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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

Published in partnership with The University of New South Wales

1234567890():,;

fxigNd

i¼1RD, a feature space mapping ϕ(x) constructs the data in

the feature space fϕðxiÞgNd

i¼1RDf. 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,

UxðÞ¼eiHðxDÞτeiHðx2ÞτeiHðx1Þτ;(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

HðxjÞ¼eixjIzX

μ<ν

dμν Iy;μIy;νIx;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

UyðxjÞUðxiÞ¼eiHðxjÞτeiH ðxiÞτ¼eixjIzeiH ð0ÞτeiðxixjÞIzeiHð0Þτeix iIz.

Since the initial state has only I

z

element and the ﬁnal observable

is also I

z

, we can omit eixiIzand eixjIzoperations 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

eiðxixjÞIzafter the forward dynamics e

−iH(0)τ

induces a phase

eim0ðxixjÞto terms in the form of Im

±which appears in the density

matrix of the system, where mand m0are integers, jm0jm, 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

expðm2=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

npj Quantum Information (2021) 94 Published in partnership with The University of New South Wales

1234567890():,;

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 aexpðm2=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

Published in partnership with The University of New South Wales npj Quantum Information (2021) 94

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

npj Quantum Information (2021) 94 Published in partnership with The University of New South Wales

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 eiHðxDÞτfor 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,

eiHðxÞτeixIzY

μ<ν

eiτdμ;νðIy;μIy;νIx;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 γkxixjk

, 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

i¼1maxf1λiyi;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.

5

Published in partnership with The University of New South Wales npj Quantum Information (2021) 94

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

xx02

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

kðfx1x0;x2x0g;fx0x0;0gÞ. With this in mind, we deﬁne

P1¼x1x0,P2¼x2x0,P3¼x0x0. 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;

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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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