BENENTI, GIULIANOG. Strinimostra contributor esterni nascondi contributor esterni 

Simple representation of quantum process tomography

Giuliano Benenti

* and Giuliano Strini

1

CNISM, CNR-INFM, and Center for Nonlinear and Complex Systems, Università degli Studi dell’Insubria, via Valleggio 11, 22100 Como, Italy

2

Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milano, Italy

3

Dipartimento di Fisica, Università degli Studi di Milano, via Celoria 16, 20133 Milano, Italy sReceived 5 May 2009; published 13 August 2009 d

We show that the Fano representation leads to a particularly simple and appealing form of the quantum process tomography matrix x

, in that the matrix x

is real, the number of matrix elements is exactly equal to the number of free parameters required for the complete characterization of a quantum operation, and these matrix elements are directly related to evolution of the expectation values of the system’s polarization mea- surements. These facts are illustrated in the examples of one- and two-qubit quantum noise channels.

DOI: 10.1103/PhysRevA.80.022318 PACS numberssd: 03.67.Lx, 03.65.Wj, 03.65.Yz

I. INTRODUCTION

The characterization of physical, generally noisy pro- cesses in open quantum systems is a key issue in quantum information science f1,2g. Quantum process tomography sQPTd provides, in principle, full information on the dynam- ics of a quantum system and can be used to improve the design and control of quantum hardware. Several QPT meth- ods have been developed including the standard QPT f2–5g, ancilla-assisted QPT f6–8g, and direct characterization of quantum dynamics f9g. In recent years QPT has been experi- mentally demonstrated with up to three-qubit systems in a variety of different implementations including quantum op- tics f8,10–16g, nuclear magnetic resonance quantum proces- sors f17–19g, atoms in optical lattices f20g, trapped ions f21,22g, and solid-state qubits f23,24g.

Any quantum state r can be expressed in the Fano form f25–27g salso known as Bloch representationd. Since the den- sity operator r is Hermitian, the parameters of the expansion over the Fano basis are real. Furthermore, due to the linearity of quantum mechanics, any quantum operation r → r 8

= Esrd is represented, in the Fano basis, by an affine map.

In this paper, we point out that in standard QPT it is convenient to compute the QPT matrix in the Fano basis.

Such process matrix, x

, has the following advantages: sid the matrix elements of x

are real and siid the number of matrix elements in x

is exactly equal to the number of free parameters needed in order to determine a generic quantum operation. Furthermore, the x

-matrix elements are directly related to the modification, induced by the quantum opera- tion E, of the expectation values of the system’s polarization measurements. We will illustrate our results in the examples of one- and two-qubit quantum noise. In particular, we will determine in the x

matrix the specific patterns of various quantum noise processes. Finally, we will discuss the number of free parameters physically relevant in determining a quan- tum operation for a two-qubit system exposed to weak local noise.

II. FANO REPRESENTATION OF THE STANDARD QPT To simplify writing, we discuss the Fano representation of the standard QPT only for qubits, even though the obtained results can be readily extended to qudit systems. Any n-qubit state r can be written in the Fano form as follows f25–27g:

r = 1

N o

a₁,. . .,a_n=x,y,z,I

c

1. . .a_n

s

1^

¯

s

n

, s1d

where N = 2

, s

, s

, and s

are the Pauli matrices, s

;1, and

c

1. . .a_n

= Trss

¯

s

n

rd. s2d

Note that the normalization condition Trsrd=1 implies c

= 1. Moreover, the generalized Bloch vector b

= hb

j

is real due to the hermiticity of r. Here b

;c

, with a ;o

i

4

, where we have defined i

= 1, 2, 3, and 4 in correspondence to a

= x, y, z, and I. Note that from 1 to n qubits run from the most significant to the least significant. For instance, for two qubits sn=2d, the N

− 1

= 15 components of vector b are ordered as follows:

b

= sb

,b

, . . . ,b

d

= sc

,c

,c

,c

,c

,c

,c

,c

,c

,c

,c

,c

,c

,c

,c

d.

s3d Due to the linearity of quantum mechanics any quantum operation r → r 8 = Esrd is represented in the Fano basis hs

1

^

. . .

s

n

j by an affine map:

F ^b 1 ⁸ G ^{= M} F ^b 1 G ⁼ F ^{M a} 0

1 GF ^b 1 G ^{, s4d}

where M is a sN

− 1d3sN

− 1d matrix, a is a column vector of dimension N

− 1, and 0 is the null vector of the same dimension.

All information about the quantum operation E is con- tained in the N

− N

free elements of matrix M, namely, in the matrix

* [email protected]

†

[email protected]

x

= fM a g. s5d To obtain the QPT matrix x

from experimental data, one needs to prepare N

linearly independent initial states hr

j, let them evolve according to the quantum operation E, and then measure the resulting states hr

8 ^{= Esr}

dj. If we call R the N

3 N

matrix whose columns are given by the Fano repre- sentation of states r

and R 8 the corresponding matrix con- structed from states r

8 ^{, we have}

R 8 ^{= MR, s6d}

and therefore

M = R 8 ^R

^{. s7d}

As it is well known f2g, the standard QPT can be per- formed with initial states being product states and local mea- surements of the final states. As initial states hr

j we choose the 4

tensor-product states of the four single-qubit states

u0l, u1l, 1

Î 2 su0l + u1ld, 1

Î 2 su0l + iu1ld. s8d To estimate R 8 , one needs to prepare many copies of each initial state r

, let them evolve according to the quantum operation E, and then measure observables s

1^

¯

s

n

. Of course, such measurements can be performed on the com- putational basis hu0l,u1lj

provided each measurement is preceded by suitable single-qubit rotations.

III. SINGLE-QUBIT SYSTEMS The matrix R corresponding to basis s8d reads

R = 3 ^{0 0 1 − 1 0 0 1 0 0 1 1 0 0 1 1 1} 4 ^{. s9d}

Therefore,

R

= 3 ^{− − 1 0 1 2 1 2 − − 0 1 1 2 1 2 − 1 2 0 0 1 2 1 2 1 2 0 0} 4 ^{. s10d}

The coefficients sc

, c

, c

d in the Fano form s1d are the Bloch-vector coordinates of the density matrix r in the Bloch-ball representation of single-qubit states. We need N

− N

= 12 parameters to characterize a generic quantum op- eration acting on a single qubit. Each parameter describes a particular noise channel ssuch as bit flip, phase flip, ampli- tude damping, etc.d and can be most conveniently visualized as associated with rotations, deformations, and displace- ments of the Bloch ball f1,2,28g. Here we point out that these noise channels lead to specific patters in the state process matrix x

.

For instance, for the phase-flip channel,

r 8 = Esrd = ps

rs

+ s1 − pdr, S ^{0 # p # 1} 2 D ^{, s11d}

we have

R 8 ⁼ 3 ^{0 0 1 − 1 1 0 0 1 1 − 2p 0 0 1 1 − 2p 0 0 1} 4 ^{. s12d}

We can then compute M = R 8 ^R

and the first three lines of M correspond to the state matrix

x

= 3 ^{1 − 2p 0 0 1 − 2p 0 0 0 0 0 0 1 0} 4 ^{. s13d}

Therefore, the Bloch ball is mapped into an ellipsoid with z as symmetry axis:

c

→ c

8 ^{= s1 − 2pdc}

,

c

→ c 8

^{= s1 − 2pdc}

,

c

→ c

8 ^{= c}

. s14d As a further example, we consider the amplitude damping channel,

r 8 ⁼ o

k=0 1

E

rE

, s15d

with the Kraus operators

E

= u0lk0u + Î 1 − pu1lk1u, E

= Î pu0lk1u, s0 # p # 1d.

s16d In this case we obtain

x

= 3 ^{Î 1 − p 0 0 Î 1 − p 0 0 1 − p p 0 0 0 0} 4 ^{. s17d}

The Bloch ball is deformed into an ellipsoid, with its center displaced along the z axis:

c

→ c

8 ⁼ Î 1 − pc

, c

→ c

8 ⁼ Î 1 − pc

,

c

→ c

8 ^{= s1 − pdc}

+ p. s18d

IV. TWO-QUBIT SYSTEMS

Matrices R and R

corresponding to the 16 tensor-product states of single-qubit states fEq. s8dg read as follows:

R =



 

  ^{0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0} 1 − 1 0 0 0 ^{0 0 1 0 0 0 1 0 0 0} 0 ^{0 0 0 0} 0 0

0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0 0 0 0 1 − 1 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1

0 0 1 0 0 0 − 1 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 − 1 0 0 0 0 0 0 0 0

1 − 1 0 0 − 1 1 0 0 0 0 0 0 0 0 0 0

1 1 1 1 − 1 − 1 − 1 − 1 0 0 0 0 0 0 0 0

0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0

0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1

1 − 1 0 0 1 − 1 0 0 1 − 1 0 0 1 − 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1     

, s19d

R

= 1 4

 



  ^{1 1 − 1 − 1 1 1} − 1 − 1 − 1 − 1 1 1 − 1 − 1 1 1 1 1 1 − 1 1 1 1 − 1 − 1 − 1 − 1 1 − 1 − 1 − 1 1

− 2 0 0 0 − 2 0 0 0 2 0 0 0 2 0 0 0

0 − 2 0 0 0 − 2 0 0 0 2 0 0 0 2 0 0

1 1 − 1 − 1 1 1 − 1 − 1 1 1 − 1 − 1 − 1 − 1 1 1

1 1 1 − 1 1 1 1 − 1 1 1 1 − 1 − 1 − 1 − 1 1

− 2 0 0 0 − 2 0 0 0 − 2 0 0 0 2 0 0 0

0 − 2 0 0 0 − 2 0 0 0 − 2 0 0 0 2 0 0

− 2 − 2 2 2 0 0 0 0 0 0 0 0 0 0 0 0

− 2 − 2 − 2 2 0 0 0 0 0 0 0 0 0 0 0 0

4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 − 2 − 2 2 2 0 0 0 0 0 0 0 0

0 0 0 0 − 2 − 2 − 2 2 0 0 0 0 0 0 0 0

0 0 0 0 4 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 4 0 0 0 0 0 0 0 0 0 0     

. s20d

Note that the 16 columns of R correspond, from left to right, to the states u0l

u0l,u0l

u1l, ... ,

su0l+iu1ld

su0l + u1ld,

su0l+iu1ld

su0l+iu1ld, that is, states are ordered with the first qubit being the most significant one.

The coordinates hc

j in the Fano form s1d are the expectation values of the polarization measurements hs

s

2

j. The coefficients in the state matrix x

represent-

ing a quantum operation E can therefore be interpreted in terms of modification of these expectation values.

For instance, let us assume that the two qubits are inde-

pendently exposed to pure dephasing, that is, to quantum

noise described by the phase-flip channel s11d, with the same

noise strength p for both qubits. The process matrix for such

uncorrelated dephasing channel is given by

x

=

 



  ^g

^{0 0 0 0 0} 0 0 0 0 0 0 0 0 0 0 g

0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 g 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 g 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 g

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 g

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 g 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 g 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 g 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 g 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 g 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 g 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1     

,

s21d where g;1−2p. Correspondingly, the mapping for the ex- pectation values of the polarization measurements reads

c

1a₂

8 ^{= g}

^c

, s22d

where m

= 1 for a

= x , y and m

= 0 for a

= z , I.

As an example of nonlocal quantum noise, we consider a model of fully correlated pure dephasing. We model the in- teraction of the two qubits with the environment as a phase kick rotating both qubits through the same angle u about the z axis of the Bloch ball. This rotation is described in the hu0l,u1lj basis by the unitary matrix

R

s u d = F ^e

0 e

⁰ G

F ^e

0 e

⁰ G ^{. s23d}

We assume that the rotation angle is drawn from the random distribution

ps u d = 1

Î 4pl e

. s24d Therefore, the final state r 8 , obtained after averaging over u , is given by

r 8 ⁼ E

^{d u ps u dR}

^{s u drR}

^{s u d. s25d}

For this correlated dephasing channel we obtain the process matrix

x

=

 



  ^{h 0 0 0 0 k} 0 0 0 0 0 0 0 0 0 0 h 0 0 − k 0 0 0 0 0 0 0 0 0 0

0 0 g 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 g 0 0 0 0 0 0 0 0 0 0 0

0 − k 0 0 h 0 0 0 0 0 0 0 0 0 0

k 0 0 0 0 h 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 g 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 g 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 g 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 g 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 g 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 g 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 1     

,

s26d where g;e

, h;

s1+g

d, and k;

s1−g

d. It is clear that process matrix s26d for correlated dephasing has a pattern that allows to clearly distinguish it from process matrix s21d for the uncorrelated dephasing.

It is also obvious that, if there exists partial previous knowledge of the dominant noise sources, it is not necessary to construct the whole state process matrix x

in order to characterize the quantum operation. For instance, if we know a priori that dephasing is the main source of noise and we wish to estimate its degree of correlation, it is sufficient to prepare, for instance, the initial state r =

su0l+u1ld

and measure the x and y polarizations of both qubits for the final state r 8 . The initial state is fully polarized along x and there- fore

c

= 1,

c

= 0. s27d

For the final state, in the case of fully correlated dephasing sc

8 ^d

^{= h = 1}

2 f1 + g

g,

sc

8 ^d

^{= k = 1}

2 f1 − g

g, s28d while the expectation values of the xx- and yy-polarization measurements are remarkably different for uncorrelated dephasing:

sc

8 ^d

^{= g}

^,

sc

8 ^d

^{= 0. s29d}

V. DISCUSSION

While in general two-qubit quantum operations depend on N

− N

= 240 real parameters, an important question is how many parameters are physically significant. The answer of course depends on the specific noise processes. However, a clear answer can be given assuming that external noise is weak and local, that is to say, it acts independently on the two qubits. In this case, local noise is described by 24 pa- rameters, 12 for each qubit. Undesired coupling effects scross-talkd between qubits can be characterized with only three additional parameters, u

, u

, and u

. Indeed, any two- qubit unitary transformation U can be decomposed as f29–31g

U = sA

B

de

sA

B

d, s30d with A

, A

, B

, and B

appropriate single-qubit unitaries. In the limit of weak noise, the state matrix x

is simply given by the sum of the contributions of each noise channel. There- fore, the local unitaries A

, A

, B

, and B

only change the 24 local noise parameters and overall we need 24+ 3 = 27

! 240 parameters to describe the quantum noise. In the sym- metric case in which the local noise parameters are the same for both qubits the number of free parameters further reduces to 12+ 3 = 15. The above argument can be easily extended to many-qubit systems. Due to the two-body nature of interac- tions, we need to determine N = 12n + 3

parameters to characterize noise. Note that N = Oslog Nd!N

− N

. Of course, cases with strong or nonlocal noise would require a larger number of free parameters.

It might be useful to briefly compare our approach with other representations of standard QPT f2–5g, which are based on the operator-sum representation of quantum operations:

E srd = o

m,n=1 N²

x

E

rE

, s31d where hE

j forms a basis for the set of operators acting on the N-dimensional Hilbert space of the system. In standard QPT usually E

= uilkju is chosen, with huilj orthonormal basis of the system. Such choice does not exploit the fact that r is Hermitian and therefore leads to complex matrix ele- ments x

. In short, x is a N

3 N

complex matrix, while in our approach based on the Fano representation we deal with N

− N

independent real parameters. Finally, we note that our approach requires inversion of the N

3 N

matrix R, while the standard QPT algorithm described in Ref. f2g in- volves the calculation of a generalized inverse for a N

3 N

matrix.

To summarize, we have shown that the Fano representa- tion of the standard QPT is convenient, since the process matrix x

is real and the number of matrix elements is ex- actly equal to the number of free parameters required for the complete characterization of a generic quantum operation.

Moreover, the matrix elements of x

are directly related to the evolution, induced by the quantum operation, of the sys- tem’s polarization measurements. We have also shown that quantum noise channels have specific patterns in the Fano representation of x

. Finally, we have shown that in the case, of interest for quantum information processing, of weak and local noise the number of relevant noise parameters is N

= Oslog Nd!N

− N

, that is, much smaller than the number of parameters needed to determine a generic quantum opera- tion. In this case, the x

matrix is very sparse and therefore the number of polarization measurements needed to recon- struct it is much smaller than for a generic quantum opera- tion, thus considerably reducing the QPT complexity.

f1g G. Benenti, G. Casati, and G. Strini, Principles of Quantum Computation and Information sWorld Scientific, Singapore, 2004d, Vol. I; Principles of Quantum Computation and Infor- mation sWorld Scientific, Singapore, 2007d, Vol. II.

f2g M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information sCambridge University Press, Cam- bridge, 2000d.

f3g I. L. Chuang and M. A. Nielsen, J. Mod. Opt. 44, 2455 s1997d.

f4g J. F. Poyatos, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 78, 390 s1997d.

f5g A. G. Kofman and A. N. Korotkov, e-print arXiv:0903.0671.

f6g G. M. D’Ariano and P. Lo Presti, Phys. Rev. Lett. 86, 4195 s2001d.

f7g D. W. Leung, J. Math. Phys. 44, 528 s2003d.

f8g J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G. Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, and A. G. White, Phys. Rev. Lett. 90, 193601 s2003d.

f9g M. Mohseni and D. A. Lidar, Phys. Rev. Lett. 97, 170501 s2006d; M. Mohseni, A. T. Rezakhani, and D. A. Lidar, Phys.

Rev. A 77, 032322 s2008d.

f10g M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M. Stein-

berg, Phys. Rev. Lett. 91, 120402 s2003d.

f11g F. De Martini, A. Mazzei, M. Ricci, and G. M. D’Ariano, Phys. Rev. A 67, 062307 s2003d.

f12g J. L. O’Brien, G. J. Pryde, A. Gilchrist, D. F. V. James, N. K.

Langford, T. C. Ralph, and A. G. White, Phys. Rev. Lett. 93, 080502 s2004d.

f13g Y. Nambu and K. Nakamura, Phys. Rev. Lett. 94, 010404 s2005d.

f14g N. K. Langford, T. J. Weinhold, R. Prevedel, A. Gilchrist, J. L.

O’Brien, G. J. Pryde, and A. G. White, Phys. Rev. Lett. 95, 210504 s2005d.

f15g N. Kiesel, C. Schmid, U. Weber, R. Ursin, and H. Weinfurter, Phys. Rev. Lett. 95, 210505 s2005d.

f16g Z.-W. Wang, Y.-S. Zhang, Y.-F. Huang, X.-F. Ren, and G.-C.

Guo, Phys. Rev. A 75, 044304 s2007d.

f17g A. M. Childs, I. L. Chuang, and D. W. Leung, Phys. Rev. A 64, 012314 s2001d.

f18g N. Boulant, T. F. Havel, M. A. Pravia, and D. G. Cory, Phys.

Rev. A 67, 042322 s2003d.

f19g Y. A. Weinstein et al., J. Chem. Phys. 121, 6117 s2004d.

f20g S. H. Myrskog, J. K. Fox, M. W. Mitchell, and A. M. Stein-

berg, Phys. Rev. A 72, 013615 s2005d.

f21g M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt, T. K.

Korber, W. Hansel, H. Haffner, C. F. Roos, and R. Blatt, Phys.

Rev. Lett. 97, 220407 s2006d.

f22g T. Monz, K. Kim, W. Hansel, M. Riebe, A. S. Villar, P. Schin- dler, M. Chwalla, M. Hennrich, and R. Blatt, Phys. Rev. Lett.

102, 040501 s2009d.

f23g M. Howard et al., New J. Phys. 8, 33 s2006d.

f24g N. Katz et al., Phys. Rev. Lett. 101, 200401 s2008d.

f25g U. Fano, Rev. Mod. Phys. 29, 74 s1957d; 55, 855 s1983d.

f26g F. T. Hioe and J. H. Eberly, Phys. Rev. Lett. 47, 838 s1981d.

f27g J. Schlienz and G. Mahler, Phys. Rev. A 52, 4396 s1995d.

f28g G. Benenti, S. Felloni, and G. Strini, Eur. Phys. J. D 38, 389 s2006d.

f29g N. Khaneja, R. Brockett, and S. J. Glaser, Phys. Rev. A 63, 032308 s2001d.

f30g B. Kraus and J. I. Cirac, Phys. Rev. A 63, 062309 s2001d.

f31g M. A. Nielsen, C. M. Dawson, J. L. Dodd, A. Gilchrist, D.

Mortimer, T. J. Osborne, M. J. Bremner, A. W. Harrow, and A.

Hines, Phys. Rev. A 67, 052301 s2003d.