |e+ (s) = ⎣ ⎦ . (4) |e− (s) = ⎣ sin θ2 cos θ2 As represented by a Bloch vector in Fig. 1, the input state |ϕ in = |e− (0) is a vector with azimuthal angle tan θ √ = (2 − N )/2 N − 1. The target state |w = |e− (1) points
FIG. 1. Upper panel (a): Trajectories of a Bloch vector r(t) on a Bloch sphere for various running times (1) T = 10, (2) T = 100, (3) T = 300. (4) The blue longitudinal line represents the adiabatic path. Here, N = 4 is set. If N is large, the initial Bloch vector becomes closer to |w⊥ . Lower panel (b): Stereographic projection of the Bloch sphere to the plane tangent to the north pole. It maps the Bloch vector r = (x, y, z) on the Bloch sphere to the point 2y 2x P = (X, Y) on the plane where X = 1+z and Y = 1+z . Circles π /4 and π /2 correspond to latitude of 45◦ N and the equator, respectively.
to the north pole. Thus, like the Landau-Zener-MajoranaStückelberg problem,4–7 Grover’s adiabatic search algorithm is just a rotation of a single qubit driven by time-dependent Hamiltonian (3). B. Non-adiabatic quantum dynamics
To understand non-adiabatic effects, we solve numerically the time-dependent Schrödinger equation d |ψ(t) = HT (t)|ψ(t) , (5) dt where the time-dependent Hamiltonian HT (t) is given by Eq. (3). As illustrated in Fig. 1, a quantum state |ψ(t) = α(t)|w + β(t)|w⊥ is visualized by a Bloch vector r(t) ≡ ψ(t)|σ |ψ(t) with Pauli √ matrices σ k for k = x, y, z. In the adiabatic limit of T N , an evolved quantum state remains in the instantaneous ground state, that is, |ψ(t)
|e− (t) up to the dynamical and geometric phase factors. So, the Bloch vector rad (s) = e− (s)|σ |e− (s) for the adiabatic state travels to the north pole along the longitude line on a Bloch sphere, the blue line in Fig. 1. The adiabatic path is a good approximation to the exact evolution if the running time T is large enough, that is, the Hamiltonian changes slowly enough. For finite running time T, however, a real path deviates from the adiabatic path as illustrated in Fig. 1. To see this in detail, we examine how a quantum state |ψ(t) deviates from the instantaneous eigenstate |e− (t) as the adiabatic parameter T is varied. The evolved state |ψ(t) is written in terms of instantaneous eigenstates as |ψ(s) = a(s)|e− (s) + b(s)|e+ (s) . Fig. 2 plots the i¯
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C. Change in transition probability from exponential to power-law decays
FIG. 2. Transition probability P(s) of being in an instantaneous eigenstate |e+ (s) as a function of s for (a) T = 20, (b) T = 200, (c) T = 500, (d) T = 1000. Here, the system size N = 10 is taken. P(s) is magnified by 10, 300, 2000, 10 000 times, respectively.
transition probability P(s) = 1 − |a(s)|2 of being in an instantaneous ground state |e+ (s) for various running time T. For the short running time T, as shown in Fig. 2(a), the maximum of P(s) does not coincide with the location of the minimum energy gap. As depicted in Figs. 2(b)–2(d), P(s) becomes smaller and more symmetric and reaches at its peak at s = 1/2 as the running time T is increased. It should be noted that the oscillations in Figs. 2(c) and 2(d) are not numerical errors which are in the order of O(10−13 ). The oscillations are consistent with the oscillations of the non-adiabatic transition probability in Fig. 3.
The non-adiabatic transition probability P(1) at s = 1 indicates the error of adiabatic quantum computation. The asymptotic form of P(1) for the Landau-Zener-MajoranaStückelberg problem is known to decrease exponentially.4–7 The non-adiabatic transition probability, however, changes in the asymptotic limit. Santoro et al. reported the 1/T2 decay in the simulated annealing system.22 Suzuki and Okada23 calculated numerically the residual energy, the difference between the energy expectation E(s) = ψ(s)|H(s)|ψ(s) and the instantaneous ground energy e− (s), for a modified LandauZener-Majorana-Stükelberg problem. They showed the transition of the residual energy from exponential decay only for the short running time to the inverse-square decay for the long running time. A similar result for the adiabatic quantum algorithm was obtained by Rezakhani et al.24 The power decay for the adiabatic quantum teleportation was obtained by Oh et al.25 This phenomenon is ubiquitous. The change in the transition probability from the exponential decay at very short time to the power law decay was analyzed in connection with the wave packet localization or Anderson localization.26, 27 While it would be interesting to explore where these could be explained on the equal footing. As illustrated in Fig. 3, we calculate numerically the nonadiabatic transition probability P(1) as a function of the running time T and find P (1) ∼
exp (−A T )
for T < Tc
B/T 2
for T > Tc
.
(6)
The coefficients A, B, and the transition time Tc depend on the system size N, as shown in Fig. 4. The numerical data show A ∼ π /4N and B ∼ 4/N. The critical running time Tc can be defined by a solution of the transcendental equation e−A T = B/T2 in Eq. (6). It is given by √
π 2 A B 8N W − √ , Tc = − W−1 − ∼ A 2 π −1 4 N3
(7)
where W−1 is the lower branch of the Lambert W function.28, 29
FIG. 3. Log-log plots of the non-adiabatic transition probability P as a function of the running time T for (a) N = 2, (b) N = 10, and (c) N = 20. The green dashed line is exp ( − AT) and the black dotted line is B/T2 . The arrows indicate the critical running time Tc .
FIG. 4. Coefficients A and B in Eq. (6), and 1/Tc as a function of the system size N. The cyan solid line is a plot with Eq. (7).
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D. Transitionless driving
As shown before, the adiabatic quantum evolution would be successful only when the time-dependent Hamiltonian changes slowly enough. This means the running time should be long enough. To overcome the slowness of the adiabatic evolution, some techniques, called shortcuts to adiabaticity, have been developed. Demirplak and Rice,30 and Berry31 showed that a time-dependent Hamiltonian HD (t), called the counter or transitionless driving term, in addition to the original time-dependent Hamiltonian makes a quantum state follow the original adiabatic state exactly. The other methods include the dynamic invariants of a time-dependent system32–34 and fast-forward technique.35 The similarities and differences among the three methods are discussed in Ref. 34. Here, we use the transitionless driving technique to accelerate the adiabatic quantum algorithm. We would like to point out that while there are many ways (for example, quantum Brachistochone,36 quantum optimal control,37 and quantum spline38 ) for getting a target state, adiabatic quantum computation is different. If a target state were known, it could be obtained by rotating an initial state around the axis, perpendicular to the plane formed by the initial and target states, by angle between them. However, in adiabatic quantum computation the target state, the solution of the problem, is assumed to be unknown, so the non-adiabatic rotation based on the knowledge of the target state must be ruled out. The main idea of the transitionless driving method is to make the driving Hamiltonian cancel the non-adiabatic term seen in the adiabatic frame. The driving Hamiltonian HD (t) for Hamiltonian (3) reads θ˙ ∂U † (t) U (t) = −¯ σy , (8) ∂t 2 where the unitary operator U(t) is composed of instantaneous eigenstates |e± (t), ⎤ ⎡ cos θ2 − sin θ2 ⎦, (9) U (t) = ⎣ sin θ2 cos θ2 HD (t) = i¯
ds = T1 dθ . Note Pauli operator σ y is repreand θ˙ = dθ ds dt ds sented by σy = −i|ww⊥ | + i|w⊥ w|. For linear interpo√ ˙ = 2 N−1 [(1 − 2s)2 + 4 (1 − s)s]−1 . As lation, one has θ(t) NT N expected, the driving Hamiltonian goes to zero in the adiabatic limit, T 1. While the driving Hamiltonian HD (t) makes a quantum state evolve exactly along the longitudinal line (adiabatic path) regardless of T, it seems to be difficult to control the strength θ˙ even in linear interpolation case. So, we investigate whether an approximate but constant driving Hamiltonian, instead of the exact time-dependent driving Hamiltonian (8), could reduce some errors. We consider two constant driving Hamiltonians which are the minimum and maximum values of HD , respectively, √ √ ¯ N −1 ¯ N −1 min max σy , HD = − σy . (10) HD = − NT T Fig. 5 shows how the instantaneous eigenvalues change when the driving Hamiltonian HD (s) is added to H(s). The role of
FIG. 5. Instantaneous eigenvalues E0, 1 (s) of H(s) + HD (s) as a function of s for T = 10, 20, 30, 50, HDmin at T = 10, and HD = 0. The size of the system is N = 100.
HD is to make the gap at the avoided crossing wider. While the approximate driving Hamiltonian HDmin seems to make a very little change in adiabatic energy levels and the trajectory as shown in Fig. 6, it produces drastic change in the non-adiabatic transition probability for the long running time, from O(1/T2 ) to O(1/T4 ) as depicted in Fig. 7. Let us take a close look at it in connection with the adiabatic condition T
maxs |e+ (s)| dH |e− (s)| ds , mins E(s)2
(11)
where E is the energy gap. For two Hamiltonians H(s) and D with T = 40, while the numerators in Eq. (11) H (s) + Hmin are same, the denominators change slightly, to be more specific, from 0.01 to 0.010396. Although the right-hand side of the inequality (11) changes very little, P(1) for the long running time changes from the inverse square to fourth power decays. Note that HDmin also reduces P(1) for the short running time. From Figs. 1(b) and 6(b), one might guess why
FIG. 6. Upper panel (a): Trajectories of Bloch vectors on a Bloch sphere when the quantum evolution is driven (1) by adiabatically or exactly HD (t), (2) by HDmin , (3) by HDmax , and (4) without driving. Here, N = 4 and T = 40 are taken. Lower panel (b): Stereographic project of the trajectories on the plane as in Fig. 1.
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FIG. 7. Non-adiabatic transition probability P(1) as a function of the running time T with HDmin (blue) and without driving Hamiltonian (red). Here, N = 10 is taken.
the approximate driving Hamiltonian HDmin speeds up the adiabatic evolution. The non-adiabatic effect pulls the trajectories to west. The corresponding non-adiabatic Hamiltonian is proportional to σ y . The approximate driving Hamiltonian proportional to −σ y reduces its strength so that it makes the trajectories close to the adiabatic path. III. CONCLUSION
We studied quantum dynamics of Grover’s adiabatic search algorithm as a time-dependent two-level system. The transition from the non-adiabatic and adiabatic quantum evolutions were visualized by changes in trajectories of Bloch vectors on a Bloch sphere. We found a drastic change in the non-adiabatic transition probability from well-known exponential decay for the short running time to the inverse-square decay for the longer running time. The dependence of the critical running time on the problem size is obtained with Lambert W function. We showed an approximate but constant driving Hamiltonian could reduce the non-adiabatic transition probability significantly which becomes the inverse fourth power decay for the long running time. It would be interesting to see whether the results obtained in this paper could be applied to other quantum system, for example, a quantum Ising model,39 quantum optimization problems,40 or PT-symmetric quantum systems.41 While our results were obtained by numerical calculations, it would be interesting to seek an exact analytic solution. Another interesting question is whether the speeding up by the transitionless driving would be applicable for systems with many minima, for example, Ising spin glasses or potential energy surfaces for the analysis of chemical reaction dynamics. According to the numerical studies in Ref. 42, systems with many minima show an inverse logarithmic decay with T except for asymptotically large T, in contrast with the inverse square decay behavior of the systems without local minima.23 Also, Mukherjee and Chakrabarti,43 following result of Ray et al.,44 presented the simple argument on the same inverse logarithmic decay of residual energy with the annealing time T. Clearly, it would be interesting to investigate how significantly the approximate transitionless driving, obtained here, could modify the logarithmic decay behavior of a system with many minima.
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