PHYSICAL REVIEW LETTERS
PRL 97, 050504 (2006)
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Limitations of Quantum Simulation Examined by Simulating
a Pairing Hamiltonian Using Nuclear Magnetic Resonance
Kenneth R. Brown, Robert J. Clark, and Isaac L. Chuang
Center for Bits and Atoms, Research Laboratory of Electronics, and Department of Physics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA
(Received 4 January 2006; published 4 August 2006)
Quantum simulation uses a well-known quantum system to predict the behavior of another quantum
system. Certain limitations in this technique arise, however, when applied to specific problems, as we
demonstrate with a theoretical and experimental study of an algorithm proposed by Wu, Byrd, and Lidar
[Phys. Rev. Lett. 89, 057904 (2002).] to find the low-lying spectrum of a pairing Hamiltonian. While the
number of elementary quantum gates required scales polynomially with the size of the system, it increases
inversely to the desired error bound . Making such simulations robust to decoherence using fault
tolerance requires an additional factor of 1= gates. These constraints, along with the effects of control
errors, are illustrated using a three qubit NMR system.
DOI: 10.1103/PhysRevLett.97.050504
PACS numbers: 03.67.Lx, 71.10.Li, 76.60.k
The unknown properties and dynamics of a given quantum system can often be studied by using a well-known and
controllable quantum system to mimic the behavior of the
original system. This technique of quantum simulation is
one of the fundamental motivations for the study of quantum computation [1–3] and is particularly of interest because it offers the possibility of solving computationally
hard problems without requiring the resources necessary
for algorithms such as factoring [4] and searching [5].
Experimental results have demonstrated simulations using
a nuclear magnetic resonance (NMR) quantum computer
[6 –11], which allows for the efficient implementation of
the quantum dynamics [12,13] despite being unable to
produce entangled states at low polarization [14]. Interest
has also extended to simulating complex condensed matter
systems with quantum optical systems [15,16].
Often overlooked in the discussion of quantum simulations, however, is the question of desired precision (or error
) in the final measurement results. Current quantum simulation techniques generally scale poorly with desired precision; they demand an amount of space or time which
increases as 1=, broadly translating into a number of
quantum gates which grows exponentially with the desired
number of bits in the final answer. Why is this scaling
behavior so poor, and what is its physical origin?
Consider as a specific example the problem of calculating the energy gap between the ground state jGi and the
first excited state jE1 i, of a Hamiltonian H. The gap can
be found using the following steps: (i) Map the Hilbert
space of the system to be simulated to n qubits, (ii) prepare the computer in the state jI i cG jGi cE jE1 i,
(iii) evolve under the Hamiltonian for times ti , and
(iv) extract the phase difference as a function of time
between the evolution of the ground and first excited states.
Two methods exist for calculating the phase difference
and, thus, . The first method uses the phase estimation
algorithm [17,18], which relies on the quantum Fourier
transform (QFT). The second method instead simulates the
0031-9007=06=97(5)=050504(4)
Hamiltonian for times tk kt0 , for integer k from 0 to Q,
and then measures any operator M such that hGjMjE1 i 0. After calculating hMtk i, one classically Fourier transforms (FT) over the averaged values, yielding a spectrum
hM!i with peaks at and 0.
The challenge of designing efficient quantum simulations is choosing a property that can be efficiently extracted. No general measurement method is known which
allows to be measured efficiently with respect to the
precision using such quantum simulations. For error , the
number of digits of precision in the result is log1=;
1= steps (or gates) are required to obtain this precision.
In contrast, an efficient algorithm would require only a
number of steps polynomial in log1=. The origin of this
limitation lies not only in the inability to design efficient
measurements but also in the accumulation of errors which
occurs in the course of performing a quantum simulation.
Here we consider these limitations in the context of a
specific algorithm for the simulation of pairing models, as
proposed by Wu, Byrd, and Lidar (WBL) [19], which
follows the framework of the two methods described
above. We present a study of the errors in its discrete
time step implementation and experimental results from a
realization using a 3 qubit NMR quantum computer, answering three questions: (i) What are the theoretical
bounds on the precision of the quantum simulation?
(ii) How do faulty controls affect the accuracy of a simulation? (iii) Can the theoretical bounds on precision be
saturated by an NMR implementation?
The WBL algorithm uses the classical FT algorithm
described above to compute the low-lying energy gap in
pairing Hamiltonians, which are used to describe both
nuclear dynamics and superconductivity. WBL map a pairing Hamiltonian onto the qubit Hamiltonian
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Hpair n
X
X Vml
m
Zm Xm Xl Ym Yl :
m1 2
m<l 2
© 2006 The American Physical Society
PRL 97, 050504 (2006)
PHYSICAL REVIEW LETTERS
The number of modes that can be simulated equals n,
and the number of pairs equals the total number of qubits in
the state j1i. WBL show that, for a specific number of pairs,
one can approximately prepare the state jI i by quasiadiabatic evolution. Since Hpair is a 2-body Hamiltonian, the
system’s evolution can be efficiently simulated on a quantum computer for any number of qubits [2]. WBL propose
implementation of their algorithm using an NMR quantum
computer, in which the operator M is simply Z for a single
spin. An advantage of the ensemble nature of NMR is that a
measurement at simulated time t yields the ensemble average hMti. Fixing a maximum energy width and desired
precision makes the FT independent of the number of
qubits.
Let us begin by addressing the first question posed
above: How does the number of gates scale with the error
? The WBL method requires constructing an operator that
approximates the simulated Hamiltonian for times tk . The
FT then yields an error of 2Emax =Q, where Emax is the
largest detectable energy 1=t0 (@ 1). Given that simulation for time Qt0 generally takes Q times longer than
simulation for time t0 , the number of gates scales inversely
with the error. A similar problem faced in Shor’s algorithm
is overcome by a clever way to perform the modular
exponentiation [4].
A second bound on the number of gates required arises
in calculating the time required to perform the algorithm.
Quantum simulations typically employ a Trotter formula to
approximate a Hamiltonian from combinations of noncommuting Hamiltonians [18]. For example, given the
ability to evolve under Hamiltonians HA and HB , one can
approximate evolution under HA HB with bounded error.
To lowest order, expitHA HB expitHA =k expitHB =kk , where for k HA ; HB k t2 1, the
error is Ot2 =k. Higher-order techniques can yield an
error Otm1 =km at the cost of needing O2m more gates
[20]. The Trotter formula evidently demands an exponential increase in the number of discrete gates for an exponential decrease in the error. However, the total time
needed to implement UA t=k expitHA =k is 1=k
the time needed to implement UA t. Therefore,
‘‘Trotterization’’ of UA t requires only time 2t, independent of k.
Unfortunately, the fault-tolerant implementation of
UA t=k [21,22] takes approximately the same amount of
time as the gate UA t, whether using teleportation [23] or
the Solovay-Kitaev approximation [24]. Consequently,
fault-tolerant simulations using the Trotter formula and
the FT-QFT require a number of gates and amount of
time that scales as 1=r , where r 2.
We turn now to the second question, which concerns the
impact of faulty controls in a real physical implementation
of the WBL algorithm. Recall that the foundation of the
WBL algorithm is approximation of the unitary evolution
under Hpair , Upair qt0 expiHpair qt0 . An ideal NMR
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implementation accomplishes this by a repeatable pulse
sequence Vpair t0 , where Upair qt0 Vpair t0 q . The
Hamiltonian Hpair contains three noncommuting parts:
P
P
H0 P
m m =2Zm , HXX m<l Vml =2Xm Xl , and
HYY m<l Vml =2Ym Yl . Assuming that the corresponding unitary operators U0 t, UXX t, and UYY t can be
implemented, Vpair t0 can be constructed using the thirdorder Trotter-Suzuki formula [20,25]
Vpair t0 U0 t0 =2kUXX t0 =2kUYY t0 =k
UXX t0 =2kU0 t0 =2k
k ;
yielding an expected error k Upair t0 Vpair t0 k
Ot30 =k2 .
^ the unitary evolution of a
In a static magnetic field B0 z,
typically used spin system P
[10,26] in the rotating frame is
given by UZZ t expi ij 2 Jij Zi Zj t. The time t required for an rf pulse to rotate individual spins by radians
is much smaller than the typical delay times td during
which no rf is applied, t td 1=Jij . Thus, ordinarily,
the rf pulses are approximated as functions in time,
implementing perfect single qubit rotations Ri expi2 Xi cos Yi sin
. This approximation breaks
down as td becomes comparable to t , leading to control
errors which, for a small number of qubits, can be mitigated using optimal control techniques [9,27].
The impact of such control errors in an NMR implementation of the WBL algorithm can be studied by comparing a baseline realization with no control error compensation (denoted W1) versus another with simple, scalable error compensation (denoted W2). Method W1 implements U0 using composite
pulses to create rotations about
Q
m Rm
the z^ axis, U0 m Rm
=2R
m =2 =2. UXX
0
=2
and UYY are generated by applying single qubit pulses to
rotate the scalar coupling from the z^ axis to the x^ and y^
axes, using the quantum circuit in Fig. 1. Control errors in
W1 are small only when delays needed to generate UXX are
long compared to the time required to perform single qubit
gates but also short enough that the Trotter error is small.
Method W2 accounts for unwanted two qubit coupling
during single qubit gates by reducing delay times during
which coupling is desired. Specifically, every instance of
FIG. 1. Three qubit quantum circuit for the unitary UXX ,
implemented using method W1 and simulating H2 (see text).
UYY is done by replacing rotations about y^ with rotations about
^ For H1 , 2 pulses are applied to qubit c in parallel with those on
x.
a and b, and the decoupling X pulse is omitted.
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R1 1 UZZ tR2 2 is replaced with R1 1 UZZ t R2 2 , where t =21 2 .
Numerical simulations comparing W1 and W2 show that
the effect of such control errors on the WBL algorithm is a
shift in the estimated gap value from the expected value.
This shift can be quite significant, as shown in Fig. 2, and
indeed can dominate other errors. Compensating for unwanted scalar couplings in NMR implementations of quantum simulations is thus vital for obtaining correct results;
implementations with other physical systems will similarly
have to deal with faulty controls.
Finally, we consider the third question: saturation of
these predicted precision bounds. An NMR system is
described by a characteristic decoherence time ; it is
convenient to work with a small number of qubits for times
shorter than 3, such that quantum error correction is
unnecessary. We implement two specific instances of the
pairing Hamiltonian Hpair with n 3 and with two pairs,
leading to a 3-dimensional Hilbert space spanned by j101i,
j110i, and j011i. WBL estimated the expected size of the
system that could be simulated without error correction by
choosing k=t0 0:1 and FT . They found that the
number of gates required scales as 3n4 =FT , including the
necessary decoupling pulses. The gate time tg is assumed
equal to 105 , and for up to n 10, can be found to
precision . Here we find to precision =100
with n 3, consistent with the WBL bound n 4 for
these parameter choices.
Preparation of our chosen initial state j011i and adiabatic evolution into jI i are done according to a procedure previously demonstrated [28], but with the discrete
step adiabatic evolution sped up according to WBL [19],
thereby exciting both jGi and jE1 i of Hpair . The
Hamiltonian used at each discrete step s is Had s 1 s=SH0 s=SHpair . We find S 4 total steps, and
time steps of tad 1=700 s are sufficient. Note that for
150
Intensity (arb. units)
W2
Exact
W1
100
50
0
300
400
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PHYSICAL REVIEW LETTERS
PRL 97, 050504 (2006)
500
Frequency (2π Hz)
600
k HXX HYY kk H0 k there can be a phase transition
as s is changed [29]; as the gap goes to zero at the phase
transition, this can be problematic, since the number of
steps required for successful quasiadiabatic evolution
grows inversely with the gap.
The second stage of the algorithm is evolution of the
state jI i under the pairing Hamiltonian for Q time steps
of duration t0 with k 2. These parameters are chosen
such that 1=Qt0 =100, Qt0 < , and k=t0 > 0:1.
Note that many Q and t0 yield the same FT ; this is used
to our advantage below.
We performed our experiments using a 500 MHz Varian
UNITY INOVA spectrometer and 13 C-labeled CHFBr , with
2
coupling strengths JHC 224 Hz, JHF 50 Hz, and
JCF 311 Hz. The two pairing Hamiltonians simulated
were H1 , in which V12 JHC , V13 JHF , and V23 JCF , and a harder case H2 , in which Vab JHC and
Vac Vbc 0. For both Hamiltonians, 1 150 Hz,
2 100 Hz, and 3 50 Hz. The experimental result exp was determined by a least-squares fit of the
highest-signal 1 H NMR peak to a damped sinusoid with
frequency exp and decay rate 1=e .
Ideally, the result should find 218 2 Hz for H1
and 452 2 Hz for H2 , as determined by direct
diagonalization. Note that, for H2 , is the energy difference between jGi and jE2 i, since jE1 i is not connected by
usual adiabatic evolution. For the experimental result (see
Fig. 2 and Table I), we expect that exp sys FT ,
where sys is an offset due to Trotterization and/or faulty
controls.
The impact of systematic and random errors was investigated by simulating H1 with W1 for FT 2:5 2 Hz
at two different simulation times t0 1 ms and t0 2 ms.
As expected, the random error for both cases is FT .
Note that the systematic error increases with smaller t0 .
This signals that the error due to unwanted scalar coupling
becomes larger than the errors due to the Trotter approximation. Consequently, a slightly longer t0 yields a systematic error that is within FT of the exact answer, saturating
the predicted theoretical bounds on precision.
While the results for Hamiltonian H1 were good even
without control error compensation, the effects of control
errors were very evident in the results for H2 . Hamiltonian
H2 was implemented using both W1 and W2 for FT TABLE I. Experimental results for gaps found for
Hamiltonians H1 and H2 . Estimated gaps (exp ) and effective
coherence times (e ) for given time steps t0 and number of steps
Q are given.
700
FIG. 2. Frequency-domain spectra of Hamiltonian H2 obtained
using methods W1 (marked by circles) and W2 (diamonds). The
solid lines are fits to the time dependent data. The width of the
exact curve is taken to be the dephasing rate (1=) of the 13 C
nucleus.
Model
=Hz
H1
218 2
H2
452 2
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Method exp =2 Hz e =ms t0 =ms Q
W1
W1
W1
W2
227 2
220 2
554 10
440 5
180
250
30
80
1
2
0.5
0.5
400
200
200
200
PRL 97, 050504 (2006)
PHYSICAL REVIEW LETTERS
10 2 Hz and t0 0:5 ms. The shorter time step was
necessary because the larger made the simulation more
sensitive to Trotter errors. Comparing the W2 and W1
results shows that, with no control error compensation, a
gap exp is found that is =5 away from the actual value. In
contrast, with simple error compensation exp is FT from
the actual value. Future implementations should strive to
detect and bound control errors by verifying that exp
converges as t30 for small values of t0 , as theoretically
expected.
In conclusion, we have shown that an NMR implementation of WBL’s algorithm, using the smallest problem
instance that requires both adiabatic evolution and
Trotterization, saturates the theoretical bounds presented
in this Letter. We have also found, however, that simulations of this type are particularly sensitive to systematic
errors in the applied Hamiltonian and that fault-tolerant
implementations are inefficient with respect to precision
using current Trotter approximation methods. Nevertheless, in practice, when only limited precision is desired
and for a sufficiently large system, quantum simulation
may still outperform classical numerical simulation, as
demonstrated for molecular energies [30]. Avoiding the
cost of precision is desirable and can be done by designing
quantum simulations to explore questions that are insensitive to the microscopic details of the Hamiltonian [31].
How to develop quantum simulations for faulty small scale
(10 –20 qubit) quantum computers that can outperform
classical computations remains an open question.
This work was supported in part by DARPA, under the
quantum architectures research project, by ARDA Contract
No. DAAD19-03-1-0075, and by the HP-MIT Alliance.
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