Experimental quantum kernel trick with nuclear spins in a solid

Abstract

The kernel trick allows us to employ high-dimensional feature space for a machine learning task without explicitly storing features. Recently, the idea of utilizing quantum systems for computing kernel functions using interference has been demonstrated experimentally. However, the dimension of feature spaces in those experiments have been smaller than the number of data, which makes them lose their computational advantage over explicit method. Here we show the first experimental demonstration of a quantum kernel machine that achieves a scheme where the dimension of feature space greatly exceeds the number of data using ¹ H nuclear spins in solid. The use of NMR allows us to obtain the kernel values with single-shot experiment. We employ engineered dynamics correlating 25 spins which is equivalent to using a feature space with a dimension over 10 ¹⁵ . This work presents a quantum machine learning using one of the largest quantum systems to date.

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ARTICLE OPEN
Experimental quantum kernel trick with nuclear spins
in a solid
Takeru Kusumoto
1,7
, Kosuke Mitarai
1,2,3,4,7
✉, Keisuke Fujii
1,2,4,5
, Masahiro Kitagawa
1,2
and Makoto Negoro
2,4,6
The kernel trick allows us to employ high-dimensional feature space for a machine learning task without explicitly storing features.
Recently, the idea of utilizing quantum systems for computing kernel functions using interference has been demonstrated
experimentally. However, the dimension of feature spaces in those experiments have been smaller than the number of data, which
makes them lose their computational advantage over explicit method. Here we show the first experimental demonstration of a
quantum kernel machine that achieves a scheme where the dimension of feature space greatly exceeds the number of data using
1
H nuclear spins in solid. The use of NMR allows us to obtain the kernel values with single-shot experiment. We employ engineered
dynamics correlating 25 spins which is equivalent to using a feature space with a dimension over 10
15
. This work presents a
quantum machine learning using one of the largest quantum systems to date.
npj Quantum Information (2021) 7:94 ; https://doi.org/10.1038/s41534-021-00423-0
INTRODUCTION
Quantum machine learning is an emerging field that has attracted
much attention recently. The major algorithmic breakthrough was
an algorithm invented by Harrow–Hassidim-Lloyd
1
. This algorithm
has been further developed to more sophisticated machine
learning algorithms
2,3
. However, a quantum computer that is
capable of executing those algorithms is yet to be realized. At
present, noisy intermediate-scale quantum (NISQ) devices
4
, which
consist of several tens or hundreds of noisy qubits, are the most
advanced technology. Although their performance is limited
compared to the fault-tolerant quantum computer, simulation of
the NISQ devices with 100 qubits and sufficiently high gate fidelity
are beyond the reach for the existing supercomputer and classical
simulation algorithms
5–7
. This fact motivates us to explore its
power for solving practical problems.
Many NISQ algorithms for machine learning have been
proposed in recent works
8–17
. Almost all of the algorithms require
us to evaluate an expectation value of an observable, which is
sometimes troublesome to measure by sampling, for example
with superconducting or trapped-ion qubits. On the other hand,
NMR can evaluate the expectation value with a one-shot
experiment owing to its use of a vast number of duplicate
quantum systems. It is, therefore, a great testbed for those
algorithms. A major weakness of NMR is that its initialization
fidelity is quite low; at the thermal equilibrium of room
temperature, the proton spins can effectively be described with
a density matrix ρeq ¼1
2ðIþϵIzÞwith ϵ≈10
−5
. Nevertheless,
ensemble spin systems can exhibit complex quantum dynamics
that are classically intractable. For example, the dynamical phase
transition between localization and delocalization has been
observed in polycrystalline adamantane along with tens of
correlated proton spins
18
. Discrete time-crystalline order has been
observed in disordered
19
and ordered
20,21
spin systems. Also, it
has been shown that general dynamics of NMR are classically
intractable under certain complexity conjecture
22,23
. These facts
strengthen our motivation to use NMR for NISQ algorithms.
In this work, we employ NMR for machine learning. Specifically,
we implement the kernel-based algorithm which utilizes the
quantum state as a feature vector and is a variant of theoretical
proposals
8,9,24
. Ref.
25
has recently proved that this approach has a
rigorous advantage over classical computation in a specific task.
The experimental verification has been provided in refs.
15,26
using
either superconducting qubits or the photonic system. Our
strategy to use the NMR is advantageous in that we can estimate
the value of the kernel, which is the inner product of two quantum
states, by single-shot experiments. Moreover, the dimension of the
Hilbert space employed in this work greatly exceeds the number
of training data. We perform simple regression and binary
classification tasks using the dynamics of nuclear spins in a
polycrystalline adamantane sample, and observe that the
performance of the trained model becomes better as more spins
are involved in the dynamics. Also, to carry out the performance
analysis of our approach without the inevitable effect of noise in
experiments, we present numerical simulations of 20 spin
dynamics, which well agree with the experimental results. We
employ one of the largest quantum systems to date for a quantum
machine learning experiment in this work with the single-shot
setting that enables the use of a large quantum system.
RESULTS
Kernel methods in machine learning
In machine learning, one is asked to extract some patterns, or
features, in a given dataset
3,27
. It is sometimes useful to pre-
process them beforehand to achieve the objective. For example, a
speech recognition task might become easier when we work in
the frequency domain; in this case, the useful pre-processing
would be the Fourier transform. The space in which such pre-
processed data live is called feature space. For a given set of data
1
Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, Japan.
2
Center for Quantum Information and Quantum Biology, Osaka University, Toyonaka,
Osaka, Japan.
3
QunaSys Inc., Bunkyo, Tokyo, Japan.
4
JST, PRESTO, Kawaguchi, Saitama, Japan.
5
RIKEN Center for Quantum Computing (RQC), Wako, Saitama, Japan.
6
Institute for
Quantum Life Science, National Institutes for Quantum and Radiological Science and Technology, Inage-Ku, Chiba, Japan.
7
These authors contributed equally: Takeru Kusumoto,
Kosuke Mitarai. ✉email: mitarai@qc.ee.es.osaka-u.ac.jp
www.nature.com/npjqi
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fxigNd
i¼1RD, a feature space mapping ϕ(x) constructs the data in
the feature space fϕðxiÞgNd
i¼1RDf. The feature map has to be
carefully taken to maximize the performance in e.g.
classification tasks.
Kernel methods are a powerful tool in machine learning. It uses
a distance measure of two inputs defined as a kernel,
k:RDf´RDf!R. For example, a kernel can be defined as an
inner product of two feature vectors:
kðxi;xjÞ¼ϕðxiÞTϕðxjÞ:(1)
Many machine learning models, such as support vector
machine or linear regression, can be constructed using the kernel
only, that is, we do not have to explicitly hold ϕ(x). This
dramatically reduces the computational cost when the feature ϕ
(x) lives in a high-dimensional space. The direct approach that
computes and stores ϕ(x) for all training data xwould demand the
computational resource that is at least proportional to N
d
and
D
f
≫N
d
just for storing them, whereas the kernel method only
requires us to store a N
d
×N
d
kernel matrix with its (i,j) elements
being k(x
i
,x
j
).
Implementing kernel by NMR
In NMR, we can prepare a data-dependent operator A(x
i
)by
applying a data-dependent unitary transformation U(x
i
) on the
initial z-magnetization Iz¼Pn
μ¼1Iz;μ, that is, A(x
i
)=U(x
i
)I
z
U
†
(x
i
).
Here, I
α,μ
(α=x,y,z) is the α-component of the spin operator of the
μ-th spin and nrepresents the number of spins. A(x
i
) with a
sufficiently large nis generally intractable by classical compu-
ters
22,23
. We employ this operator A(x
i
) as a feature map ϕ
NMR
(x
i
).
A(x
i
) can be regarded as a vector, for example, by expanding A(x
i
)
as a sum of Pauli operators. For an n-spin-1/2 system, A(x
i
)isa
vector in R4n. The dynamics of NMR can involve tens of spins
maintaining its coherence
18,28–30
, which means we can employ an
approximately 4
O(10)
dimensional feature vector for machine
learning. Although the high-dimensional feature space does not
always mean superiority in machine learning tasks, the fact that
we can work with the feature space which has been intractable
with a classical computer motivates us to explore its power.
The kernel method opens up a way to exploit A(x
i
) directly for
machine learning purposes. While we cannot evaluate each
element of A(x
i
) because it takes an exponential amount of time,
we can evaluate the inner product of two feature vector A(x
i
) and
A(x
j
)efficiently. The Frobenius inner product between two
operators A(x
i
) and A(x
j
)isTrAðxiÞAðxjÞ

, which we employ as
our NMR kernel function k
NMR
(x
i
,x
j
) in this work. Noting that,
Tr AðxiÞAðxjÞ

¼Tr UyðxjÞUðxiÞIzUyðxiÞUðxjÞIz

;(2)
and also at the thermal equilibrium, the density matrix of spin
systems is ρeq 1
2nIþϵIz
ðÞassuming ϵ≪1, we obtain,
Tr AðxiÞAðxjÞ

/Tr UyðxjÞUðxiÞρeqUyðxiÞUðxjÞIz

:(3)
The right hand side can be measured experimentally by first
evolving the system at thermal equilibrium with U(x
i
) and then
with U
†
(x
j
), and finally measuring I
z
. A similar protocol is also used
for measuring out-of-time-ordered correlator (OTOC)
31,32
, which is
considered as a certain complexity measure of quantum many-
body systems.
Experimental
We propose to use U(x) for an input x¼fxjgD
j¼1that takes the
form of,
UxðÞ¼eiHðxDÞτeiHðx2ÞτeiHðx1Þτ;(4)
where τis a hyperparameter that determines the performance of
this feature map, and H(x
j
) is an input-dependent Hamiltonian
(Fig. 1b). In this work, we choose H(x
j
)tobe
HðxjÞ¼eixjIzX
μ<ν
dμν Iy;μIy;νIx;μIx;ν

eixjIz;(5)
where I
α
=∑
μ
I
α,μ
. The Hamiltonian H(0) can approximately be
constructed from the dipolar interaction among nuclear spins in
solids with a certain pulse sequence
18,33,34
shown in Fig. 1b. See
“Methods”for details. Shifting the phase of the pulse by xprovides
us H(x) for general x. This Hamiltonian with x
j
=0 created in
adamantane has been shown to have a delocalizing feature in
refs.
18,28–30
, which makes it appealing as we wish to involve as
many spins as possible in the dynamics.
To illustrate character of the kernel function, we show the shape of
the kernel for one-dimensional input xobtained with this sequence
setting τ=Nτ
1
where τ
1
=60 μsandN=1, 2, ⋯, 6 as Fig. 1c. In this
experiment, the overall unitary dynamics applied to the system is
UyðxjÞUðxiÞ¼eiHðxjÞτeiH ðxiÞτ¼eixjIzeiH ð0ÞτeiðxixjÞIzeiHð0Þτeix iIz.
Since the initial state has only I
z
element and the final observable
is also I
z
, we can omit eixiIzand eixjIzoperations in the experiment.
We, therefore, show the value of the kernel as a function of x
i
−x
j
in Fig. 1c. The decay of the intensity of the signal with increasing N
is due to decoherence. The number of spins involved in this
dynamics can be inferred by the Fourier transform of the signal in
Fig. 1c with respect to x
i
−x
j
. This is because the z-rotation
eiðxixjÞIzafter the forward dynamics e
−iH(0)τ
induces a phase
eim0ðxixjÞto terms in the form of Im
±which appears in the density
matrix of the system, where mand m0are integers, jm0jm, and
I
±
=I
x
±iI
y
. This Fourier spectra is called multiple-quantum
spectra
18,30,31
, from which we can extract a cluster size K, which
can be considered as the effective number of spins involved in the
dynamics, by fitting it with Gaussian distribution
expðm2=KÞ
18,30,33
. We show the Fourier transform of the
measured NMR kernel (Fig. 1c) along with corresponding Kin
Fig. 1d. We can conclude that, for N> 5, 6, the dynamics involve
25 spins, which corresponds to a feature space dimension of 4
25
≈
10
15
.
One-dimensional regression task
As the first demonstration, we perform the one-dimensional
kernel regression task using the kernel shown in Fig. 1c. To
evaluate the nonlinear regression ability of the kernel, we use
y¼sinð2πx=50Þand y¼sinð2πx=50Þ
2πx=50 , which will be refered to as sine
and sinc function, respectively. We randomly drew 40 samples of x
from [−45, 45] (in degrees) to construct the traning data set which
consists of the input fxjg40
j¼1and the teacher fyjg40
j¼1calculated at
each x
j
. The NMR kernel k
NMR
(x
i
,x
j
) is measured for each pair of
data to construct the model by kernel ridge regression
27
. We let
the model predict yfor 64 x’s including the training data. The
regularization strength was chosen to minimize the mean squared
error of the result at the 64 evaluation data.
The result for the sine function is shown in Figure 2a, b. That for
the sinc function is shown in “Methods”. Figure 3a shows the
accuracy of learning evaluated by the mean squared error
between the output from the trained model and true function.
We see that the regression accuracy tends to increase with a larger
N. For all N, the dimension of the feature space exceeds the
number of data as the dynamics involve over 5 spins which
corresponds to the Hilbert space dimension of 4
5
(see Fig. 1d).
However, because of the deteriorating signal-to-noise ratio, the
result also gets noisy with increasing N.
To certify the trend without the effect of noise, we conducted
numerical simulations of 20-qubit dynamics with all-to-all random
interaction in the form of Eq. (5), whose coupling strength, d
ij
, are
drawn uniformly from [−1, 1]. We set the evolution time τ=0.01,
0.02, ⋯, 0.06. Details of the numerical simulations are described in
Methods. The result for the sine function is shown as Fig. 2g–l. For
T. Kusumoto et al.
2
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the sinc function, we place the result in “Methods”. The mean
squared error of the prediction evaluated in the same manner is
shown in Fig. 3b. We can see the performance gets better with
increasing τ, which corresponds to increasing Nin the experiment.
This certifies the trend observed in the NMR experiment.
Two-dimensional classification task
As the second demonstration, we implement two-dimensional
classification tasks. We employ the hard-margin kernel support
vector machine
27
and its implementation in scikit-learn
35
for this
task. We use two data sets which we call “circle”and “moon”data
set generated by functions available in scikit-learn package
35
.
They are depicted as dots in Fig. 4. A data set consists of pairs of
input data point (x
1
,x
2
)∈[−45, 45]
2
and its label y∈{−1, +1}. For
experimental simplicity, we have modified the generated (x
1
,x
2
)to
be on a lattice. The NMR kernels with N=1, 2, 3 are utilized in this
task. We again conducted numerical simulations with the same
setting as the previous section along with the experiment with
τ=0.03, 0.06, 0.09. The values of this numerical kernel are shown
in the “Methods”.
The results are shown in Fig. 4. Also for this classification task,
the dimension of feature space exceeds the number of data
already at N=1. Note that the evolution time, in this case, is
doubled compared to the previous demonstration where the
input is one-dimensional (see Eq. (4)), that is, the evolution time of
N=1 for two-dimensional input is equivalent to that of N=2 for
one-dimensional input. We also note that, for the moon dataset
with N=1 experimental NMR kernel, the kernel matrix was
singular, and we did not obtain a reliable result. We reason this to
the broadness of the kernel at N=1.
For the circle data set with the experimental kernel, the best
performance is achieved when N=1. We believe this is because
the task of classifying the circle dataset was too easy and did not
need large feature space. This dataset can be completely
classified, for example, by merely mapping two-dimensional data
(x
1
,x
2
) to three-dimensional feature space ðx1;x2;x2
1þx2
2Þ. As for
the moon dataset, the performance increased with increasing N.
This can be attributed to the increasing dimension of the
feature space.
DISCUSSION
In the one-dimensional regression task, we observed the trend of
better performance with longer evolution time. This can be
explained by the shape of the kernel generated by the NMR
dynamics, which is shown in Fig. 1c. As mentioned earlier, this
experiment is essentially the Loschmidt echo, and the shape of
the signal sharpens as the evolution time increases. The sharpness
of the kernel can directly be translated to the representability of
the model as it can be in the popular Gaussian kernel because this
property allows the machine to distinguish different data more
clearly. However, it also causes overfitting problems if the data
points are sparse. The most extreme case is when we use a delta
function as a kernel, where every training point is learned with the
perfect accuracy while the trained model fails to predict for
unknown inputs. In our experimental case, we did not observe any
overfitting problem, which means that our training samples were
Fig. 1 Experimental setups. a Adamantane molecule. The white and pink balls represent hydrogen and carbon atoms, respectively.
bQuantum circuit and pulse sequence employed in this work to realize evaluation of k
NMR
(x
i
,x
j
). The top right shows a pulse sequence for
x
1
=0, where X and Xrepresent +π/2 and −π/2 pulses along x-axis, respectively. In the experiment, Δand Δ0are set 3.5 μs and 8.5 μs,
respectively. cNMR kernel employed in this work. dFourier transform of (c) which corresponds to the obtained
1
H multiple-quantum spectra
for N=1toN=6. Red line is Gaussian aexpðm2=KÞfitted to the spectra using aand Kas fitting parameters. The Gaussian is fitted only to
even mdata points. Kis the cluster size of the system
18,30,33
.
T. Kusumoto et al.
3
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dense enough for the sharpness of the kernel utilized in the
model, and thus we observed an increasing performance from the
improved representability of the kernel with longer evolution
time. For the classification task, we observed the dataset-
dependent trend with respect to the evolution time in the
performance. We suspect that there is an optimal evolution time
for this kind of task, which should be explored in future works. We
believe the performance can also be improved by employing
ensemble learning approach
36
.
We note that the shape of the kernel resembles the Gaussian
kernel which is widely employed in many machine learning tasks.
In fact, we can obtain better results using the Gaussian kernel as
shown in “Methods”. More concretely, it can achieve mean
squared error less than 10
−5
for the regression tasks and hinge
loss of order 10
−5
for the classification tasks without fine tuning of
a hyperparameter. While this fact casts a shadow on the
usefulness of the NMR kernel used in this work, it is also true
that the NMR dynamics evolved by general Hamiltonian cannot be
simulated classically under certain complexity conjecture
22,23
. This
leaves a possibility that the NMR kernel performs better than
classical kernel in some specific cases. We also note that there is a
quantum kernel that is rigorously advantageous over classical
ones in a certain machine learning task
25
. More experiments using
different Hamiltonians are required to test whether the "quantum”
kernel has any advantage in machine learning tasks over widely
used conventional kernels.
The previous research
18,30,33
has shown that the cluster size can
be brought up to over 1000 with more precise experiments. This
indicates that the solid-state NMR is a platform well-suited to
experimentally analyze the performance of kernel-based quantum
machine learning. To extend the experiment to larger cluster sizes,
we have to resolve the low initialization fidelity of nuclear spin
Fig. 2 Demonstration of one-dimensional regression task of y¼sinð2πx=50Þ.a,bshows results for NMR kernel and numerically simulated
kernel, respectively. The blue dots, green lines, red dashed lines represent the training data, the prediction of the trained model, and the
function from which the function the training data are sampled.
Fig. 3 Mean squared error of the trained models for the regression task of sine and sinc functions. a,bshows results for experimental
NMR kernel and numerically simulated kernel, respectively.
T. Kusumoto et al.
4
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systems, which is a major challenge faced by the NMR quantum
information processing in general. We believe this can be
overcome by the use of e.g., a technique called dynamic nuclear
polarization
37,38
where we initialize the nuclear spin qubits by
transferring the polarization from initialized electrons.
To conclude, we proposed and experimentally tested a
quantum kernel constructed by data-dependent complex
dynamics of a quantum system. Experimentally, we used complex
dynamics of 25
1
H spins in adamantane to compute the kernel.
This allowed us to achieve a scheme where the dimension of
feature space greatly exceeds the number of data. We also stress
that the spin dynamics of NMR is generally intractable by a
classical computer
22,23
. Machine learning models for one-
dimensional regression tasks and two-dimensional classification
tasks were constructed with the proposed kernel. The experi-
mental and numerical results showed similar results. Experiments
along with numerical simulation also showed that the perfor-
mance of the model tended to increase with longer evolution
time, or equivalently, with a larger number of spins involved in the
dynamics for certain tasks. It would be interesting to export this
method to more quantum-oriented machine learning tasks. For
example, one may be able to distinguish two dynamical phases of
spin systems, such as localized and delocalized phases demon-
strated in ref.
18
, with the kernel support vector machine
employed in this work. More experiments are needed to verify
the power of this "quantum kernel”approach, but our results can
be thought of as one of the baselines of this emerging field.
Note added—After the initial submission of this work, we
became aware of related works on quantum kernel methods
39,40
.
METHODS
Experimental details
A pulse sequence to realize H(0) is given in Supplementary Fig. 1. In the
experiment, we set the length of π/2 pulse, τ
p
, to 1.5 μs. For the waiting
period, we used Δ0¼2Δþτpwith Δ=3.5 μs, which makes the evolution
time for a cycle, τ
1
,60μs. By repeating the sequence for Ntimes, we can
effectively evolve the spins with eiHðxDÞτfor τ=Nτ
1
. NMR spectroscopy
with polycrystalline adamantane sample was performed at room
temperature with OPENCORE NMR
41
, operating at a resonant frequency
of 400.281 MHz for
1
H nucleus observation.
Detail of numerical simulation
We drew the interaction strength, d
μν
, from uniform distribution on [−1, 1]
for all μ,ν. The evolution according to the Hamiltonian H(x)is
approximated by the first-order Trotter formula, that is,
eiHðxÞτeixIzY
μ<ν
eiτdμ;νðIy;μIy;νIx;μIx;νÞ=M
"#
M
eixIz:(6)
We set τ=0.01, 0.02, ⋯, 0.06 and τ/M=0.001 in the simulation. In order
to reduce the computational cost, we set AðxÞ¼UðxÞQμI
2þIz;μ

UyðxÞ,
which allows us to evaluate Tr(A(x
i
)A(x
j
)) by computing 0
hj
UyðxjÞUðxiÞ0
ji

2,
where 0
ji
is the ground state of I
z
. Since we can compute this quantity by
simulating dynamics of a 2
20
-dimensional state vector, it is significantly
easier than computing A(x)=U(x)I
z
U
†
(x) where we would need to simulate
dynamics of 2
20
×2
20
matrices. All simulations are performed with a
quantum circuit simulator Qulacs
42
.
One-dimensional regression task
We show the results for the regression task of sinc function performed with
the experimental NMR kernel and that of numerical simulations in
Supplementary Fig. 2.
Comparison with Gaussian kernel
Here, we perform the same machine learning tasks with Gaussian kernel
kðxi;xjÞ¼exp γkxixjk

, where γ> 0 is a hyperparameter.
First, we perform kernel regressions of sine and sinc functions using a
Gaussian kernel with γ=0.02. Results are shown as Supplementary Fig. 3.
Mean squared errors of the results are 1.3 × 10
−6
and 6.1 × 10
−8
for sine
and sinc functions, respectively.
Secondly, we perform kernel support vector machine on circle and
moon datasets using a Gaussian kernel with γ=0.5. Results are shown as
Supplementary Fig. 4. Hinge losses of the results are 7.9 × 10
−5
and 5.2 ×
10
−5
for circle and moon datasets, respectively.
Fig. 4 Demonstration of simple classification tasks. a Classification with the NMR kernel. Left and right panels respectively show the results
for “circle”and “moon”data set. The red and blue dots represent inputs (x
1
,x
2
) with label y=+1 and y=−1, respectively. The background
color indicates the decision function of the trained model. The number in each figure is the hinge loss after training defined as
1
NsPNs
i¼1maxf1λiyi;0gwhere N
s
is the number of training data, and λ
i
and y
i
are the output from the trained model and the teacher data
corresponding to a training input x
i
. Top, middle, and bottom panels are the results with N=1, 2, 3 NMR kernels, respectively. bResults from
the numerically simulated quantum kernel. Top, middle, and bottom panels are the results with simulated kernel with τ=0.03, 0.06, 0.09. All
the other notations follow that of (a).
T. Kusumoto et al.
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Kernel from the numerical simulations
The kernel computed from numerical simulations for one-dimensional
input is shown as Supplementary Fig. 5. It is shown as a function of
xx02
π
2;π
2

. We can observe the similar features as the experimental
one, such as the sharpening of the kernel with increasing evolution time.
Kernel for two-dimensional data
Let x=(x
1
,x
2
) and x0¼ðx0;x0Þbe two data points with which we wish to
evaluate the kernel kðx;x0Þ¼kðfx1;x2g;fx0;x0gÞ. We note that our
experimental kernel satisfies the equality: kðfx1;x2g;fx0;x0gÞ ¼
kðfx1x0;x2x0g;fx0x0;0gÞ. With this in mind, we define
P1¼x1x0,P2¼x2x0,P3¼x0x0. The value of the experimental
and simulated kernel are sliced by the value of P
3
and shown in
Supplementary Figs. 6–8 and Supplementary Figs. 9–11, respectively.
DATA AVAILABILITY
Data are available upon reasonable requests.
CODE AVAILABILITY
Program codes are available upon reasonable requests.
Received: 8 June 2020; Accepted: 1 May 2021;
REFERENCES
1. Harrow, A. W., Hassidim, A. & Lloyd, S. Quantum algorithm for linear systems of
equations. Phys. Rev. Lett. 103, 150502 (2009).
2. Biamonte, J. et al. Quantum machine learning. Nature 549, 195 (2017).
3. Schuld, M. & Petruccione, F. Supervised Learning with Quantum Computers
(Springer, 2018).
4. Preskill, J. Quantum Computing in the NISQ era and beyond. Quantum 2,79
(2018).
5. Boixo, S. et al. Characterizing quantum supremacy in near-term devices. Nat. Phys.
14, 595–600 (2018).
6. Villalonga, B. et al. Establishing the quantum supremacy frontier with a 281 Pflop/
s simulation. Quantum Sci. Technol. 5, 034003 (2020).
7. Arute, F. et al. Quantum supremacy using a programmable superconducting
processor. Nature 574, 505–510 (2019).
8. Mitarai, K., Negoro, M., Kitagawa, M. & Fujii, K. Quantum circuit learning. Phys. Rev.
A98, 032309 (2018).
9. Schuld, M. & Killoran, N. Quantum machine learning in feature hilbert spaces.
Phys. Rev. Lett. 122, 040504 (2019).
10. Benedetti, M. et al. A generative modeling approach for benchmarking and
training shallow quantum circuits. npj Quantum Inf. 5, 45 (2019).
11. Dallaire-Demers, P.-L. & Killoran, N. Quantum generative adversarial networks.
Phys. Rev. A 98, 012324 (2018).
12. Cong, I., Choi, S. & Lukin, M. D. Quantum convolutional neural networks. Nat. Phys.
15, 1273–1278 (2019).
13. Liu, J.-G. & Wang, L. Differentiable learning of quantum circuit Born machines.
Phys. Rev. A 98, 062324 (2018).
14. Huggins, W., Patil, P., Mitchell, B., Whaley, K. B. & Stoudenmire, E. M. Towards
quantum machine learning with tensor networks. Quantum Sci. Technol. 4,
024001 (2019).
15. Havlícek, V. et al. Supervised learning with quantum-enhanced feature spaces.
Nature 567, 209–212 (2019).
16. Fujii, K. & Nakajima, K. Harnessing disordered-ensemble quantum dynamics for
machine learning. Phys. Rev. Appl. 8, 024030 (2017).
17. Ghosh, S., Opala, A., Matuszewski, M., Paterek, T. & Liew, T. C. H. Quantum
reservoir processing. npj Quantum Inf. 5, 35 (2019).
18. Álvarez, G. A., Suter, D. & Kaiser, R. Localization-delocalization transition in the
dynamics of dipolar-coupled nuclear spins. Science 349, 846–848 (2015).
19. Choi, S. et al. Observation of discrete time-crystalline order in a disordered dipolar
many-body system. Nature 543, 221–225 (2017).
20. Rovny, J., Blum, R. L. & Barrett, S. E. Observation of discrete-time-crystal signatures
in an ordered dipolar many-body system. Phys. Rev. Lett. 120, 180603 (2018).
21. Rovny, J., Blum, R. L. & Barrett, S. E.
3
1P NMR study of discrete time-crystalline
signatures in an ordered crystal of ammonium dihydrogen phosphate. Phys. Rev.
B97, 184301 (2018).
22. Morimae, T., Fujii, K. & Fitzsimons, J. F. Hardness of classically simulating the one-
clean-qubit Model. Phys. Rev. Lett. 112, 130502 (2014).
23. Fujii, K. et al. Impossibility of classically simulating one-clean-qubit model with
multiplicative error. Phys. Rev. Lett. 120, 200502 (2018).
24. Farhi, E. & Neven, H. Classification with quantum neural networks on near term
processors. arXiv:1802.06002. Preprint at http://arxiv.org/abs/1802.06002 (2018).
25. Liu, Y., Arunachalam, S. & Temme, K. A rigorous and robust quantum speed-up in
supervised machine learning. arXiv:2010.02174. Preprint at http://arxiv.org/abs/
2010.02174 (2020).
26. Bartkiewicz, K. et al. Experimental kernel-based quantum machine learning in
finite feature space. Sci. Rep. 10, 12356 (2020).
27. Bishop, C. M. Pattern Recognition and Machine Learning (Information Science and
Statistics) https://www.xarg.org/ref/a/0387310738 (Springer, 2011).
28. Krojanski, H. G. & Suter, D. Scaling of decoherence in wide NMR quantum reg-
isters. Phys. Rev. Lett. 93, 090501 (2004).
29. Krojanski, H. G. & Suter, D. Decoherence in large NMR quantum registers. Phys.
Rev. A 74, 062319 (2006).
30. Álvarez, G. A. & Suter, D. NMR quantum simulation of localization effects induced
by decoherence. Phys. Rev. Lett. 104, 230403 (2010).
31. Gärttner, M. et al. Measuring out-of-time-order correlations and multiple quan-
tum spectra in a trapped-ion quantum magnet. Nat. Phys. 13, 781–786 (2017).
32. Wei, K. X., Ramanathan, C. & Cappellaro, P. Exploring localization in nuclear spin
chains. Phys. Rev. Lett. 120, 070501 (2018).
33. Baum, J., Munowitz, M., Garroway, A. N. & Pines, A. Multiple-quantum dynamics in
solid state NMR. J. Chem. Phys. 83, 2015–2025 (1985).
34. Warren, W. S., Weitekamp, D. P. & Pines, A. Theory of selective excitation of
multiple-quantum transitions. J. Chem. Phys. 73, 2084–2099 (1980).
35. Pedregosa, F. et al. Scikit-learn: machine learning in Python. J. Mach. Learn. Res.
12, 2825–2830 (2011).
36. Nakajima, K., Fujii, K., Negoro, M., Mitarai, K. & Kitagawa, M. Boosting computa-
tional power through spatial multiplexing in quantum reservoir computing. Phys.
Rev. Appl. 11, 034021 (2019).
37. Tateishi, K. et al. Room temperature hyperpolarization of nuclear spins in bulk.
Proc. Natl Acad. Sci. USA 111, 7527–7530 (2014).
38. Henstra, A., Lin, T.-S., Schmidt, J. & Wenckebach, W. High dynamic nuclear
polarization at room temperature. Chem. Phys. Lett. 165,6–10 (1990).
39. Otten, M. et al. Quantum machine learning using gaussian processes with per-
formant quantum kernels. arXiv:2004.11280. Preprint at http://arxiv.org/abs/
2004.11280 (2020).
40. Peters, E. et al. Machine learning of high dimensional data on a noisy quantum
processor. arXiv:2101.09581. Preprint at http://arxiv.org/abs/2101.09581 (2021).
41. Takeda, K. OPENCORE NMR: Open-source core modules for implementing an
integrated FPGA-based NMR spectrometer. J. Magn. Reson. 192,218–229
(2008).
42. Suzuki, Y. et al. Qulacs: a fast and versatile quantum circuit simulator for research
purpose. arXiv:2011.13524. Preprint at https://arxiv.org/abs/2011.13524 (2020).
ACKNOWLEDGEMENTS
K.M. is supported by JSPS KAKENHI No. 19J10978, 20K22330, and JST PRESTO Grant
No. JPMJPR2019. K.F. is supported by KAKENHI No.16H02211, JST PRESTO
JPMJPR1668, JST ERATO JPMJER1601, and JST CREST JPMJCR1673. M.N. is supported
by JST PRESTO JPMJPR1666. This work is supported by MEXT Quantum Leap Flagship
Program (MEXT Q-LEAP) Grant Number JPMXS0118067394 and JPMXS0120319794.
We also acknowledge support from JST COI-NEXT program.
AUTHOR CONTRIBUTIONS
T.K. and K.M. contributed equally to this work. T.K. and M.N. conducted the
experiments. K.M. conceived the idea and designed the experiments. K.F. provided
the program code for simulations. M.K. and M.N. supervised the project. All authors
contributed to the manuscript.
COMPETING INTERESTS
K.M., K.F., M.K., and M.N. own stock/options of QunaSys Inc.
ADDITIONAL INFORMATION
Supplementary information The online version contains supplementary material
available at https://doi.org/10.1038/s41534-021-00423-0.
Correspondence and requests for materials should be addressed to K.M.
T. Kusumoto et al.
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