Diabatic Quantum Gates

INVESTIGACION´ Revista Mexicana de F´ısica 58 (2012) 428–437 OCTUBRE 2012

Diabatic quantum gates

P.J. Salas Peralta Departamento de Tecnolog´ıas Especiales Aplicadas a la Telecomunicacion,´ Universidad Politecnica´ de Madrid, Ciudad Universitaria s/n, 28040 Madrid (Spain). e-mail: [email protected] Recibido el 12 de enero de 2012; aceptado el 30 de julio de 2012

At present, several models for quantum computation have been proposed. Adiabatic quantum computation scheme particularly offers this possibility and is based on a slow enough time evolution of the system, where no transitions take place. In this work, a new strategy for quantum computation is provided from the opposite point of view. The objective is to control the non-adiabatic transitions between some states in order to produce the desired exit states after the evolution. The model is introduced by means of an analogy between the adiabatic quantum computation and an inelastic atomic collision. By means of a simple two-state model, several quantum gates are reproduced, concluding the possibility of diabatic universal fault-tolerant quantum computation. Going a step further, a new quantum diabatic computation model is glimpsed, where a carefully chosen Hamiltonian could carry out a non-adiabatic transition between the initial and the sought final state.

Keywords: Quantum gates; fault-tolerance; adiabatic quantum computation.

En la actualidad se han propuesto varios modelos de computacion´ cuantica.´ Concretamente, se ha introducido el esquema de computacion´ cuantica´ adiabatica,´ basado en una lenta evolucion´ temporal del sistema empleado, lo que impide la aparicion´ de transiciones entre estados. En este trabajo, se propone una nueva estrategia para la computacion´ cuantica,´ basada en el punto de vista opuesto. El objetivo es controlar las transiciones no-adiabaticas´ entre estados que den lugar a la evolucion´ deseada. Se introduce el modelo por medio de una analog´ıa entre la computacion´ adiabatica´ y una colision´ inelastica.´ Mediante un simple modelo con dos estados, es posible reproducir varias puertas cuanticas,´ concluyendo la posibilidad de una computacion´ cuantica´ diabatica´ universal tolerante a fallos. Dando un paso mas,´ se vislumbra un nuevo modelo de computacion´ cuantica´ diabatico,´ en el que las transiciones entre el estado inicial y final deseado, se controlan mediante una adecuada eleccion´ del Hamiltoniano.

Descriptores: Puertas cuanticas;´ tolerancia a fallos; computacion´ cuantica´ adiabatica.´

PACS: 03.67.Lx; 03.67.Pp; 03.67.Hk

1. Introduction main close to the state |ψk(t = T )i at the end of the process. Starting in the ground state |ψ0(t=0)i , the adiabatic evolution The adiabatic theorem has a long history in quantum me- will guarantee that the system will be (with a great probabil- chanics [1]. It was Farhi et al., [2] who applied it to quan- ity) in the same state at the end. More precisely, defining the tum computation by suggesting a new way of solving classic minimum gap between the lowest two energy eigenvalues as: problems, such as SAT, by means of an algorithm based on ∆Emin = min (E1(t) − E0(t)) (1) adiabatic quantum computation (AQC). After the first specu- 0≤t≤T lation about the promising power of this novel way of com- putation [3], finally demonstrated its polynomial equivalence and the maximum value of the operator dH(t)/dt between with the conventional circuit model [4]. In spite of that, it these two states ¯ ¯ has some advantages such as an inherent robustness to some ¯ dH ¯ D = max ¯hψ0(t)| |ψ1(t)i¯ (2) kinds of noise. AQC is robust against dephasing in the ground max 0≤t≤T ¯ dt ¯ state or thermal errors [5,6]. Several error correction strate- gies have also been proposed [7]. From the experimental if the initial state is |ψ0(0)i, provided the condition point of view, some quantum adiabatic algorithms have al- D max ≤ ε (3) ready been implemented experimentally by means of NMR ∆E2 techniques [8]. min In the following, the adiabatic quantum computation is fulfilled, then model is briefly described, and the details may be re- 2 2 |hψ0(T )|ψ(T )i| ≥ 1 − ε (4) viewed elsewhere [9]. Consider a state evolving according the Schrodinger¨ equation and a time-depending Hamiltonian If H(T ) = H(0), then the final state |ψ(T )i is ε2-close H(t) (t ∈ [0,T], T being the total computation time), having (according (4)) to the |ψ0(T )i state, except for an overall a set of eigenstates and eigenvalues {|ψk(t)i, Ek(t)}. The phase. Roughly speaking, the meaning of this statement is: adiabatic theorem states that if the initial state is the kth- the state of the system will be kept close to the instanta- eigenvector |ψk(t=0)i and H(t) varies slowly enough, the neous state |ψ0(t)i if it is weakly coupled with the remain- instantaneous state |ψ(0≤ t ≤ T )i of the system will re- ing states (small Dmax) and is kept well separated in energy DIABATIC QUANTUM GATES 429 from them (large ∆Emin). In this situation, there are nei- used in atomic collisions, although not exactly with the same ther crossings nor avoided-crossings between the considered meaning. In this context, the word diabatic refers to a new states and, consequently, the transition probabilities to the ex- set of basis states providing smaller coupling values. This cited states are very small. new basis includes states that go smoothly across an avoided- Despite the above criterion not actually being necessary crossing and are more suitable to describe inelastic processes nor sufficient in general, it has been widely used because of at medium and higher energies. its simplicity. Work has recently been carried out in order As a result of the above ideas on non-adiabatic transitions to study the consistency of the theorem [10], and replace the (diabatic behaviour) between quantum states, the construc- previous formulation with a rigorous statement [11], even in- tion of quantum diabatic gates will be established. The main cluding noise [12]. framework of AQC can be used but with some new charac- The adiabatic theorem can be used to design a new teristics. In the AQC, by choosing the ending Hamiltonian paradigm of quantum computation. The model is specified properly, the final correct state is identified with the ground by two Hamiltonians (in the simplest case), the initial H0 and state. Now, in the diabatic behaviour, the correct state will final H1. The initial state is the ground state of H0 that is re- be the first excited one of a suitable Hamiltonian. In order quired to be an-easy-to-prepare state whereas the solution of to get this diabatic evolution, the speed of the computation the computation is the ground state of H1. The problem could is not restricted to being a sufficiently small evolution speed be to work out the structure of H0 and H1. The time evolution as in AQC. In contrast, the final state will be reached with is controlled by means of the total Hamiltonian (H) prepared high probability if the computation speed is high enough. as the interpolation of the previous Hamiltonians, depending This diabatic behaviour will not limit the computation time on a parameter s(t): H(t) = f(s(t))H0 + g(s(t))H1, with as in AQC. Another advantage of the diabatic computation s ∈ [0,1] and s = t/T , T being the total computation time. The is its robustness against errors shown as an intrinsic fault- local condition is usually necessary for the Hamiltonian (re- tolerance, as established in Sec. 3.1. In addition, the concept quiring that its implementation only involve a constant num- of diabatic computation as a sequence of diabatic gates is ber of particles), in order to be realistically implemented. suitable to be extended to a full diabatic computation, un- Initially, the system is synthesized in the ground state of derstood as a single evolution carried out by means of an ap- H0, |ψ0(0)i, and then evolves according to H(t). If the evo- propriate Hamiltonian. The only remaining work is to char- lution rate (ds(t)/dt) is slow enough (adiabatic evolution), the acterise the physical processes by implementing this evolu- intermediate state (|ψ(t)i) will not produce any transition to tion. These Hamiltonians could involve non-local interac- (possible) higher energy states, and will end in the ground tions of more than two or three qubits. This could be seen as state of H1, that has been chosen as the solution of the prob- a heavy restriction nowadays, but is not forbidden by the laws lem. of Quantum Mechanics. Providing suitable quantum systems The question addressed in this work is: could the main as well as general non-local Hamiltonians is not the scope of framework of AQC be used to reach the solution of a problem the paper. by taking advantage of the possible transitions between the The paper is structured as follows. In Sec. 2 the main states? In fact, this is what is happening in the well-known ideas about inelastic atomic collisions are reviewed and are context of quantum chemistry when two atoms come close related to the diabatic computation model. In Sec. 3, a sim- to form a molecule, if the Born-Oppenheimer approxima- ple two-state model will be used to implement several dia- tion [13] is not fulfilled. This behaviour is crucial in the con- batic quantum gates, demonstrating its general intrinsic fault- text of inelastic atomic collisions [14], where the studied pro- tolerance and estimating the error gate probability for each cesses are, specifically, those producing outgoing states dif- one. ferent from the ingoing ones. The transitions are produced by the breakdown of the Born-Oppenheimer approximation, in- volving states whose energies cross or pseudocrosse (avoided crossing). The molecular Hamiltonian depends on the atomic 2. Diabatic quantum computation model separation R(t) that can be seen as a time-dependent param- eter of the total Hamiltonian. The starting point for study- The adiabatic computation involves the slow enough evolu- ing these systems is to find the instantaneous (for R con- tion of a time-dependent Hamiltonian, this fact ensures the stant) adiabatic states and then solving the time-independent lack of transitions. The opposite behaviour is sought in the Schrodinger¨ equation. The dynamic is included by expanding present method. The goal is to develop the model through a the total collision state in the adiabatic basis set. If, through parallelism with an inelastic atomic collision. the evolution, a transition probability from the initial state Consider an inelastic atomic collision where an atom A to some of the final states is not negligible, it is said that a crosses an interaction region occupied by another atom B or non-adiabatic transition may have occurred. In this work, the a field. The collision frame is shown in Fig. 1. The incoming strongly non-adiabatic behaviour of the states is proposed as atom comes from zmin → −∞ (equivalently t → −∞, be- a possible mechanism for quantum computation, calling it di- cause the time origin is situated on the coordinate z = 0) to abatic quantum computation (as opposed to adiabatic); a term zmax → +∞ (or t → +∞). If the atomic speed (v) is large

Rev. Mex. Fis. 58 (2012) 428–437 430 P.J. SALAS PERALTA

eter [16] is interesting in classifying the processes accord- ing the transition probabilities. It comes from very general considerations. The non-adiabatic behaviour (large transition probability) between two states is expected to be important when the typical collision frequency written as the inverse of the collision time (ω = 1/τc), matches the frequency of the energy splitting ∆E/(h/2π). Taking into account that τc ∼ a/v (a being a characteristic z-region of interaction), FIGURE 1. Reference framework to describe an atomic collision the constraint ξ = a∆E/v ∼ 1 (with h/2π = 1) is obtained. between two atoms A and B. In the process, the ingoing atom A The condition of adiabaticity comes from a small coupling goes from z → −∞ to z → +∞ through a straight line trajectory. (low transition probability), implying a large value of ∆E and small v, then ξ À 1. The speed for the maximum tran- sition probability can be estimated be means of this Massey enough, the trajectories can be taken as straight lines and parameter ξ. The previous behaviour is closely related to the characterized by an impact parameter b. These are (roughly) coupled system (6), if the following change in the variable the assumptions made in the Impact Parameter Method [15], z = vt is introduced, and an energy phase is extracted: used to describe atomic collisions in the medium energy   range. Notice that the parameter R(t) provides a radial Zz speed as R˙ (t) = zv/R, with z = vt fulfilling the equation i 0 0 ak(z) = bk(z) exp − Ek(R(z ))dz  (7) R(t)2 = z(t)2 + b2. v −∞ The total Hamiltonian H describing the collision, as well as the states are time-dependent through the implicit R(t) pa- The new system of equations is: rameter dependence. To describe the dynamic, the collision db (z) X 1 ∂ state |ψ(t)i is developed in a basis set {|µk(R(t))i}: k = − b (z) hφ | |φ i dz n v k ∂t n X n6=k |ψ(t)i = ak(t)|µk(R(t))i (5)   Zz k i × exp − [E (R(z0))−E (R(z0))dz0] (8) It is evident that in the case of a “real” atomic collision, v n k all of the states must depend on the electronic coordinates; −∞ however these are not shown in the equations in order to in- The coupling hφk|∂/∂t|φni should be expressed in terms troduce a closer notation to that used below. of z, with two new couplings appearing: the radial coupling H The Hamiltonian is not (in general) diagonal in the ba- hφ |∂/∂R|φ i and a rotational one. The radial coupling orig- {|µ i} k n sis set k , then, in the Molecular Model of atomic col- inates in the case of dealing with states of the same molecular lisions [14,15], a new and more appropriate adiabatic (or- symmetry and appears, for example, in an avoided-crossing. {|φ (R)i} thonormal) basis set k is provided as the instanta- The rotational coupling comes out between the states of dif- H neous eigenvectors of . This process is static since the diag- ferent molecular symmetry and will not be taken into account H|φ (R)i E (R)|φ (R)i onalizing process ( k = k k ) is carried out in the following, because the states considered will have the R for each constant value. The dynamic is included by solv- same symmetry and display an avoided-crossing. ing the standard time-dependent Schrodinger¨ equation for the The system of equations has, in general, to be solved collision state of the system |ψ(t)i expanded in the adiabatic numerically. The behaviour of the adiabaticity follows the basis set. Introducing |ψ(t)i into the Schrodinger¨ equation, tracks of the Massey parameter. If the initial conditions are the following (general) differential coupled equation system b (0) = δ , a simple first order estimation for the transi- is obtained: k k0 tion probability will depend on how quickly the phase os- X · ¸ dak(t) ∂ cillates. An estimation of this phase could be done through = − an(t) hφk| |φni + ihφk|H|φni (6) dt ∂t the ξ parameter. The coefficient bm6=0(+∞) will be small n if ξ À 1 [17] and the behaviour is adiabatic. In the case The system involves the dynamic coupling hφk|∂/∂t|φni of ξ ∼ 1, a large value of bm6=0(+∞) is expected, and the and the electrostatic coupling hφk|H|φni. The index k behaviour is non-adiabatic. runs over the coupled states. Two properties simplify the In principle, the most appropriate choice as a basis set to aforementioned equations: using an adiabatic basis set, develop the collision state is the adiabatic one. To describe hφk|H|φni = En δkn, and the matrix elements hφk|∂/∂t|φki an inelastic collision adequately, all coupled states must be = [h∂φk/∂t|φki + hφk|∂/∂t|φki]/2 = (1/2) ∂hφk|φki/∂t = 0 included in order to account for the possible transitions. Un- (bearing in mind that the states |φki are real functions). Tak- fortunately, in some cases, when the states are highly coupled ing into account the relationship z = vt, the above system of or when the speed of the colliding systems is high enough, equations can be transformed into another equivalent depend- the couplings are very active to produce transitions. In this ing on the z coordinate. In this context, the Massey param- case the number of states to be included in the system (8)

Rev. Mex. Fis. 58 (2012) 428–437 DIABATIC QUANTUM GATES 431 is so huge that it makes it impossible to carry out the de- set {|φki, k = 1, .., n > 2}, but, with an adequate choice of scription. In this situation, a new (and smaller) set of states, the Hamiltonian and the parameters (v, b), only a two-state called diabatic, could help to decrease the dimension of the model could be enough. This would be case if the two cou- problem. Diabatic states were introduced in the early 60’s by pled states considered are far separated in energy from the Lichten [18] and generalized by Smith [19]. Roughly speak- remaining excited states or if they are not coupled by sym- ing, they have the property of running smoothly through an metry considerations. In this case the process of computation avoided-crossing, having smaller couplings. Intensive work involves only two eigenstates {|χ0i, |χ1i} of the Hamilto- has been carried out in the past on the adequate definition of nian H0: diabatic states (see [14] and references therein). Trying to introduce the flavour of an atomic collision in H0|χ0i =∈0 |χ0i H0|χ1i =∈1 |χ1i (12) the present computation model, a parameter s(t) similar to with H providing the coupling between them. These R(t), is included, allowing the computation to be seen as a W states are not (in general) eigenstates of the total Hamiltonian kind of “collision”. Defining T as the total computation time, H = f(s)H + g(s) H , so the energies ∈ and ∈ could the parameter s is defined as s(t) = t/T . The states for the 0 W 1 2 cross. Diagonalizing H in the basis {|χ i, |χ i} a new adia- incoming particle (for R large in the atomic collision model), 0 1 batic basis set {|φ i, |φ i} is obtained whose new eigenval- would now correspond to s = 1 and the closest atomic dis- 0 1 ues (E (s)) could show an avoided crossing (pseudocross- tance (equivalent to R small), to s = 0. Suppose the searched 0,1 ing). The adiabatic basis is used to develop the evolution of for state is one of the eigenstates |χ i of a non-degenerate m the total state vector |ψ(t)i: Hamiltonian H0. The diabatic quantum computation would be like a kind of “collision” depending on the s(t) parameter. |ψ(t)i = a0(t)|φ0(s(t))i + a1(t)|φ1(s(t))i (13) By keeping the same framework as in the AQC scheme, two time-independent Hamiltonians are taken into account: H0 The coefficients {a0(t), a1(t)} are provided by the gen- (carrying out the information of the initial and/or final possi- eral equation system (6). Changing to the z variable through ble states) and HW (including a coupling). The HW includes z = vt, the two-state coupled equation system is: a coupling between the basis eigenstates of H0, allowing the da0(z) E0(s(z)) z possibility of transitions between them. The evolution of the = −ia0(z) − a1(z) W (s(z)) system is produced according the Hamiltonian: dz v s(z) H(s(t)) = f(s(t))H + g(s(t))H (9) da1(z) E1(s(z)) z 0 W = −ia (z) − a (z) W (s(z)) (14) dz 1 v 0 s(z) The functions f(s) and g(s) are chosen to fulfil the gen- eral properties: W (s(z)) = hφ0(s)|∂/∂s|φ1(s)i being a kind of “radial cou- 0 ←−−−|s→0 f(s)| −−−→s→1 1 pling” in the parameter s (having considered the property hφ0(s)|∂/∂s|φ1(s)i = −hφ1(s)|∂/∂s|φ0(s)i). The first terms 1 ←−−−|s→0 g(s)| −−−→s→1 0 (10) on the right hand side of (14) describe the adiabatic evolution (keeping the populations and, perhaps, changing the phases) The behaviour reflects the introduction of the perturba- and, the second terms include the non-adiabatic transitions by tion HW , when s goes from 1 to 0. The state vector of this means of the W coupling. The integration of this system is “collision” is expanded in the adiabatic basis states (eigen- 2 1/2 carried out according to take s = 1: from zmin = -(1 − b ) vectors) {|φk(s)i} of H(s): 2 1/2 X to zmax = (1−b ) , b being the impact parameter, seen now |ψ(t)i = ak(t)|φk(s(t))i (11) as a parameter to be adjusted. k The time-evolution is controlled by a system of coupled 3. Diabatic gates equations similar to (6) or (8) if the z variable is taken. The diabatic computation is seen as a process in which the syn- The first step in reaching quantum computation is to describe thesis of the appropriate ingoing state (expanded in the adi- the quantum diabatic gates. Each gate can be seen as a black abatic basis of H0), is introduced for the s(0) = 1 black box inside which a quantum evolution, according to a Hamil- box wire, and then the “collision” run evolving forward and tonian, takes place in some ingoing state in order to produce backward from s = 1 → 0 → 1, and get the outgoing state an outgoing one. Unlike the adiabatic computation, some |φout(s → 1)i. For s → 1, the coupling goes to 0 because transitions are required and, to reach them, two parameters g(s → 1) → 0, then |φout(s → 1)i could be very close to (v, b) are free to be adjusted. the sought eigenvector |χmi of H0 (or perhaps some linear Working in the computation basis set {|χ0i = |0i, combination of them), if values of the parameters (v, b) are |χ1i = |1i}, the initial (s = 1) state is introduced into the carefully chosen. black-box-gate. The whole evolution as a “collision” varying s = 1 → 0 → 1, is described through an evolution operator 2.1. Two-state model U(t, −t) (remember the time origin t = 0 is in z = 0) from t = −T to t = T in the adiabatic basis set [20]: In general, the diabatic quantum computation could involve ingoing and outgoing states developed in the adiabatic basis Rev. Mex. Fis. 58 (2012) 428–437 432 P.J. SALAS PERALTA

à p ! i2α00 i2α01 (1 − p)e + pe −2i (1 − p)p sin(α00 − α01) U(T, −T ) = p (15) −i2α00 −i2α01 −2i (1 − p)p sin(α00 − α01) (1 − p)e + pe p being the probability of a non-adiabatic transition through the avoided-crossing and αij some phases, both depending parameters (v, b) selected experimentally. Calling |Ψi the on the (v, b) values. The probability that the system exits initial vector state, the gate error probability (Pe(Ψ)) asso- through a particular outgoing channel can be calculated by ciated is defined as the orthogonal component of the vector applying the evolution operator U(T, −T ) to the incoming |ξψi = (Ua-Ut)|Ψi = D|Ψi: state. The different quantum diabatic gates can be repre- ­ ¯ ® ⊥¯ ⊥ sented by choosing the values of the parameters (v, b) ade- Pe(Ψ) = ξΨ ξΨ (16) quately as will be shown throughout paragraphs 3.2-3.5. First of all the fault tolerance of the diabatic gates (15) will be con- The error probability for the gate is defined as: sidered. Pe = max(Pe(Ψ)) (17) ∀|Ψi 3.1. Fault-tolerance of diabatic gates and fulfils the condition: The features providing the power to quantum computers are P (Ψ) = hξ⊥|ξ⊥i ≤ hξ |ξ i = tr(ρ P ) (18) parallelism and interference, which are intrinsically quantum e Ψ Ψ Ψ Ψ Ψ properties. The implementation of a quantum algorithm re- P being the positive and hermitian operator D+D and quires a quantum computer to keep working on the qubits ρΨ = |ΨihΨ|. By diagonalizing P and taking into account for a long time. The computation requires the creation and that tr{ρΨP } is upper bounded by dm = max{di, eigenvalues manipulation of entangled states involving large ensembles of P } for every state vector |Ψi, the error probability for the of qubits. Unfortunately the quantum states are necessar- gate is Pe ≤ dm. As the P operator is positive, the condition ily coupled with the environment producing the qubit deco- dm ≤ tr{P } is also fulfilled, then: herence. This process introduces errors into the computa- tion, making it useless. To fight against error accumulation, Pe ≤ dm ≤ tr(P ) (19) Shor [21,22] and Steane [23] introduced in the mid 90’s, the concept of quantum error correcting codes, capable of keep- In some cases the tr{P } is much easier to be calculated ing the quantum decoherence under control. Unfortunately, than dm and it will be used as an upper bound of Pe. error-correcting methods are not strong enough to achieve a The next step will be to check the error propagation when total control of error spreading through a quantum algorithm. the gate is described by the evolution operator U(T, −T ). As In trying to solve this problem, Shor introduced fault-tolerant was mentioned before, the error affecting the operator Ua methods [24] in quantum computation. There are several comes from an error in the (v, b) parameters. If the target evo- basic ideas involved in it: applying quantum gates directly lution operator corresponds to the (v0, b0) values, the Ua will to the encoded qubits, correcting the errors periodically, a include some small error and it will correspond to (v0 + δv, carefully designed encoded gates in order to avoid the error b0 + δb). Assuming the errors δv and δb are small enough, spreading and the use of a concatenated quantum code struc- their effect on U(T, −T ) could be considered as lineal (by ture with a hierarchical encoding [25]. Roughly speaking, a means of ε) in the phases and the probability p. The Ua could fault-tolerant recovery method would introduce fewer errors be written as: than those it is able to eliminate. The fusion of fault-tolerant ∗ encoded quantum gates and concatenated codes has estab- [Ua(T, −T )]00 = [Ua(T, −T )]11 lished the existence of an error threshold. If the evolution i2(α00+c0ε) = (1 − p + cpε)e and gate errors are below this threshold, quantum states will remain stabilized for a time long enough to carry out the com- + pei2(α01+c1ε) putation. Several estimations for the value of this threshold [U (T, −T )] = [U (T, −T )] have been published [26-31]. From these works, it is possible a 01 a 10 q to establish an approximate error-gate-probability-threshold = −2i (1 − p + cpε)(p + cpε) (Perr) to carry out a long enough quantum computation, as P ≤ 10−4. err × sin(α00 + c0ε − α01 − c1ε) (20) Following the same argument used in Ref. 32, the gate error probability is defined in the following way: consider c’s being real numbers. + the target unitary operation to be implemented as represented By means of the Ua(T, −T ), the matrix P = (Ua-Ut) by the evolution operator Ut and the approximate one by Ua. (Ua-Ut) is calculated and its trace is developed in a ε-power The error included on Ua could come from an error in the series. For a small enough ε, the error propagation provides a

Rev. Mex. Fis. 58 (2012) 428–437 DIABATIC QUANTUM GATES 433

2 gate error probability behaviour as Pe ∼ O(ε ), showing the |0i↔|1i, meaning that if the ingoing state is |0i, the outgoing intrinsic fault-tolerance of the diabatic gates built. The coef- must be |1i and vice versa, both with certainty. 2 ficient of ε is complicated and depends on p and the phases Suppose the initial (s = 1) state |χ0i = |0i is introduced αij , and will be shown for each particular gate. into the NOT-gate black box, after the evolution as a “col- In the following, the set of one-qubit Pauli gates lision” varying s = 1 → 0 → 1, a probability of one is {X = NOT, Z, Y = -iXZ}, T (called π/8 gate), H (Hadamard), required for the process |χ0i → |χ1i = |1i (the same for CNOT and Toffoli gate will be considered. Through a pos- the reverse process). Two Hamiltonians are introduced: H0 sible Hamiltonian, the parameters (v, b) will be numerically having the eigenstates {|χ0i = |0i, |χ1i= |1i} and H with estimated to implement the gates as well as the gate error an adiabatic basis {|φ0i, |φ1i}, fulfilling |φ0(s = 1)i = |0i probabilities. and |φ1(s = 1)i = |1i. The |0i state evolves in the NOT-gate black box through the evolution operator (15), and the prob- 3.2. Pauli gates ability that the system exits through the |φ1(s = 1)i = |1i channel (or vice versa) is: Working in the computation basis set {|0i, |1i}, the looked for evolution inside the black box for the NOT gate is

FIGURE 2. (a) Energy of adiabatic eigenstates {|φ0i, |φ1i} of the total Hamiltonian describing a NOT-gate versus the parameter s. The initial states for s = 1 are shown on the right and the avoided-crossing will provide a radial coupling between them with a maximum at s ∼ 0.5. The possible paths interfering to produce the |φ1i ≡ |1i state are shown. (b) Radial coupling W = hφ0|∂/∂s|φ1i. (c) and (d) Coefficients for the adiabatic states |φi,ki = c(iork),0|0i + c(iork),1|1i.

Rev. Mex. Fis. 58 (2012) 428–437 434 P.J. SALAS PERALTA

Figure 2 shows the results for the eigenvalues, “radial” 2 coupling and coefficients {cij(s), i, j = 0, 1}. The ingoing P|0i↔|1i = 4(1 − p)p sin (α00 − α01) (21) state of the system is |φ0(s = 1)i = |0i, and the objective is To get the state |1i with certainty, a probability p ∼ 0.5 to reach an appropriate evolution (by adjusting the parame- and α00 − α01 ∼ mπ/2 (m an integer), are needed. Notice ters v and b) to get a transition probability P|0i→|φ1i ∼ 1 and the probability p as well as the phase difference depends on U(T, −T ) ∝ X. Optimizing the parameters (v, b), the condi- the collision parameters (v, b), producing a structure of the tion α00 −α01 = m π/2 (m an integer) is fulfilled and the gate probability P|0i↔|1i similar to the well known Stuckelberg¨ U(T, −T ) = ±i X can be reached, that is a NOT-gate except oscillations [33] appearing in atomic collisions. The oscilla- an unimportant global phase. Carrying out the numerical cal- tion came from the two-way interference, producing the exit culation, the optimized values are (v = 0.2547, b = 0). Fig- state as is shown in Fig. 2(a). ure 3 details the transition probability P|0i→|φ1i versus v, for

The next step must be the estimation of the (v, b) b = 0, showing a P|0i→|φ1i = 0.99992 for v ∼ 0.2547. The parameters to produce this gate. The impact parameter required value of p ∼ 0.5, reached for this speed, is shown (0 ≤ b ≤ s = 1) is related to how active the coupling in Fig. 4. In the same Fig. 3 the parameter η is included to W is in the collision, because it controls the minimum ap- appreciate its capability of estimating the speed (v ∼ 0.249) proach distance to the z = 0 point. The value b → 0 will for the maximum of the transition probability. For v < 0.1, be chosen to make the coupling completely active, and the typical Stuckelberg¨ oscillations appear as a consequence of a most important parameter becomes the speed v. In the low- two-path interference producing the same outgoing state (see speed region, the adiabatic term in the system (14) will dom- Fig. 2(a)). The envelope of the first oscillating part does inate the non-adiabatic one and, the evolution will be mainly not reach the value P|0i→|φ1i ∼ 1 because of the probability adiabatic. In this region the adiabatic behaviour for quan- p < 0.5 for this speed. tum computation is recovered, whereas in the higher speed Consequently, it is possible to get a high fidelity NOT- range, the non-adiabatic transitions will take place, thus rais- gate by means of this simple Hamiltonian involving only two ing the diabatic behaviour. In order to estimate the speed for coupled states. Because what is looked for is the transition which the maximum of P|0i→|1i appears, the Massey param- between the adiabatic states (including the appropriate rel- eter could be used. Instead of this, and supposing a first order ative phases), the process could be called diabatic, and the solution for the initial condition a1(0) = δ01, a more accurate gate, quantum diabatic NOT-gate. estimation comes from the parameter: The gate error probability, calculated by means of tr{P} ¯  ¯2 has the explicit structure: ¯ Z ¯ ¯ Zmax ZZ ¯ z i 2 2 2 3 ¯  0¯ Pe(NOT) = (2(c0 − c1) + 8c )ε + O(ε ) (26) η(v, b) = ¯ W exp  (E1 − E0)dz ¯ (22) p ¯ s v ¯ ¯ ¯ 2 Zmin Zmin The O(ε ) behaviour for Pe when v or b are changed has 2 been checked. For instance, if the collision speed is changed providing an approximation to |b1(zmax)| . A possible Hamiltonian describing the “collision” could be as follows: 4 H = f(s)H0 + g(s)HW = s (−|0ih0| + |1ih1|) − (1 − s)4(|0ih1| + |1ih0|) (23) The functions f(s) = s4 and g(s) = −(1−s)4 are chosen only as an example to reach the objectives. By diagonalizing this Hamiltonian in the computation basis set {|0i, |1i}, the adiabatic basis {|φ0i, |φ1i} is obtained, that will be used to expand the total state describing the dynamic:

|φ0(s)i = c00(s)|0i + c01(s)|1i

|φ1(s)i = c10(s)|0i + c11(s)|1i (24)

The coupling Hamiltonian introduced HW , provides a ra- dial coupling whose surface is π/4 and the adiabatic states comply with the asymptotic behaviour: FIGURE 3. Continuous line: Transition probability |0i → |1i 1 s→0 s→1 (P|0i→|φ1i) in a two-state NOT-gate versus the speed v. The maxi- |+i = √ (|0i + |1i) ←−−−|φ0i −−−→|0i 2 mum value of the vertical axis is 1. The impact parameter b is taken as 0. Dashed line: η(v, b = 0), parameter providing an estimation 1 s→0 s→1 −|−i = √ (−|0i + |1i) ←−−−|φ1i −−−→|1i (25) of the maxima. In this case the vertical axis goes on to 2.5 arbitrary 2 units.

Rev. Mex. Fis. 58 (2012) 428–437 DIABATIC QUANTUM GATES 435

To obtain the gate, the conditions α00 = 15π/16 and α01 = −π/16 must be fulfilled by optimizing (v, b). The general error gate probability is given by:

2 2 2 2 3 Pe(T ) = (8cp + 4c0 + 4c1)ε + O(ε ) (30) If the values of (v, b) are adequately tuned, the T-gate does not depend on the probability p, nor the value of Pe. The numerical calculation for (v = 0.0337, b = 0.2164) give −6. the value Pe ≤ dm(T ) ∼ 3×10 The behaviour for Pe when v is changed by keeping b = 0.2164 constant provides 5 2 an error dependence Pe ∼ 2×10 εv, εv being the error for the speed.

3.4. Hadamard gate

A Hadamard (in fact the gate iH) gate can be obtained by tun- ing the (v, b) parameters to fulfil the conditions α00 = 3π/8 and α01 = -7π/8 and p ∼ 0.5 (shown in Fig. 4). The numeri- cal calculation provides (v = 0.2249, b = 0.2677). For these FIGURE 4. Probability p versus v and b. Several symbols represent −5 values, the Pe ≤ dm(H) ∼ 3.5×10 . The behaviour for the values for the optimized parameters (v, b) used for the quantum Pe when v is changed by keeping b = 0.2677 constant gives gates considered. 2 an error dependence Pe ∼ 27 εv, εv being the error for the speed. around the optimum value v0 in some ε = εv, the behaviour 2 is Pe ∼ 40 εv. For (v0 = 0.2547, b = 0), the gate error prob- ability (Eq. (19)) given as the maximum eigenvalue of the P 3.5. Control-Not and Toffoli gates −5 matrix, is dm(NOT) = 8×10 ≥ Pe. Surprisingly, the same kind of Hamiltonian can be used to im- A similar study can be carried out with the remain- plement several other (more complicated) gates as the CNOT ing Pauli gates. For the Z gate, the chosen parameters gate. In this case the sought transitions are |10i ↔ |11i, (v = 0.051, b = 0.1094) fulfil the condition α00 = π/4 whereas the states |00i and |01i should not be affected. A and α01 = 5π/4 and U(T, −T ) = iZ with Pe(Z) ≤ dm(Z) Hamiltonian representing this behaviour could be: = 5×10−6. The general behaviour for the error gate proba- bility is: H = s4(−|10ih10| + |11ih11|)

2 2 2 2 2 3 − (1 − s)4(|10ih11| + |11ih10|) (31) Pe(Z) = 8(cp − p(c0 − c1) + c0)ε + O(ε ) (27)

In this case, if the (v, b) parameters are correctly tuned; The same discussion is appropriate for the Toffoli-gate, the gate does not depend on the probability p, so the condi- in this case, the total Hamiltonian could be: tion p ∼ 0.5 is not strictly needed, although the value of Pe H = s4(−|110ih110| + |111ih111|) depends on p. The value of p for (v = 0.051, b = 0.1094) is 4 shown in Fig. 4. The behaviour for Pe when v is changed by − (1 − s) (|110ih111| + |111ih110|) (32) keeping b = 0.1094 constant provides an error dependence 3 2 All the conclusions reached in the case of the NOT-gate Pe ∼ 4×10 εv, εv being the error for the speed. The gate Y can be implemented by applying two succes- are completely valid. Although in the case of Toffoli gate the sive gates according Y = -i XZ. Hamiltonian would involve three qubit interactions, there is no fundamental reason to not be considered. In fact, to imple- 3.3. T gate ment the gate only an eight state system is needed in which the behaviour was according (32). Another interesting one-qubit gate is the T -gate defined as: µ ¶ 1 0 4. Is there a diabatic quantum computation T = (28) 0 eiπ/4 model?

This gate is also referred to as π/8 because, up to an unim- Several one and two qubits gates have been represented as portant eiπ/8 phase, it is equivalent to π/8-gate-phase, with: diabatic gates, identifying a simple theoretical Hamiltonian µ ¶ carrying out the work. In this situation, the concept of dia- e−iπ/8 0 batic computation considered has been as a sequence of dia- T = eiπ/8 (29) 0 eiπ/8 batic gates. Perhaps a further step could be made: could this

Rev. Mex. Fis. 58 (2012) 428–437 436 P.J. SALAS PERALTA diabatic gate model be extended to a full diabatic computa- foli, the sets being {CNOT, T ,H} as well as {CNOT, H, Tof- tion understood as a single evolution carried out by means of foli} universal. These gates have the advantage of an intrinsic a suitable Hamiltonian? I guess the answer to this question fault-tolerance because the error gate probability behaves as is yes, because the general form of the evolution operator in- O(ε2), ε being some error with respect to the correct param- volving two-states (Eq. (15)) is completely determined pro- eters (v0, b0). For each gate considered, the parameters (v, b) viding three parameters (n(n+1)/2 with n=2), in our case p, have been obtained in order to get an error gate probability −4 α00 and α01. The only remaining (perhaps hard) work is to smaller than 10 , this value is considered as the threshold characterise the physical processes by implementing the re- for doing fault-tolerant quantum computation. Evidently, if quired Hamiltonian. These Hamiltonians could involve non- the experimental implementation is able to tune these param- local interactions of more than two or three qubits, and this eters with higher precision, the gate error will reach smaller could be seen as a heavy restriction nowadays, but is not for- values. bidden at all by the laws of Quantum Mechanics. In addi- As the present strategy looks for large transition proba- tion, and as a consequence of the previous sections, this quan- bilities, it could be called diabatic quantum computation, in- tum diabatic computation should show some kind of intrinsic volving states going smoothly through an avoided-crossing. fault-tolerance against noise. The conclusion is the possibility of doing universal diabatic fault-tolerant quantum computation; opening up a new pos- 5. Conclusions sibility of implementing quantum gates and quantum algo- rithms experimentally. A new model for quantum computation is provided based Finally, and going a step further, a new quantum dia- on controlling the transition probabilities in a strong non- batic computation model is glimpsed, where a carefully cho- adiabatic evolution represented by an avoiding crossing. The sen Hamiltonian could carry out a non-adiabatic transition model is introduced by means of the analogy with an evo- between the initial and the sought final state. The problem lution in an inelastic atomic collision and is described by would be to get a physical system evolving according this means of a Hamiltonian having an implicit time-dependence highly non-local Hamiltonian. through an s(t) parameter closely related to the internuclear atomic distance. A simple two-state model is enough to re- Acknowledgment produce the quantum gates. The evolution operator is char- acterized by two parameters, the collision speed v and the This work has been supported by the research project Quan- impact parameter b, that must be adjusted in order to repro- tum Information Technologies (QUITEMAD), P2009/ESP- duce the desired quantum gate. 1594 of Comunidad Autonoma´ de Madrid in Spain. By using this method, several gates have been con- structed, particularly the CNOT, Pauli, T , Hadamard and Tof-

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