On extrema of the objective functional for short-time generation of single qubit quantum gates
Alexander Pechen Nikolay Il'in




On extrema of the objective functional for short-time generation of single qubit quantum gates
    Alexander Pechen^1[Corresponding author. E-mail: apechen@gmail.comapechen@gmail.com; Webpage: http://www.mathnet.ru/eng/person17991mathnet.ru/eng/person17991]   and Nikolay Il'in^2
    
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^1Department of Mathematical Methods for Quantum Technologies,

Steklov Mathematical Institute of Russian Academy of Sciences
 Gubkina str., 8, Moscow 119991, Russia

^2The National University of Science and Technology MISiS

Leninsky Prospekt, 4, Moscow 119049, Russia




In the present work the extrema of the objective functional for the problem of  generation of quantum gates (logical elements for quantum computations) for two–level systems are investigated for short duration of the control. The problem of  existence of local but not global extrema, the so called traps, is considered. In prior works the absence of traps was proved for a sufficiently long control duration. In this paper we prove that for almost all target unitary operators and system Hamiltonians traps are absent for an arbitrarily small control duration. For the remainder target unitary operators and Hamiltonians we obtain a new estimate for the lower boundary of the control duration which guarantees the absence of traps.





§ INTRODUCTION

In the present paper we study the problem of controlling a qubit, i.e. a two-level quantum
system, by using coherent control pulses (electromagnetic
field). Qubit is one of basic elements for realization of quantum
computing and for creation of quantum computer. An important problem is to generate
single qubit gates (logical elements for quantum computation) <cit.>.

The dynamics of a qubit interacting with coherent control f(t) under the assumption 
of good enough isolation of the qubit from the environment is described by the Schrödinger equation for unitary evolution U_t (a 2× 2 unitary matrix):

    i dU_t/dt=(H_0+f(t)V)U_t,     U_t=0=𝕀.

Here H_0 and V are 2× 2 Hermitian matrices. In order to make the control problem non-trivial we assume that [H_0,V]0. In the presence of the environment the reduced dynamics of the qubit is described by various master equations <cit.> and by a quantum channel instead of a unitary transformation <cit.>. The control f belongs to some set of admissible controls 𝒰, f∈𝒰. In the applications one considers sets of admissible controls 𝒰=L^1([0,T];ℝ), 𝒰=L^2([0,T];ℝ) and others. Here T>0 is the fixed control duration. The problem of optimal performance is also considered <cit.>. In this work we consider the set of control 𝒰=L^1([0,T];ℝ). Matrix elements [U_t]_ik are assumed to be absolutely continuous functions on the interval [0,T], [U_t]_ik∈AC[0,T].  In this case the equation (<ref>) has a unique solution for every control f∈𝒰 <cit.>.

An important problem in quantum information is generation of quantum gates (special unitary (2× 2)–matrices) W∈SU(2), i.e. the search for such a control f that U_T=W, perhaps up to a phase factor.
This problem can be formulated as the problem of finding a control f which maximizes the objective functional

    𝒥_W[f] = 1/4|Tr(W^† U_T)|^2.

The objective functional 𝒥_W reaches its maximum value
𝒥_W^max=1 on a unitary matrix of the form  U_T=We^iω,
where ω∈ℝ is an arbitrary phase. The global minimum of the objective 𝒥_W is equal to zero, 𝒥_W^min=0.
Examples of the objective matrix W which are important for applications include the Hadamard gate
W=ℍ,

    ℍ=1/√(2)[  1  1;  1 -1 ],

the phase shift gate W=U_ϕ, where ϕ∈(0,2π),

    U_ϕ=[    1    0;    0 e^iϕ ]

and other.

In the present work we consider the problem of possible existence of local but not global maxima for the objective functional 𝒥_W for short T. Such local maxima are called traps <cit.>–<cit.>.
We prove  the absence of traps for almost all objective unitary operators and system Hamiltonians for an arbitrarily small control duration. For the remainder  set of objective unitary operators and  Hamiltonians we obtain the new estimate for the lower boundary of the control duration, which guarantees the absence of traps.




§ THE ABSENCE OF TRAPS FOR CONTROLLING A QUBIT AT LONG TIMES


If traps would exist they would become the obstacle for the search of globally optimal control by local search algorithms. In works <cit.> the  absence of traps was conjectured  for typical control problems for systems which are isolated from the environment, i.e. for closed quantum systems <cit.>. In works <cit.> the absence of traps was proved for two–levels closed quantum systems in the case of sufficiently long T.

Define the special control f_0  and time T_0: 

    f_0   :=-TrH_0TrV+2Tr(H_0V)/(TrV)^2-2TrV^2,
    
    
    T_0   :=π/H_0-𝕀TrH_0/2+f_0(V-𝕀TrV/2).

Here and below the norm of a matrix A is the operator norm

    A=sup_𝐚=1A𝐚.

Note that T_0<∞, because if H_0=𝕀TrH_0/2-f_0V
then [H_0,V]=0, that contradicts the assumption of non-triviality of the system Hamiltonian.

If Tr H_00 and Tr V0, then replacing
H_0 and V by H_0=H_0-Tr H_0/2
and V=V-Tr V/2 we can  transform the free Hamiltonian 
to the form with TrH_0=0 and TrV=0.
Such a replacement does not affect the existence of traps, because the evolution operator
U_T, which is determined by the solution of the equation (<ref>) for the pair
(H_0,V), is related to the evolution operator U_T, which is determined by the solution of the equation (<ref>) for the pair (H_0,V), by the equality
U_T=U_Te^-iλ(T)𝕀. Here
λ(T)=(TTr H_0+Tr V∫_0^Tf(t) dt)/2.
Hence, U_T differs from U_T by a phase and the objective value under such  replacement 
does not change:

    |Tr(W^†U_T)|^2=|e^-iλ(T)Tr(W^† U_T)|^2=|Tr(W^† U_T)|^2.

Unless otherwise stated, without loss of generality below we assume that the matrices H_0 and V are traceless. We will also use the Pauli matrices σ_x,
σ_y and σ_z:

    σ_x=[ 0 1; 1 0 ],   σ_y=[  0 -i;  i  0 ],   σ_z=[  1  0;  0 -1 ].


In <cit.> the following statement is proved.



If [H_0,V]0 and T≥ T_0, then all maxima of the objective functional
𝒥_W are global. Any control f f_0 is not a trap  for any T>0.


From the results of the paper <cit.> it follows that for small T only the special control
f=f_0 may be a trap. Below in this work we will show that for almost all
W the control f_0 is not a trap for any T>0.




§ THE ABSENCE OF TRAPS FOR SMALL T


According to Theorem <ref>, the only potential trap can be the
control f=f_0. Therefore, to explore the possibility of the existence of traps
for small T it is sufficient to investigate the behaviour of the objective  at this point.

Theorem <ref> states the absence of traps for the objective 𝒥_W for almost all W. Let
d=H_0+f_0V-𝕀Tr H_0/2-f_0𝕀Tr V/2.
Note, that any matrix W such that [H_0+f_0V,W]=0 has the form 

    W=e^iα_W (H_0+f_0V)+iβ_W,     α_W∈(0,π/d],  β_W∈[0,2π).




Let in the equation (<ref>) be [H_0,V] 0. Let
[H_0+ f_0V,W]0.
Then for any T>0 all maxima of the objective functional 𝒥_W are global.
Let [H_0+f_0V,W]=0. If α_W∈(0,π/(2d)), then all maxima
of the objective functional 𝒥_W are global for any T>0. If
α_W∈[π/(2d),π/d], then all maxima
of the objective functional 𝒥_W
are global for any T>π/d-α_W.


Proof of the theorem <ref> is based on the lemmas <ref>–<ref>.

We will use the expansion of the objective functional 𝒥_W in Taylor series
up to the second-order term <cit.>.



There is an asymptotic expansion

    𝒥_W[f+δ f]   =𝒥_W[f]+∫_0^Tδ𝒥_W/δ
    f(t)δ f(t) dt
    
           +1/2∫_0^T∫_0^Tδ^2𝒥_W/δ f(t_2)δ f(t_1)δ f(t_1)δ f(t_2) dt_1 dt_2+o(δ f^2_L^1).

Here

    δ𝒥_W/δ f(t)=1/2Im(Tr
    Y^†Tr(YV_t)),     Y=W^† U_T,   V_t=U_t^† V U_t,
    
    δ^2𝒥_W/δ f(t_2)δ f(t_1)=12
    Re(Tr(Y V_t_1)Tr(Y^† V_t_2)
    -Tr(Y V_t_2V_t_1)Tr Y^†),    t_2≥ t_1,
    
    12Re(Tr(Y V_t_2)Tr(Y^† V_t_1)
    -Tr(Y V_t_1V_t_2)Tr Y^†),    t_2<t_1.

The linear map A L^1([0,T];ℝ)↦ℝ which is defined as

    Ag=∫_0^Tδ𝒥_W/δ f(t)g(t) dt,

is the Frechet differential of the map f→𝒥_W[f].



The evolution operator U_t^f induced by the control f
satisfies the Schrödinger equation

    i dU_t^f/dt=(H_0+fV)U_t^f,

The evolution operator U_t^f+g induced by the control f+g satisfies the  equation

    i dU_t^f+g/dt=(H_0+fV)U_t^f+g+gVU_t^f+g.

Making the replacement U_t^f+g=U_t^fZ_t in the equation (<ref>), we obtain

    i dZ_t/dt=gV_tZ_t,     V_t=U_t^f†VU^f_t.

Now let represent (<ref>) in the integral form

    Z_t=𝕀-i∫_0^tV_t_1Z_t_1g(t_1) dt_1.

Iterating the expression (<ref>) and multiplying by  U^f_t on the left, we obtain

    U_T^f+g   =U_T^f -i∫_0^TU_T^fV_t
    g(t) dt-∫_0^T∫_0^t_1U_T^fV_t_1V_t_2g(t_1)g(t_2) dt_2 dt_1
    
           +i∫_0^T∫_0^t_1∫_0^t_2U_T^fV_t_1V_t_2Z_t_3 g(t_1)g(t_2)g(t_3) dt_3 dt_2 dt_1.

Because U_T^f=Z_t= 1 and V_t=V, for the last summand in (<ref>) we obtain the estimate 

    ∫_0^T∫_0^t_1∫_0^t_2U_T^fV_t_1V_t_2Z_t_3
    g(t_1)g(t_2)g(t_3) dt_3 dt_2 dt_1≤g^3_L^1([0,T];ℝ)V^2.

For the first and second order variations we have the estimates  

    ∫_0^TU_T^fV_t g(t_1) dt_1≤g_L^1([0,T];ℝ)V,
    
    ∫_0^T∫_0^t_1U_T^fV_t_1V_t_2g(t_1)g(t_2) dt_2 dt_1
    ≤g^2_L^1([0,T];ℝ)V^2.

Replacing in the objective  

    𝒥_W[f+δ f]=1/4Tr(W^† U_T^f+δ f)
    Tr(WU_T^† f+δ f)

the expression (<ref>), we obtain the asymptotic expansion (<ref>).
From (<ref>), because δ𝒥_W/δ f(t)
and δ^2𝒥_W/δ f(t_2)δ f(t_1) are bounded functions, we obtain

    𝒥_W[f+g]=𝒥_W[f]+Ag+o(g_L^1),
    
    
    Ag=∫_0^Tδ𝒥_W/δ f(t)g(t) dt.

Therefore bounded linear operator A L^1([0,T];ℝ)↦ℝ
is the Frechet differential of the map f→𝒥_W[f].
This proves the lemma.


Necessary conditions for the point f=f_0 to be a maximum or a minimum of the objective  functional 𝒥_W
are determined by vanishing of the gradient
δ𝒥_W/δ f|_f=f_0=0 and by semi-definiteness of the quadratic form

    ∫_0^T∫_0^THess(τ_2,τ_1)f(τ_2)f(τ_1) dτ_1 dτ_2≥0,
      where  Hess(t_1,t_2)=δ^2 𝒥_W/δ f(t_2)δ f(t_1)|_f=f_0.

A sufficient condition for the control  f=f_0 to be a saddle point is determined by vanishing of the gradient δ𝒥_W/δ f|_f=f_0=0
and by alternating quadratic form

    ∫_0^T∫_0^THess(τ_2,τ_1)f(τ_2)f(τ_1) dτ_1 dτ_2.


For the analysis of properties of the control problem in a neighbourhood of the special control f_0
it is convenient to choose a special basis in the space of (2×2)-matrices, for which
the equation (<ref>)  has a simple form.
Recall that, without loss of generality, we can set
TrH_0=TrV=0.
Therefore there exists a unitary matrix S such that S(H_0+f_0V)S^†=hσ_z,
where h=H_0+f_0V and SVS^†=v_xσ_x+v_yσ_y+v_zσ_z.
Note that v_z=0, because
v_z=Tr(SVS^†σ_z)/2=Tr[V(H_0+f_0V)]/2h=0,
since
f_0=-Tr(H_0V)/TrV^2 under condition
TrH_0=TrV=0.
If we make the replacement U→ SUS^† and the replacement of time t→ t/h, then
the equation (<ref>) takes the form

    i dU_t/dt=(σ_z+g(t)(v_xσ_x+v_yσ_y))U_t.

Here g(t)=f(t)-f_0, v_x=v_x/h, v_y=v_y/h. For the system (<ref>) we have f_0=0 and T_0=π.
In the transition to a new basis the matrix W undergoes transformation W→ SWS^†
which does not change the objective  values, so that
|Tr(W^† U_T)|^2/4=|Tr(SW^† U_TS^†)|^2/4.
In particular, critical points of the control problem for the system (<ref>) correspond to critical points  for (<ref>). Therefore in lemmas <ref>–<ref> we can consider maximization of the objective functional |Tr(W^† U_T)|^2/4 for the system (<ref>).



Let [W,σ_z]0. Then for any T>0 the control g=0 is not a trap for maximization of the objective  functional  𝒥_W for the system (<ref>), and all maxima of the objective  functional are global.



The solution of the equation (<ref>) with constant control g=0
has the form U_t=e^-iσ_z t. The matrix V_t=e^iσ_z
tVe^-iσ_z t takes the form

    V_t=vcos(2t-ϕ)σ_x-vsin(2t-ϕ)σ_y.

Here v=√(v_x^2+v_y^2) and ϕ=arctan(v_y/v_x).

The matrix Y=W^† U_T is unitary. We can parametrize it by the angles  φ, ψ, θ and by phase ω (see <cit.>) as

    Y=e^iω[   e^iφcosθ   e^iψsinθ; -e^-iψsinθ  e^-iφcosθ ].

The angles ψ, φ and θ are expressed in terms of the Euler angles φ',
θ' and ψ', which belong to the domain 0≤φ'<2π,
0<θ'<π, -2π<ψ'<2π, in the following way: φ=(φ'+ψ')/2,
ψ=(φ'-ψ'+π)/2 and θ=θ'/2. The overall phase ω can be omitted,
since in the expression for the gradient (<ref>) and Hessian (<ref>) the matrices Y
and Y^† appear in pairs, so that the gradient and the Hessian are phase invariant.
The expression for the objective at the point g=0 through angles φ, ψ and
θ has the form

    𝒥_W[0]=1/4|Tr Y|^2=cos^2φcos^2θ.

We introduce the notation

    L(X):=1/2Im(Tr Y^†Tr(YX)).

If g=0 is an extrema for (<ref>), then at this point the gradient of the objective  functional
must vanish identically:

    δ𝒥_W/δ f(t)|_g=0=
    vcos(2t-ϕ)L(σ_x)-vsin(2t-ϕ)L(σ_y)=0  ∀ t∈[0,T].

The equality (<ref>) can be satisfied only if L(σ_x)=0 and
L(σ_y)=0. This imposes the restrictions on the angles φ, ψ and θ
at the extremal points:

    L(σ_x)   =2cosφcosθsinθsinψ=0
    ,
    
    
    L(σ_y)   =2cosφcosθsinθcosψ=0.

If cosφcosθ=0, then according to (<ref>) the objective
reaches its global minima 𝒥_W=0. If cosφcosθ0,
then the satisfaction of the equalities (<ref>) and (<ref>) requires that
sinθ=0. Under this condition cosθ=± 1 and up to a phase multiplier we have
 Y=e^iσ_zφ. Then

    W=U_TY^†=e^-iσ_z TY^†=e^-iσ_z (T+φ).

Thus, in this case [W,σ_z]=0 and, therefore, the assumption of the
lemma is not satisfied. This proves the lemma.


Recall that for the system (<ref>) the special time is T_0=π and for T≥ T_0 according to theorem <ref> traps are absent. The lemma <ref> states that if
[W,σ_z] 0 then traps are absent for any T>0.
Here we show that if [W,σ_z]=0 then the lower bound for the time T, for which there are no trap, can be reduced.



For T≥π/2,  for any W the control g=0 is not a trap for maximization of the objective functional 𝒥_W for the system (<ref>).



Under the condition sinθ=0 the Hessian has the form

    Hess(t_2,t_1)=-2v^2cosφcos(2|t_2-t_1|+φ).

We introduce the auxiliary function

    δ_ε (t)=
    0,     |t|≥ε2,
    
    1ε,     |t|<ε2.

Then for all f∈C[0,T] and t∈(ε/2,T-ε/2) we have

    ∫_0^Tδ_ε(τ-t)f(τ) dτ=f(t)+O(ε).

Let

    f_ε(t)=λδ_ε(t-t_1)+μδ_ε (t-t_2),
        ε/2<t_1<t_2<T-ε/2,  ε<t_2-t_1.

Substituting the function f_ε(t) in the expression for the second variation of the objective functional, we obtain

    (f,Hessf)   =
    ∫_0^T∫_0^THess(τ_2,τ_1)f(τ_2)f(τ_1) dτ_1 dτ_2
    
       =-2v^2cosφ G(λ,μ)+O(ε),

where

    G(λ,μ)=λ^2cosφ+2λμcos(2|t_2-t_1|+φ)+μ^2cosφ.

Bilinear form G(λ,μ) is alternating if and only if its discriminant D is positive:

    D=D(|t_2-t_1|)=cos^2(2|t_2-t_1|+φ)-cos^2φ>0.

If cos^2φ=0 or cos^2φ=1, then the objective functional has the global extrema 𝒥_W=0 or 𝒥_W=1. Hence,
trap may correspond  only to those angles φ for which
0<cos^2φ<1. Then D as function of the difference  |t_2-t_1| takes positive values at some points in the interval  [0,π/2], and because it has a period π/2, its maximum values are positive and minimum values are negative.
Let t_1,t_2∈ [0,π/2] be such that G(λ,μ) is an alternating form. For such t_1 and t_2 choose λ_1, λ_2, μ_1 and
μ_2 so that G(λ_1,μ_1)>0 and G(λ_2,μ_2)<0. Let

    f_1,ε(t)   =λ_1δ_ε (t-t_1)+μ_1δ_ε
    (t-t_2),
    
    
    f_2,ε(t)   =λ_2δ_ε (t-t_1)+μ_2δ_ε
    (t-t_2),

where ε is such that signs of (f_j,ε,
Hessf_j,ε), j=1,2, coincide with signs of
lim_ε→0(f_j, ε,Hessf_j,ε).
Then (f_1,ε,Hessf_1,ε) and
(f_2,ε,Hessf_2,ε) will have
opposite signs, i. e. Hessian at the point f=0 is not sign definite.
This proves the lemma.


Consider matrix W of the form

    W=e^iσ_zφ_W.

Note that we can put φ_W∈(0,π]. In the opposite case
φ_W=φ_W'+π k, φ_W'∈[0,π] and

    W=e^iσ_zφ_W=e^iσ_zφ_W'(-1)^k=W'e^iπ k,

that differs by a non-significant phase factor. The corresponding matrix
Y has the form

    Y=e^-iσ_z(φ_W+T),

i. e. φ=-φ_W-T.



If φ_W∈(0,π/2), then for any T>0 the control g=0 is not a trap for maximization of the objective functional 𝒥_W for the system (<ref>). If φ_W∈[π/2,π], then for any
T>π-φ_W the control g=0 is not a trap for maximization of the objective functional 𝒥_W for the system (<ref>).



The Hessian of the objective functional 𝒥_W[f] at the point g=0 has the form (<ref>),
where the bilinear form G(λ,μ) is defined by the equality (<ref>). The Hessian is sign definite if the discriminant D, which is defined by the formula (<ref>),
is positive. Let us demonstrate that there exist 0<t_1 and t_2<T such that D>0.

The expression (<ref>) for D  can be rewritten as

    D=sin2(φ_W+T-|t_2-t_1|)sin2|t_2-t_1|.

If φ_W∈(0,π/2), then for any T, π/2>T>0 chose
|t_2-t_1|=T-ε, where ε is sufficiently small such that
(φ_W+ε)∈(0,π/2) and T-ε>0. Then

    D=sin2(φ_W+ε)sin2(T-ε)>0,

because  (φ_W+ε)∈(0,π/2) and T<π/2.

If φ_W∈[π/2,π), then in order to obtain D>0 it is sufficient to satisfy the
inequalities

    π<φ_W+T-|t_2-t_1|<3π/2,    |t_2-t_1|<π/2.

For any T which satisfies the  inequality

    π<φ_W+T<3π/2,

choose such a small ε that

    π<φ_W+T-ε<3π/2.

Choose t_1 and t_2 such that |t_2-t_1|=ε.
Then for such ε the first inequality in (<ref>) holds by the virtue of (<ref>). The right inequality in (<ref>), i.e.,
T<3π/2-φ_W, is a consequence of the inequality T<π/2, but for T≥π/2 traps are absent according to the lemma <ref>.
Hence, for φ_W∈[π/2,π] and T>π-φ_W
there exist t_1 and t_2 such that D>0. This proves the lemma.



Let us make the inverse transformation from the system (<ref>) to the system (<ref>) U→ S^† US, W→ S^† WS, replace time t→ th and take into account that
g(t)=f(t)-f_0. In this case T→ Th and φ_W→α_W h. Then  we obtain the statement of the Theorem as a corollary of the lemmas <ref>–<ref>.





§ NUMERICAL ANALYSIS OF THE BEHAVIOUR OF THE OBJECTIVE FUNCTIONAL IN THE NEIGHBOURHOOD OF THE SPECIAL CONTROL F_0


The absence of traps for the objective functional 𝒥_W for the system (<ref>)
when T>π/2 may be illustrated by the numerical analysis. Fig. <ref>
shows the plots for the objective value   J_0=𝒥_W[0] calculated at the special control
f=0, and the probability P that a random control f in a sufficiently small neighbourhood of a special control satisfies the
inequality 𝒥_W[f]<J_0, both as  functions of (α,φ_W). Here
α is the angle between the vector
𝐯=Tr(σV)/2 and the axis Ox and the angle φ_W
parametrizes the matrix 

    W=[  e^iφ_W       0;       0 e^-iφ_W ].

The matrix  W of this form determines the phase shift gate 
U_ϕ=e^-iϕ/2W, where φ_W=-ϕ/2. At each point (α,φ_W)
the probability P(α,φ_W) is estimated as the proportion of realizations of the inequality 
𝒥_W[f]<J_0 for values of the objective functional  𝒥_W calculated for  M=10^3 randomly chosen controls in the neighbourhood of the special control f=0:

    P(α,φ_W)=#(f𝒥_W[f]<J_0)/M.




The random controls are generated as piecewise constant functions
f=∑_i=1^100a_iχ_i, where χ_i is the characteristic function of the interval [(i-1)T/100, iT/100],
and each a_i has normal distribution with unit variance.
Fig. <ref> shows that for T>π/2 the maxima of the probability P=P(α,φ_W)
coincide with the points where J_0(α,φ_W)=1, i. e. where f=0 is a global maximum and, therefore, at this point where are no traps.
In the remaining points P<1 and, therefore, there are also no traps. On Fig. <ref>
the probability P  does not depend of the angle α, i. e. on the direction of the vector 
𝐯=Tr(σV)/2. This is due to the fact that
values of the objective functional do not depend on this vector, as is stated 
in the following lemma.



Let [W,σ_z]=0. Then for any vectors 𝐯 and 𝐯' determined by the interaction Hamiltonians
V=(v_xσ_x+v_yσ_y) and
V'=(v'_xσ_x+v'_yσ_y) in the equation (<ref>), the values of the objective functional 𝒥_W^𝐯[f] and 𝒥_W^𝐯'[f], calculated at point  f for the V and V', are the same:

    𝒥_W^𝐯[f]=𝒥_W^𝐯'[f].




Let U_T^𝐯,f and U_T^𝐯',f satisfy the equation (<ref>) with Hamiltonians
V=v_xσ_x+v_yσ_y and V'=v'_xσ_x+v'_yσ_y. Since

    e^-iσ_zϑ/2(v_xσ_x+v_yσ_y)e^iσ_zϑ/2
    =v'_xσ_x+v'_yσ_y,

where ϑ=α'-α is the angle between the vectors  𝐯' and 𝐯, then the matrix

    Z_t=e^-iσ_zϑ/2U_t^𝐯,fe^iσ_zϑ/2

satisfies the equation (<ref>) with the potential V'=v'_xσ_x+v'_yσ_y.
Since Z_0=𝕀, then

    Z_t=e^-iσ_zϑ/2U_t^𝐯,fe^iσ_zϑ/2=U_t^𝐯',f.

Then

    𝒥_W^𝐯[f]=1/4|Tr(W^† U_t^𝐯,f)|^2
    =1/4|Tr(e^-iσ_zϑ/2W^†
    e^iσ_zϑ/2 U_t^𝐯',f)|^2=𝒥_W^𝐯'[f],

because in the considered case [W,σ_z]=0 and, therefore, 

    e^-iσ_zϑ/2W^† e^iσ_zϑ/2=W^†.

This proves the lemma.




Fig. <ref> shows the probability P=P(α) that
J_0>𝒥_ℍ[f], where the matrix

    ℍ=1/√(2)[  1  1;  1 -1 ]

describes Hadamard gate. From Fig. <ref> it is clear that P(α)≈ 1/2.
Therefore, where are no traps for any α. It is in accordance with Theorem <ref>,
because [ℍ,σ_z] 0.




§ CONCLUSION
 
In this paper  we prove the Theorem <ref> about the absence of traps for maximizing the objective functional 

    𝒥_W[f]=1/4|Tr(W^† U_T)|^2

for a qubit for small T for almost all W. If [H_0+f_0V,W]0, then for any T>0 all  maxima of the objective functional 𝒥_W are global.
If [H_0+f_0V,W]=0, then the matrix W has the form (<ref>). In this case if

    α_W∈(0,π/2d),
    
    
    d=H_0+f_0V-1/2𝕀Tr H_0-1/2f_0𝕀Tr V,

then all  maxima of the objective functional 𝒥_W are global for any T>0.
If α_W∈[π/(2d),π/d] then all  maxima of the objective functional
𝒥_W are global for any T>π/d-α_W.



§ ACKNOWLEDGEMENTS
 The authors thank Corresponding Member of RAS I. V. Volovich for  discussion and useful comments.


99

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