representations of the qubit states · iulian petrila and vasile manta 62 it should be noted that...

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI Publicat de

Universitatea Tehnică „Gheorghe Asachi” din Iaşi Tomul LIX (LXIII), Fasc. 2, 2013

SecŃia AUTOMATICĂ şi CALCULATOARE

REPRESENTATIONS OF THE QUBIT STATES

BY

IULIAN PETRILA1,2∗∗∗∗

and VASILE MANTA2

1“Alexandru Ioan Cuza” University of Iaşi,

Department of Physics and RAMTECH 2“Gheorghe Asachi” Technical University of Iaşi,

Faculty of Automatic Control and Computer Science Received: May 13, 2013 Accepted for publication: June 17, 2013

Abstract. In this article we present the relevant characteristics of the qubit and its various representations and descriptions. We give both the state vector and matrix density formulations and follow a proper graphic illustration of the qubit states and transformations. Besides the new Bloch Sphere representations, a new Cartesian Sphere representation is proposed for both state vector and matrix density formulations. In this representation the fundamental states are considered in the xOy plane and imaginary combinations of the fundamental states are placed on a direction orthogonal to this plane. This representation is useful for the qubits identified in quantum systems with certain symmetry or asymmetry.

Key words: quantum logic, quantum algorithms and complexity, quantum computation.

2010 Mathematics Subject Classification: 03G12, 68Q12, 81P68.

1. Introduction

One of the present challenges in computer science is to include, at least partially, the computing facilities provided by the quantum systems through processing the information directly, at the quantum level (Prakash & Vidyarthi, ∗Corresponding author; e-mail: [email protected]

Iulian Petrila and Vasile Manta

58

2013; Chen et al., 2013; Zhang et al., 2012; Nielsen & Chuang, 2000). This would greatly improve the computing power of some critical devices in that it will be used quantum laws that are applicable to the isolated or interacting quantum systems (atoms, photons etc.) instead of using many such systems just to get and used (through a mediation) a conventional information unit (bit), as used in current (classical) computing systems (Gonzalez-Arroyo & Nuevo, 2012; Knuth, 1997). There are many problems to be solved in quantum computing (the existence of the quantum states for only the fractions of a second, keeping quantum information intact for long periods etc.) before a purely quantum computing system to be efficacious (Zhang, 2013; Yamamoto et al., 2012; Morimae, 2012; Zuniga-Hansen et al., 2012). Until then, the efforts of describing the systems and processes in order to handle the quantum information in an appropriate way are not unimportant (Venegas-Andraca, 2012; Jones et al., 2012; Cleve et al., 2013). The non intuitive characteristics of quantum theory makes it difficult to be used in computation. The different representations of quantum states and transformations can help to better understand and exploit the powerful information processing facilities provided by the quantum world. In this respect, the analysis of different forms of representation of the quantum information unit (qubits), as Bloch sphere (Lou & Cong, 2011; Kim, 2011), is useful for a better description of the involved processes (Blasiak, 2013). The following sections describe the relevant characteristics of the qubit using new Bloch Sphere representations for the state vector, matrix density and their transformations, and also the extension of these descriptions to new representations such as the Cartesian Sphere.

2. Qubit States and Transformations

The quantum bit (qubit) is the smallest unit of quantum information that can be identified in quantum systems. A qubit can be compound by a two level quantum system whose two basis states are conventionally labeled 0 and 1

(Podoshvedov, 2013). A general qubit state vector Ψ is an arbitrary linear

superposition of the basis states through α and β complex coefficients (probability amplitudes) and written as:

10 βα +=Ψ . (1)

The probability preservation of the state vector that describes a quantum system

12

=Ψ⋅Ψ=Ψ (2)

is reflected in a constraint on to the probability amplitudes coefficients as

Bul. Inst. Polit. Iaşi, t. LIX (LXIII), f. 2, 2013

59

122=+ βα . (3)

This constraint outlines the probabilistic nature of the parameters involved in the description of a quantum system and implicitly of a qubit (Kumar & Skinner, 2013). The transformations that occur at the quantum level are described through unitary and linear operators U that act on the state vector

Ψ=Ψ U' (4)

changing its state from Ψ to 'Ψ while preserving its norm. The evolution of

the state vector and implicitly the unitary operators U can be obtained by integrating the Schrödinger equation

Ψ=Ψ∂∂

Ht

iℏ (5)

where H is the Hamiltonian operator associated with the total energy of the system. In the case of a constant energy system (Hamiltonian operator H is not time-dependent), the unitary operator U that describes the system transition between any two moments 1t and 2t , is given by

2 1( )

1 2( , )it t H

U t t e− −

= ℏ . (6)

Some quantum systems (and implicitly some qubits) can be found as a mixture or ensemble of quantum states. In such cases these states can be elegantly described by the density operator (Bruning et al., 2012)

∑=

ΨΨ=ni

iiip

,1

ρ , 10 ≤≤ ip , 1,1

=∑= ni

ip (7)

where ip is the probability (weight) of the pure state iΨ . Like the state vector

Ψ , the density operator ρ contains all the relevant information of the

quantum system. The transformations performed on the quantum systems can be described by the evolution of the density operator given by

+= UUρρ ' (8)

where the unitary operators U can be obtained from the Liouville-von Neumann equation

],[ ρρ Ht

i =∂∂ℏ (9)


60

where HHH ρρρ −=],[ is the commutator operator. In the case of constant energy system (the Hamiltonian H is time-independent) the differential eq. (9) can be easily solved as )/exp()0()/exp()( ℏℏ iHtiHtt ρρ −= , the result that can also be obtained from (6) and (8). From the quantum computing perspective, the problem consists in the identification of quantum systems which are able of being subjected to transformations in terms of quantum operations (Nielsen & Chuang, 2000).

3. Qubit State Vector Representations in

Bloch Sphere Diagram

The probabilistic characteristic of the quantum world is reflected in a

non intuitive description of states and transformations associated to quantum systems. In this respect, the use of different representations provides a better understanding and characterization of quantum systems. In the qubit case, the state vector which is compound by a combination of two states 0 and 1 , is

usually represented as a point on the Bloch Sphere (Lou & Cong, 2011), as can be seen in Fig.1.

Fig. 1 – Bloch Sphere representation of qubit state vector.

The representation of a qubit as points on a unit sphere (Bloch Sphere) is allowed by the reparameterization of the state vector as:

12

sin02

cos ϕθθ je+=Ψ (10)


61

with πθ ..0= and πϕ 2..0= spherical angles. The probability amplitudes coefficients, up to a phase multiplicative term (physically insignificant because they describe the same qubit state), are )2/cos(θα = and )2/sin(θβ ϕje= .

In essence, a transformation U performed on a qubit state vector can be

represented as a rotation on the Bloch Sphere ),(),( 2211 ϕθϕθ →U , as shown in Fig. 2. This transformation (rotation in Bloch Sphere space) can be obtained by successive rotations on the Ox , Oy or Oz axes using the corresponding rotation operators

=

−=

−

−=

−

2

2

0

0)(,

2cos

2sin

2sin

2cos

)(,

2cos

2sin

2sin

2cos

)( θ

θ

θθθ

θθ

θθθ

θθ

θi

i

ZYX

e

eRR

i

iR (11)

Generally, it is not necessary to use rotations in relation to all axes, therefore, the unitary operator U can be expressed through

),( YZ RR decomposition

( ) ( )

( ) ( )

−

==+−

−−+−

2cos

2sin

2sin

2cos

)()()(22

22

γγ

γγ

δγβδβδβ

δβδβ

ααjj

jj

jZYZ

j

ee

eeeRRReU (12)

where ),,,( δγβα are general parameters which can be identified depending on the particularities of the transformation.

Fig. 2 – Bloch Sphere representation of qubit state vector transformation.


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It should be noted that the pure qubits (described by pure states) can be represented by vectors that point on the surface of the Bloch Sphere, while the mixed states qubits are adequately described by the density operator. An important operation involved in the process of extracting information from a quantum system is the measurement operation. The specificity of quantum systems is that the measurement process is probabilistic. This means that an operation M of measuring the qubit state described by state vector

10 βα +=Ψ will involve collapsing to one of the two qubit basis states

1

01

0

→

→Ψ

p

p

M (13)

where 20 α=p is the probability to get the 0 state and 2

1 β=p is the

probability to find the system in state 1 after measurement (Fig.3).

Fig. 3 – Bloch Sphere representation of qubit state vector measurement transformation.

As can be seen in Fig.3, the measurement process constrains the system to switch to one of the two basis states 0 or 1 , which implies (in terms of

Bloch Sphere representation) the alignment of the state vector along the Oz axis.

4. Qubit Density Matrix Representations in Bloch Sphere Diagram

The systems with mixed states can be managed properly with density operators instead of state vectors. However, the pure state is a particular case of mixed states that can be represented by density operators as:


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ΨΨ=ρ (14)

Like the state vector, the density operator associated with a qubit can be parameterized using the Pauli operators (matrices):

−≡≡

−≡≡

≡≡

≡≡

10

01,

0

0,

01

10,

10

013210 zyx

i

iI σσσσσσσ (15)

The density operator associated with a qubit developed in relation to the Pauli basis becomes

( )zzyyxx rrr σσσσρ +++= 02

1 (16)

where ( ) r=zyx rrr ,, are Bloch coordinates (parameters). Like the state vector

Ψ , the density operator ρ can be represented geometrically through the

Bloch Sphere, as shown in Fig. 4.

Fig. 4 – Bloch Sphere representation of qubit density operator.

However, unlike the state vector Ψ that pointed just the surface of the

Bloch Sphere, the density operator can cover the whole sphere, representing both mixed ( 1|| <r ) and pure ( 1|| =r ) states. Consequently, the transformations performed on the density operator lead to changes in its parameters

22221111 ),,(),,( rr =→= zyxU

zyx rrrrrr which are represented both by a rotation

and a scaling in Bloch Sphere representation, as can be seen in Fig. 5.


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Fig. 5 – Bloch Sphere representation of qubit density operator transformation.

It should be noted that the measurements M performed on a qubit

cause a collapse of the system with probability ( )+= iii MMtrp ρ to certain values of the density operator (Distler & Paban, 2013)

( )+

+

=ii

iii

MM

MM

ρ

ρρ

tr (17)

where iM is a measurement operator and tr is the trace operation.

5. Qubit Representations in Cartesian

Sphere Diagram

Besides the Bloch Sphere representation there are also other intuitive ways of describing the qubit states and transformations that can be identified. In the following, we propose a Cartesian representation of the qubit by associating the 0 state to the Ox axis and 1 to the Oy axis. To keep the imaginary

parameter on state 1 , as in the case of the Bloch Sphere representation, the

qubit state vector can be described by

1)cossin(sin0cossin θϕθϕθψ i++= (18)

where ),( ϕθ are the usual spherical coordinates, and the corresponding diagram can be seen in Fig. 6.


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Fig. 6 – Cartesian Sphere representation of qubit state vector with 1i state on Oz axis.

An alternative representation would be one in which imaginary

parameters ponder symmetrically both basis states by

+++= θϕθϕθψ cos1sinsin0cossin i (19)

where ( )102/1 +=+ . This parametrization, as can be seen in Fig. 7, is

useful for systems whose basis states have similar contributions.

Fig. 7 – Cartesian Sphere representation of qubit state vector with +i state on Oz axis.


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In a similar manner the asymmetric parameterization with

( )102/1 −=− replacing + in (19) may be used for the asymmetric

systems. It may be noted that, in the above parametrizations, the state vector in the xOy plane containing the combinations of basis states 0 and 1 is

weighted by real parameters.

6. Conclusions

The graphical representation of unintuitive states and transformations that characterize quantum systems and especially qubits, represents an important step for understanding and operating with quantum information. By detailing the qubit state vector representations on the Bloch Sphere and inclusion of the qubit density operator in these representations have created the conditions for characterizing any type of qubit, including those formed from the mixture or ensemble of quantum states. The proposed Cartesian representations outline the versatility of the mathematical tools that can characterize the quantum systems and the multiple representations in which these systems can be described. These Cartesian representations are useful for qubits states and transformations that require highlighting of certain symmetry or asymmetry.

REFERENCES

Blasiak P., Quantum Cube: A Toy Model of a Qubit. Physics Letters A, 377, 847−850,

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Matrices. Journal of Modern Optics, 59, 1−20, 2012. Chen J., Wang L., Charbon E., Wang B., Programmable Architecture for Quantum

Computing. Physical Review A, 88, 022311, 2013. Cleve R., Dam W., Nielsen M., Tapp A., Quantum Entanglement and the

Communication Complexity of the Inner Product Function. Theoretical Computer Science, 486, 11–19, 2013.

Distler J., Paban S, Uncertainties in Successive Measurements. Physical Review A, 87, 062112, 2013.

Gonzalez-Arroyo A., Nuevo F., Real-Time Quantum Evolution in the Classical Approximation and Beyond. Physical Review D, 86, 103504, 2012.

Jones N.C., Whitfield J.D., McMahon P.L., Yung M.H., Van Meter R., Aspuru-Guzik A., Yamamoto Y., Faster Quantum Chemistry Simulation on Fault-Tolerant Quantum Computers. New Journal of Physics, 14, 115023, 2012.

Kim S., Distances of Qubit Density Matrices on Bloch Sphere. Journal of Mathematical Physics, 52, 102303, 2011.

Knuth D.E., The Art of Computer Programming, Seminumerical Algorithms. Vol. 2, Addison-Wesley, 1997.


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Kumar P., Skinner S.R., Using Non-Ideal Gates to Implement Universal Quantum Computing Between Uncoupled Qubits. Quantum Information Processing, 12, 973–996, 2013.

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Morimae T., Continuous-Variable Blind Quantum Computation. Physical Review Letters, 109, 230502, 2012.

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Prakash S., Vidyarthi D.P., A Novel Scheduling Model for Computational Grid Using Quantum Genetic Algorithm. The Journal of Supercomputing, 65, 742–770, 2013.

Venegas-Andraca S.E., Quantum Walks: a Comprehensive Review. Quantum Information Processing, 11, 1015–1106, 2012.

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REPREZENTĂRI ALE STĂRILOR QUBITULUI

(Rezumat)

Descrierea proceselor şi a transformărilor sistemelor cuantice, deşi extrem de

facilă datorită aparatului matematic complet oferit de mecanica cuantică, este îngreunată adesea de faptul că aceasta este neintuitivă, fiind mediată de accesul probabilistic la informaŃia sistemelor în cauză. Aplicarea directă a sistemelor cuantice în procesarea informaŃiei este destul de problematică în special din cauza modului dificil de operare cu sisteme pentru care un simplu proces de observare (citire de informaŃie) de exemplu reprezintă o perturbare ireversibilă (transformare predictibilă doar probabilistic). Astfel, chiar şi operarea cu cele mai mici entităŃi cuantice informaŃionale (qubit), formată din un sistem cuantic cu două stări distincte, din punct de vedere tehnic este dificilă. Până la realizarea fizică concretă a unor sisteme cuantice de calcul, eforturile sunt axate mai ales pe descrierea cât mai relevantă a entităŃilor ce ar compune astfel de sisteme. În această categorie se înscrie şi articolul prezent. Mai întâi sunt expuse succint caracteristicile bitului cuantic (qubitului) şi transformările acestuia. Apoi, plecând de la reprezentarea uzuală a vectorului de stare asociat unui qubit în coordonate Bloch, sunt


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introduse şi reprezentări mult mai generale date de operatorul densitate asociat qubitului. De asemenea, în vederea conturării caracterului simetric sau asimetric al stărilor qubitului asociat unor sisteme cuantice, sunt introduse noi parametrizări şi reprezentări, numite Carteziene sferice, în care stările de bază sunt plasate pe axele Ox şi Oy iar pe axa Oz sunt reprezentate componentele (ponderile) imaginare simetrice sau asimetrice ale stărilor de bază. Reprezentările stărilor si transformărilor qubiŃilor prezentate în articol sunt utile demersului de identificare şi caracterizare cât mai adecvată a unor sisteme cuantice particulare (ce prezintă diferite simetrii) ce pot fi utilizate ca elemente de calcul cuantic.

representations of the qubit states · iulian petrila and vasile manta 62 it should be noted that...

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