We consider a qubit subject to various independent control mechanisms and
present a general strategy to identify both the internal Hamiltonian and the
interaction Hamiltonian for each control mechanism, relying only on a single,
fixed readout process such as $\sigma_z$ measurements.Comment: submitted to Proceedings of the QCMC04 (4 pages RevTeX, 5 figures

Cole, Jared H.

Kolli, Avinash

Oi, Daniel K. L.

Schirmer, Sonia G.

English

arXiv

Sonia G. Schirmer

Crossref

Experimental Hamiltonian Identification for Qubits subject to Multiple Independent Control Mechanisms

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Experimental Hamiltonian Identification for Qubits subjec t
to Multiple Independent Control Mechanisms.
Sonia G. Schirmer∗, Avinash Kolli∗, Daniel K. L. Oi∗ and Jared H. Cole†
∗Dept of Applied Maths and Theoretical Physics, University of Cambridge,
Wilberforce Rd, Cambridge, CB3 0WA, United Kingdom
†Centre for Quantum Computer Technology, School of Physics, University of Melbourne, Melbourne, Australia
Abstract. We consider a qubit subject to various independent control mechanisms and present a general strategy to identify
both the internal Hamiltonian and the interaction Hamiltonian for each control mechanism, relying only on a single, fixed
readout process such asσ̂z measurements.
1. INTRODUCTION
Realizing the ultimate goal of quantum information processing, namely building a working quantum computer, is to a
large extent a problem of finding ways to control the dynamicsof a quantum system precisely. A crucial prerequisite
for this task is one’s ability to accurately determine of thedynamics of the physical system and its response to external
(control) fields. Quantum process tomography (QPT), by providing a general procedure to identify the unitary (or
completely positive) processes acting on a system, addresses thi problem but does not solve it completely.
One problematic aspect of QPT is the assumption that one can experimentally determine the expectation values
of a complete set of observables, or at least performa bitrary single qubit measurements on a register ofn qubits.
Most QIP proposals rely on a single readout process, i.e., measur ment in a fixed basis. For example, qubits encoded
in internal electronic states of trapped ions or neutral atoms are read out by quantum jump detection via a cycling
transition; readout for solid-state qubits based on Cooper-pair boxes, Josephson junctions, or electrons in double-well
potentials usually involves charge localization using single electron transistors or similar devices. Finally, solid state
architectures based on electron or nuclear spin qubits are exp cted to be limited toσz measurements via spin-charge
transfer.
It is usually assumed that local projective measurements ina fixed basis are sufficient since arbitrary single qubit
measurements can then be realized by performing a local unitry transformation before measuring to achieve a change
of basis. However, implementing such a basis change requires precise knowledge of the dynamics of each individual
qubit and its response to control fields in the first place, thevery information we seek to determine experimentally, and
which may be difficult topredict precisely based on theoretical models and computer simulations alone, in particular
for systems that are sensitive to fabrication variance. A possible solution to this seemingly intractable problem was
described in [1]. In the following we outline the general strategy, discuss various ways of extracting the system
parameters from noisy experimental data, and illustrate the key steps using examples with simulated measurement
data.
2. GENERAL STRATEGY FOR HAMILTONIAN IDENTIFICATION
The state of a two-level system can be mapped to a Bloch vectors, i.e., a real vector inR3, with pure states
corresponding to points on the Bloch sphere, i.e., the surface of the unit ball. On timescales sufficiently short compared
to the decoherence time, the evolution of the system is governed by a Hamiltonian, which can be written in terms of
the Pauli matriceŝσ∗ for ∗ ∈ {x,y,z}, 2Ĥ = d0Î+dxσ̂x +dyσ̂y +dzσ̂z, whered0, dx, dy anddz are real constants. If the
Hamiltonian remains constant fort0 ≤ t ≤ t1, the Bloch vector undergoes a rotation about the axisd = (dx,dy,dz).
FIGURE 1. Left: Declinationsθ and relative azimuthal anglesφ for two rotation axes. Right: Determination of the rotation
frequency and angleθ of the rotation axis by mapping the precession of the state|0〉.
The length||d|| of this vector determines rotation frequency; the rotationaxis can be specified by a unit vector
d̂ = (sinθ cosφ ,sinθ sinφ ,cosθ )T , i.e., by two anglesθ andφ as shown in Fig. 1. To identify the parametersdx,
dy anddz of the Hamiltonian it therefore suffices to determine the rotati n frequency and the anglesθ andφ . Sinced0
results only in an unobservable global phase factor, it can be ignored.
If the system can be repeatedly initialized in a known state,e.g., one of the measurement basis states|0〉 or |1〉,
and then measured after having evolved for progressively longer time periods, we can map the trajectory of thez-
component of the Bloch vector, and extract the frequency andangleθ of the rotation as shown in Fig. 1 (right). For
some systems such as NMR-based schemes this may be sufficientas the phase relationship betweendx anddy is fixed
by the phase relationship between the control fields. In general, however, a second series of measurements is necessary
to determine the horizontal angleφ of the rotation axis with respect to a reference axisdr. This procedure is given in
Ref. [1] and involves using the values ofθ and||d|| determined in the first step to select a different initial state, nd
mapping its precession about the desired rotation axis.
For a system subject to various control fieldsfm (e.g., fields associated with different control electrodes) in addition
to its free evolution, we must determine both its internal HamiltonianĤ0 and the interaction Hamiltonian̂Hm for each
independent control mechanism. Although we usually cannotdetermine the interaction Hamiltonians directly since
we cannot switch off the internal dynamics, we can identify the rotation axisd(k)0+m = d0 + f
(k)
m dm corresponding to
the evolution of the system under the HamiltonianĤ0 + f
(k)
m Ĥm for a fixed control settingfm = f
(k)
m . Repeating this
procedure for each available control fieldfm with several control field settingsf
(k)
m then allows us to extract both the
internal and interaction parts of the Hamiltonian providedthat the dependence of the Hamiltonian on the control fields
is linear. (Nonlinear field effects require additional correction terms, and we will exclude this case in this paper.)
3. MAPPING z(t) AND EXTRACTING THE RELEVANT DATA
A crucial factor in the Hamiltonian identification strategyoutlined above is the mapping of the evolution ofz(t) as
the system precesses around a fixed rotation axis. The accuracy with which we can identify the relevant parameters
such as the rotation frequency and angleθ depends on the total length of timet f for which z(t) is mapped, the time
resolution∆t and the uncertainty of each data pointz(tk) = 〈σz(tk)〉, which depends on the number of timesNe each
experiment is repeated to obtain the ensemble average (in add tion to the frequency of measurement errors etc). The
total number of measurements required to mapz(t) is thusNT = Net f /∆t. The choice of∆t, t f andNe will depend both
on the system to be characterized and the method of data extraction to be used.
One possible approach is to use a small time step∆t and a large number of repetitionsNe to obtain a dense set of
accurate data points for a period of time covering at least one quarter of the rotation period, and fit a cosine segment
to the data. Since the time we have to monitor the evolution ofz(t) is limited by the decoherence time of the system,
this approach may be useful for systems that decohere rapidly because it requires only sampling over a short period of
time.
0 50 100 150 200 250 300
−0.5
0
0.5
1
Time (arb. units)
<
σ z
 >
 (
av
e.
 o
ve
r 
10
 m
ea
s.
)
280 285 290 295 300
0
10
20
30
40
Total sampling time t
f
 (arb. units)
P
(t
f )
t
F
 = 288.50, P(t
F
 ) = 33.89
0 0.2 0.4 0.6 0.8
0
0.1
0.2
0.3
0.4
0.5
Frequency (arb. units)
F
ou
rie
r 
co
ef
f. 
|F
t F
(ω
n)
|
||d
0+2
(2) || = 0.28
Declination θ = 0.80
t
F
 
Figure 2: The top graph to the left shows an
example of noisy (simulated) measurement
data for rotations aboutd0 + f
(2)
2 d2 with
f (2)2 = 0.1. The dash-dot line indicates the
optimal sampling time tF determined by
finding the maximum of the functionP(t f )
(bottom-left). The Fourier transform of the
(truncated) data for 0≤ t ≤ tF is shown in
the bottom-right graph. Its zero-frequency
component determinesθ = arccos
√
|F(0)|;
and the location of the second peak gives the
rotation frequencyω = ||d0 + f
(2)
2 d2||. The
estimates obtained,θ = 0.80 andω = 0.28,
are close to the actual valuesθ = 0.7854 and
ω = 0.2828.
Another approach, explained in detail in [1], is to use relatively coarse time sampling, with a moderate number
of repetitions for each data point, over several rotation periods to obtain a rough estimate of the rotation frequency
using the discrete Fourier transform, followed by a second step of acquiring more accurate additional data points in
the region where the first minimum ofz(t) is expected based on the first estimate, and fitting a parabolato the new
data to find the rotation frequency and declination of the rotati n axis.
A third alternative [2] is to eliminate curve-fitting altogeth r and extract all the required information directly from
the Fourier transform. This method is rather elegant and does n t require the high time resolution∆t and measurement
repetition ratesNe usually necessary for curve-fitting methods. All we need to avoid aliasing is that∆t be less than half
the rotation periodT . Ensuring that this condition is satisfied requires a rough apriori estimate ofT but this should
normally not be a problem. It also permits easy estimation ofthe accuracy of the parameters. However, since the
frequency resolution∆ω = 2π/t f , we must be able to map the data for a least two complete cyclesto be able to extract
the rotation frequency, and more cycles will be required to obtain a clearly defined peak in the frequency spectrum.
Hence, this method will be most suited to systems whose decoherence time is sufficiently long to allow mapping of
the evolution ofz(t) over several cycles.
4. ILLUSTRATIVE EXAMPLE
To demonstrate the procedure, we choose a test system withd0 = (0.2,0,0.1)T , d1 = (1,1,0)T andd2 = (0,0,1)T .
Fig. 2 illustrates how we identify the rotation frequenciesand declination angles forf1 = 0 and f2 = 0.1 following the
3rd approach outlined in Sec. 3. We samplez(t) over several rotation periods with an intermediate time step (∆t = 0.25)
and a small number of repetitions (Ne = 10) for each measurement. We then obtain an estimate of the rotation frequency
by taking the Fourier transform of the data and finding the frequencyωp such that|F(ωp)| = maxn>0 |F(ωn)|, where
F(ωn) is thenth Fourier coefficient. Since the sampling period is usuallynot an integer multiple of the rotation period,
the peak in the Fourier spectrum will tend to be unsharp, and our estimate inaccurate. To improve it, we compute
the functionP(t f ) = [Ft f (ωp)−Ft f (ωp−1)−Ft f (ωp+1)]/[Ft f (ωp−1)+Ft f (ωp+1)] whereFt f is the Fourier transform
of the truncated data for 0≤ t ≤ t f andωp is the frequency where the first peak in the spectrum (excl.F(0)) occurs,
|Ft f (ωp)| = maxn>0 |Ft f (ωn)|. P(t f ) assumes a maximum whent f is an integer multiple of the rotation period, thus
allowing us to find the optimal sampling timetF . Fig. 3 illustrates how we can find the horizontal anglesφ , having
identified the rotation frequencies and anglesθ for all control settings, using a local curve fitting approach similar to
the second strategy outlined in Sec. 3 and described in Ref. [1]. Finally, Fig. 4 shows how we can extractd0 anddm
by plotting thex, y andz-components of the rotation axesd0+ f
(k)
m dm versusf
(k)
m for m = 1,2 and fitting straight lines
to the data.
Figure 3: The system is initialized in states1 =
(cosβ ,sinβ ,0)T (here β = −1.0446) by rotatings0 =
(0,0,1)T about the reference axisdr (hered0) by a suitable
angleψ , and the precession ofs1 aboutd0+ f
(1)
1 d1, whose
frequency andθ are already known, is mapped. Using a
small number of data points (stars) we find thex-intercept
of z(t) (circle), which allows us to estimate the location of
the extrema ofz(t). Additional data points (dots) are then
acquired in these regions, and parabolas fitted to the data.
The desired angleφ =−β −arcsin(γ cosδ/sinθr), where
γ = (zmax − zmin)/2 andδ = π − (αmin +αmax)/2 are de-
termined by the vertices of the parabolas, and the valuesβ
andθr (known from part 1). We obtainφ = 0.34, which is
close to the actual valueφ = 0.3218.
0 1 2 3 4 5 6 7
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Angle α of 2nd rotation about  d
0
+f
1
(1)  d
1
<
σ z
>
 (
av
er
ag
e 
ov
er
 4
00
 m
ea
su
re
m
en
ts
)
z
min
=−0.50
α
min
=1.27
z
max
=0.93
α
max
=4.39
0 0.1 0.2 0.3 0.4 0.5
0
0.2
0.4
0.6
0.8
1
f
1
 (arb. units)
 d
0+
 f 1
  d
1
x−comp.
y−comp.
z−comp.
0 0.1 0.2 0.3 0.4 0.5
−0.5
0
0.5
1
f
2
 (arb. units)
 d
0+
 f 2
  d
2
x−comp.
y−comp.
z−comp.
Figure 4: Having determined the rotation frequen-
cies ω(k)0+m and the anglesθ
(k)
0+m and φ
(k)
0+m of the
rotation axesd0 + f
(k)
m dm for various values of the
controls f1 and f2, we convert the data into Carte-
sian coordinates, plot the values of thex, y andz-
components of the axesd0+ f
(k)
m dm for m = 1,2, re-
spectively, and fit straight lines. They-intercepts de-
termine thex, y andz-components ofd0, the slopes
those ofdm, for m = 1,2. For the data shown, we
have
dest0 = (0.1986,0.0048,0.0979)
T
dest1 = (0.9884,1.0163,0.0087)
T
dest2 = (0.0531,0.0246,0.9819)
T
The distances||destm − d
act
m || of 0.0054, 0.0218 and
0.0613, respectively, compare favorably to the 3%
readout error rate of the simulated experiments.
ACKNOWLEDGMENTS
We thank A. D. Greentree and L. C. H. Hollenberg for helpful discussions. S.G.S and D.K.L.O acknowledge financial
support from the Cambridge-MIT Institute, Fujitsu, the UK goverment and IST grants RESQ (IST-2001-37559)
and TOPQIP (IST-2001-39215). J.H.C acknowledges the support of the Australian Research Council, the Australian
government, the US National Security Agency, The Advanced Rsearch and Development Activity and the US Army
Research Office (DAD19-01-1-0653). D.K.L.O also thanks Sidney Sussex College for support.
REFERENCES
1. Schirmer, S. G., Kolli, A., and Oi, D. K. L.,Phys. Rev. A, 69, 050603(R) (2004).
2. Cole, J. H.,et al., In preparation.


We consider a qubit subject to various independent control mechanisms and present a general strategy to identify both the internal Hamiltonian and the interaction Hamiltonian for each control mechanism, relying only on a single, fixed readout process such as o - z measurements

Schirmer, S

Kolli, A

Oi, D

Cole, J

RMIT Research Repository

Experimental Hamiltonian identification for qubits subject to multiple independent control mechanisms

Experimental Hamiltonian Identification for Qubits subject to Multiple
  Independent Control Mechanisms

Experimental Hamiltonian Identification for Qubits subject to Multiple Independent Control Mechanisms

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