Single-Shot Quantum Process Tomography

Abhishek Shukla and T. S. Mahesh∗

Department of Physics and NMR Research Center,Indian Institute of Science Education and Research, Pune 411008, India

The standard procedure for quantum process tomography (QPT) involves applying the quantumprocess on a system initialized in each of a complete set of orthonormal states. The correspond-ing outputs are then characterized by quantum state tomography (QST), which itself requires themeasurement of non-commuting observables realized by independent experiments on identically pre-pared system states. Thus QPT procedure demands a number of independent measurements, andmoreover, this number increases rapidly with the size of the system. However, the total number ofindependent measurements can be greatly reduced with the availability of ancilla qubits. Ancilla as-sisted process tomography (AAPT) has earlier been shown to require a single QST of system-ancillaspace. Ancilla assisted quantum state tomography (AAQST) has also been shown to perform QSTin a single measurement. Here we combine AAPT with AAQST to realize a ‘single-shot QPT’(SSPT), a procedure to characterize a general quantum process in a single collective measurementof a set of commuting observables. We demonstrate experimental SSPT by characterizing severalsingle-qubit processes using a three-qubit NMR quantum register. Furthermore, using the SSPTprocedure we experimentally characterize the twirling process and compare the results with theory.

PACS numbers: 03.67.Lx, 03.67.Ac, 03.65.Wj, 03.65.TaKeywords: state tomography, process tomography, ancilla register, twirling

I. INTRODUCTION

An open quantum system may undergo an evolutiondue to intentional control fields as well as due to unin-tentional interactions with stray fields caused by envi-ronmental fluctuations. Even the carefully designed con-trol fields may be imperfect to the extent that one mightneed to characterize the overall process acting on thequantum system. Such a characterization, achieved bya procedure called quantum process tomography (QPT),is crucial in the physical realization of a fault-tolerantquantum processor [1, 2]. QPT is realized by consider-ing the quantum process as a map from a complete setof initial states to final states, and experimentally char-acterizing each of the final states using quantum statetomography (QST) [3]. Since the spectral decompositionof a density matrix may involve noncommuting observ-ables, Heisenberg’s uncertainty principle demands multi-ple experiments to characterize the quantum state. ThusQST by itself involves the measurement of a series ofobservables after identical preparations of the system inthe quantum state. Hence, QPT in general requires anumber of independent experiments, each involving ini-tialization of the quantum system, applying the processto be characterized, and finally QST. Furthermore, thetotal number of independent measurements required forQPT increases exponentially with the size of the systemundergoing the process.

The physical realization of QPT has been demon-strated on various experimental setups such as NMR[4, 5], linear optics [6–9], ion traps [10, 11], supercon-ducting qubits [12–17], and NV center qubit [18]. Several

∗ mahesh.ts@iiserpune.ac.in

developments in the methodology of QPT have also beenreported [19, 20]. In particular, it has been shown thatancilla assisted process tomography (AAPT) can charac-terize a process with a single QST [6, 7, 21, 22]. However,it still requires multiple measurements each taken over aset of commuting observables. On the other hand, if suffi-cient ancilla qubits are available, QST can be carried outwith a single joint measurement over the entire system-ancilla space. This procedure, known as ancilla assistedquantum state tomography (AAQST), has been studiedboth theoretically and experimentally [23–26]. Here wecombine AAPT with AAQST and realize a ‘single-shotquantum process tomography’ (SSPT), which can char-acterize a general process in a single collective measure-ment of the system-ancilla state.

In the next section, after briefly revising QPT andAAPT, we describe SSPT procedure. In section III, weillustrate SSPT using a three-qubit nuclear magnetic res-onance (NMR) quantum register. We characterize cer-tain unitary processes corresponding to standard quan-tum gates. We also characterize a nonunitary process,namely twirling operation. Finally we conclude in sec-tion IV.

II. THEORY

A. Quantum Process Tomography (QPT)

A process ε maps a quantum state ρ to another stateε(ρ). Here we consider an n-qubit system with N2(=22n)-dimensional Liouville space S. In order to charac-terize ε, we let the process act on each linearly indepen-dent element of a complete basis set {ρ1, ρ2, · · · , ρN2}.

arX

iv:1

404.

7830

v2 [

quan

t-ph

] 2

7 M

ay 2

014

2

1 0

0 0

0 1

0 0

0 0

0 1

0 0

1 0

(S)

(S)

(S)

(S)

QST

QST

QST

QST

{1k}

{4k}

{2k}

{3k}

1 =

4 =

2 =

3 =

1 0

0 0

0 1

0 0

0 0

0 1

0 0

1 0

(S)1(A1)

{jk}S,A1 =

1 0

0 0

0 1

0 0

0 0

0 1

0 0

1 0S,A1,A2 =

1 0

0 0

0 1

0 0

0 0

0 1

0 0

1 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 0

0 0

0 0

0 0

(a)

(b)

(c)

QST

{jk}AAQST

½

¼ (S)1(A1)

1(A2)

(2 meas.)

(2 meas.)

(2 meas.)

(2 meas.)

(2 meas.)

(1 meas.)

FIG. 1. (Color online) Illustrating (a) single-qubit QPT re-quiring a total of 8 NMR measurements, (b) AAPT requiring2 NMR measurements, and (c) SSPT requiring a single NMRmeasurement.

Expressing each output state in the complete basis weobtain

ε(ρj) =∑k

λjkρk, (1)

where the complex coefficients λjk can be extracted afterQST.

The outcome of a trace-preserving quantum process εalso has an operator-sum representation

ε(ρ) =∑i

EiρE†i , (2)

where the Kraus operators Ei satisfy the completeness

relation∑iE†iEi = I. To assist experimental character-

ization of the process, we utilize a fixed set of basis oper-ators {Em}, and express Ei =

∑m eimEm. The process

is now described by

ε(ρ) =∑mn

EmρE†nχmn, (3)

where χmn =∑i eime

∗in form a complex matrix which

completely characterizes the process ε. Since the set {ρk}forms a complete basis, it is also possible to express

EmρjE†n =

∑k

βmnjk ρk, (4)

n MQPT MAAPT (nA1) MSSPT (nA1, nA2)

1 8 2 (1) 1 (1, 1)2 32 4 (2) 1 (2, 2)3 192 11 (3) 1 (3, 3)4 1024 32 (4) 1 (4, 5)5 7168 103 (5) 1 (5, 6)

TABLE I. Comparison of number of independent measure-ments and number of ancilla qubits (in parenthesis) requiredfor n-qubit QPT, AAPT, and SSPT.

where βmnjk can be calculated theoretically. Eqns. 1, 3and 4 lead to

ε(ρj) =∑k

λjkρk =∑k

∑mn

βmnjk χmnρk. (5)

Exploiting the linear independence of {ρk}, one obtainsthe matrix equation

βχ = λ, (6)

from which χ-matrix can be extracted by standard meth-ods in linear algebra.

For example, in the case of a single qubit,one can choose the linearly independent basis{|0〉〈0|, |0〉〈1|, |1〉〈0|, |1〉〈1|} (see Fig. 1a). Whilethe middle-two elements are non-Hermitian, they canbe realized as a linear combination of Hermitian densityoperators [27]. A fixed set of operators {I,X,−iY, Z}can be used to express the χ matrix. Thus the stan-dard single-qubit QPT procedure requires four QSTexperiments.

QPT on an N -dimensional system requires N2-QSTexperiments, and a single QST involves several quantummeasurements each taken jointly over a set of commut-ing observables. The exact number of measurements maydepend on the properties of available detectors. In thecase of an n-qubit NMR system with a well resolved spec-trum [28], QST requires '

⌈Nn

⌋measurements, where dc

rounds the argument to next integer [26]. Therefore ann-qubit QPT needs a total of MQPT ' N2

⌈Nn

⌋mea-

surements. Estimates of M for a small number of qubitsshown in the first column of Table 1 illustrate the expo-nential increase of MQPT with n.

B. Ancilla-Assisted Process Tomography (AAPT)

If sufficient number of ancillary qubits are available,ancilla assisted process tomography (AAPT) can be car-ried out by simultaneously encoding all the basis ele-ments onto a higher dimensional system-ancilla Liouvillespace S ⊗ A1 [6, 7, 21, 22]. AAPT requires a single fi-nal QST, thus greatly reducing the number of indepen-dent measurements. For example, a single-qubit pro-cess tomography can be carried out with the help ofan ancillary qubit by preparing the n−qubit cat state|ψn〉 = (|00 · · · 0〉 + |11 · · · 1〉)/

√2, applying the process

3

on the system-qubit, and finally carrying out QST of thetwo-qubit state (see Fig. 1b). Although only two inde-pendent measurements are needed for a two-qubit QST,this number grows exponentially with the total numberof qubits.

All the N2 basis elements of the n-qubit system canbe encoded simultaneously in independent subspaces of asingle N2 ×N2 Liouville operator belonging to 2n-qubitspace S ⊗ A1. Thus exactly n-ancilla qubits are neededto carry out AAPT on an n-qubit system. Therefore

AAPT requires MAAPT '⌈N2

2n

⌋independent measure-

ments. The minimum number of experiments for a fewsystem-qubits are shown in the second column of Table I.While AAPT requires significantly lesser number of mea-surements compared to QPT, it still scales exponentiallywith the number of system-qubits.

C. Single-Shot Process Tomography (SSPT)

It had been shown earlier that, if sufficient numberof ancillary qubits are available, QST of a general den-sity matrix of arbitrary dimension can be performed witha single joint measurement of a set of commuting ob-servables [23–26]. This method, known as ancilla as-sisted quantum state tomography (AAQST) is based onthe redistribution of all elements of the system densitymatrix on to a joint density matrix in the combinedsystem-ancilla Liouville space. Initially ancilla registerfor AAQST is prepared in a maximally mixed state thuserasing all information in it and redistribution of matrixelements is achieved by an optimized joint unitary oper-ator [26].

By combining AAPT with AAQST, process tomogra-phy can be achieved with a single joint measurement ofall the qubits (see Fig. 1c and 3rd column of Table I).If AAQST is carried out with an ancilla space (A2) ofnA2-qubits, the combined space S⊗A1⊗A2 correspondsto n = 2n + nA2 qubits. A single joint measurementsuffices if the total number of observables is equal to orexceeds the number of real unknowns (i.e., N4−1) in the

2n-qubit density matrix, i.e., if nN ≥ (N4 − 1), where

N = 2n [26]. The number of ancillary qubits nA1 andnA2 required for SSPT are shown in the third column ofTable I.

The complete circuit for SSPT is shown in Fig. 2.It involves two ancilla registers, one for AAPT and theother for AAQST. Initially AAQST register is preparedin a maximally mixed state and the other two registersare set to |0⊗n〉 states. Hadamard gates on the AAPTancilla followed by C-NOT gates (on system qubits con-trolled by ancilla) prepare state |ψn〉, which simultane-ously encodes all the basis elements required for QPT. Asingle application of the process ε, on the system qubits,acts simultaneously and independently on all the basiselements {ρj}. The final AAQST operation allows esti-mation of all the elements of the 2n-qubit density matrix

AAQST

|0n+2⟩

|0n+1⟩

|0n⟩

|02⟩

|01⟩

|02n⟩ H

H

H

1 (A2)

System (S)

AAPTAncilla (A1)

AAQSTAncilla (A2)

jointmeasurement

FIG. 2. (Color online) Quantum circuit for SSPT.

∑j ε(ρj)⊗A1(j), where A1(j) identifies the jth subspace.

The output of each subspace ε(ρj) can now be extracted,

and the coefficients λjk = Tr[ε(ρj)ρ†k] can be calculated.

III. EXPERIMENTS

We used iodotrifluoroethylene (C2F3I) dissolved inacetone-D6 as a 3-qubit system. The molecular structureand labelling scheme are shown in Fig. 3 (a). All the ex-periments described below are carried out on a Bruker500 MHz NMR spectrometer at an ambient temperatureof 300 K using high-resolution NMR techniques. TheNMR Hamiltonian in this case can be expressed as

H = −π3∑i=1

νiσiz + π

3,3∑i=1,j>i

Jijσizσ

jz/2 (7)

F3

F1

F2

I

(a)

(b)

Qubit (Hz) T1 (s) T2*(s)

1 -17323 6.2 0.6

2 0 7.5 0.8

3 11833 6.9 0.8

J12 = -128.3 Hz

J13 = 47.4 Hz

J23 = 69.9 Hz

π/4y

1/4J 1/4J

π/4x

π/3x πx πx

F2

F1

G2 G3PFG

F3

F2

F1G1

PFG

H

H

(c)

PPS

π/2x π/2y

1 2

AAQST

FIG. 3. (Color online) Molecular structure of iodotrifluo-roethylene (a), and the table of Hamiltonian and relaxationparameters (b), NMR pulse-sequence to demonstrate SSPT(c). Pulse sequence for preparing |00〉 pseudopure state isshown in the inset of (c).

4

E X -iYZ

E X

-iYZ

0

0.5

1

E X -iYZ

E X

-iYZ

-0.5

0

0.5

E X -iYZ

E X

-iYZ

0

0.5

1

E X -iYZ

E X

-iYZ

-0.5

0

0.5

E X -iYZ

E X

-iYZ

0

0.5

1

E X -iYZ

E X

-iYZ

-0.5

0

0.5

E X -iYZ

E X

-iYZ

0

0.5

1

E X -iYZ

E X

-iYZ

-0.5

0

0.5

E X -iYZ

E X

-iYZ

0

0.5

1

E X -iYZ

E X

-iYZ

-0.5

0

0.5

E X -iYZ

E X

-iYZ

0

0.5

1

E X -iYZ

E X

-iYZ

-0.5

0

0.5

NOP (0.99)

NOT-Y (0.99)

Hadamard (0.95) Phase-π/4 (0.97)

Phase-π (0.99)

NOT-X (0.99)

FIG. 4. (Color onlinee) The barplots showing experimental χ-matrices for various quantum processes obtained using SSPT.In each case, the left and right barplots correspond to thereal and imaginary parts respectively, and the fidelities areindicated in parenthesis.

where σiz and σjz are Pauli z-operators of ith and jthqubits [28]. The chemical shifts νi, coupling constantsJij , and relaxation parameters (T1 and T∗2) are shownin Fig. 3 (b). All the pulses are realized using gradientascent pulse engineering (GRAPE) technique [29] andhad average fidelities above 0.99 over 20% inhomogeneousRF fields.

We utilize spins F1, F2, and F3 respectively as the sys-tem qubit (S), AAPT ancilla (A1), and AAQST ancilla(A2). The NMR pulse-sequence for SSPT experimentsare shown in Fig. 3(c). It begins with preparing A2qubit in the maximally mixed state by bringing its mag-netization into transverse direction using a Hadamardgate, and subsequently dephasing it using a PFG. Theremaining qubits are initialized into a pseudopure |00〉state by applying the standard pulse-sequence shown inthe inset of Fig. 3c [30]. The Bell state |ψ2〉 prepared us-ing a Hadamard-CNOT combination had a fidelity of over0.99. After preparing this state, we applied the process εon the system qubit. The final AAQST consists of (π/2)xand (π/2)y pulses on all the qubits separated by delaysτ1 = 6.7783 ms and τ2 = 8.0182 ms [26]. A single jointmeasurement of all the qubits now leads to a complex sig-nal of 12 transitions, from which all the 15 real unknownsof the 2-qubit density matrix ρS,A1 =

∑j ε(ρj) ⊗ A1(j)

of F1 and F2 can be estimated [26]. In our choice of fixedset of operators and basis elements

ρS,A1 =

λ11 λ12 λ21 λ22

λ13 λ14 λ23 λ24

λ31 λ32 λ41 λ42

λ33 λ34 λ43 λ44

(8)

[31]. The χ matrix characterizing the complete processcan now be obtained by solving the eqn. 6.

A. SSPT of quantum gates

We now describe experimental characterization of sev-eral single-qubit unitary processes. The quantum gatesto be characterized are introduced as process ε on F1

qubit in Fig. 3c. The experimental χ-matrices for NOP(identity process), NOT-X (e−iπX/2), NOT-Y (e−iπY/2),Hadamard, Phase−π (eiπZ), and Phase−π/4 (eiπZ/8) areshown in Fig. 4. Starting from thermal equilibrium, eachSSPT experiment characterizing an entire one-qubit pro-cess took less than four seconds. A measure of overlapof the experimental process χexp with the theoreticallyexpected process χth is given by the gate fidelity [18]

F (χexp, χth) =|Tr[χexpχ

†th]|√

Tr[χ†expχexp] Tr[χ†thχth]. (9)

The gate fidelities for all the six processes are indicated inFig. 4. Except in the cases of Hadamard and Phase-π/4,the gate fidelities were about 0.99. The lower fidelities inHadamard (0.95) and Phase-π/4 (0.97) are mainly dueto imperfections in the RF pulses implementing theseprocesses.

In order to study the robustness of SSPT procedurewe first considered an ideal process, simulated the corre-sponding spectral intensities, and reconstructed the fi-nal density matrix ρS,A1. Using eqn. 8 we obtainedλjk values and calculated the matrix χ0 simulating thenoise-free SSPT procedure. We then introduced noise byadding random numbers in the range [−η, η] to the spec-tral intensities and used the resulting data calculatingχη simulating the noisy SSPT procedure. The variationsof average gate fidelities F (χ0, χη) for various processesversus noise amplitude η are shown in Fig. 5. Interest-ingly, the noise has similar effects on fidelities of all thesimulated processes. We also observe that fidelities re-mained above 0.9 for η < 0.1, indicating that SSPT isfairly robust against the noise in this range.

0.01 0.02 0.05 0.1 0.20.75

0.8

0.85

0.9

0.95

1

η

F

NOP

NOT-X

NOT-Y

Phase-π

Hadamard

Phase-π/4

FIG. 5. (Color online) Simulated fidelity of various processesas a function of noise η.

5

B. SSPT of twirling process

Twirling is essentially a nonunitary process usually re-alized by an ensemble average of a set of unitary oper-ations. It was first introduced by Bennett et al [32] forextracting singlet states from a mixture of Bell states.Twirling has been studied in detail [33–37] and vari-ous modified twirling protocols have also been suggested[38, 39].

In NMR, twirling can be achieved with the help of apulsed field gradient (PFG), which produces a contin-uous space-dependent spin-rotation, such that the en-semble average effectively emulates a nonunitary pro-cess [40]. A z PFG produces a space-dependent unitary

Uφ = exp(−iφ2

∑nj=1 σjz

)with φ ∈ [−Φ,Φ], where j is

the summation index over all the qubits, and the maxi-mum phase Φ depends on the strength and duration ofPFG and the sample volume. When the z PFG acts onan initial n-qubit density matrix ρin =

∑lm ρlm|l〉〈m|,

the resultant output density matrix is,

ρout =1

2Φ

∫ Φ

−Φ

dφ UφρinU†φ

=∑lm

ρlm|l〉〈m| sinc(qlmΦ). (10)

Here sinc(x) = sin xx and qlm = 1

2

∑j [(−1)mj − (−1)nj ]

is the quantum number of the element|l1l2 · · · ln〉〈m1m2 · · ·mn|, i.e., the difference in thespin-quantum numbers of the corresponding basisstates. While the diagonal elements |l〉〈l| and otherzero-quantum elements are unaffected by twirling, theoff-diagonal elements with qlm 6= 0 undergo decayingoscillations with increasing Φ values.

SSPT of twirling process is carried out using the pro-cedure described in Fig. 3c after introducing δ-PFG-δin place of the process ε, where δ is a short delay forswitching the gradient. Applying PFG selectively on thesystem qubit is not simple, and is also unnecessary. Sincethe F3 qubit (AAQST ancilla) is already in a maximallymixed state, twirling has no effect on it. For the Bellstate |ψ2〉, applying a strong twirling on either or bothspins (F1, F2) has the same effect, i.e., a strong mea-surement reducing the joint-state to a maximally mixedstate. However, since |ψ2〉 corresponds to a two-quantumcoherence (i.e., q00,11 = 2), its dephasing is double that ofa single-quantum coherence. Assuming the initial stateρin = |ψ2〉〈ψ2|, and using expressions 1 and 10, we findthat the non-zero elements of λ are

λ11 = λ44 = 1, and, λ22 = λ33 = sinc(2Φ). (11)

Solving expression 6, we obtain a real χ matrix with onlynonzero elements

χEE =1 + sinc(2Φ)

2and χZZ =

1− sinc(2Φ)

2. (12)

In our experiments, the duration of PFG and δ are

0 0.5 1 1.5 2 2.5 3 3.50

0.25

0.5

0.75

1

Φ/π

E X -iYZ

E X

-iYZ

0

0.5

1

E X -iYZ

E X

-iYZ

0

0.5

1

E X -iYZ

E X

-iYZ

0

0.5

1

E X -iYZ

E X

-iYZ

0

0.5

1

χZZ

χEE

F(χexp

,χth

)

(a)

(b) (c) (d) AA

FIG. 6. (Color online) (a) The experimental values of |χEE|(|χZZ|) are shown by filled squares (filled circles). The solidline (χEE) and the dashed line (χZZ) illustrate theory. Thebarplots correspond to experimental |χ| matrices at (b) Φ = 0,(c) Φ = 0.64π, (d) Φ = π, and (e) Φ = 3.43π.

set to 300 µs and 52.05 µs respectively, such that thechemical shift evolutions and J-evolutions are negligible.The strength of twirling is slowly varied by increasing thePFG strength from 0 to 2.4 G/cm in steps of 0.05 G/cm.The results of the experiments are shown in Fig. 6. Thefilled squares (circles) in Fig. 6(a) correspond to experi-mentally obtained values for |χEE| (|χZZ|). Small imagi-nary parts observed in experimental χmatrices are due tominor experimental imperfections. The smooth (dashed)lines indicate corresponding theoretical values obtainedfrom eqns. 12. The crosses indicate the gate fidelitiesF (χexp, χth) calculated using eqn 9. The barplots showexperimental |χ| matrices for (b) Φ = 0, (c) Φ = 0.64π,(d) Φ = π, and (e) Φ = 3.43π, and χEE and χZZ valuesin Fig 6(a) corresponding to these Φ values are circledout.

At zero twirling, the process is essentially a NOP pro-cess as is clear from the bar plot in Fig. 6(b), wherein|χEE| ≈ 1 and |χZZ| ≈ 0. When Φ = kπ/2 with aninteger k, the ensemble initially prepared in state |ψ2〉undergoes an overall phase distribution over [−kπ, kπ],and at this stage χEE = χZZ = 0.5 (eg. Fig. 6 d). Fur-ther increase in Φ leads to oscillations of χEE and χZZ

about 0.5, and for large Φ values, both of these elementsdamp towards 0.5 and all other elements vanish (eg. Fig.6 e). The errors in experimental χEE and χZZ values wereless than 8 %. The good agreement of the experimentalvalues with theory indicates the overall success of SSPTprocedure. The average of the gate fidelities was over0.96. Small deviations of the experimental values fromtheory are due to nonlinearities in PFG profile as wellas due to imperfections in RF pulses implementing theSSPT procedure.

6

IV. CONCLUSIONS

Information processing requires two important phys-ical resources, namely, the size of the register and thenumber of operations. Often there exists an equivalencebetween these two resources which allows trading one re-source with another. Likewise, in the present work weshow that, if some extra qubits are available, it is pos-sible to carry out quantum process tomography of thesystem qubits with a single joint measurement over aset of commuting observables. We have illustrated thismethod on a single system qubit and two ancillary qubitsusing NMR quantum computing methods. In particular,we extracted the χ matrices characterizing certain quan-tum gates and obtained their gate fidelities with the helpof a single measurement of a three qubit system in eachcase. We studied the robustness of SSPT procedure us-ing numerical simulations. We also characterized twirling

operation which is essentially a nonunitary process. Theoverall procedure of SSPT is general and can be appliedto other fields such as optical qubits, trapped ions, orsuperconducting qubits.

A potential application of single-shot process tomog-raphy could be in high throughput characterization ofdynamic processes. The standard methods require re-peated applications of the same process either to collectindependent outputs from all the basis states or to al-low quantum state tomography. However, the presentmethod requires only one application of the process forthe entire characterization.

ACKNOWLEDGEMENTS

The authors are grateful to Prof. Anil Kumar, SwathiHegde, Hemant Katiyar, and Rama Koteswara Rao fordiscussions. This work was partly supported by DSTproject SR/S2/LOP-0017/2009.

[1] I. L. Chuang and M. A. Nielsen, Journal of Modern Op-tics 44, 2455 (1997).

[2] J. F. Poyatos, J. I. Cirac, and P. Zoller, Phys. Rev. Lett.78, 390 (1997).

[3] M. G. K. I. L. Chuang, N. Gershenfeld and D. W. Leung,Proc. R. Soc. Lond.,Ser A 454, 447 (1998).

[4] A. M. Childs, I. L. Chuang, and D. W. Leung, Phys.Rev. A 64, 012314 (2001).

[5] Y. S. Weinstein, T. F. Havel, J. Emerson, N. Boulant,M. Saraceno, S. Lloyd, and D. G. Cory, The Journal ofChemical Physics 121, 6117 (2004).

[6] F. De Martini, A. Mazzei, M. Ricci, and G. M. D’Ariano,Phys. Rev. A 67, 062307 (2003).

[7] J. B. Altepeter, D. Branning, E. Jeffrey, T. C. Wei, P. G.Kwiat, R. T. Thew, J. L. O’Brien, M. A. Nielsen, andA. G. White, Phys. Rev. Lett. 90, 193601 (2003).

[8] 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 (2004).

[9] M. W. Mitchell, C. W. Ellenor, S. Schneider, and A. M.Steinberg, Phys. Rev. Lett. 91, 120402 (2003).

[10] M. Riebe, K. Kim, P. Schindler, T. Monz, P. O. Schmidt,T. K. Korber, W. Hansel, H. Haffner, C. F. Roos, andR. Blatt, Phys. Rev. Lett. 97, 220407 (2006).

[11] D. Hanneke, J. P. Home, J. D. Jost, J. M. Amini,D. Leibfried, and D. J. Wineland, Nature Phys. 6, 13(2010).

[12] M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz,N. Katz, E. Lucero, A. O/’Connell, H. Wang, A. N. Cle-land, and J. M. Martinis, Nature Phys. 4, 523 (2008).

[13] J. M. Chow, J. M. Gambetta, L. Tornberg, J. Koch,L. S. Bishop, A. A. Houck, B. R. Johnson, L. Frunzio,S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. Lett.102, 090502 (2009).

[14] R. C. Bialczak, M. Ansmann, M. Hofheinz, E. Lucero,M. Neeley, A. D. O/’Connell, D. Sank, H. Wang, J. Wen-ner, M. Steffen, and J. M. Cleland, A. N.and Martinis,Nature Phys. 6, 409 (2010).

[15] T. Yamamoto, M. Neeley, E. Lucero, R. C. Bialczak,J. Kelly, M. Lenander, M. Mariantoni, A. D. O’Connell,D. Sank, H. Wang, M. Weides, J. Wenner, Y. Yin, A. N.Cleland, and J. M. Martinis, Phys. Rev. B 82, 184515(2010).

[16] J. M. Chow, A. D. Corcoles, J. M. Gambetta, C. Rigetti,B. R. Johnson, J. A. Smolin, J. R. Rozen, G. A. Keefe,M. B. Rothwell, M. B. Ketchen, and M. Steffen, Phys.Rev. Lett. 107, 080502 (2011).

[17] A. Dewes, F. R. Ong, V. Schmitt, R. Lauro, N. Boulant,P. Bertet, D. Vion, and D. Esteve, Phys. Rev. Lett. 108,057002 (2012).

[18] J. Zhang, A. M. Souza, F. D. Brandao, and D. Suter,Phys. Rev. Lett. 112, 050502 (2014).

[19] A. Shabani, R. L. Kosut, M. Mohseni, H. Rabitz, M. A.Broome, M. P. Almeida, A. Fedrizzi, and A. G. White,Phys. Rev. Lett. 106, 100401 (2011).

[20] Z. Wu, S. Li, W. Zheng, X. Peng, and M. Feng, TheJournal of Chemical Physics 138, 024318 (2013).

[21] A. Mazzei, M. Ricci, F. De Martini, and G. D’Ariano,Fortschritte der Physik 51, 342 (2003).

[22] G. M. D’Ariano and P. Lo Presti, Phys. Rev. Lett. 91,047902 (2003).

[23] A. E. Allahverdyan, R. Balian, and T. M. Nieuwen-huizen, Phys. Rev. Lett. 92, 120402 (2004).

[24] X. Peng, J. Du, and D. Suter, Phys. Rev. A 76, 042117(2007).

[25] Y. Yu, H. Wen, H. Li, and X. Peng, Phys. Rev. A 83,032318 (2011).

[26] A. Shukla, K. R. K. Rao, and T. S. Mahesh, Phys. Rev.A 87, 062317 (2013).

[27] Nielsen and I. L. Chuang, Quantum computationand quantum information (Cambridge university press,2010).

[28] J. Cavanagh, W. J. Fairbrother, A. G. Palmer III, andN. J. Skelton, Protein NMR spectroscopy: principles andpractice (Academic Press, 1995).

7

[29] N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbruggen,and S. J. Glaser, Journal of Magnetic Resonance 172,296 (2005).

[30] D. G. Cory, A. F. Fahmy, and T. F. Havel, Proc. Natl.Acad. Sci. USA 94, 1634 (1997).

[31] see supplementary material.[32] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher,

J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. 76,722 (1996).

[33] C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, andW. K. Wootters, Phys. Rev. A 54, 3824 (1996).

[34] J. Emerson, R. Alicki, and K. yczkowski, J. Opt. B:Quantum Semiclass. Opt. 7, S347 (2005).

[35] J. Emerson, M. Silva, O. Moussa, C. Ryan, M. Laforest,J. Baugh, D. G. Cory, and R. Laflamme, Science 317,1893 (2007).

[36] M. Silva, E. Magesan, D. W. Kribs, and J. Emerson,Phys. Rev. A 78, 012347 (2008).

[37] C. C. Lopez, A. Bendersky, J. P. Paz, and D. G. Cory,Phys. Rev. A 81, 062113 (2010).

[38] C. Dankert, R. Cleve, J. Emerson, and E. Livine, Phys.Rev. A 80, 012304 (2009).

[39] O. Moussa, M. P. da Silva, C. A. Ryan, and R. Laflamme,Phys. Rev. Lett. 109, 070504 (2012).

[40] M. Anwar, L. Xiao, A. Short, J. Jones, D. Blazina,S. Duckett, and H. Carteret, Physical Review A 71,032327 (2005).