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Dynamics of adiabatic quantum search

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Dynamics of adiabatic quantum search

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Content DYNAMICS OF ADIABATIC QUANTUM SEARCH by Artem K. Pimachev A Thesis Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF SCIENCE (CHEMISTRY) December 2009 Copyright 2009 Artem K. Pimachev Table of Contents List of Figures iii Abstract iv Preface v Introduction 1 States and Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Adiabatic Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Adiabatic Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Chapter 1: Isospectral Hamiltonians 7 Projective Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Chapter 2: The Rotating Axes Representation 14 Chapter 3: Performance 20 Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 Constant Energy Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Chapter 4: Grover’s Hamiltonian 24 Uniform Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Non-uniform Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Essential Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Conclusion 41 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Bibliography 43 ii List of Figures Figure 1: Uniform rotation withr = 64. TOP: (T) obtained by solv- ing numerically the system of equations (solid) along with inte- gral representation (108) (dashed). Dashed asymptote is 2'/T + sech[ T 4' √ r ]/2, solid asymptote is 2'/T + sech[ T 4' √ r ]. BOTTOM: Difference between numerical calculation of(T) and integral rep- resentation (108). Dashed line isc/T 2 . . . . . . . . . . . . . . . . 31 Figure 2: Uniform rotation. TOP: ∣ℐ()∣ as a highly oscillatory func- tion, along with the simplified integral (111). The dashed line is the asymptote2/T +sech[ T 4' √ r ]/(2'). BOTTOM:(T) obtained by solving numerically the system of equations. The dashed line is 2'/T +sech[ T 4' √ r ]. . . . . . . . . . . . . . . . . . . . . . . . . . 33 Figure 3: (T) obtained by solving numerically the system of equations. TOP: withk = 1. Dashed line is2'∣ (2) 1 (1)∣/T 2 +sech[ C 1 T ' √ r ]. BOT- TOM: with k = 2. Dashed line is 2'∣ (3) 2 (1)∣/T 3 + sech[ C 2 T ' √ r ], C 2 <C 1 </4. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Figure 4: (T) obtained by solving numerically the system of equations for r = 2 12 to 2 14 , = 1. TOP: with = 0.3. Dashed line is sech[C 3 T/(' √ r)]. BOTTOM: with = 0.05. Dashed line is sech[C 4 T/(' √ r)],C 3 <C 4 </4. . . . . . . . . . . . . . . . . . 40 iii Abstract We describe a class of the Hamiltonians, for which the local adiabaticity condi- tion is satisfied automatically. We consider the controllable adiabatic interpolations between the initial and final Hamiltonians. We show that the quantum search by the adiabatic quantum computation can be mapped into this class. Assuming that the interpolation is infinitely differentiable and the target adiabatic ground state is non- degenerate and separated by a gap from the rest of the spectrum, it is shown that one can obtain an exponentially small error in the algorithm run time T between the finale adiabatic eigenstate and the actual state of the system. The optimal for quantum search scaling ofT as square root of the system size is also satisfied. iv Preface In Chapter 1 the Adiabatic Theorem is discussed. Following the basic approaches adopted from standard Quantum Mechanics textbooks (e.g. Messiah), we describe global and local adiabaticity conditions, which are extremely important for the Adi- abatic Quantum Computation (AQC). Chapter 2 deals with the special class of the Hamiltonians, which we call gener- alized isospectral Hamiltonians. This is the generalization of the unitary evolution proposed by Stewart Siu. This class is found to be very reach in the context of AQC. Chapter 3 considers another class of the Hamiltonians, which we call projective. Under certain conditions, they fall into the class, described in Chapter 2. We also discuss a peculiar property they have, directly related to the Adiabatic Theorem. Chapter 4 describes the way of measuring the distance between two given states of a system. We define an error of the quantum algorithm. This defined error will be used later as the computational error of the Adiabatic Quantum Search. In Chapter 5, we formulate the Adiabatic version of the quantum algorithm for searching in a unstructured database discovered by Lov K. Grover. In particular, we show its relation to the class of the discussed Hamiltonians. At the end, we solve the Schrodinger equation, providing its asymptotic solution for a large run time of the algorithm. v Introduction One may address the question why we need to appeal to Natural Sciences in order to do computation, while the Computer is described well mathematically. Quantum Computation relies on the assumption, that a computer is a physical system. Hence the laws of nature should be applied. The development of Quantum Mechanics was based on the fact, that we cannot determine certain parameters on small scales. It is then crucial to understand these limitations in order to perform a valid computation. We will review the basic laws governing the quantum mechanical processes. States and Evolution We associate to every physical system a complex vector space, a Hilbert spaceℋ, with inner product⟨ 1 ∣ 2 ⟩. It is postulated, that the state of a system is completely defined by a unit vector∣ ⟩ of the spaceℋ. Exact form, basis and dimensionality of a Hilbert space are not specified and vary for the different physical systems. We will restrict ourselves to a finite dimensional Hilbert space. For any to states∣ 1 ⟩ and ∣ 2 ⟩ we can construct a linear combinationa∣ 1 ⟩+b∣ 2 ⟩, which is also an element ofℋ. Normalization require that∣a∣ 2 +∣b∣ 2 = 1. Moreover, we can add the phases and still keep this state in a Hilbert space. For example,e i 1 (a∣ 1 ⟩+e i 2 b∣ 2 ⟩) is in the sameℋ. The phase 1 is called ”global” and does not affect on the evolution or 1 the measurement, associated with a state. However, the relative phase 2 dramati- cally changes the state and it is relevant to the evolution and therefore to quantum computation. A closed quantum system by definition does not interact with any environment. The evolution of the states is described by the Schrodinger equation: i d dt ∣ (t)⟩ =H(t)∣ (t)⟩, (1) where we set~ = 1. The operatorH(t) is hermitian, i.e.,H † (t) =H(t), where we use† for a complex conjugated and transposed operator. H(t) is called the Hamil- tonian of the system, which needs to be specified individually in each problem we will consider. In the case when the Hamiltonian is time independent, the Schrodinger equation can be integrated exactly: ∣ (t 1 )⟩ =e i(t 1 −t 0 )H ∣ (t 0 )⟩ =U(t 0 ,t 1 )∣ (t 0 )⟩, (2) defining the time evolution of the state∣ (t)⟩ fromt 0 tot 1 . The operatorU(t 0 ,t 1 ) is called the evolution operator. Such an operator can always be defined even if the Hamiltonian is time dependent. Form unitarityU † (t 0 ,t 1 ) =U −1 (t 0 ,t 1 ) it is follows that every quantum process can be reversed. So far we described the system without any interaction, however, if we want to perform a measurement on the system, we must interact with it. We introduce the orthogonal operatorsP m , such thatP m P m ′ = mm ′P m , acting in the Hilbert space. They satisfy a completeness relation X m P m =I. (3) 2 By the indexm we understand the results obtained in the experiment. Each physical property, which can be measured corresponds to a Hermitian operator M, called observable. It has the following spectral decomposition: M = X m mP m , (4) where P m is a projector on the eigenspace with a corresponding eigenvalue m. Quantum Mechanics postulates, that as a result of the measurement, performed on the observable M, we would obtain its eigenvalues m, and that the state of the system after the measurement will belong to the eigenspace ofM corresponding to m. Adiabatic Evolution When the Hamiltonian of the system is time dependent, the Schrodinger equation cannot in general be analytically integrated. One standard approach to finding an approximate solution is to assume, that the Hamiltonian varies slowly with time. The adiabatic process will take the system, if it is initially prepared in the ground state, to to a state that is arbitrary close to the instantaneous ground state of the Hamiltonian. This approach was found to be extremely useful in a wide range of fields such as atomic and molecular physics. In quantum chemistry, for example, it is basis of the Born-Oppenheimer approximation. Adiabatic Theorem The adiabatic theorem was formulated in the early years of quantum mechanics [2] and is known to be an important concept, helped to get deeper insight into physical phenomena, such as quantum Hall effect [1], etc. In order for the dynamics 3 to follow spectral subspaces it is necessary for the energy bands to be separated by gaps, however the spectrum does not need to be discrete. Kato [9] proved an adiabatic theorem for Hamiltonians with continuous spectra, for a system starting in a spectral subspace of a finitely degenerate discrete eigenvalue. Later Garrido and Sancho [5] extended Kato’s results to higher orders in the inverse of the evolution time. Recently the adiabatic theorem has garnered renewed attention and has become a field for discussions. Its validity was doubted and standard treatment of it was proven not to guarantee the correct result [11]. We find the solution to the Schrodinger equation asymptotically for large evolution time. In combination with the standard adiabatic theorem, we obtain and analyze the results. The system is described by its quantum state∣ ()⟩, where =t/T is a scaled time (0≤t≤ 1) and quantum evolution proceeds according to i d d ∣ ()⟩ =TH()∣ ()⟩, (5) whereH() is the Hamiltonian of the system. LetH 0 andH 1 be initial and final Hamiltonians of the system, respectively. Adopting and generalizing [4] we choose an interpolation path, continuously connectingH 0 andH 1 : H(t) =x 1 ()H 0 +x 2 ()H 1 . (6) The boundary conditions require thatx 1 (0) = x 2 (1) = 1 andx 1 (1) = x 2 (0) = 0. This describes a very broad class of the Hamiltonians. For example, in the Ising model one would haveH 0 as a global field, or a sum over i (the Pauli matrices) andH 1 as an interaction, or a sum over the nearest neighbors i j with a coupling constant. 4 We define∣E 0 ,⟩ and∣E 1 ,⟩ to be the ground state and the first excited state ofH() at time with corresponding eigenvaluesE 0 () andE 1 (). Let∥dH/d∥ be the operator norm of the derivative of the Hamiltonian. By the operator norm we mean its largest eigenvalue (more generally, its largest singular value). The adiabatic theorem [13] states that for 0<"≪ 1 ∣⟨E 0 ,1∣ (1)⟩∣ 2 ≥ 1−" 2 , (7) provided that dH d max TΔ 2 min ≤", (8) whereΔ min is a minimal gap betweenE 0 () andE 1 (). This is the so called ”global adiabatic condition”, applicable for the entire evolution intervalT . While the condition (8) imposes a limit on the evolution rate during the whole computation, this limit is only important when the energy gap attains its minimum. It is then necessary to divideT into infinitesimal intervalsdt and apply the condition on each interval. This will allow us to continuously vary the evolution rate and speed up the computation. When the Hamiltonian is defined by one interpolating functions(), the local adiabaticity condition defines a new evolution rate ds/dt. When the condition (8) is applied to every infinitesimal interval, we obtain the computation rate: ds d ≤" TΔ 2 dH ds (). (9) We can generalize this approach further. Let() is some arbitrary function of time, satisfying the boundary conditions (0) = 0 and (1) = 1. This modified 5 scaled time allows us to manipulate with the evolution rate. Excluding the non- physical cases we restrict ourselves to ˙ ()≥ 0. The adiabaticity condition is now applied similarly to the infinitesimal intervalsd: ds d ≤" TΔ 2 dH ds (). (10) Along with computer science and algorithm analysis, the adiabatic theorem gave birth to a new powerful tool - adiabatic quantum computation. Evolution of a quantum algorithm is governed by a Hamiltonian, varying continuously and slowly. The standard approach is to connect initial and final Hamiltonians using some interpolation. It is convenient to choose a simple initial Hamiltonian with known ground state. If we encode the solution of the algorithm in the ground state of the final Hamiltonian and let system evolve sufficiently slowly then the adiabatic theorem ensures that the final state of a system will be arbitrarily close to the ground state of the final Hamiltonian. An important example of adiabatic quantum evolu- tion was proposed by Roland and Cerf [16], building on an algorithm due to Grover [7]. 6 Chapter 1: Isospectral Hamiltonians To perform an adiabatic quantum computation, one should specify the Hamiltonian, bringing the system to the desired state. For instance, the ground eigenstate of the system at the final time instance is the solution to the problem. We find appealing the class of the Hamiltonians, constructed by the means of a time dependent similarity transformation: H() = A −1 ()H (A) A(), whereH (A) is the initial Hamiltonian. This is a spectrum-preserving evolution, i.e., the eigen- values of H() are constant during the entire period and equal to those of H (A) . These Hamiltonians were found to be useful in the context of an adiabatic quantum computation [17]. We generalize this class by allowing the spectral gap to change with time. We write: H() =A −1 ()H (A) ()A(), (11) where H (A) () = diag{E 0 (),E 1 (),...}. (12) In other words, the transformation (11) not only takes the eigenstates of theH (A) from the beginning to the end of an evolution, but also adjusts the distance between the corresponding eigenvalues{E 0 (),E 1 (),...} in some manner. We will show that some Hamiltonians can be mapped into this class. One of them is of a particular interest - quantum search algorithm [7],[16]. 7 We define the energy distance between ground state and first excited state to be E 1 ()−E 0 () ≡ Δ(). For simplicity, we assume thatE 0 () andE 1 () are not degenerate and the rest of the spectrum is degenerate. Also, we request, that the total energy stays constant during the entire evolution. LetE 1 ()+E 0 ()≡ 1 and E j () ≡ 1, forj > 2. Under these conditions the Hamiltonian can be written as follows: H() =A −1 () 1 2 [I 2×2 +Δ() z ]⊕I 2,...,N−1 A(). (13) Effectively, this Hamiltonian is two-dimensional and the dynamics is governed by a 2×2 part of the unitary operator. H() = 1 2 I 2×2 +Δ()A −1 2×2 () z A 2×2 () ⊕I 2,...,N−1 . (14) We will restrict ourself to the case of the real-valued Hamiltonians. It implies that the operator A 2×2 () is orthogonal with detA 2×2 () = 1. In general, such two- dimensional operator can be written as: A 2×2 () = ⎛ ⎝ a() p 1−a 2 () − p 1−a 2 () a() ⎞ ⎠ . (15) Following the prescription given by the adiabaticity condition, we calculate dH d Δ 2 () = q Δ 2 1−a 2 da d 2 + 1 4 dΔ d 2 Δ 2 (). (16) Requesting the right hand side of the equation to be time-independent, we satisfy the local adiabaticity condition along the entire period of evolution. The solution yields: dH d Δ 2 () = ' , (17) 8 witha() = cos['()] and Δ() = sec['(1− 2())] or Δ() = , where' and are some arbitrary constants. We see from here and the condition (10), that the run-timeT has the following complexity: T ≥ ' " . (18) In a more compact notation the operatorA() can be rewritten as a rotation: A() =e i'()y . (19) Hence, the evolution of the system can be described as a rotation by an angle2' in theXZ plane. Projective Hamiltonians A useful and remarkable subclass of the generalized isospectral Hamiltonians can be constructed as an interpolation between two projectors: H() =x 1 ()P a +x 2 ()P b , (20) where P a = I −∣a⟩⟨a∣, and P b = I −∣b⟩⟨b∣ for some normalized vectors ∣a⟩ and∣b⟩ ∈ ℋ, dimℋ = N. The boundary conditions on the Hamiltonian require that x 1 (0) = x 2 (1) = 1 and x 1 (1) = x 2 (0) = 0. This is a so called ”oracular” Hamiltonian. The oracle which checks, whether the current state of the system is a solution, is already included in the Hamiltonian. The cost of that is that we have to wait for the algorithm to work for a runtime T in order to obtain the solution with some error. We are not supposed to know what is the solution∣b⟩ a priori, so this vector is arbitrary. The final Hamiltonian, which is a projector on the subspace 9 orthogonal to∣b⟩ will have a ground state energy zero, when a state of the system approaches the solution. Hence, by measuring whether the system is in its ground state it is possible to determine when a solution is obtained. It is possible to show, that H [Eq. (20)] is effectively two-dimensional. For a given∣a⟩ we are free to choseN−1 vectors{∣a i ⟩} N−1 i=1 such that⟨a∣a i ⟩ = 0 and⟨a i ∣a j ⟩ = ij which span an orthonormal basis forℋ. This way we can decompose∣b⟩ = 0 ∣a⟩+ P N−1 i=1 i ∣a i ⟩. Using the freedom of choice for{∣a i ⟩}, by rotation we can make i>1 = 0, so that ∣b⟩ = 0 ∣a⟩+ 1 ∣a 1 ⟩, (21) with 0 = ⟨a∣b⟩ and 1 = ⟨a 1 ∣b⟩ = e i p 1−∣⟨a∣b⟩∣ 2 for an arbitrary ∈ [0,2) (for simplicity we choose = 0). Using the completeness of the{∣a⟩,∣a i ⟩} N−1 i=1 basisI =∣a⟩⟨a∣+ P N−1 i=1 ∣a i ⟩⟨a i ∣ andI 2,...,N−1 = P N−1 i=1 ∣a i ⟩⟨a i ∣ we now can rewrite the Hamiltonian: H() = ⎛ ⎝ x 2 ()∣ 1 ∣ 2 −x 2 () 0 ¯ 1 −x 2 ()¯ 0 1 x 1 ()+x 2 ()∣ 0 ∣ 2 ⎞ ⎠ ⊕[x 1 ()+x 2 ()]I 2,...,N−1 , (22) where the first matrix is written in the effective{∣a⟩,∣a 1 ⟩} basis. We can find a similarity transformation, which diagonalizes the two-dimensional matrix in (22), i.e., brings it to the formA −1 DA, whereD = diag{E − (),E + ()}. We find: A = ⎛ ⎝ a − a + −a + a − ⎞ ⎠ , (23) where a ∓ = 1 r 1+ E ∓ ()−x 2 ()(1−∣ 0 ∣ 2 ) x 2 ()∣ 0 ∣∣ 1 ∣ 2 , (24) 10 with E ∓ () = 1 2 x 1 ()+x 2 ()∓ p (x 1 ()−x 2 ()) 2 +4∣⟨a∣b⟩∣ 2 x 1 ()x 2 () . (25) It is easy to see from (20) and (25), that the operator norm ofdH/d is just dH d () = 1 2 ˙ x 1 ()+ ˙ x 2 ()+ p (˙ x 1 ()− ˙ x 2 ()) 2 +4∣⟨a∣b⟩∣ 2 ˙ x 1 ()˙ x 2 () . (26) One can show that, regardless of a chosen interpolationx 1 (),x 2 (),A is orthog- onal, i.e., A −1 = A T , and det(A) = 1. The Hamiltonian (22) has three different eigenvalues: {x 1 () +x 2 (),E ∓ }. The gap between the ground state and the first excited state is as follows: Δ() = p (x 1 ()−x 2 ()) 2 +4∣⟨a∣b⟩∣ 2 x 1 ()x 2 (). (27) With everything said above we can rewrite (22): H() = 1 2 (x 1 ()+x 2 ())I−Δ()A −1 () z A() ⊕[x 1 ()+x 2 ()]I 2,...,N−1 . (28) The evolution of the Hamiltonian can be forced to be a spectrum-preserving unitary evolution by demanding that x 1 () +x 2 () = 1. Let x 2 () ≡ s(), which we call an interpolating function. Using local adiabaticity condition (10), we solve for the optimal computational paths(). By optimal we mean such path, along which the evolution stays in the adiabatic regime. In principle, we allow s() to be a composite function, i.e.,s()≡s(()). For the Hamiltonian above we have: dH d Δ 2 () = p 1−∣ 0 ∣ 2 1−4(1−∣ 0 ∣ 2 )s()(1−s()) s() d . (29) 11 The energy gap has its minimum whens() = 1/2. The global adiabaticity condi- tion (8) assumes constant evolution rates() =. T , the run time is then T ≥ p 1−∣ 0 ∣ 2 "∣ 0 ∣ 2 . (30) In order to satisfy the local adiabaticity condition, we solve fors() the following differential equation: p 1−∣ 0 ∣ 2 1−4(1−∣ 0 ∣ 2 )s()(1−s()) ds() d =C, (31) whereC is constant, which can be found from the boundary conditions. The solu- tion is: s() = 1 2 − ∣ 0 ∣ 2 p 1−∣ 0 ∣ 2 tan['(1−2())], (32) with ' = arctan " p 1−∣ 0 ∣ 2 ∣ 0 ∣ # . (33) Similar interpolation path can be found in [15]. We have obtained a generalized form for an arbitrary function(). Plugging this solution into Eq. (29) we find the complexity of the evolution timeT as: dH d Δ 2 () = ' ˙ () ∣ 0 ∣ . (34) The condition (10) with the modified time() gives the complexity of the run time as T ≥ ' "∣ 0 ∣ . (35) In comparison with the result (30) of the global condition, the local adiabaticity condition gives a quadratic speed up of the run time depending on the size of the system (∣ 0 ∣≪ 1). As we discussed earlier, the Hamiltonian has the form of unitary 12 evolution, described by the operatorA(). Now we can find the explicit form of this operator with the interpolation (32): H() = 1 2 I +Δ()A −1 () z A() ⊕I 2,...,N−1 , (36) with A() =e i'()y , (37) and Δ() =∣ 0 ∣sec['(1−2())]. (38) The unitary transformation, which satisfies the adiabaticity is then a rotation in the XZ plane with a time dependent spectral gap. We see that the projective Hamilto- nian fits in the class (11). Another important result is that for theΔ() andA() we have found, the local adiabaticity condition (10) is satisfied automatically, without imposing it on the evolution of the system. The Hamiltonians of such form always stay in adiabatic regime. 13 Chapter 2: The Rotating Axes Representation The evolution of the quantum states of a system with the isospectral Hamiltonian is determined by the unitary operator A(). It is convenient to change the frame of reference such way, that the vectors, corresponding to the quantum states, are intact in the new representation. Simultaneously, this change of the representation diagonalizes the Hamiltonian. The Schrodinger equation for the evolution operatorU() acting on a state of the system, described by the HamiltonianH() is as follows: i dU() d =TH()U(), (39) with H() =A −1 ()H (A) ()A(), (40) where A() is a unitary operator with initial conditionA(0) = I andH (A) () = diag{E − (),E + ()}. We are interested in the case of large evolution timeT , while the scaled time changes from = 0 to = 1. We require the eigenvalues of the Hamiltonian, E ∓ (), to be discrete and non-degenerate. In the computa- tional basis the corresponding eigenvectors at the initial time are Φ − (0) = {0,1} and Φ + (0) = {1,0}. The projectors of the corresponding eigenstates are P ∓ (). 14 We suppose that the eigenvalues and the projectors are continuous and infinitely differentiable functions of . The Hamiltonian can be expanded as a sum of the subspaces, corresponding to the different eigenvalues: H() =E − ()P − ()+E + ()P + (). (41) From here we see that the projectors are as follows: P − () =A −1 () ⎛ ⎝ 1 0 0 0 ⎞ ⎠ A(), (42) P + () =A −1 () ⎛ ⎝ 0 0 0 1 ⎞ ⎠ A(). (43) In the trivial case, when the subspaces do not change with time, we would immedi- ately obtain for the evolution operator U() =P − e −iT − () +P + e −iT + () , (44) where ∓ () = Z 0 E ∓ ( ′ )d ′ . (45) In general case, however, the subspaces, corresponding to the eigenvalues of H() change with time. In our case, they rotate in the Hilbert space according to the unitary transformationA(). It is convenient [13] to work in the rotating axes representation, where the time variation of the eigenstates of the Hamiltonian is canceled. In the new representation the projectors evolve as follows: P ∓ () =A −1 ()P ∓ (0)A(). (46) 15 Let us introduce the evolution operator in the new representation: U (A) () = A()U(). LetK() be a Hermitian operator, found from the equation: i dA −1 () d =K()A −1 (), (47) and with an arbitrary condition P ∓ ()K()P ∓ () =O. (48) In the rotating axes representation the HamiltonianH() is diagonal, and we can rewrite equation (39) as follows: i dU (A) () d = TH (A) ()−A()K()A −1 () U (A) (). (49) To approximately solve this equation, we introduce an interaction picture, where H 0 () = TH (A) () is an unperturbed Hamiltonian and H 1 () = −A()K()A −1 () is a perturbation. The solution is written as U (A) () =V 0 ()V 1 (). (50) Plugging it back to the Schrodinger equation we obtain: i dV 0 () d =H 0 ()V 0 (), (51) i dV 1 () d =V −1 0 ()H 1 ()V 0 ()V 1 (), (52) 16 whereV 0 () is the evolution operator for the unperturbed motion. Since the Hamil- toniansH 0 ( 1 ) andH 0 ( 2 ) commute for all times, we can integrate the equation: V 0 () = ⎛ ⎝ e −iT − () 0 0 e −iT + () ⎞ ⎠ . (53) The perturbed solution with the operator K[] = −V −1 0 ()H 1 ()V 0 () satisfies following integral equation: V 1 () = 1+i Z 0 K[ ′ ]V 1 ( ′ )d ′ , (54) which is solved by iteration. The result can be written as follows: V 1 () =I +i R 0 K[ ′ ]d ′ +i 2 R 0 d ′ R ′ 0 d ′′ K[ ′ ]K[ ′′ ]+ ... (55) ...+i n R 0 d ′ R ′ 0 d ′′ ... R ′(n−1) 0 d ′(n) K[ ′ ]...K[ ′(n) ]+ .... (56) The operatorK[] explicitly can be written as K[] =e i( + ()− − ())T A()K ∓ ()A −1 ()+e i( − ()− + ())T A()K ± ()A −1 (), (57) where K ∓ () =P ∓ ()K()P ± (). (58) We now can apply this method to the Hamiltonian (36). The Hermitian operator K() is found from equation (47). Explicitly: K() =' ˙ () y . (59) 17 It is easy to see that the operatorsA() andK() commute. As a consequence of that, the operatorK() is not changed in the rotating axes representation: K() =A()K()A −1 (). (60) From equation (57) with the use of the invariance ofK() we obtain the form of K[]: K[] =i' ˙ () ⎛ ⎝ 0 −e i( + ()− − ())T e i( − ()− + ())T 0 ⎞ ⎠ . (61) The operatorsK[ 1 ] andK[ 2 ] at different times do not commute, in order to obtain an approximate solution, we have to truncate the expansion (56) and analyze the error of the approximation. To simplify notation let us define the following integral transform: ℐ() = Z 0 d ′ ˙ ( ′ )exp " −iT Z ′ 0 Δ( ′′ )d ′′ # , (62) where Δ() is the spectral gap between the eigenvaluesE + () andE − (). In this notation, the first terms of the solutionV 1 () can be written as: V 1 ()≈I +' ⎛ ⎝ 0 ℐ ∗ () −ℐ() 0 ⎞ ⎠ . (63) In order to transform the evolution operator back to the original representation, we apply the operator A() to the solution. The actual state Ψ() of the system, governed by the Schrodinger equation is obtain by applying the evolution operator on the starting state. Combining all the terms we get: Ψ() =U()Φ − (0) =A −1 ()V 0 ()V 1 ()Φ − (0). (64) 18 Summarizing the results of this chapter, we have obtained an expansion of a solution to the Schrodinger equation with the interpolating Hamiltonian (36) in the interaction picture. The approximate solution is found by truncating the expansion at the desired place. 19 Chapter 3: Performance We are now interested how the actual state of the system, given by the solution of the Schrodinger equation, is different from the adiabatic state, which is an eigenstate of the Hamiltonian at the final time instance. Let∣ 0 (1)⟩ denotes the ground eigenstate ofH(1) and∣ 0 (1)⟩ denotes the solution after some evolution periodT , where the system started from the initial state∣ 0 (0)⟩, which is a ground eigenstate ofH(0). The distance∥∣ 0 (1)⟩−∣ 0 (1)⟩∥ is the error of computation. In principle, we cannot measure the state of the system directly. The indirect measurement of the error can be performed be evaluating the observables⟨ 0 (1)∣A∣ 0 (1)⟩ and⟨ 0 (1)∣A∣ 0 (1)⟩, from which the error can be deduced. Error Estimates In order to estimate the error of adiabatic computation after the intervalT we define the error of the state in the following way: (T) = p 1−∣⟨ 0 (1)∣ 0 (1)⟩∣ 2 . (65) This definition of(T) is helpful in the sense that it cancels the difference between the states due to the phase. Indeed, when the vectors are different only by the phase factor, the error approaches zero. We are interested in the envelope of (T) as a function of the evolution timeT . For the Hamiltonian (36) the effective eigenstate 20 ∣ 0 (1)⟩ = A −1 (1)∣ 0 (0)⟩, while∣ 0 (1)⟩ can be obtained from (64) neglecting all the phase-like prefactors. The resulting error is as follows: (T)≈'∣ℐ(1)∣. (66) The third term of the expansion (56) produces the integral transform: ℐ 1 () = Z 0 d ′ ˙ ( ′ )ℐ( ′ )exp " iT Z ′ 0 Δ( ′′ )d ′′ # . (67) Simple integration by parts is helpful for an analysis of a validity of the approxima- tion. We rewrite the integral in the following way: ℐ 1 () = ˙ ()ℐ() exp iT R 0 Δ( ′ )d ′ iTΔ() − Z 0 d h ˙ ( ′ )ℐ( ′ ) i exp h iT R ′ 0 Δ( ′′ )d ′′ i iTΔ( ′ ) . (68) The contribution of the first term is of the higher order in1/T then the one ofℐ(). The integration by parts may go on, expanding the last term. It is useful for evalu- ating the magnitude of the integralℐ 1 (1) when the condition of zero derivatives is applied. Indeed, if the derivative of the Hamiltonian at the initial and the final time is zero, then ˙ (1) = 0 and the first non-integral term of the sum vanishes. Constant Energy Gap We have shown in Chapter 2, that the Hamiltonians (14) with the rotation inXZ plane stay in the adiabatic regime if the energy gap between the ground state and the first excited state is constant. In particular, we letA() =e i'y and Δ() =. In 21 this case the Schrodinger equation can be integrated exactly. Following the method of changing the representation, we obtain: K[] =i' ˙ () ⎛ ⎝ 0 −e iT e −iT 0 ⎞ ⎠ . (69) Equation (52) can be solved analytically. We are interested in the estimation of the computational error(T). Following the developed procedure we obtain: (T) =' sin q 2 T 2 4 +' 2 q 2 T 2 4 +' 2 . (70) In order to evaluate the complexity ofT and relate to the result, obtained from the adiabaticity condition, we expand(T) in the powers of 1/T : (T)≤ 2' T +4 ' T 3 +O 1 T 5 . (71) The evolution time of the adiabatic regime is then T ≥ 2' , (72) which matches with the result (18). So far we assumed, that the unitary operationA() linearly rotates the states in a Hilbert space. This was due to the choice() =. For an arbitrary() this is not the case. Introducing non-linear rotation, we gain more control on the system. We can show, how the use ofk first zero derivatives of the Hamiltonian at the initial and 22 final time instances modifies the scaling of the evolution timeT . Using the method developed in [8], which we will discuss later, we obtain the following result: (T)≤ 2'∣ (k+1) (1)∣ (T) k+1 +O 1 T k+2 . (73) We have demonstrated, that for the class of isospectral Hamiltonians with a constant energy gap the local adiabaticity condition gives a good estimate for the evolution timeT . Slightly modified, the result can be generalized for the case of ”non-linear rotation”. Therefore we have confirmed the validity of the adiabatic theorem in the powers of1/T , as it was proved in [5]. As was showed in Chapter 2, the solution with a non-constant gap also satisfies the adiabaticity. It turns out, that the same time-dependent gap appears in the important problems of AQC. Therefore we will analyze in details the approximate solution on a specific example. 23 Chapter 4: Grover’s Hamiltonian We now can apply the theory developed above to an important case - adiabatic quantum search algorithm. First, we start from the simple case of a ”linear rota- tion”, which yields a similar interpolating function as in [16]. However, solving the Schrodinger equation we obtain the result, different from the one proposed by the adiabaticity condition. In particular, after some time of evolution, time does not scale with system size in the expected Grover-like (square-root) manner. Second, we demonstrate the use of k first zero derivatives of the Hamiltonian at the end- points as in [5] and apply the method from [8] to support the result. The last part deals with the case of essential singularities at the endpoints so that the expansion as in [8] is not valid. We demonstrate the way to improve the rate of convergence and extend the exponentially decaying regime. We use the method of steepest descent to obtain the power of exponential term. The time-dependent Schrodinger equation (5) describes the evolution of a sys- tem. We are interested in the solution, or the state of the system after some timeT , not necessarily large. The initial state is prepared as an equal superposition of all basis states ∣ 0 ⟩ = 1 √ N N−1 X i=0 ∣i⟩. (74) The initial Hamiltonian is constructed such that∣ 0 ⟩ is a ground state. As in [16] we adopt H 0 = 1−∣ 0 ⟩⟨ 0 ∣, (75) 24 as the initial Hamiltonian and H m = 1− X m∈ℳ ∣m⟩⟨m∣, (76) as the final Hamiltonian, where∣m⟩ is some ”marked” state andℳ is the set of solutions. Let dimℳ = M. In the computational basis the matrix elements of the HamiltoniansH 0 andH m are: {H 0 } ij = ij − 1 N , (77) {H m } ij = ij − X m∈ℳ mj ij . (78) The time-dependent Hamiltonian of the system is an interpolation between two specified Hamiltonians: H() =x 1 ()H 0 +x 2 ()H m , (79) with the boundary conditionsx 1 (0) = x 2 (1) = 1 andx 1 (1) = x 2 (0) = 0. There are three different eigenvalues:E() =x 1 ()+x 2 () has degeneracyN−2 and E ∓ () = 1 2 x 1 ()+x 2 ()∓ r (x 1 ()−x 2 ()) 2 + 4M N x 1 ()x 2 () ! , (80) are singly degenerate. Similarly the operator norm of the time derivative of the Hamiltonian is found to be ∥ ˙ H∥() = max(˙ x 1 ()+ ˙ x 2 (), (81) 1 2 (˙ x 1 ()+ ˙ x 2 ()+ r (˙ x 1 ()− ˙ x 2 ()) 2 + 4M N ˙ x 1 ()˙ x 2 ())). 25 The energy gap between the non-degenerate eigenvalues is then Δ() = r (x 1 ()−x 2 ()) 2 + 4M N x 1 ()x 2 (). (82) We can make all degenerate energies become constant by takingx 1 ()+x 2 () = 1 and, to simply the notation, redefine x 2 () ≡ s(). In accordance with the local adiabaticity condition (10) we demand: dH d Δ 2 () =C, (83) for some constantC, which in general may depend on the size of the systemN,M. Now we have only one parameters() to solve for: r N−M N 1 1−4(1−M/N)s()(1−s()) ds d =C, (84) with the boundary conditionss(0) = 0 ands(1) = 1. We find s() = 1 2 − √ M 2 √ N−M tan['(1−2())], (85) where ' = arctan " r N M −1 # . (86) We can think of ' as an angle, by which the system rotates during the evolution. 0 < ' < /2 for anyN > M. With the particular interpolation functions() we obtain: Δ() = r M N sec['(1−2())], (87) and dH d () =' r M N sec 2 ['(1−2())]. (88) 26 It is now easy to see that dH d Δ 2 () =' r N M . (89) Here the adiabatic condition comes into play. According to the local adiabatic theorem, we are guaranteed to have the evolution time scaling as p N/M which is known to be optimal for a quantum search algorithm [7],[16]. We can check that such an interpolating function brings our Hamiltonian into the form of a unitary rotation. The time-dependent Hamiltonian of the system is a linear interpolation between two specified Hamiltonians: H() = (1−s())H 0 +s()H m . (90) In terms of the matrix elements in the computational basis, the Schrodinger equation becomes the following coupled system of equations: i d d i () =T N X j=1 " f 1 () ij − 1 N +f 2 () ij − X m∈ℳ mj ij !# j (). (91) After evaluating the sum overj we obtain: i d d i () =T " f 1 ()+f 2 ()−f 2 () X m∈ℳ mi ! i ()− f 1 () N N X j=1 j () # . (92) We note that there are two sets of different equations here, one corresponds to the the unmarked states, and other to the marked states. Since the initial conditions are the same for each equation, the solution is the same at any time instance. Based on that we can sum over the same solutions: N X i=1 i () = (N−M) u ()+M m (), (93) 27 where the indexesu andm represent the unmarked and marked states. This reduces our problem to two dimensions. We now need to solve only for the unmarked and marked states, i.e., i d d u () =T f 2 ()+f 1 () M N u ()−f 1 () M N m () , (94) i d d m () =T f 1 () M−N N u ()+f 1 () N−M N m () . (95) The reduced pseudo-Hamiltonian in the basis{ u (), m ()} is now a2×2 matrix: ˜ H() = f 1 () N ⎛ ⎝ M −M M−N N−M ⎞ ⎠ +f 2 () ⎛ ⎝ 1 0 0 0 ⎞ ⎠ . (96) The Schrodinger equation for the reduced two dimensional problem is i d d ∣ ˜ ()⟩ =T ˜ H()∣ ˜ ()⟩, (97) where∣ ˜ ()⟩ = u () m () and the initial condition u (0) = m (0) = 1/ √ N. The solution of the non-reduced N dimensional Grover’s problem at the final time instance is constructed from the solution to the reduced problem as fol- lows: ∣ (1)⟩ = u (1) ... u (1) m (1) ... m (1) , (98) where the number of the unmarked and marked basis elements isN −M andM respectively, and∥∣ (1)⟩∥ = 1. In order to find the computational error we com- pare this solution to the ground state of the Hamiltonian at the final time instance: ∣ m ⟩ = 0 ... 0 1/ √ M ... 1/ √ M , where the number of entries is as 28 above and ∥∣ m ⟩∥ = 1. The difference between the eigenstate of the Hamilto- nian at the final time and the actual solution of the Schrodinger equation after the evolution timeT is (T) = p 1−∣⟨ (1)∣ m ⟩∣ 2 = q 1−∣ √ M m (1)∣ 2 = q 1−∣ ˆ m (1)∣ 2 , (99) where ˆ m (1) is the second element of the solution of the Schrodinger equation i d d ∣ ˆ ()⟩ =T ˜ H()∣ ˆ ()⟩, (100) with the initial conditions∣ ˆ (0)⟩ = p M/N p M/N . We have already found the form of the interpolating functions() in Eq. (85). Plugging this into the expression for ˜ H() we get ˜ H() = 1 2 ⎛ ⎝ 1+M/N −M/N −1+M/N 1−M/N ⎞ ⎠ + (101) + √ M √ N−M 2N tan['(1−2())] ⎛ ⎝ −1 −M/(N−M) −1 1 ⎞ ⎠ . We can extract a symmetric form of this pseudo-Hamiltonian by the similarity trans- formation with S = ⎛ ⎝ p 1−M/N − p 1−M/N p N/M(1−M/N) p M/N ⎞ ⎠ . (102) After some algebra the pseudo-Hamiltonian can be written in the form: ˜ H() =S −1 I 2 + 1 2 Δ()A −1 () z A() S, (103) 29 with A() = exp[i'() y ], (104) and Δ() is given by Eq. (87). This is in the form of the unitary rotation with a time-dependent gap. We can separate the rotation and similarity transformation and plug it back into the Schrodinger equation. With the Hamiltonian ˜ H rot () = I 2 + 1 2 Δ()A −1 () z A(), (105) we have i d d S∣ ˆ ()⟩ =T ˜ H rot ()S∣ ˆ ()⟩, (106) with the same initial conditions as before. Everywhere in the problem N and M occur only as a ratio, so we do not need to take care of them independently, let r =N/M. We can analyze the solution of this equation in a similar manner to Eq. (62), evaluating the dominant term of the expansion for large parameterT . ℐ() = R 0 ˙ (s)exp −iT R s Δ(s ′ )ds ′ ds, (107) (T)≈'∣ℐ(1)∣. (108) We plot the numerical value of(T), found from the Schrodinger equation, and its approximation given by (108) (FIG. 1). Uniform Rotation This assumes that () changes linearly with time, i.e, () = and ˙ () = 1. After evaluating the integral inℐ() overΔ() defined in (87), we have: ℐ() = Z 0 1−sin(')+cos(')tan('s) 1+sin(')−cos(')tan('s) −iT/(2' √ r) ds. (109) 30 r=64 r=64 100 50 200 30 150 70 T 0.50 0.20 0.10 0.05 0.02 r=64 r=64 100 1000 500 200 300 150 700 T 5´ 10 -5 1´ 10 -4 5´ 10 -4 0.010 0.005 0.001 Figure 1: Uniform rotation with r = 64. TOP: (T) obtained by solving numerically the system of equations (solid) along with integral representation (108) (dashed). Dashed asymptote is 2'/T +sech[ T 4' √ r ]/2, solid asymptote is 2'/T +sech[ T 4' √ r ]. BOTTOM: Difference between numerical calculation of(T) and integral representation (108). Dashed line isc/T 2 . To simplify this expression we evaluate only the contribution of the term in the numerator (FIG. 2); for largeT this integral can be bounded by: ∣ℐ()∣⪅ R 0 [tan('s)] −iT/(2' √ r) ds , (110) ∣ℐ(1)∣⪅ 1 2' exp h − T 4' √ r i B −tan 2 (') 1 2 − iT 4' √ r ,0 , (111) where B z (a,b) is an incomplete beta function [6]. For Re(a) > 0, Re(b) > 0 and ∣z∣< 1 B z (a,b) = Z z 0 t a−1 (1−t) b−1 dt. (112) 31 A complete beta function B(a,b) stays for the case whenz = 1. Using the following identity for the beta function and the asymptotic expression for large argument: B z (a,0) = (−1) 2a B1 z (1−a,0)+(−1) a B(a,1−a) (113) B z (a,0) = z a a(1−z) +O z a+1 a 2 (1−z) 2 (114) we can estimate the value of the integral (111) for large evolution timeT as follows: (T)≤ 2' T + 2 sech T 4' √ r +O 1 T 2 . (115) The exponentially decaying sech term becomes suppressed very fast compared to the first order polynomially decaying term in the evolution timeT . From here we see that ifT approaches infinity as √ r or faster, the error approaches zero, and the adiabatic theorem is satisfied. If T approaches infinity slower than √ r, the error does not converge to zero. This approximate calculation gives us two main results: the coefficient of the polynomially decaying term, and the power of the exponentially decaying term. We will justify these results later. The coefficient of the exponentially decaying term may be dropped, as we will demonstrate later (FIG. 1), (FIG. 2). Non-uniform Rotation We have assumed that the rate of a unitary operation is constant. In principle, we are free to choose any(), satisfying the boundary conditions. Regardless of our choice, the dependence of the computation time T on the size of the system r is supposed to be optimal. However, such freedom affects the bound on the compu- tational error (T). When () is some non-decreasing function of time, as was mentioned earlier, the error is given by equation (108). 32 r=1024 r=1024 100 1000 500 200 2000 300 150 1500 700 T 0.500 0.100 0.050 0.010 0.005 0.001 r=2080 r=2080 r=64 r=64 10 20 50 100 200 500 1000 T 10 -4 0.001 0.01 0.1 1 ΔHTL Figure 2: Uniform rotation. TOP: ∣ℐ()∣ as a highly oscillatory function, along with the simplified integral (111). The dashed line is the asymptote 2/T + sech[ T 4' √ r ]/(2'). BOTTOM: (T) obtained by solving numerically the system of equations. The dashed line is 2'/T +sech[ T 4' √ r ]. Consider the following example: k () = B (1+k,1+k) B(1+k,1+k) . (116) Whenk = 0 we have the case of uniform rotation described above. Ifk is an integer greater than zero, then for all integersn≤k d n k () d n ={0,1} = 0, (117) where the time derivative is computed at the initial and the final time instances. From [5] we expect that(T)∼O(T −k−1 ). 33 We can find the coefficient of the leading term by using the approach described in [8]. Let Φ() be a smooth normalized eigenvector of H() corresponding to the ground state energy E(). We assume that its phase satisfies the condition ⟨Φ(), ∂ ∂ Φ()⟩ = 0 for each. Adopting the technique of Hagedorn and Joye, we construct an expansion of the solution to the Schrodinger equation i ∂Ψ ∂ =TH()Ψ, (118) in powers of1/T : Ψ(,T) =e −iT R 0 E(s)ds 0 ()+ 1 () T + 2 () T 2 + . . .+ n () T n + ⊥ n+1 () T n+1 . (119) This expansion is finite and becomes more precise for larger n. It needs to be normalized for every given number of termsn. In order to obtain the form of the terms n (), we plug this expansion into the Schrodinger equation and compare the terms of the same order of1/T . Terms of zero order require that [H()−E()] 0 () = 0. (120) To satisfy this equation we can take 0 () = f 0 ()Φ(), where the functionf 0 () is yet to be determined. Terms of order 1/T give the following equation: i ∂ 0 () ∂ = [H()−E()] 1 (), (121) or explicitly: i ∂f 0 () ∂ Φ()+if 0 () ∂Φ() ∂ = [H()−E()] 1 (). (122) 34 This equation is solved by separately examining the components of the equation, that are proportional toΦ() and those that are perpendicular to it. From there with a use of the phase condition on Φ() we obtain: i ∂f 0 () ∂ = 0, (123) and i ∂Φ() ∂ = [H()−E()] 1 (). (124) The first equation implies thatf 0 () is constant. The second equation forces us to choose 1 () =f 1 ()Φ()+ ⊥ 1 (), (125) with yet to be foundf 1 (), and ⊥ 1 () =i[H()−E()] −1 r ∂Φ() ∂ . (126) We define [H()−E()] −1 r to be a reduced resolvent operator ofH on the orthog- onal complement of the span of Φ(). For the higher order terms we have: n+1 () =f n+1 ()Φ()+ ⊥ n+1 (), (127) with f n+1 () =− Z 0 Φ(s), ∂ ⊥ n+1 (s) ∂s ds, (128) and ⊥ n+1 () =i[H()−E()] −1 r f n () ∂Φ() ∂ +P ⊥ () ∂ ⊥ n () ∂ , (129) whereP ⊥ () = 1−∣Φ()⟩⟨Φ()∣. 35 We are interested in the term k+1 ()T −k−1 , which is the leading term, provided that the condition on the firstk zero derivatives is satisfied [5]. Now we can apply the described method to solve equation (106). Explicitly: E() = 1 2 (1−Δ()), (130) is the ground state energy of the system with the Hamiltonian ˜ H rot (). The corre- sponding eigenvector is: Φ() = sin[' k ()] cos[' k ()] . (131) The reduced resolvent operator is as follows: [ ˜ H rot ()−E()] −1 r = P ⊥ () Δ() = 1 2Δ() (1+A −1 () z A()). (132) We are interested in evaluating the magnitude of a vector k+1 (1) fork≥ 0, which is a prefactor ofT −k−1 term of the expansion. From the equations above we get: S −1 n+1 (1) = 0 f n+1 (1) , (133) for 0<n<k, and S −1 k+1 (1) = −2i k+1 ' (k+1) k (1) f k+1 (1) . (134) Althoughf k (1) can be large, its contribution to the k-th order vanishes when we normalize Ψ(1,T). Using a definition for the distance between the adiabatic and exact solutions as: (T) = s 1− ∣⟨Ψ(1,T)∣ 0 (1)⟩∣ 2 ∥Ψ(1,T)∥ 2 (135) 36 we find that (T) = s 1− ∣1+f 1 (1)/T + ...+f k (1)/T k ∣ 2 ∣1+f 1 (1)/T + ...+f k (1)/T k ∣ 2 +(2' (k+1) k (1)/T k+1 ) 2 = (136) = 2'∣ (k+1) k (1)∣ T k+1 +O 1 T k+2 . This result matches with the one, obtained for the case of uniform rotation(k = 0) (115). We can show numerically, that this bound works fork > 0. We plot (FIG. 3) the polynomially decaying term along with the exponentially decaying term, similarly to the case of the uniform rotation. r=2080 r=2080 r=64 r=64 r=4096 r=4096 0 1000 2000 3000 4000 5000 T 10 -8 10 -6 10 -4 0.01 ΔHTL r=2080 r=2080 r=64 r=64 r=4096 r=4096 0 2000 4000 6000 8000 T 10 -12 10 -10 10 -8 10 -6 10 -4 0.01 ΔHTL Figure 3:(T) obtained by solving numerically the system of equations. TOP: with k = 1. Dashed line is 2'∣ (2) 1 (1)∣/T 2 +sech[ C 1 T ' √ r ]. BOTTOM: withk = 2. Dashed line is 2'∣ (3) 2 (1)∣/T 3 +sech[ C 2 T ' √ r ],C 2 <C 1 </4. 37 We see, that increasing number of first zero derivatives of the Hamiltonian at initial and final time instances, two regimes of(T) become strongly pronounced. Before some critical valueT c (r,k) the error decays exponentially, as the evolution goes on. In terms of the algorithm complexity,T ≈C(k)log[1/(T)] √ r, forT < T c (r,k) andr >> 1. However, whenT >T c (r,k) the computational error decays polynomially, T ≈ ˜ C(k)(T) −1/k , and the optimal scaling with the size of the database is vanished. Therefore, it is important to extend the first regime of the evolution. Essential Singularities The previous case assumed, that () has a Taylor series expansion everywhere around the endpoints. However, this expansion would not be possible to have if we introduce essential singularities. For example, at the point = 0 we want to have a singularity of the type()≈ exp(−/∣ 0 −∣ ). In order to satisfy the boundary conditions, () = R 0 f(s,,)ds R 1 0 f(s,,)ds . (137) As an example we can choose a symmetric ”bump” function: f(,,) = exp − (1−) . (138) We can estimate the rate of convergence by the method of steepest descent. Let for simplicity = 1. Again, with the use of the approximate integral (111) assuming that the main contribution comes from the region near the endpoints. We calculate integral transform of ˙ (): Z 1 0 ˙ ()[tan(')] −iT/(2' √ r) d. (139) 38 We are looking for a saddle point of the function g() = − (1−) −iT/(2' √ r)log[tan(')]. (140) With the assumption∣∣ ≪ 1, which turns out to be valid at our saddle point we write g()≈ − −−iT/(2' √ r)log[']. (141) At a saddle point 0 , dg()d = 0. We find that 0 = 2'/(iT √ r). Due to the imaginary component of 0 the argument of the logarithm gives a prefactor /2 at the power of the exponent. So when → 0 the leading term becomes exp[−T/(4' √ r)]. This matches with the exponentially decaying term obtained earlier for the case of a uniform rotation (115). 39 æ æ æ æ æ æ æ æ æ æ æ à à à à à à à à à à à ì ì ì ì ì ì ì ì ì ì ì ò ò ò ò ò ò ò ò ò ò ò ô ô ô ô ô ô ô ô ô ô ô ç ç ç ç ç ç ç ç ç ç ç 3000 4000 5000 6000 7000 T 10 -11 10 -9 10 -7 10 -5 0.001 ΔHTL æ æ æ æ æ æ à à à à à à ì ì ì ì ì ì ò ò ò ò ò ò ô ô ô ô ô ô ç ç ç ç ç ç 2500 3000 3500 4000 T 10 -12 10 -10 10 -8 10 -6 10 -4 ΔHTL Figure 4:(T) obtained by solving numerically the system of equations forr = 2 12 to 2 14 , = 1. TOP: with = 0.3. Dashed line is sech[C 3 T/(' √ r)]. BOTTOM: with = 0.05. Dashed line issech[C 4 T/(' √ r)],C 3 <C 4 </4. 40 Conclusion Satisfying the adiabaticity condition is important to improve the quantum algo- rithms, based on AQC. There is a rich class of Hamiltonians, which always stay in the adiabatic regime. Grover’s adiabatic quantum search can be represented by one of them. The dynamics of the adiabatic quantum search algorithm was described. For a path, analytic everywhere on the the interval[0,T], the error has two decaying regimes: exponential and polynomial, and the exponential regime can be extended by increasing the power of the polynomial decay by setting the first derivatives of the Hamiltonian to be zero at the initial and final instances of the evolution. We have showed, that one can improve the rate of the exponentially decaying errors by introducing a non-analyticity at the endpoints of an evolution interval. Outlook We have demonstrated, that for the adiabatic quantum search there is a regime, when the error of the algorithm is bounded as: (T)≤e −C TΔ 2 min ∥ dH d ∥ max . (142) This behavior may be very general and should be studied in other examples of the complex quantum systems. 41 We have showed, that for an analytic Hamiltonian the regime of an exponen- tially small error can be extended, by increasing the number of zero derivatives. However the rate of this decay will be decreasing. The recipe on optimal rate of exponential convergence may be found. The exponentially small error was predicted in [10]. However, no real example of a system was studied so far. The system, proposed in the thesis is an important example. The differential-geometric approach was recently introduced in [15]. The geodesic path gives better performance of the algorithms based on AQC. The approach and the methods developed in this thesis may find a great use in the quan- tum geometric picture. 42 Bibliography [1] J. E. Avron, R. Seiler, and L. G. Yaffe, Commun. Math. Phys. 110, 33-49 (1987). [2] M. Born and V . A. Fock, Zeitschrift fr Physik a Hadrons and Nuclei 51 (3-4): 165180 (1928). [3] H. Cheng, Advanced Analytic Methods in Applied Mathematics, Science, and Engineering. Boston: LuBan Press, 2006. [4] E. Farhi, J. Goldstone, S. Gutmann, and M. Sipser, e-print quant-ph/0001106. [5] L.M. Garrido, F.J. Sancho, Physica 28, 553 (1962). [6] I. M. Gel’fand and G. E. Shilov, Generalized Functions. New York: Academic Press, 1964. [7] L. K. Grover, Phys. Rev. Lett. 79, 4709 (1997). [8] G.A. Hagedorn, A. Joye, J. Math. Anal. Appl. 267, 235246 (2002). [9] T. Kato, J. Phys. Soc. J. Jpn. 5, 435-439( 1950). [10] D.A Lidar, A.T. Rezakhani, A. Hamma, ”Adiabatic approximation with better than exponential accuracy for many-body systems and quantum computation”, arXiv:0808.2697v2 (2008). [11] K.P. Marzlin and B.C. Sanders, Phys. Rev. Lett. 93, 160408 (2004). [12] K. O. Mead and L. M. Delves, ”On the convergence rate of generalized Fourier expansions,” IMA J. Appl. Math., vol. 12, no. 3, pp. 247-259, 1973. [13] A. Messiah, Quantum Mechanics (Dover Publication, New York, 1999). [14] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Infor- mation (Cambridge University Press, Cambridge, UK, 2000). 43 [15] A.T. Rezakhani, W.-J. Kuo, A. Hamma, D.A. Lidar, P. Zanardi ”Quantum Adiabatic Brachistochrone”, arXiv:0905.2376v1 (2009). [16] J. Roland and N. J. Cerf, Phys. Rev. A 65, 042308 (2002). [17] M.S. Siu, Phys. Rev. A 71, 062314 (2005). 44

Abstract (if available)

Abstract We describe a class of the Hamiltonians, for which the local adiabaticity condition is satisfied automatically. We consider the controllable adiabatic interpolations between the initial and final Hamiltonians. We show that the quantum search by the adiabatic quantum computation can be mapped into this class. Assuming that the interpolation is infinitely differentiable and the target adiabatic ground state is nondegenerate and separated by a gap from the rest of the spectrum, it is shown that one can obtain an exponentially small error in the algorithm run time T betweenthe finale adiabatic eigenstate and the actual state of the system. The optimal for quantum search scaling of T as square root of the system size is also satisfied.

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Creator Pimachev, Artem K. (author)

Core Title Dynamics of adiabatic quantum search

School College of Letters, Arts and Sciences

Degree Master of Science

Degree Program Chemistry

Publication Date 11/06/2009

Defense Date 08/18/2009

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag adiabatic,adiabaticity,computer,Grover,OAI-PMH Harvest,quantum,search

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Lidar, Daniel A. (committee chair), Haas, Stephan (committee member), Wittig, Curt (committee member)

Creator Email pimachev@gmail.com,pimachev@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m2718

Unique identifier UC1461775

Identifier etd-Pimachev-3226 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-270462 (legacy record id),usctheses-m2718 (legacy record id)

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Dmrecord 270462

Document Type Thesis

Rights Pimachev, Artem K.

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu