Error characterization in quantum information processing:
A protocol for analyzing spatial correlations and its
experimental implementation
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López, Cecilia C., Benjamin Lévi, and David G. Cory. “Error
characterization in quantum information processing: A protocol
for analyzing spatial correlations and its experimental
implementation.” Physical Review A 79.4 (2009): 042328. ©
2009 The American Physical Society.
As Published
http://dx.doi.org/10.1103/PhysRevA.79.042328
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American Physical Society
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Thu May 26 09:47:07 EDT 2016
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http://hdl.handle.net/1721.1/51777
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Detailed Terms
PHYSICAL REVIEW A 79, 042328 共2009兲
Error characterization in quantum information processing: A protocol for analyzing spatial
correlations and its experimental implementation
Cecilia C. López, Benjamin Lévi, and David G. Cory
Department of Nuclear Science and Engineering, MIT, Cambridge, Massachusetts 02139, USA
共Received 30 May 2008; published 22 April 2009兲
We present a protocol for error characterization and its experimental implementation with four qubits in
liquid state NMR. The method is designed to retrieve information about spatial correlations and scales as
O共nw兲, where w is the maximum number of qubits that have non-negligible interaction. We discuss the
practical aspects regarding accuracy and implementation.
DOI: 10.1103/PhysRevA.79.042328
PACS number共s兲: 03.67.Ac, 03.65.Yz, 33.25.⫹k
I. INTRODUCTION
S共␳0兲 =
Precise and reliable control of the system constituting a
quantum information processor 共QIP兲 remains one of the
biggest challenges in the quantum information field. In order
to assess the reliability of a device and to tailor quantum
error correction schemes to a faulty one, we need to characterize the errors occurring in the system. Quantum process
tomography 共QPT兲 关1兴 presented a first answer to this problem, providing full characterization of the process under
analysis. Experimental implementations of QPT have been
already conducted in a variety of small systems 关2–5兴. Nevertheless, QPT becomes impractical beyond a few qubits, as
it requires O共24n兲 experiments for a system of n qubits. In
recent years, the idea of getting less information at a lower
cost has become a popular strategy in tackling error characterization, and several works have been devoted to the subject 关6–12兴. Our proposal fits in this context, providing a
subset of information yet still using scalable resources.
The scheme we present selectively keeps information
about the spatial correlations of the errors occurring in the
process under study 共a gate, a noisy channel, etc.兲. Both the
magnitude and the structure of the errors are relevant to
evaluate fault-tolerance. In particular, fault-tolerance threshold theorems are designed for certain conditions of spatial
correlation 共also termed range or locality兲 关13兴. So even
when it is experimentally determined that only up to w qubits
are involved in an error process, we have to further establish
in which way the 共 wn 兲 possible sets are being affected.
We have implemented our protocol in a liquid state NMR
four-qubit QIP. The core mathematical work for this protocol
was introduced in 关7兴. Here we extend our proposal to a more
general setting and include an experimental realization. Our
basic method, like others proposed 关1,6–8,10,11兴, assumed
error-free implementation stages. This idea is of course unrealistic in practice, and implementation errors complicate
the task of reliable error characterization. Thus here we have
included an analysis of their effect.
冕
ជ 兲E共␩ជ 兲␳0E†共␩ជ 兲d␩ជ .
p共␩
ជ denotes D2 complex coefficients 兵␩0 , ␩l , l
The vector ␩
2
= 1 , . . . , D − 1其 that parametrize E, an arbitrary operator in
the Hilbert space HD, as
n
D2−1
E = ␩ 0I +
兺
l=1
␩ lO l,
Ol =
丢O
1050-2947/2009/79共4兲/042328共6兲
共j兲
l ,
共2兲
j=1
is an element of the Pauli group
where each O共j兲
l
兵I , ␴x , ␴y , ␴z其 but at least one factor in each Ol is a Pauli
ជ 兲 is a 共real兲
matrix. Notice that Tr关OlOl⬘兴 = D␦l,l⬘. When p共␩
non-negative distribution, Eq. 共1兲 describes an arbitrary comជ 兲兩␩ជ 兩2d␩ជ
pletely positive 共CP兲 map, and the condition 兰p共␩
= 1 guarantees the preservation of Tr关␳兴. This representation
of a CP map is an operator-sum representation with continuous parameters. We prefer this form as it is more suitable to
describe nonunitary dynamics arising from stochastic Hamilជ , are trivially related to the
tonians. Our parameters, the ␩
parametrizations of the operator-sum representations used in
other works 关8,10兴. Also, for small ␩l, it is possible to relate
Eq. 共1兲 to a description of the noise in terms of generators
rather than propagators 关7兴.
When necessary, we shall denote the ␩l in more detail as
..
␩ p,q.
j,k. . . , where j ⬎ k ⬎ . . . label qubits, and p , q , . . . = x , y , z.
..
Therefore ␩ p,q.
j,k. . . labels a term in Eq. 共2兲 that is a product of
␴ p for qubit j, ␴q for qubit k, etc., and I for the qubits absent
in the subscript. Notice that the number of qubits in the subscript gives the Pauli weight 共also Hamming weight兲 of the
term 关14兴.
In our protocol we measure a subset of m qubits at a time,
which will allow us to extract the magnitude of the errors
involving that subset. So we now break the system in two: m
qubits belonging to the Hilbert space HM , which will be the
ones measured, and m̄ = n − m qubits belonging to the
complementary space HM¯ , HD = HM 丢 HM¯ . We require the
initial state to be separable in these two spaces, and within
HM , thus
II. THEORY BEHIND THE PROTOCOL
We start by describing the action of a general map on the
state of an n-qubit system 共D = 2n兲, described by an initial
state ␳0, as
共1兲
共M̄兲
␳0 = ␳共M兲
0 丢 ␳0
and
␳共M兲
0 =
丢␳
共j兲
0 .
共3兲
j苸M
We now perform a U共2兲 丢 m twirl on the target map S,
042328-1
©2009 The American Physical Society
PHYSICAL REVIEW A 79, 042328 共2009兲
LÓPEZ, LÉVI, AND CORY
Km
1
␳1 = m 兺 C†k S共Ck␳0C†k 兲Ck ,
K k=1
共4兲
where the Ck are m-fold tensor products of the twirl operators
on one qubit, for the m qubits in HM . For the task we propose, a subset of K = 6 operators from the Clifford group will
suffice. We refer to this minimum set of operators required to
apply the twirl 关Eq. 共4兲兴 as the 6m-Clifford element pool 共see
Appendix A for references and some mathematical details on
this twirling兲.
¯
The reduced density matrix ␳共M兲
1 = TrM 关␳1兴 is the state of
the m qubits we measure. In general, the fidelity decay,
which gives a measure of how ␳1 differs from the initial ␳0,
encodes information about the map S we are characterizing
关6,7兴. In particular, considering only the qubits we will mea共M̄兲
丢 m̄
+ ␳dev
兲 / 2m̄, we obtain
sure and expressing ␳共M̄兲
0 = 共I
共M兲 共M兲
共M兲
2
共M兲
Tr关共␳共M兲
− ⍀dev
,
0 兲 兴 − TrM 关␳0 ␳1 兴 = ␥
共5兲
冉兿
共6兲
␥共M兲 = 兺 具兩␩l兩2典
l
Pj −
j苸M
兿 C j共l兲冊 ,
j苸M
共M兲
共M̄兲
共M兲
共M̄兲
= 0 if ␳dev
= 0 共⍀dev
for ␳dev
⫽ 0 can be exactly comand ⍀dev
puted if desired兲. The quantity defined in the left-hand side
of Eq. 共5兲 is the fidelity decay rate 关6,7兴 we will analyze in
ជ 兲 . . . d␩ជ ,
this work. In Eq. 共6兲 we have denoted 具. . .典 = 兰p共␩
2
兲
兴
the
initial
purity
of
each
of
the
M-qubits,
and
P j = Tr关共␳共j兲
0
C j共l兲 =
再
共2/3兲共1 − P j/2兲 if O共j兲
l = ␴ x, ␴ y , ␴ z
if O共j兲
l = I.
Pj
冎
共7兲
For the derivation of Eqs. 共5兲 and 共6兲 we use the equivalence
between a Clifford twirl and a Haar twirl 关15兴, and apply the
tools developed in 关16兴. See also Appendix A.
In sum 共6兲, a l-term vanishes when Ol is the identity operator for the m qubits being measured. So by choosing different sets of M-qubits, it is possible to leave out certain ␩l in
a given ␥. Note however that C j does not distinguish the
direction of the Pauli matrices, so there is an implicit coarsegraining of all the Ol that have the identity I for the same
subset of m qubits. This is how we get the “collective coefficients” ␩col
l ,
2
兩␩col
j 兩 =
兺
p=x,y,z
兩␩ pj 兩2,
2
兩␩col
j,k 兩 =
兺
2
兩␩ p,q
j,k 兩 ,
etc. 共8兲
p,q=x,y,z
共3兲
共1兲 共3兲
For example, the terms ␴共1兲
x ␴x and ␴x ␴z contribute colcol
lectively to ␩1,3.
By combining the ␥’s from different M-sets it is possible
to further isolate the collective coefficients. If we prepare a
and b in a pure state so Pa = Pb = 1, we obtain
9 共a兲
col 2
2
共␥ + ␥共b兲 − ␥共a,b兲兲 = 具兩␩a,b
兩 典 + 兺 具兩␩col
a,b,j兩 典 + ¯ . 共9兲
4
j
共Please refer to Appendix B for the formulas of the ␥’s involved in this example.兲 If three-body and higher multibody
col 2
兩 典. Similarly, the
terms can be neglected, Eq. 共9兲 gives 具兩␩a,b
共j兲
共j,j⬘兲
combination of the seven ␥ , ␥
, and ␥共j,j⬘,j⬙兲 for a set of
col 2
兩 典, and so on. The
three qubits a , b , c would return 具兩␩a,b,c
collective coefficients then report on the spatial correlations
of errors.
III. PROTOCOL
We present now a systematic protocol for measuring the
collective coefficients involving the m qubits of a particular
subset.
共i兲 Prepare each of the qubits to be measured in the initial
state 兩0典. Prepare each of the other m̄ qubits in the maximally
mixed state I / 2.
共ii兲 Apply one of the m-fold Clifford operators from the
6m-Clifford element pool.
共iii兲 Implement the target gate or noise under study.
共iv兲 Invert the Clifford operator applied in 共ii兲.
共v兲 Measure the projection of the resulting state on the
initial state 兩0典, for each of the qubits being measured.
To implement the twirl, we repeat 共i兲–共v兲 each time taking
a different operator from the 6m-Clifford element pool, and
average the results.
In a canonical QIP, the implementation of this protocol to
measure the decay rates involving m qubits will require N
realizations. This will take care of: 共1兲 preparing the desired
initial state as in step 共i兲 共starting from the 兩0典 丢 n state, and
randomly flipping the m̄ qubits we do not measure兲; 共2兲 measuring the ␥’s through repeated projective measurements of
the m qubits as prescripted in step 共v兲 关notice Eq. 共5兲 be共M兲
共M兲
for the proposed initial state兴;
comes 1 − TrM 关␳共M兲
0 ␳1 兴 = ␥
共3兲 implementing the twirl approximately, by randomly
drawing the twirl operators for steps 共ii兲 and 共iv兲 from the
corresponding 6m-Clifford element pool 共or from an infinite
pool of one-qubit random rotations兲.
With this strategy, the outcome of the measurement step
共v兲 is a Bernoulli variable, and thus N can be estimated from
usual statistics. More precisely, the standard error in our estimation of the decay rate will be ␴␥ ⱕ 1 / 冑N 共following the
central limit theorem兲 so for a desired precision ␦ we must
have N ⱖ ␦−2. On the other hand, the Chernoff bound tells us
that, for a desired precision ␦ and an error probability ⑀N
Ⰶ 1, we must have N = log共2 / ⑀N兲 / 共2␦2兲, which is a stronger
requirement when ␦ ⬍ 2e−2. In any case, N is independent of
the number of qubits; thus the efficiency of the protocol is
independent of the size of the system.
In the case of liquid state NMR ensemble QIP,
pseudopure state preparation allows for initialization in the
I / 2 state over the ensemble of molecules, and ensemble measurements avoid the need of repeated realizations in order to
perform step 共v兲 by quantum state tomography 共QST兲. This
is the case in our experiment.
The measurement step 共v兲 retrieves the information to calculate the decay rates for the m qubits and for any smaller
subset of qubits within those. For example, twirling qubits a
and b only 共m = 2兲, we can obtain ␥共a兲, ␥共b兲, and ␥共a,b兲 in one
col 2
兩 典 as in Eq. 共9兲 neglecting the higher
shot, and calculate 具兩␩a,b
multibody terms. This procedure can be repeated for the
共 2n 兲 = n共n − 1兲 / 2 pairs of qubits, and by doing so all the collective coefficients for one- and two-body terms can be extracted.
The scalability of the method goes as follows. If we can
neglect the multi-body terms above a certain Pauli weight w,
042328-2
PHYSICAL REVIEW A 79, 042328 共2009兲
ERROR CHARACTERIZATION IN QUANTUM-…
and N is the number of realizations required to measure the
fidelity ␥ for w qubits, then with N共 wn 兲 ⱕ Nnw / w! experiments we can estimate all the non-negligible coefficients.
This should be compared against N24n, the overhead in the
number of experiments required for QPT. We emphasize here
that our proposal seeks to characterize the correlations
among up to w qubits in order to establish the range of the
noise, not to characterize the process fully.
To use our protocol, the negligibility of multi-body terms
above a certain Pauli weight w must be established a priori.
In a canonical QIP we could apply our protocol to measure
all the n qubits, obtain all the decay rates
␥共j兲 , ␥共j,k兲 , . . . , ␥共1,. . .,n兲, and extract all the collective coefficients after only N experiments 共independently of n兲. In this
way we can handle all the Pauli weights, from zero to n. But
the error in the decay rates ␴␥ propagates into the ␩col for m
qubits inefficiently, roughly as ␴␩2 ⬀ 冑兺wj=0共 nj 兲␴␥. Therefore
our strategy is to look for spatial correlations after establishing a cutoff Pauli weight w: to our knowledge, neither QPT
nor other proposals so far are able to make use of the negligibility of high order correlations in order to gain further
insight. In general, even after it has been established that a
subset of parameters is null, it is not trivial to direct the
measurement procedure to target the non-null ones exclusively.
An approach to establishing this cutoff w, demanding the
same resources as our protocol, is to apply the method developed by Emerson et al. 关8兴 which gives the probability of
errors happening, distinguishing them only by Pauli weight
关this is an average of all the 共 wn 兲 collective coefficients having Pauli weight w兴. It is worth pointing out that both methods require the same experimental work: the algorithms retrieve different information because the measurement and
processing of the data is different, but actually both protocols
can be implemented simultaneously and used complementarily.
TABLE I. Measured 共meas兲 collective coefficients for selected
pairs of qubits, for the various gates studied. theo are the theoretical
values for these gates as described in the text. The sim values come
from numerical simulation: simG are the values obtained by the
calculation of the propagator as given from the rf pulse sequence
corresponding to the gate 共G兲 alone; simE are the values obtained
by completely mimicking the experiment 共E兲, considering the average of 36 simulations, each applying rf pulses to prepare the initial
state and implement the Cliffords gates and the gate under study.
Gate
We implemented our protocol in a liquid state NMR QIP
using the four 13C-labeled carbons of crotonic acid 关17兴 in a
400 MHz Bruker spectrometer. The initial state preparation
and all the gates required by the protocol were implemented
with rf pulse sequences engineered using either gradient ascent pulse engineering 共GRAPE兲 关18兴 or strongly modulating
pulses 共SMP兲 关17共c兲兴 methods. Their simulated gate fidelities
Fg were on average 0.98. The typical experimental performance of one-qubit gates on the spectrometer is 1%–2% below their simulated fidelities. See 关2兴 for details on the model
used in the simulation.
We studied the following processes 共see Table I for more
details兲: 共i兲 A time suspension sequence IE since it is important to study our ability to “do nothing” in a system with a
natural Hamiltonian that is always on. 共ii兲 An engineered
error creating a coupling between qubits 1 and 2, of the form
C12共␤兲 = exp共−i␤␴z1␴z2兲. We chose ␤ = 0.1, and ␤ = 0.4 共the
previous one applied consecutively four times兲. 共iii兲 A
controlled-NOT 共CNOT兲 gate between qubits 1 and 2: CNOT
共1兲 共2兲
= 0.5共I + ␴z共1兲 + ␴共2兲
x − ␴z ␴x 兲. Also, this gate applied twice:
2
具兩␩col
2,3兩 典
2
具兩␩col
1,4兩 典
IE
12.2 ms
Fg = 0.96
meas
theo
simG
simE
0.02
0.00
0.01
0.01
0.02
0.00
0.02
0.03
0.01
0.00
0.00
0.00
C12共0.1兲
4.88 ms
Fg = 0.99
meas
theo
simG
simE
0.02
0.01
0.02
0.02
0.02
0.00
0.00
0.00
0.01
0.00
0.00
0.00
C12共0.4兲
19.52 ms
Fg = 0.87
meas
theo
simG
simE
0.26
0.15
0.23
0.24
0.03
0.00
0.01
0.02
0.03
0.00
0.00
0.02
CNOT
meas
theo
simG
simE
0.32
0.25
0.25
0.28
0.01
0.00
0.01
0.02
0.03
0.00
0.00
0.00
meas
theo
simG
simE
0.07
0.00
0.01
0.01
0.05
0.00
0.01
0.02
0.04
0.00
0.00
0.00
11.88 ms
Fg = 0.99
CNOT2
23.76 ms
Fg = 0.97
CNOT2 = I.
IV. EXPERIMENTAL RESULTS
2
具兩␩col
1,2兩 典
These gates are more complex than one-qubit operations 共which are typically less than 1 ms long兲 and they all
involve refocusing idle times 共periods of free evolution under the internal Hamiltonian兲 in their pulse sequences.
The results on the measurement of the collective coefficients for the qubit pairs 共1,2兲, 共2,3兲, and 共1,4兲 are presented
in Table I. The pair 共1,2兲 is the one targeted by the C12 and
CNOT gates while the other two are the pairs involving qubits
1 or 2 with the highest J-coupling 关17兴. We expect the errors
for the three chosen pairs to be larger 共due to internal evolution that is not perfectly refocused兲. See Appendix C for
more details on the experiment and simulations using liquid
state NMR QIP.
col 2
兩 典 for the various pairs of qubits
The resulting 具兩␩a,b
shown in Table I exhibit good agreement with the predicted
ones. Notice that the largest differences between measured
and predicted appear on the pair 共1,2兲, and on the most complex gates: CNOT2, CNOT, and C12共0.4兲. This indicates that
these deviations are due to the errors expected from spurious
processes in our QIP, particularly an imperfect refocusing
during the gate sequence rather than from an imperfect
implementation of the protocol.
042328-3
PHYSICAL REVIEW A 79, 042328 共2009兲
LÓPEZ, LÉVI, AND CORY
The simulations account for well-known sources of error
in liquid state NMR QIP 共imperfect pulse design 关17共c兲兴, rf
field inhomogeneities 关2兴, and the presence of the magnetically active hydrogens in crotonic acid 关17兴兲. It is worth
mentioning that the main errors occurring in one-qubit gates
are correlated one-body errors 共this is, a one-qubit rotation
that is slightly off兲, and they do not introduce two-body errors, which are the main target of a spatial correlation analysis. This can be noticed also in the fact that the simG values
are similar to the simE values 共a change of 0.00–0.01, except
in two cases where we found 0.02 and 0.03兲. Moreover, there
is a contribution arising from T2 relaxation. However, a calculation over the theoretical propagator and the numerics
over the simulated one show a change on the order of 0.01.
Elements outside our model of the system, which would
explain further the gap between theory and experiment, are
B0 共static兲 field inhomogeneities, the presence of transients
and residual nonlinearities in the spectrometer circuitry, and
an imperfect spectral fitting of the measured signal. These
are well-known issues in liquid state NMR QIP, whose effect
falls within the 1%–2% error.
We must differentiate between the implementation errors
in the protocol 共initial state preparation, one-qubit twirling,
and readout兲, and the errors in the gate under study. The
former ones affect the accuracy of protocol, which accounts
for the measured non-null coefficients that are expected to be
zero. As discussed, rf field inhomogeneities, the presence of
hydrogen, and T2 relaxation already give an error bar ␴␩2
⬇ 0.03. There are still other sources of error mentioned
above that could make ␴␩2 larger, but within that order.
Moreover, a fiducial initial state preparation is critical to
the success of the algorithm. We can quantify this as follows.
If we call ␧0 the error in the initial state preparation, and
similarly we call ␧1 the error in the implementation of the
Clifford gates, an error propagation in the formula for the ␥’s
gives ␴␥2 ⱕ ␧20共1 + 4␥兲 + ␧21, and then it is simple to propagate
this into the formulas for the 兩␩2l 兩: for example, for a pair of
qubits a and b, we follow Eq. 共9兲 and obtain ␴␩2
= 49 冑␴␥2 + ␴␥2 + ␴␥2 . These ␧’s account for nonstatistical era
b
a,b
rors 共typically correlated one-body errors兲 and set the accuracy of the method. Given the low complexity of initial state
preparation and one-qubit operators, these ␧’s are smaller
than the errors in target operations 共a fact reflected, for example, in the gate fidelities兲. This is why even though the
theory was developed for error-free initial state preparation
and twirl, the actual implementation can still retrieve information about the target operations.
V. DISCUSSION ON THIS AND OTHER PROPOSALS
Our protocol belongs to the family of characterization
strategies based on the use of twirling to coarse-grain the
original O共24n兲 complex parameters to a poly共n兲 number of
parameters 共cf. the work by Emerson and co-workers in
关6,8兴兲. In our case, we gain detail about the process under
study 共spatial correlations兲. Note also that the required resources in our proposal are within the minimal performance
expected from a functional QIP: fiducial state preparation
and readout, and a set of 6n one-qubit gates.
One of the protocols presented by Bendersky et al. in 关10兴
also returns similar information 共although our method performs a coarse graining of the directions兲. Nevertheless, their
proposal requires more demanding resources 关although still
scaling as poly共n兲兴: a full twirl on U共D兲 and the implementation of the Ol operators.
Other characterization methods include the ancillaassisted ones in 关10,11兴 but unfortunately they have a rather
strong requirement: one or more clean error-free qubits
within the system. Contrasting with all the proposals discussed so far, the method developed by Knill et al. in 关9兴
does not require error-free stages, allowing for certain type
of errors to occur during the whole computation. Unfortunately it is not yet evident how to take their scheme beyond
one-qubit QIP 关12兴.
On a different note, we would like to point out a particular
feature of the variables 具兩␩l兩2典, which are the diagonal elements of the so-called ␹ matrix in the Ol basis 共cf. the ai in
关8兴, or the ␹m,m in 关10兴兲. When the error E acts on a short
time we can expect the coefficients to be small and Eq. 共2兲
can be taken instead as a first-order Taylor expansion of E,
where the ␩l with l ⱖ 1 play the role of a generator of the
error. This idea was originally developed in 关7兴 under the
same setting but aiming to study the generators directly led
to limitations in the error model. Nevertheless, drawing the
connection between the two opens the possibility of identifying a generator, which is essentially a Hamiltonian with
varying parameters ␩l whose dynamics we can only observe
on average through the 具兩␩l兩2典. This interpretation allows a
different insight into the dynamics of the system, as many
physical processes are better described by the action of a
stochastic Hamiltonian rather than by an operator-sum representation arising from a system+ bath picture.
In conclusion, we have presented a method to characterize
the spatial correlations occurring in a gate or process under
study, showing its potential through a liquid state NMR QIP
experiment. Second, we have pointed out the need of experimental feedback in order to arrive to a not only scalable but
also feasible protocol, as the one we introduced. Finally, we
have analyzed the relevance of implementation errors, showing the need for strategies that are not only scalable but also
robust.
ACKNOWLEDGMENTS
We thank J. Emerson, M. Silva, and L. Viola for useful
discussions, and J. S. Hodges and T. Borneman for invaluable assistance with the experiments. This work was supported in part by the National Security Agency 共NSA兲 under
Army Research Office 共ARO兲 Contract No. W911NF-05-10469.
APPENDIX A: ON THE CLIFFORD TWIRL
Consider the twirling of the map S for m qubits in the
Hilbert space HM , of dimension 2m. As shown in 关15兴, a Haar
twirl is equivalent to Clifford twirl as follows
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␳1 =
冕
U共M兲
dUU†S共U␳0U†兲U
共A1兲
PHYSICAL REVIEW A 79, 042328 共2009兲
ERROR CHARACTERIZATION IN QUANTUM-…
Km
1
= m 兺 C†k S共Ck␳0C†k 兲Ck ,
K k=1
共A2兲
where dU denotes the Haar measure on the unitary group
U共2m兲. The sum on Eq. 共A2兲 runs over all the elements of the
Clifford group for U共2m兲. In particular, this equation holds
for a twirl on U共2兲 丢 m, where the Ck are m-fold tensor products of the K = 24 elements of the Clifford group for one
qubit, for the m qubits in HM .
We can write the Clifford operators for one qubit as
products SP of an element S of the symplectic
group
兵exp关−i␯共␲ / 3兲共␴x + ␴y + ␴z兲 / 冑3兴 , ␯ = 0 , 1 , 2 ;
exp关−i共␲ / 4兲␴ p兴 , p = x , y , z其 and an element P of the Pauli
group 兵I , ␴x , ␴y , ␴z其. Due to some redundancy in the symplectic twirling, the minimum number of elements to achieve
the two-design property 关Eq. 共A1兲 and 共A2兲, as defined in
关15兴兴 is actually half of the whole Clifford group: S can be
taken from either S1 = 兵exp关−i␯共␲ / 3兲共␴x + ␴y + ␴z兲 / 冑3兴 , ␯
= 0 , 1 , 2其 or S2 = 兵exp关−i共␲ / 4兲␴ p兴 , p = x , y , z其.
This number can be further decreased to K = 6 if we take
an initial state ␳0 as described in the main text 关Eq. 共3兲兴 and
we remark that we will be only interested in the projection of
␳共M兲
onto given initial states of the form ␳共j兲
1
0 = 共I + ␴z兲 / 2.
There is then an additional redundancy in the values of these
projections which enables us to chose P in the set 兵P1 , P2其
with P1 = I or ␴z and P2 = ␴x or ␴y. Here we have chosen ␳共j兲
0
along z for definiteness 共which is what we use in the experi-
冉
col 2
2
␥共a兲 = Aa 具兩␩col
a 兩 典 + 兺 具兩␩a,j 兩 典 +
␥共a,b兲 =
8PaPb + 2共Pa + Pb兲 − 4
冉
9
2
+ PaAb 具兩␩col
b 兩 典+
兺
j⫽a,b
j⫽a
冉␩
具兩
col 2
a,b兩 典
2
具兩␩col
b,j 兩 典 +
+
兺
j⫽a,b
兺
兺
k⬎j⫽a
ment兲 but an equivalent result can be obtained for the x or y
directions. Notice that the number of elements K required in
Eq. 共A2兲 depends on what quantity we will measure. In
U共2兲 丢 m 共as it is in our case兲, whether some elements are
redundant can be proved simply by straightforward calculation.
Implementing a Haar twirl as defined on Eq. 共A1兲 would
require sampling over an infinite pool of random rotations
Uk. The equivalence with the Clifford twirl allows us to
implement the twirling with a finite set of gates: the Km
Clifford operators in U共D兲. Nevertheless, although not infinite, the pool is of exponential size in m, thus again calling
for a sampling strategy to implement the twirl approximately.
Of course, for m small enough, it may be possible to implement the twirl exactly 共as in our experiment, where m = 2 and
the Clifford pool has size 36兲. But on general grounds we can
take on any of the two approaches.
Depending on the experimental setup in question, one
twirl may be more robust than the other 共depending on what
type of errors are expected for one-qubit gates兲, and also one
pool may be easier to construct than the other.
APPENDIX B: COMBINING ␥(a), ␥(b), AND ␥(a,b)
2
TO RETRIEVE Š円␩col
a,b円 ‹
To illustrate the mechanism of combining different ␥’s,
consider measuring one and two qubits as follows. With A j
= 共4P j − 2兲 / 3:
2
具兩␩col
a,j,k兩典 + ¯
冊
冊
冉
and similarly ␥共b兲 ,
冉
col 2
2
具兩␩col
a,b,j兩 典 + ¯ + AaPb 具兩␩a 兩 典 +
k⬎j⫽a,b
兺
j⫽a,b
兺
2
具兩␩col
a,j 兩 典 +
k⬎j⫽a,b
2
具兩␩col
a,j,k兩 典 + ¯
冊
2
具兩␩col
b,j,k兩 典 + ¯ ,
4
where . . . denote the corresponding higher order multi-body
terms. If we prepare the qubits in a pure state so Pa = Pb = 1,
the combination
␥共a兲 + ␥共b兲 − ␥共a,b兲 =
冊
4
col 2
2
具兩␩a,b
兩 典 + 兺 具兩␩col
a,b,j兩 典 + ¯
9
j⫽a,b
冊
leaves only the collective coefficients involving both the qubits a and b.
APPENDIX C: DETAILS ON THE EXPERIMENTAL
IMPLEMENTATION
The internal Hamiltonian of the system in the rotating
frame is given by
Hint = ប 兺
j=1
4
␲J j,k 共j兲 共k兲
␻␦,j 共j兲
␴ +ប 兺
␴z ␴z ,
2 z
k⬎j=1 2
共C1兲
where the chemical shifts, at our 9.4 T spectrometer, are of
the order of kHz: ␻␦,1 = 6650.6 Hz, ␻␦,2 = 1695.8 Hz, ␻␦,3
= 4210.0 Hz, and ␻␦,j = −8796.7 Hz. The J-couplings are
J12 = 72.6 Hz, J23 = 69.8 Hz, and J14 = 7.1 Hz while J24
= 1.6 Hz, J13 = 1.3 Hz, and J34 = 41.6 Hz 共according to the
characterization of the sample we used; see also 关17兴兲.
The experimental initial state preparation over the four
qubits reported a correlation with the theoretical one that was
on average 0.99 共0.98 the lowest兲. The correlation for the
targeted qubits 共a pair of qubits兲 was, in each case, between
1.00–0.99 共the pseudopure state preparation was designed to
optimize this correlation兲.
042328-5
PHYSICAL REVIEW A 79, 042328 共2009兲
LÓPEZ, LÉVI, AND CORY
We implemented the twirl of pairs of qubits exactly using
36 Clifford operators. We had to pick one of the eight available six-element subsets of Clifford operators for each qubit,
and we chose the subset that performed best experimentally:
we applied each candidate to the thermal equilibrium state
and compared the experimental performance with the theoretical one. This criterion coincided with choosing the subset
that best took the equilibrium state to the I / 2 state for the
qubit being twirled.
To perform QST on four qubits with a liquid state NMR
QIP, we used a set of 18 readout pulses. Therefore the number of experiments required to measure one collective coefcol 2
兩 典 for a given pair of targeted qubits 共a , b兲 for a
ficient 具兩␩a,b
particular gate under study was 648, plus 18 experiments to
characterize the initial state 兩00典具00兩共I / 2兲 丢 2 corresponding to
preparing that pair in a pseudopure state. We performed QST
of the full system therefore having a broader knowledge of
the experimental performance but this is not required by the
protocol: only the target qubits must be measured.
The negligibility of higher order multibody terms in the
gates under study is to be expected in liquid state NMR QIP.
In a simple model, these gates consist basically of periods of
free evolution of length ␶ 关the corresponding propagator is
U␶ = exp共−iHint␶ / ប兲兴 separated by ␲-pulses on some of the
j
兴, already accounting for
qubits 共so U␲ = exp关−i共␲ / 2 + ⑀兲␴x,y
some error ⑀兲. For example, the sequence for the gate IE is
2
3,4
1,4
3,4
2
3,4
1,4
共␶ − ␲兲3,4
x − 共␶ − ␲兲x − 共␶ − ␲兲x − 共␶ − ␲兲x − 共␶ − ␲兲−x − 共␶ − ␲兲−x − 共␶ − ␲兲−x − 共␶ − ␲兲−x ,
where ␶ denotes free evolution for a time ␶, and 共␲兲 pj denotes a ␲ pulse 共180° rotation兲 around the p axis for the qubits j.
Using the Baker-Campbell-Hausdorff 共BCH兲 formula 关19兴 is straightforward to see that in the building block U␶U␲,
␶
three-body and higher order terms will appear with a factor at least ␲J/2
smaller with respect to any possible one-body and
␶
two-body terms. For the values of J j,k␶ involved in our experiment, we have ␲J/2
ⱕ 0.14. This means that any possible
three-body and four-body terms would appear with a coefficient ten times smaller than the ones for one-body and two-body
terms.
On the other hand, the simulation of the engineered pulse sequences used in the experiment showed that all the three-body
col
2
2
and four-body terms appear with collective coefficients 兩␩col
j,k,j⬘兩 , 兩␩ j,k,j⬘,k⬘兩 ⬍ 0.005 for the CNOT and C12共0.4兲 gates, and
col 2
兩 , which can
⬍0.002 for the rest. These are much smaller than the differences between measured and predicted values of 兩␩a,b
be better explained as implementation errors in the protocol or genuine gate errors arising from imperfect refocusing.
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