Bang-Bang Control Design for Quantum State Transfer based on Hyperspherical Coordinates and Optimal Time-energy Control
BBang-Bang Control Design for Quantum StateTransfer based on Hyperspherical Coordinates andOptimal Time-energy Control
Weiwei Zhou , S. G. Schirmer , Ming Zhang , Hong-YiDai Department of Automatic Control, College of Mechatronics and Automation,National University of Defense Technology, Changsha 410073, People’sRepublicof China Department of Applied Mathematics and Theoretical Physics, University ofCambridge, Cambridge, CB3 0WA, UK Department of Physics, College of Science, National University of DefenseTechnology, Changsha 410073, People’s Republic of ChinaE-mail: [email protected], [email protected], [email protected],[email protected]
Abstract.
We present a constructive control scheme for solving quantum stateengineering problems based on a parametrization of the state vector in termsof complex hyperspherical coordinates. Unlike many control schemes based onfactorization of unitary operators, the scheme gives explicit expressions for allgeneralized Euler angles in terms of the hyperspherical coordinates of the initialand final states. The factorization, when applicable, has added benefits that phaserotations can be combined and performed concurrently. The control procedurecan be realized using simple bang-bang or square-wave-function controls. Optimaltime-energy control is considered to find the optimal control amplitude. Theextension of the scheme to implement arbitrary unitary operators is also discussed.
Keywords: quantum systems, Bang-Bang control, geometric parametrization,controllability, optimal control
1. Introduction
Control of phenomena governed by the laws of quantum mechanics is increasinglyrecognized as a crucial task and prerequiste to realizing promising new technologiesbased on quantum effects from the use of photonic reagents in chemistry [1] to quantummetrology and quantum information processing [2] to mention only a few examples.From early beginnings in the 1980s (see e.g., [3, 4, 5, 6, 7]), there has been considerablerecent progress in both theory and experiment of quantum control [8, 9]. Among thecore tasks for quantum control are quantum state and operator engineering. In theformer case the main objective is to prepare the system in a desired state, which isusually a pure state | ψ f (cid:105) represented by a unit vector in a Hilbert space H associatedwith the system. The task can take various forms, from state transfer, i.e., steeringthe system from a known initial state | ψ (cid:105) to the target state [10, 11, 12, 13, 14],to purification or state reduction, i.e., preparation of a desired pure state startingwith a mixed or unknown initial state, usually involving some form of feedback from a r X i v : . [ qu a n t - ph ] F e b ang-Bang Control Design for Quantum State Transfer λ whichrepresents the ratio of costs of time and energy, and further explore the trade- ang-Bang Control Design for Quantum State Transfer J = (cid:82) t f [ λ + E ( t )] dt , where E ( t ) is energy cost of Bang-Bang control at t , t f isterminal time. It is shown that the product of the terminal time t ∗ f and the energycost E ∗ for optimal bounded or unbounded piecewise constant controls only dependson the geometric parameters of the initial and target states and is independent of λ but λ determines the optimal field strength of the controls, L ∗ = √ λ . The schemecan be generalized to implement arbitrary unitary operators, and we again find thatthe resulting decomposition has some advantages in that many operations commuteand can be performed in parallel.
2. Pure-state Transfer by Bang-Bang Control
Pure-states | ψ (cid:105) of a quantum system defined on a complex Hilbert space H withdim H = N < ∞ can be represented by complex vectors (cid:126)c ∈ C N by choosing asuitable basis {| n (cid:105)} Nn =1 for H , | ψ (cid:105) = N (cid:88) n =1 c n | n (cid:105) . (1)The modulus squared | c n | of the coordinates can be interpreted in terms ofprobabilities provided (cid:126)c is a unit vector. For most applications the global phase of thestate is irrelevant, i.e. we can further identify | ψ (cid:105) ∼ e iφ | ψ (cid:105) . Given these considerations,physically distinguishable pure states can be uniquely identified with elements in thecomplex projective space CP N − = S N − /S , and we can uniquely represent purestates by unit vectors in C N if we fix the complex phase of one coordinate.Pure-state transfer is the task of transforming a given pure quantum state | ψ (0) (cid:105) to a desired pure quantum state | ψ ( s ) (cid:105) and is one of the most fundamental tasksin control of quantum systems. Many of the control strategies mentioned in theintroduction have been applied to this problem, including constructive control schemesbased on the Lie group decomposition. Indeed, it is quite straightforward to see howto solve the state transfer problem for an N -level system in principle, if we are able toimplement unitary gates on a sequence of connected two-level subspaces. Assume, e.g.,that SU (2) operations can be implemented on the subspaces spanned by {| (cid:105) , | (cid:105)} , {| (cid:105) , | (cid:105)} , . . . , {| N − (cid:105) , | N (cid:105)} . We can decompose any unitary operator in SU ( N ) intoa sequence of SU (2) rotations on these two-dimensional (2D) subspaces. Each of thesecan be further decomposed into a sequence of three rotations about two orthogonalaxes using the Euler decomposition. It therefore suffices if we can implement rotationsabout two fixed orthogonal axes on each of the 2D subspaces. Applied to the problemof quantum state transfer, it is not difficult to see that we can transform any complexunit vector (cid:126)c (0) into any other complex unit vector (cid:126)c ( s ) by a sequence of N − (cid:126)c ( s ) = U ( N − ,N ) . . . U (2 , U (1 , (cid:126)c (0) (2)where U ( n,n +1) indicates a complex rotation on the subspace spanned by {| n (cid:105) , | n +1 (cid:105)} .Decomposing each U ( n,n +1) further into three rotations about two fixed orthogonalaxes, U ( n,n +1)1 ( α ) and U ( n,n +1)2 ( β ), by suitable angles γ k , U ( n,n +1) = U ( n,n +1)1 ( γ ) U ( n,n +1)2 ( γ ) U ( n,n +1)1 ( γ ) , (3) ang-Bang Control Design for Quantum State Transfer N −
1) such rotations are required to transform a given initialstate to a target state using a sequence of elementary unitary transformations, (cid:126)c ( s ) = U ( N − ,N )1 ( γ N − ) U ( N − ,N )2 ( γ N − ) U ( N − ,N )1 ( γ N − ) × . . . × U (1 , ( γ ) U (1 , ( γ ) U (1 , ( γ ) (cid:126)c (0) . (4)It is easy to see how to transform pure states in principle, but it is not obvious howto derive the correct rotation angles γ k in the sequence, which is what matters inpractice. Although it is possible to constructively compute the γ k , the dependence of γ k on the state vectors (cid:126)c (0) and (cid:126)c ( s ) is complicated.
3. Bang-Bang Control Scheme based on Hyperspherical Parametrization
In this section we discuss how to obtain explicit expressions for the rotation angles γ k and show that it can be easily solved by parameterizing the initial and target statesin terms of complex hyperspherical coordinates. Any complex unit vector (cid:126)c can be parametrized in terms of complex hypersphericalcoordinates ( (cid:126)θ, (cid:126)φ ), c c ... c N − c N = e iφ cos θ e iφ sin θ cos θ ... e iφ N − sin θ . . . sin θ N − cos θ N − e iφ N − sin θ . . . sin θ N − (5)where (cid:126)θ and (cid:126)φ are vectors in R N − with 0 ≤ θ n ≤ π and − π ≤ φ n ≤ π , and e iφ is a global phase factor, which is usually negligible. Thus, assuming normalizationand neglecting global phases, any pure state is uniquely determined by its complexhyperspherical coordinates ( (cid:126)θ, (cid:126)φ ) which can be calculated easily by Algorithm 1.Although there are many equivalent parameterizations of pure state vectors, thebeauty of complex hyperspherical coordinates is that we can easily give an explicitconstructive bang-bang control scheme for state transfer | ψ (0) (cid:105) (cid:55)→ | ψ ( s ) (cid:105) such that allcontrol pulses are determined directly by the coordinates of the initial and final states( (cid:126)θ (0) , (cid:126)φ (0) , (cid:126)θ ( s ) , (cid:126)φ ( s ) ). The following scheme is based on the assumptions that (a) we can neglect free evolution H = 0; (b) we have local phase control, i.e., we can implement control operators thatintroduce a local phase shift, Z n = Π n , n = 2 , . . . , N (6)where I N is the identity on H and Π n is the projector onto the subspace of H spannedby the basis state | n (cid:105) ; and (c) we can individually control transitions between adjacent ang-Bang Control Design for Quantum State Transfer θ, φ ) ← HyperCoord ( c ) Compute complex hyperspherical coordinates
In: c complex vector/pure state Out: θ, φ hyper-spherical coordinates1: N ← length ( c )2: c ← c/ norm ( c )3: c ← exp( − i * angle ( c )) ∗ c φ ← angle ( c N )5: a ← abs ( c )6: θ ← arccos( a )7: s ← sin( θ )8: for n ← , . . . , N − θ n ← arccos( a n /s n − )10: s n ← s n − sin( θ n )Algorithm 1: Computation of complex hyperspherical coordinatesenergy levels, i.e. that we can realize control Hamiltonians of the form X n or Y n , X n = ( | n + 1 (cid:105)(cid:104) n | + | n (cid:105)(cid:104) n + 1 | ) , n = 1 , . . . , N − . (7a) Y n = i ( | n + 1 (cid:105)(cid:104) n | − | n (cid:105)(cid:104) n + 1 | ) , n = 1 , . . . , N − . (7b)The evolution of the system under any Hamiltonian H is governed by the Schr¨odingerequation i (cid:126) ˙ U ( t ) = HU ( t ) , U (0) = I N , (8)and we choose units such that the Planck constant (cid:126) = 1. This shows that theevolution under the control Hamiltonian H ∈ { LX n , LY n , LZ n } is given by the one-parameter groups exp( − iLtX n ), exp( − iLtY n ) and exp( − iLtZ n ), respectively. Theevolution is unitary as the operators X n , Z n and Y n are Hermitian. In particular, thismeans that we can implement the complex rotations U Xn ( α ) = exp( − iαX n ) , U Yn ( α ) = exp( − iαY n ) , U Zn ( α ) = exp( − iαZ n ) , (9)by applying the control Hamiltonians LX n , LY n and LZ n respectively for some time t = α/L . In the following only two types of control operations { X n , Z n } or { Y n , Z n } are required.The assumptions on the control Hamiltonian are somewhat demanding, althoughno more so than the control requirements for the standard geometric decompositionEq.(4). While these requirements cannot always be satisfied, there are systems forwhich these control operations are quite natural such as a charged particle trapped ina multi-well potential created and controlled by surface control electrodes as shown inFig. 1. A physical realization of such a system could be a multi-well potential created ina 2D electron gas in a semiconductor material by surface control electrodes. Changingthe voltages applied to different control electrode enables us to vary the depth ofindividual wells as well as the height of the potential barrier between adjacent wellsand thus the tunnelling rate, giving raise Z n and Y n rotations, respectively. ang-Bang Control Design for Quantum State Transfer Control electrodesPotential well Tunnel barrier
Figure 1.
Charged particle trapped in a multi-well potential created by controlelectrodes. Red electrodes allow control of potential barriers and thus tunnellingrates, while blue electrodes control depths of the wells 1 to 3 and thus their energylevels. We can choose default voltage settings such that all wells have the samedepth and there is no tunnelling, so that we effectively have H = 0. Then byraising or lowering the voltage of electrode 1 we can introduce a relative phaseshift between the ground state | (cid:105) of the first well and the other two ground states,and by changing the voltage applied to the red electrode between wells 1 and 2we can induce tunnelling between the first two wells, etc. To illustrate the constructive procedure, let us consider the case N = 3 with Y, Z controls. In this case the control operators take the explicit form Z = , Z = , Y = − i i , Y = − i i . and the corresponding evolution operators are U Z ( α ) = e − iα
00 0 1 , U Z ( α ) = e − iα ,U Y ( α ) = cos α − sin α α cos α
00 0 1 , U Y ( α ) = α − sin α α cos α , Given these control operators and the hyperspherical coordinate representation of theinitial and target states, it is now very easy to see how to steer an arbitrary initialstate to an arbitrary target state in the following seven steps:
Step 1. ( θ (0)1 , θ (0)2 ; φ (0)1 , φ (0)2 ) → ( θ (0)1 , θ (0)2 ; φ (0)1 , U Z ( φ (0)2 ) e − iφ (0)2 cos θ (0)1 e iφ (0)1 sin θ (0)1 cos θ (0)2 e iφ (0)2 sin θ (0)1 sin θ (0)2 = cos θ (0)1 e iφ (0)1 sin θ (0)1 cos θ (0)2 sin θ (0)1 sin θ (0)2 . Step 2. ( θ (0)1 , θ (0)2 ; φ (0)1 , → ( θ (0)1 , θ (0)2 ; 0 , U Z ( φ (0)1 ) e − iφ (0)1
00 0 1 cos θ (0)1 e iφ (0)1 sin θ (0)1 cos θ (0)2 sin θ (0)1 sin θ (0)2 = cos θ (0)1 sin θ (0)1 cos θ (0)2 sin θ (0)1 sin θ (0)2 ang-Bang Control Design for Quantum State Transfer Step 3. ( θ (0)1 , θ (0)2 ; 0 , → ( θ (0)1 ,
0; 0 , U Y ( − θ (0)2 ) θ (0)2 sin θ (0)2 − sin θ (0)2 cos θ (0)2 cos θ (0)1 sin θ (0)1 cos θ (0)2 sin θ (0)1 sin θ (0)2 = cos θ (0)1 sin θ (0)1 Step 4. ( θ (0)1 ,
0; 0 , → ( θ ( s )1 , ,
0; 0 , U Y ( θ ( s )1 − θ (0)1 ) cos( θ ( s )1 − θ (0)1 ) − sin( θ ( s )1 − θ (0)1 ) 0sin( θ ( s )1 − θ (0)1 ) cos( θ ( s )1 − θ (0)1 ) 00 0 1 cos θ (0)1 sin θ (0)1 = cos θ ( s )1 sin θ ( s )1 Step 5. ( θ ( s )1 ,
0; 0 , → ( θ ( s )1 , θ ( s )2 ; 0 , U Y ( θ ( s )2 ) θ ( s )2 − sin θ ( s )2 θ ( s )2 cos θ ( s )2 cos θ ( s )1 sin θ ( s )1 = cos θ ( s )1 sin θ ( s )1 cos θ ( s )2 sin θ ( s )1 sin θ ( s )2 Step 6. ( θ ( s )1 , θ ( s )2 ; 0 , → ( θ ( s )1 , θ ( s )2 ; φ ( s )1 , U Z ( − φ ( s )1 ) e iφ ( s )1
00 0 1 cos θ ( s )1 sin θ ( s )1 cos θ ( s )2 sin θ ( s )1 sin θ ( s )2 = cos θ ( s )1 e iφ ( s )1 sin θ ( s )1 cos θ ( s )2 sin θ ( s )1 sin θ ( s )2 Step 7. ( θ ( s )1 , θ ( s )2 ; φ ( s )1 , → ( θ ( s )1 , θ ( s )2 ; φ ( s )1 , φ ( s )2 ): Apply phase rotation U Z ( − φ ( s )2 ) e iφ ( s )2 cos θ ( s )1 e iφ ( s )1 sin θ ( s )1 cos θ ( s )2 sin θ ( s )1 sin θ ( s )2 = cos θ ( s )1 e iφ ( s )1 sin θ ( s )1 cos θ ( s )2 e iφ ( s )2 sin θ ( s )1 sin θ ( s )2 The generalization to
N > H = N − (cid:88) m =1 u m ( t ) H m (10)where H n − = Z n +1 , H n = Y n and u m ( t ) are controls (e.g. voltages), the bang-bang control sequence given by Algorithm 2 can be implemented by applying 4 N − k th step we apply a constant control field u S ( k ) = L k for time t k = γ k /L k , while all other controls are set to 0 (or the voltages are set to their defaultvalues). Notice that in practice we cannot apply fields for negative times, thus thesign of L k must match that of γ k . However, if γ k is negative and L k >
0, we can alsoapply a field f S ( k ) = L k for time t k = ( γ k + 2 π ) /L k as γ k + 2 π > X n control Hamiltonians are used instead of Y n control Hamiltonians, error i n − is created in the n th coordinate by the population rotations. So the algorithmneed be slightly modified to correct phase factors of | n (cid:105) ( n = 2 , · · · , N ). We can achievethis by adding π ( n mod 4) to the phase angles φ n , noting that e iπ/ n mod 4) = i n and the phase factor of the n th coordinate is e iφ n − . ang-Bang Control Design for Quantum State Transfer N − N − S, γ ) ← StateTransfer ( c (0) , c ( s ) ) Compute sequence of rotations required for state transfer
In: c (0) , c ( s ) initial and target state vectors Out:
S, γ
Bang-bang control sequence1: ( θ (0) , φ (0) ) ← HyperCoord ( c (0) )2: ( θ ( s ) , φ ( s ) ) ← HyperCoord ( c ( s ) )3: for n ← N − , . . . ,
14: Append S by 2 n − γ by φ (0) n // Apply Phase Rotation U Zn +1 ( φ (0) n )5: for n ← N − , . . . ,
26: Append S by 2 n , γ by − θ (0) n // Apply Population Rotation U Yn ( − θ (0) n )7: Append S by 2, γ by θ ( s )1 − θ (0)1 // Apply Population Rotation U Y ( θ ( s )1 − θ (0)1 )8: for n ← , . . . , N −
19: Append S by 2 n , γ by θ ( s ) n // Apply Population Rotation U Yn ( θ ( s ) n )10: for n ← , . . . , N − S by 2 n − γ by − φ ( s ) n // Apply Phase Rotation U Zn +1 ( − φ ( s ) n )Algorithm 2: Control Scheme to achieve state transfer (cid:126)c (0) (cid:55)→ (cid:126)c ( s ) in 4 N − (cid:126)S and (cid:126)γ are vectors of length 4 N −
5, whose elements are integer labels indicating the controlHamiltonian ( m = 1 , . . . , N −
2) and rotation angle γ k , respectively. W -state As a simple application of the scheme, suppose first that we have N sites and startingwith only site 1 populated, i.e., in state | (cid:105) , we would like to prepare an equalsuperposition of all N sites | n (cid:105) for n = 1 , . . . , N : | ψ (cid:105) = √ N N (cid:88) n =1 | n (cid:105) . All we need to do is compute the hyperspherical coordinates of | ψ (cid:105) , e.g., for N = 10 (cid:126)θ = (1 . , . , . , . , . , . , . , . , . (cid:126)ϕ = (cid:126)
0, which tells us that we need to apply a sequence of 9 Y -rotations U Y ( θ ) U Y ( θ ) U Y ( θ ) U Y ( θ ) U Y ( θ ) U Y ( θ ) U Y ( θ ) U Y ( θ ) U Y ( θ )to the initial state | (cid:105) . This results in the following sequence of states being created ang-Bang Control Design for Quantum State Transfer n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 91.0000 0.3162 0.3162 0.3162 0.3162 0.3162 0.3162 0.3162 0.3162 0.31620 0.9487 0.3162 0.3162 0.3162 0.3162 0.3162 0.3162 0.3162 0.31620 0 0.8944 0.3162 0.3162 0.3162 0.3162 0.3162 0.3162 0.31620 0 0 0.8367 0.3162 0.3162 0.3162 0.3162 0.3162 0.31620 0 0 0 0.7746 0.3162 0.3162 0.3162 0.3162 0.31620 0 0 0 0 0.7071 0.3162 0.3162 0.3162 0.31620 0 0 0 0 0 0.6325 0.3162 0.3162 0.31620 0 0 0 0 0 0 0.5477 0.3162 0.31620 0 0 0 0 0 0 0 0.4472 0.31620 0 0 0 0 0 0 0 0 0.3162 Each Y n extends the superposition by one site until we are left with the desired stateafter the final step. Creating such a superposition state may not seem very interestingbut it has an interesting application in the area of entanglement creation, for instance.The Hamiltonians for many systems such as interacting quantum dots orcoupled Josephson junctions, for example, can be described to a reasonably goodapproximation by an XXZ-spin network model H = (cid:88) n α n σ Zn + (cid:88) m 4. Optimal piecewise-constant Control and Time-energy Performance The bang-bang control sequence given by Algorithm 2 leaves us considerable freedomof choice for the controls. Choosing large control amplitudes will result in short pulsedurations, thus optimizing the transfer time t f . However, large control amplitudes ang-Bang Control Design for Quantum State Transfer J = (cid:90) t f [ λ + N − (cid:88) m =1 | u m ( t ) | ] dt (14)where λ is the ratio factor of the costs of time and energy and λ > 0. Larger valuesof λ indicate a stronger emphasis on time-cost, while smaller values of λ give moreweight to the energy cost of the controls.If the controls can take values f m ( t ) ∈ { , ± L } and the pulses are applied strictlysequentially, then the total length t f of the control sequence is t f = 1 L (cid:34) N − (cid:88) n =1 | φ (0) n | + | φ ( s ) n | + N − (cid:88) n =2 ( θ (0) n + θ ( s ) n ) + | θ ( s )1 − θ (0)1 | (cid:35) ≤ L (cid:104) N − π + 2( N − π π (cid:105) = (6 N − π L (15)because of 0 ≤ θ n ≤ π and 0 ≤ | φ n | ≤ π . Noting that a + b ≥ ab , with equalityexactly if a = b , we have J = K (cid:88) k =1 ( λ + L k ) t k ≤ K (cid:88) k =1 √ λL k t k ≤ √ λt f max k L k (16)with equality if and only if L k = √ λ . This shows that the optimal choice of the fieldamplitudes is L k = √ λ , for which we have t ∗ f ≤ (6 N − π √ λ , J ∗ = min J = 2 λt ∗ f ≤ √ λ (6 N − π (17)and the corresponding optimal energy cost is E ∗ = J ∗ − λt ∗ f ≤ √ λ (6 N − π . Asexpected, as λ goes to 0, t ∗ f becomes infinite and E ∗ goes to 0, but their productremains constant t ∗ f · E ∗ = (cid:104)(cid:80) N − n =1 | φ (0) n | + | φ ( s ) n | + (cid:80) N − n =2 ( θ (0) n + θ ( s ) n ) + | θ ( s )1 − θ (0)1 | (cid:105) ≤ (6 N − π (18)and depends only on the geometric parameters of the initial state and target states.If first and last N − t (cid:48) f = 1 L (cid:34) max n | φ (0) n | + max n | φ ( s ) n | + N − (cid:88) n =2 ( θ (0) n + θ ( s ) n ) + | θ ( s )1 − θ (0)1 | (cid:35) ≤ L (cid:104) π + 2( N − π π (cid:105) = (2 N + 3) π L . (19)Setting φ (0)max = max n | φ (0) n | and φ ( s )max = max n | φ ( s ) n | , shows that we have t = φ (0)max /L and t K = φ ( s )max /L , and thus we must choose L n ≥ φ (0) n /t and L n = φ ( s ) n /t K ,respectively for the control amplitude of the first and last N − ang-Bang Control Design for Quantum State Transfer N − t or t K , respectively.Furthermore the performance index changes J ≤ t (cid:48) f √ λ max N ≤ k ≤ K +1 − N L k + N − (cid:88) k =1 L k + K (cid:88) k = K − N +2 L k , (20)which suggests that we can improve the performance index and reduce the energycost by choosing the amplitudes of the first and last N − L n = φ (0) n /t and L n = φ ( s ) n /t K , and L k = √ λ for all otheramplitudes. 5. Implementation of Unitary Operators Any N − dimensional unitary operator can be represented as follows: U = N (cid:88) j =1 e iϕ j | u j (cid:105)(cid:104) u j | (21)where {| u j (cid:105)} constructs an orthonormal basis set in N − dimensional Hilbert space.From (21), we can find that global phase factors of all | u j (cid:105) s do not affect U , so theycan be neglected. Assuming (cid:104) | u j (cid:105) is real and positive for each j , by (5) the complexhyperspherical parametrization for {| u j (cid:105) : j = 1 , · · · , N } can be given by | u (cid:105)| u (cid:105) ... | u N (cid:105) = (cid:18) I N − C (2) (cid:19) · · · (cid:18) I C ( N − (cid:19) C ( N ) | (cid:105)| (cid:105) ... | N (cid:105) (22)where C ( k ) = (cid:16) −→ c k ) ) , ( −→ c k ) , . . . , ( −−→ c k − k ) ) , ( −→ c k ( k ) ) (cid:17) T , (23)and ( −→ c i ( k ) ) T is the transpose of the vector −→ c i ( k ) and −→ c k ) = cos θ ( k )1 e iφ ( k )1 sin θ ( k )1 cos θ ( k )2 ... e iφ ( k ) k − (cid:81) k − l =1 sin θ ( k ) l cos θ ( k ) k − e iφ ( k ) k − (cid:81) k − l =1 sin θ ( k ) l , (24) −→ c k ) = sin θ ( k )1 − e iφ ( k )1 cos θ ( k )1 cos θ ( k )2 ... − e iφ ( k ) k − cos θ ( k )1 (cid:81) k − l =2 sin θ ( k ) l cos θ ( k ) k − − e iφ ( k ) k − cos θ ( k )1 (cid:81) k − l =2 sin θ ( k ) l , (25) ang-Bang Control Design for Quantum State Transfer −−→ c k − k ) = e iφ ( k ) k − sin θ ( k ) k − − e iφ ( k ) k − cos θ ( k ) k − cos θ ( k ) k − − e iφ ( k ) k − cos θ ( k ) k − sin θ ( k ) k − , (26) −→ c k ( k ) = e iφ ( k ) k − sin θ ( k ) k − − e iφ ( k ) k − cos θ ( k ) k − (27)for 2 ≤ k ≤ N . Note that C ( k ) ( C ( k ) ) † = I k . Suppose that Y n and Z n controls as defined in eqs. (6) and (7b) are permitted for1 ≤ n ≤ N − ≤ n ≤ N , respectively. Then U can be realized by a sequence ofbang-bang controls U = T † (cid:32) N (cid:89) n =1 U Zn ( ϕ n ) (cid:33) T (28)where T = N (cid:89) n =2 U n , U n = n − (cid:89) j =1 U YN − n + j ( θ ( n ) j ) n − (cid:89) j =1 U ZN − n + j +1 ( φ ( n ) j ) . (29)The Z -phase rotations (underlined) commute and can be applied concurrently. Thus U n can be implemented in n steps and T in N ( N +1) / − N ( N + 1) − U ∈ SU ( N ) into a sequence of N ( N + 1) / {| n (cid:105) , | n + 1 (cid:105)} , and further decomposed each of these SU (2) rotationsinto three elementary Y n and Z n rotations using the Euler decomposition, we wouldrequire 3 N ( N + 1) / Z n and Y n operations do not commute,these could not be implemented concurrently.The proof of the result is constructive. (1) Effect of each U n . Let | e ( n ) (cid:105) be an arbitrary state in the space spanned by {| (cid:105) , | (cid:105) , · · · , | N − n (cid:105)} and | e ( n ) N − n +1 (cid:105)| e ( n ) N − n +2 (cid:105) ... | e ( n ) N (cid:105) = C ( n ) | N − n + 1 (cid:105)| N − n + 2 (cid:105) ... | N (cid:105) ang-Bang Control Design for Quantum State Transfer C ( n ) is as defined in Eq.(23). That is, | e ( n ) N − n +1 (cid:105) = cos θ ( n )1 | N − n + 1 (cid:105) + e iφ ( n )1 sin θ ( n )1 cos θ ( n )2 | N − n + 2 (cid:105) + · · · + e iφ ( n ) n − sin θ ( n )1 · · · sin θ ( n ) n − cos θ ( n ) n − | N − (cid:105) + e iφ ( n ) n − sin θ ( n )1 · · · sin θ ( n ) n − sin θ ( n ) n − | N (cid:105)| e ( n ) N − n +2 (cid:105) = sin θ ( n )1 | N − n + 1 (cid:105) − e iφ ( n )1 cos θ ( n )1 cos θ ( n )2 | N − n + 2 (cid:105)− · · · − e iφ ( n ) n − cos θ ( n )1 sin θ ( n )2 · · · sin θ ( n ) n − cos θ ( n ) n − | N − (cid:105)− e iφ ( n ) n − cos θ ( n )1 sin θ ( n )2 · · · sin θ ( n ) n − sin θ ( n ) n − | N (cid:105)· · · · · ·| e ( n ) N − (cid:105) = e iφ ( n ) n − sin θ ( n ) n − | N − (cid:105)− e iφ ( n ) n − cos θ ( n ) n − cos θ ( n ) n − | N − (cid:105) − e iφ ( n ) n − cos θ ( n ) n − sin θ ( n ) n − | N (cid:105)| e ( n ) N (cid:105) = e iφ ( n ) n − sin θ ( n ) n − | N − (cid:105) − e iφ ( n ) n − cos θ ( n ) n − | N (cid:105) .U n leaves any state | e ( n ) (cid:105) in the subspace spanned by {| (cid:105) , | (cid:105) , · · · , | N − n (cid:105)} invariant, i.e., U n | e ( n ) (cid:105) = | e ( n ) (cid:105) as U n is the identity on this subspace. Furthermore,Sec. 3.3 shows that applying U n to | e N − n +1 (cid:105) maps it to the basis state | N − n + 1 (cid:105) ,and we can verify by direct computation U n | e ( n ) N − n +2 (cid:105) = U YN − n +1 ( θ ( n )1 )(sin θ ( n )1 | N − n + 1 (cid:105) − cos θ ( n )1 | N − n + 2 (cid:105) )= −| N − n + 2 (cid:105)· · · · · · U n | e ( n ) N − (cid:105) = n − (cid:89) j =1 U YN − n + j ( θ ( n ) j )(sin θ ( n ) n − | N − (cid:105) − cos θ ( n ) n − | N − (cid:105) )= −| N − (cid:105) U n | e ( n ) N (cid:105) = n − (cid:89) j =1 U YN − n + j ( θ ( n ) j )(sin θ ( n ) n − | N − (cid:105) − cos θ ( n ) n − | N − (cid:105) )= −| N (cid:105) U n | e ( n ) (cid:105) = | e ( n ) (cid:105) | e ( n ) (cid:105) ∈ Span {| (cid:105) , · · · , | N − n (cid:105)} U n | e ( n ) N − n +1 (cid:105) = | N − n + 1 (cid:105) U n | e ( n ) j (cid:105) = −| j (cid:105) N − n + 2 ≤ j ≤ N. (30) (2) Effect of T. Using the previous result we now show that T | u n (cid:105) = ( − ( n − | n (cid:105) with | u n (cid:105) as defined in (22). Let (cid:126)a ( k ) = (0 , · · · , , ( −→ c k ) ) T ) be a row vector of length N where the coefficient vector c ( k )1 is as in Eq. (24) and the number of zeros is N − k .Eqs (22)–(29) and (30) give T | u (cid:105) = U · · · U N (cid:126)a ( N ) | (cid:105) ... | N (cid:105) = U · · · U N − U N | e ( N )1 (cid:105) = U · · · U N − | (cid:105) = | (cid:105) ang-Bang Control Design for Quantum State Transfer T | u (cid:105) = U · · · U N (cid:126)a ( N − C ( N ) | (cid:105) ... | N (cid:105) = U · · · U N (cid:126)a ( N − | e ( N )1 (cid:105) ... | e ( N ) N (cid:105) = U · · · U N − (cid:126)a ( N − | (cid:105)−| (cid:105) ... −| N (cid:105) = − U · · · U N − | e ( N − (cid:105) = −| (cid:105) Furthermore, for 3 ≤ n ≤ N − U N | u n (cid:105) = (cid:126)a ( N − n +1) N − (cid:89) k = N − n +2 (cid:18) I N − k C ( k ) (cid:19) | (cid:105)−| (cid:105) ... −| N (cid:105) = (cid:126)a ( N − n +1) N − (cid:89) k = N − n +2 (cid:18) I N − k C ( k ) (cid:19) | (cid:105)−| e ( N − (cid:105) ... −| e ( N − N (cid:105) U N − U N | u n (cid:105) = (cid:126)a ( N − n +1) N − (cid:89) k = N − n +2 (cid:18) I N − k C ( k ) (cid:19) | (cid:105)−| (cid:105) ( − | e ( N − (cid:105) ...( − | e ( N − N (cid:105) and continuing we obtain for 3 ≤ n ≤ N − U N − n +3 · · · U N | u n (cid:105) = (cid:126)a ( N − n +1) ( − | (cid:105) ...( − ( n − | n − (cid:105) ( − ( n − | e ( N − n +2) n − (cid:105) ...( − ( n − | e ( N − n +2) N (cid:105) U N − n +2 · · · U N | u n (cid:105) =( − ( n − | e ( N − n +1) n (cid:105) T | u n (cid:105) = U N − n +1 · · · U N | u n (cid:105) =( − ( n − | n (cid:105) , Finally, we have U · · · U N N − (cid:89) k =3 (cid:18) I N − k C ( k ) (cid:19) C ( N ) | (cid:105)| (cid:105) ... | N (cid:105) = ( − | (cid:105) ...( − ( N − | N − (cid:105)(cid:105) ( − ( N − | N (cid:105) ang-Bang Control Design for Quantum State Transfer T | u N (cid:105) = U (0 , · · · , , ( −→ c ) T ) ( − | (cid:105) ...( − ( N − | N − (cid:105) ( − ( N − | N (cid:105) = U ( − ( N − | e (2) N (cid:105) = ( − ( N − | N (cid:105) . Thus we finally have T U T † = N (cid:88) n =1 e iϕ n T | u n (cid:105)(cid:104) u n | T † = N (cid:88) n =1 e iϕ n | n (cid:105)(cid:104) n | = N (cid:89) n =1 U Zn ( ϕ n )and U = T † (cid:81) Nn =1 U Zn ( ϕ n ) T as claimed. 6. Discussions and Conclusion We have presented an explicit geometric control scheme for quantum state transferproblems based on a parametrization of the pure state vectors in terms of complexhyperspherical coordinates. Although it is not difficult to find constructive controlschemes for state transfer based on Lie group decompositions, most schemes do notgive explicit expressions for the rotation angles (“generalized Euler angles”) in thefactorization, and thus the rotation angles usually have to computed numerically. Byparametrizing the initial and target states in terms of hyperspherical coordinates, weobtain a factorization where all generalized Euler angles are given explicitly in termsof the hyperspherical coordinates of the initial and target states, eliminating the needfor numerical calculation of the generalized Euler angles, aside from computation ofthe hyperspherical coordinates, which is trivial in terms of computational overhead.The factorization is applicable given controls capable of implementing phaserotations and population rotations (of either X or Y type) on a collection of two-dimensional subspaces, similar to the general requirements for constructive geometriccontrol schemes. Compared to control schemes based on the standard factorization,this scheme has the additional advantages that all initial and final phase rotations canbe combined in a single step and executed concurrently, reducing the time required toachieve the state transfer. As with all bang-bang control schemes based on Lie groupdecompositions, the factorization only determines the sequence in which the controlsare applied and the pulse area (rotation angle) of the control pulses, leaving us withconsiderable freedom to choose the pulse shapes and amplitudes, which can be usedto further optimize a performance index. Here we have considered optimization of thepulse amplitudes for piecewise constant controls such as to minimize a time-energyperformance index that takes into account the competing goals of trying to minimizethe transfer time and energy cost of the controls.The scheme can be generalized to realize unitary operators. By expressing theeigenvectors of the target gate U in hyperspherical coordianates we obtain an explicitdecomposition for arbitrary unitary operators. 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