An information theoretical analysis of quantum optimal control

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by  S. Lloyd

An information theoretical analysis of quantum optimal control
S. Lloyd1, S. Montangero2
1Massachusetts Insitiute of Technology, Department of Mechanical Engineering, Cambridge MA 02139 USA,
2Institut fu¨r Quanteninformationsverarbeitung, Universita¨t Ulm, 89069 Ulm, Germany.
(Dated: January 21, 2014)
Weshowthatifan efficient classical representation ofthedynamicsexists,optimal controlprob-
lems on many-body quantum systems can be solved efficiently with finite precision. We show that
thesizeofthespaceofparametersnecessary tosolvequantumoptimalcontrolproblemsdefinedon
pure, mixed states and unitaries is polynomially bounded from the size of the of the set of reach-
able states in polynomial time. We provide a bound for the minimal time necessary to perform
the optimal process given the bandwidth of the control pulse, that is the continuous version of the
4 Solovay-Kitaev theorem. We explore the connection between entanglement present in the system
1 and complexity of the control problem, showing that one-dimensional slightly entangled dynamics
0 can be efficiently controlled. Finally, we quantify how noise affects thepresented results.
2
PACSnumbers: 03.67.-a,02.30.Yy
n
a
J
Quantumoptimalcontrolliesatthe heartofthe mod- worst case disappears in presence of perturbations [11].
0
ern quantum revolution, as it allows to match the strin- In this letter, we perform an information theoreti-
2
gent requirements needed to develop quantum technolo- cal analysis providing a first step towards the theoret-

] gies,to developnovelquantumprotocolsandto improve ical understanding of the complexity of OC problems
h their performances [1]. Along with the increased nu- in many-body quantum systems. We present a count-
p
mericalandexperimentalcapabilitiesdevelopedinrecent ing argument to bound the size of the space of parame-
-
t years, problems of increasing complexity have been ex- ters needed to solve OC problems defined overthe set of
n
plored and recently a lot of attention has been devoted time-polynomially reachable states. We explore the im-
a
u totheapplicationofoptimalcontrol(OC)tomany-body plications of this result in terms of SC identifying some
q quantumdynamics: OChasbeen appliedto information classes of problems that can be efficiently solved. We
[ processing in quantum wires [2], the crossing of quan- characterize the effects of noise in the control field and

tum phase transitions [3], the generation of many-body oftheentanglementpresentduringthesystemdynamics.
1
squeezed or entangled states [4], chaotic dynamics [5], We finally provide an information-time bound, relating
v
7 unitary transformations [6]. Recent studies have been the bandwidthofthe controlfieldwiththe minimaltime
4 devoted to the understanding of the fundamental limits necessary to achieve the optimal transformation.
0 ofOCintermsofenergy-timerelations(time-optimal)[7] AquantumOCproblemcanbestatedasfollows: given
5 and its robustness against perturbations [8, 9]. a dynamical equation
.
1
These exciting developments call for the development
0 ρ˙ =L(ρ,γ(t)); (1)
4 of a general framework to understand when and under
1 which conditions is it possible to solve a given OC prob- with boundary condition ρ(t = 0) = ρ where ρ is the
0
: leminamany-bodyquantumsystem. Indeed,duetothe density matrix describing a quantum system defined on
v
i exponentialgrowthoftheHilbertspacewiththenumber anHilbertspaceH=CN,andLtheLiouvillianoperator
X of constituents, solving an OC problem on a many-body with the unitary part generated by an Hamiltonian
r system is in general highly inefficient: the algorithmic
a
complexity (AC) of exact time-optimal problems can be H =H +γ(t)H , (2)
D C
super-exponential [6]. However, limited precision, errors
andpracticallimitationsnaturallyintroduceafinitepre- whereγ(t)isatime-dependentcontrolfield,andH and
D
cisionboth inthe functional to be minimized andonthe H the drift and control Hamiltonian respectively. For
C
total time of the transformation. The smoothed com- simplicity here we consider the case where only a single
plexity (SC) has been introduced recently to cope with control field is present (the generalization is straightfor-
this situation to describe the “practical” complexity of ward) and we work in adimensional units. From now on
solving a problem in the real world with finite precision. we focus on finite-size Hilbert space of dimension N, as
It has been shown that the SC can be drastically differ- any quantum system with limited energy and limited in
entfromtheAC:indeedtheAC–whichisdefinedbythe spaceiseffectivelyfinite-dimensional. Eq.(1)generatesa
scaling of the worstcase– might be practically irrelevant setofstatesdependingonthecontrolfieldγ(t)andonthe
as the worst case might be never found in practice [10]. initial state ρ : the manifold that is generated for every
0
Aparadigmaticcaseis thatofthe simplex algorithmap- γ(t) defines the set of reachable states W with dimen-
pliedtolinearprogrammingproblems: itischaracterized sion DW(N) [12]. If the system is controllable –i.e. the
by an exponential AC in the dimension of the searched operators H ,H generate the complete dynamical Lie
D C
space, however the SC is only polynomial, that is, the algebra–themanifoldW isthecompletespaceofdensity
2
matrix operators and its dimension is DW = N2 for an is reached also with a constant Hamiltonian. Similarly
N−dimensional Hilbert space, where for n d-level quan- to standard definitions, we define a time-polynomial
tum systems N = dn. Given a goal state ρ¯the problem reachable system if all states can be reached (with
tobe solvedisto findacontrolpulse γ¯(t)thatdrivesthe precision ε) in polynomial time by means of at least
system from a reference state ρ0 within an ǫ-ballaround one path (i.e. DW+ = DW) and a time-polynomially
the goal state ρ¯. Equivalently, the OC problem can be controllable system if W+ is equal to the whole Hilbert
expressed as a functional minimization of the form space. Notice that if the bound on the strength of the
control γmax is relaxed we have DW+ =DW. Given the
minF(ρ0,ρ¯,γ(t),[λı]), (3) above definitions, we can state the following:
γ(t)
where the functional F might also include constraints Theorem The size D of a quantum OC problem
introducedviaLagrangemultipliersλ . ThefunctionalF in W+ up to precision ε is a polynomial function of the
ı
is minimised by an (not necessary unique) optimal γ¯(t), size of the manifold of the time-polynomial reachable
that identifies a final state ρf such that ||ρf −ρ¯|| <ǫ in states DW+.
some norm ||·||.
We now recall the definition of the information con- Proof We first prove that the dimension of the problem
tent ofthe controlpulse γ(t) as we show in the following isboundedfrombelowbyDW+ andthenthatisbounded
that it is intimately related to the complexity of the OC from above by a polynomial function of DW+.
problem. Theinformation(numberofbitsb )carriedby Lower bound: We divide the complete set of time-
γ
thecontrolpulseγ(t)isgivenbytheclassicalchannelca- polynomial reachable states W+ in balls of size εDW+.
pacity C times the pulse duration T. In the simple case The number of ε-balls necessary to cover the whole set
of a noiseless channel, the channel capacity is given by W+ isε−DW+ andoneofthemidentifiesthesetofstates
Hartley’s law, thus thatlivearoundthe state ρ¯within aradiusε. The infor-
mationcontentoftheOCfieldmustbeatleastsufficient
b =T∆Ωκ (4) to specify the ε-ball surrounding the goal state, that is
γ s
bγ ≥b−S, where b−S =logε−DW+. Finally one obtains
where ∆Ω is the bandwidth, and κ = log(1+∆γ/δγ)
s
is the bit depth of the control pulse γ(t), and ∆γ = −T∆ΩκS
γ −γ and δγ are the maximal and minimal al- ε≥2 DW+ . (5)
max min
lowed variation of the field [19]. Note that given an uni-
Setting a maximal precision (e.g. machine precision) ex-
form sampling rate of the signal δt, T ∆Ω = T/δt = n
where n is the number of sampling points of the signals. pressed in bits κε =−log2 ε results in nsκs/DW+ =κε;
s and imposing κ =κ we obtain
Any optimization method of choice depends on these n ε s
s
variables,i.e. n definesthedimensionoftheinputofthe
s ns ≥DW+. (6)
optimisation problem. We thus define the dimension of
the quantum OC problemD as follows: Given a dynam- Upper bound: The goal state belongs to the set of time-
ical law of the form of Eq.(1), a reference initial state ρ0 polynomial states ρ¯ ∈ W+, thus a path of finite length
andanypossiblegoalstateinthesetreachablestatesW, L that connects the initial and goalstates in polynomial
the dimension of the quantum OC problemis defined by time exists. The maximum of (non-redundant) informa-
the minimal number of independent degrees of freedom tionthatprovidesthesolutiontotheproblemistheinfor-
D in the OC field necessaryto achievethe desiredtrans- mationneededtodescribethecompletepathb+. Setting
S
formation up to precision ε. Notice that D might be the the desired precision ε, this is equal to logε−DW+ bit of
minimal number of sampling points ns, of independent informationforeachε-ballneededtocoverthepathtimes
bang-bang controls, of frequencies present in the control the number of balls n . The latter is given by
ε
field or the dimension of the subspace of functions the
control field has non-zero projection on. nε =L/ε≤Tvmax/ε=Poly(DW+)vmax/ε (7)
Fromnowonweconsiderthephysicalsituationswhere
the control is performed in some finite time t ∈ [0,T], whereListhelengthofthepath,vmax isthemaximalal-
with bounded control field and bounded Hamiltonians, lowedvelocityalongthepathduetotheboundedenergy.
e.g. ||H ||=||H ||=1andγ(t)∈[γ :γ ]∀t. The In conclusion, we obtain that
D C min max
aforementioned physicalconstraints, naturally introduce
a new class of states, that we define as follows: The set b+ = Poly(DW+)vmax logε−DW+, (8)
of time-polynomial reachable states W+ ⊆ W is the set S ε
of states that can be reached (with finite energy) with that implies together with the condition b ≤b+
precision ε in polynomial time as a function of the set γ S
size DW+(N)≤DW(N). This is the class of interesting Poly′(DW+)vmax/ε≥ns (9)
states from the point of view of OC, as if a state can
be reached only in exponential time there is no need Asns isboundedbyapolynomialfunctionofDW+,thus
of OC at all: in exponential time any reachable state D=Poly(DW+) (cid:4)
3
Notice that the lower bound holds in general for any more generally, to this class of dynamics belongs for ex-
reachable state in W and can be saturated, as recently ample thosethatcanbe representedefficiently by means
shown in [13]. On the other hand, the upper bound di- of a tensor-network as t-DMRG [17]. We can thus state
verges for ε → 0, as finding the exact solution of the the following:
control problem might be as difficult as super exponen- 3 - The dimension D of an OC problem defined on a
tial [6]. The theorem has a number of interesting prac- dynamical process that can be described efficiently by a
tical and theoretical implications that we present in the tensor network, e.g. in one dimension a matrix product
rest of the paper. state,ispolynomialinthenumberofsystemcomponents
Complexity - The aforementioned theorem poses the n. Thedimensionofthesetofthetime-polynomialreach-
basis to set the SC of solving the OC problem. An algo- able states W+ that can be efficiently represented by a
rithmrecentlyintroducedtosolvecomplexquantumOC tensornetworkscalesasDW+ ≤DW ≤Poly(n)·T where
problems, the Chopped RAndom Basis (CRAB) optimi- T isthetotaltimeoftheevolutionandPoly(n)isthedi-
sation, builds on the fact that the space of the control mension of the biggest tensor network state represented
pulse γ¯(t) is limited from the very beginning to some duringthetimeevolution. Noticethat,althoughthepre-
(small) value D, and then solves the problem by means vious statement is in principle valid in all dimensions, it
of a direct search method as the simplex algorithm. Re- haspracticalimplicationsmostlyinone-dimensionalsys-
cently, numerical evidence has been presented that this temsasmuchlessefficientrepresentationsofthe dynam-
algorithm efficiently founds exponentially precise solu- ics are known in dimensions bigger than one [18].
tions as soon as D ≥ DW [14]. This result can be put We can now link directly the entanglement present in
now on solid ground as under fairly general conditions the system during its dynamics with the complexity of
OC problems are equivalent to linear programming [15] controlling it:
and linear programming can be solved via simplex algo-
4 - Time evolution of slightly entangled one-
rithm with polynomial SC [10]: thus, the CRAB opti-
dimensional many-body quantum systems can be ef-
misation solves with polynomial SC OC problems with
ficiently represented via Matrix Product States with
dimension D. More formally, one can make the follow- DW+ ≤ DW = O(T d22Sn) parameters, where S is
ing statement: The class of OC problems that satisfy
the maximal Von Neumann entropy of any bipartition
the hypothesis(H1-H3)ofRef.[15],ischaracterisedbya
present in the system. Thus, systems with S ∝ log(n)
polynomial SC in the size of the problem D. In conclu-
for every time can be efficiently controlled.
sion,studyingthe scalingofthe dimensionofthe control
We stress that the entanglement present in the sys-
problem D = Poly(DW+) is of fundamental interest to tem is not uniquely correlated with the complexity of
understand and classify our capability of efficiently con-
the OC problem: indeed due to the previous results, in-
trol quantum systems. The first results in this direction
tegrable systems (also highly entangled) are efficiently
can be obtained observing the influence of the integra-
controllable, as shown recently in [13]. On the contrary,
bility of the quantum system on DW+, resulting in the as said before, highly entangled dynamics of non inte-
following properties:
grable systems, for which it does not exists an efficient
1-ThesizeDofagenericOCproblemdefinedontime-
representation as S ∝ n are exponentially difficult to
polynomial controllable non-integrable n-body quantum
control. In conclusion, the size of the control problem
system is exponential with the number of constituents
dependsonthedimensionofthemanifoldoverwhichthe
n. Indeed the dynamics of a controllable non integrable
dynamicstakesplace. This canbe simply understoodby
many-body quantum system explores the whole Hilbert
considering the scenario where the dynamics over which
space, i.e. the set of time-polynomial reachable states is
the control problem is defined is restricted to the space
the whole Hilbert space, that is DW+ = N2 (DW+ = N of two eigenstates of a complex many-body hamiltonian,
for pure states).
each of them highly entangled w.r.t some local bases. If
Onthecontrary,despitethe exponentialgrowthofthe one has access to a direct coupling between them, the
Hilbertspace,the sizeofW+ forintegrablesystemsis at complexity of the OC problem is not more than that of
mostlinearinthenumbernofconstituentsofthesystem, a simple Landau-zener process (independently from the
that implies together with the theorem above that: entanglement present in the system) as the manifold is
2 - The size D of OC problems defined on time- effectively two-dimensional. However, this is not gener-
polynomially controllable integrable many-body quan- ally the case, as one has usually access to some local (or
tum system, is polynomial with n = log (N). Notice global) operator, and the dynamic of the system is not
d
that this statement generalizes a theorem that has been in general restricted to two states. In the case of non
provenforthe particularcaseoftridiagonalHamiltonian integrable systems, a generic couple of initial and goal
systems presented in [16]. states projects on exponentially many basis states inde-
Finally, there exists a class of intermediate dynamics pendently of the chosen basis, while for integrable states
that despite in principle might explore an exponentially itexistsabasewherethestateshaveasimplerepresenta-
big Hilbert space, are confined in a corner of it and can tion. Thus, the minimal amount of information needed
thus be efficiently represented. The simplest example of to solve the quantum OC problem is exponential and
this class of problems is mean-field dynamics, however polynomial respectively. In between, there is the class
4
of TN-efficiently representable dynamics, for which we and similarly
know how to build an efficient representation and cor-
respondingly we know how to efficiently solve the OC
problem. DW log(1/ε)
T ≥ . (14)
Time bounds - Manipulating Eq. (5) applied to the ∆Ω log(1+S/N)
whole set of reachable states W we achieve a bound for
the minimal time needed to achieve the desired trans-
formation as a function of the control bandwidth: The
For small noise to signal ratio (N/S ≪ 1) the previ-
minimal time needed to reach a given final state in DW ous bound results in ε & (N/S)ns/DW which together
with precision ε at finite bandwidth is
with the fact that n has to be a polynomial function of
s
DW DW+ show that the control problem is in general expo-
T ≥ log(1/ε) (10) nential sensitive to the problem dimension. However, if
∆Ωκ
S one saturates the lower bound on the complexity of the
or again, under the assumption that κ =κ : optimal field, i.e. ns = DW, the sensitivity to Gaussian
ε s
white noise become linear in the noise to signal ratio.
DW That is, the effects of the noise on the optimal trans-
T ≥ . (11)
∆Ω formation are negligible if the noise level is below the
error, N/S . ε. As requiring the optimal transforma-
The previous relation is a continuous version of the tion to be more precise than the error on the control
Solovay-Kitaev theorem: it provides an estimate of the signal is somehow unnatural, this relation demonstrate
minimaltimeneededtoperformanoptimalprocessgiven that OC transformations are in general robust with re-
afiniteband-width. Noticealsothatthebandwidthpro- spect to noise, as recently observed [20]. At the same
vides the average bits rate per second, thus this results time, for ε.N/S this results agreeswith the scaling for
coincides with the intuitive expectation that the mini- exact optimal transformations recently found in [9].
maltimeneededtoperformanoptimalquantumprocess
is the time necessary to “inform” the system about the Control of unitaries - The aforementioned statements
goalstategiventhatthecontrolfieldhasonlyafinitebit also hold for the generation of unitaries as the differen-
transmission rate. tial equation governing the evolution of the time evolu-
We recallthatthere is a time-energybound, knownas tionoperatorı~U˙(t)=H(t)U(t)isformallyequivalentto
quantum speed limit that in its general form is Eq.(1)replacingthedensitymatrixwiththetimeevolu-
tion operator U(t), the reference state with the identity
d(ρ0,ρG) operator, and the goal state with the unitary to be gen-
T ≥ , (12)
QSL Λ erated.
T Observability - As any controllable system is also ob-
where d(·,·) is the distance and Λ = R ||L|| dt/T with
0 p servable by a coherent controller [21], the previous defi-
||·|| the p-norm[7]. The bestefficientprocesssaturates
p
nitions and results can be straightforwardapplied to the
both bounds, that implies ∆Ω ∝ DW; thus the band-
complexity of observing a many-body quantum system
width of the time-optimal pulse in general should scale
with precision ε.
as the dimension of the space W, requiring exponential
higher frequencies for non integrable many-body quan- Inconclusion,wehaveshownthatifoneallowsafinite
tum systems and thus practically preventing its physical error (both in the goal state and in time) as it typically
realization. occurs in any practical application of OC, what can be
Noise - In presence of noise Eq. (4) has to be mod- efficientlysimulatedcanalsobeoptimallycontrolledand
ified: in the following we consider a common scenario that the optimal solution is in general robust with re-
however this analysis can be adapted to the specific spect to perturbation on the control field. Notice that
noise considered. For gaussian white noise, accord- the presented results are valid both for open and closed
ing to Shannon-Hartley theorem the channel capacity is loop OC.
k =log(1+S/N),whereS/N isthesignaltonoisepower
s
ratio [19]. Thus, following the same steps as before we We thank T. Calarco, A. Negretti, and P. Rebentrost
obtain that for discussions and feedback. SM acknowledge support
fromtheDFGviaSFB/TRR21andfromtheEUprojects
− ns
ε≥(1+S/N) DW, (13) SIQS and DIADEMS.
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