Simulating quench dynamics on a digital quantum computer with data-driven error mitigation
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Simulating quench dynamics on a digital quantum computer
with data-driven error mitigation
Alejandro Sopena, Max Hunter Gordon, Germán Sierra, and Esperanza López
Instituto de Física Teórica, UAM/CSIC, Universidad Autónoma de Madrid, Madrid, Spain
Error mitigation is likely to be key in ob- the noise free observable. Despite much success [14–
taining near term quantum advantage. In this 17] this technique is limited by the assumption of low
work we present one of the first implementa- hardware noise, which may not be valid in the circuits
tions of several Clifford data regression based of a size and depth necessary to demonstrate quantum
methods which are used to mitigate the effect advantage.
of noise in real quantum data. We explore the Recently, it has been shown that data sets produced
dynamics of the 1-D Ising model with trans-
arXiv:2103.12680v2 [quant-ph] 5 Apr 2021
by classically simulable quantum circuits such as near-
verse and longitudinal magnetic fields, high- Clifford circuits [18, 19], circuits based on fermionic
lighting signatures of confinement. We find linear optics or matchgate circuits [20] can be used to
in general Clifford data regression based tech- mitigate the effects of noise. In so called Clifford data
niques are advantageous in comparison with regression (CDR) the exact and noisy data from near-
zero-noise extrapolation and obtain quantita- Clifford circuits is used to learn a functional relation
tive agreement with exact results for systems between the noisy and exact observables. This rela-
of 9 qubits with circuit depths of up to 176, in- tion can then be applied to a noisy observable of inter-
volving hundreds of CNOT gates. This is the est which cannot be simulated classically. This tech-
largest systems investigated so far in a study nique can also be unified with ZNE [21]. In variable
of this type. We also investigate the two-point noise Clifford data regression (vnCDR) near-Clifford
correlation function and find the effect of noise circuits are evaluated at several controlled noise levels.
on this more complicated observable can be The exact and noisy values are then used to perform
mitigated using Clifford quantum circuit data a guided extrapolation to the zero-noise limit. In gen-
highlighting the utility of these methods. eral, these regression based methods appear advanta-
geous over ZNE due to their simplicity and scalabil-
ity. However, there are few examples of these meth-
1 Introduction ods being applied to real data from currently available
quantum computers, where noise is significantly more
The rapid progress in the field of quantum com- difficult to mitigate.
puting is encouraging, with current machines ap- One clear application of quantum technologies is
proaching the qubit qualities and system sizes ex- the simulation of quantum systems. The classical re-
pected to demonstrate some useful quantum advan- sources necessary to simulate such systems in gen-
tage. However, noise within the computation still eral scales exponentially with the system size. Spin -
1
presents a large obstacle in obtaining useful results 2 systems are particularly relevant as they map di-
as current systems cannot implement full error cor- rectly onto physical qubits, making spin chains an
rection. Therefore, it is expected that error miti- ideal testing ground for both current and future quan-
gation techniques will be essential in demonstrating tum computers [22]. A common problem to consider
useful quantum advantage. These techniques aim to in condensed matter simulations is non-equilibrium
reduce the impact of noise rather than remove its ef- dynamics. These dynamics can be induced by a
fects completely. This relatively new field is experi- global quench, which is a sudden change to the sys-
encing a period of rapid progress with novel methods tem Hamiltonian. Simulations obtaining quantita-
being developed in quick succession. Common ap- tive accuracy have been reported in ion trap archi-
proaches include quantum circuit compiling, machine tectures [23] and more recently in super-conducting
learning [1–3] and variational algorithms [4–8]. Re- architectures [24].
cent advances show phase estimation [9] and so-called In this work we provide a comparison of several
virtual state distillation [10–12] can also be used for error mitigation strategies applied to the problem of
error mitigation and show great promise. simulating a quantum quench in the one dimensional
One of the most popular techniques is so called Ising model with transverse and longitudinal magnetic
zero-noise extrapolation (ZNE) [13]. Data from an fields. To investigate how these methods perform in
observable of interest evaluated at several controlled real quantum devices we explore the behavior of sev-
noise levels is used to give an improved estimate of eral observables of interest and simulate the system
1dynamics with various circuit depths. We measure unitary folding [25, 26] and identity insertion meth-
the frequency of oscillations of the magnetisation for ods [27]. Furthermore, additional extrapolation tech-
different initial states in a system of 9 spins. Fur- niques have been proposed [14, 28].
thermore, we present one of the first measurements of Despite widespread success ZNE performance guar-
the two-site correlation function in a study of these antees are limited due to uncertainty in the extrapola-
dynamics on a superconducting device. We are able tion. Additionally, in real devices often the base-level
to mitigate the effect of noise in data produced by noise is too strong to enable an accurate extrapola-
deeper circuits and larger systems than previously ex- tion, particularly in circuits with significant depth.
plored in similar works. We find that Clifford regres-
sion based methods are able to obtain quantitative 2.2 CDR
accuracy with the exact results and consistently out-
perform ZNE. More recently, Clifford circuit quantum data has been
First, we present an overview of the techniques used used to mitigate the effect of noise [18, 19]. Quan-
in data-driven error mitigation, with particular focus tum circuits composed of mainly Clifford gates can
on CDR and vnCDR and their relation to recent ad- be evaluated efficiently on a classical computer. In
vances. A slight variation of these methods is ex- CDR near-Clifford circuits are used to construct a set
plored, namely "poor-man’s" CDR (pmCDR), which of noisy and exact expectation values for some ob-
performs well for short depth trotterised simulations servable of interest. This dataset is used to train a
of hamiltonian dynamics. We then review the theoret- simple linear ansatz mapping noisy to exact values.
ical expectations of the model and the methods used Following the presentation in [18] taking µ̂0 to be the
to implement the simulation in a super-conducting ar- observable evaluated with hardware noise, CDR uses
chitecture. We show that using error mitigation we Clifford circuits to train the following anstaz:
are able to obtain consistent quantitative accuracy in-
f (µ̂0 ) = a1 µ̂0 + a2 (1)
volving circuits of depth 110 with 160 CNOT gates
and 9 qubits, while also obtaining some results with The parameters a1 , a2 are chosen using least-squares
quantitative accuracy for depths up to 176 with 256 regression on the near-Clifford circuit dataset. For a
CNOT gates. Finally we conclude with a discussion training set of m Clifford circuits with noisy expec-
of the results presented here and an outline of future tation values {xi } and exact expectation values {yi }
directions. evaluated classically, one calculates
m
X 2
(a∗1 , a∗2 ) = argmin [yi − (a1 xi + a2 )] . (2)
2 Data-driven error mitigation (a1 ,a2 ) i=1
Data-driven error mitigation uses classical post pro- These learned parameters are then used to mitigate
cessing of quantum data to improve the zero-noise es- the effect of noise on an observable produced by a cir-
timates of some observable of interest. In this work cuit which is not classically simulable. As noted in
ZNE [13], CDR [18] and vnCDR [21] are used to ob- Ref. [18] the form of the anstaz can be motivated by
tain noise-free estimates of various observables. Fur- considering the action of a global depolarizing chan-
thermore, following the recent work showing the suc- nel. Letting ρ be the density matrix for the noise-free
cess of a simple mitigation strategy with an assumed state after some evolution. Consider the action of a
noise model [24], we demonstrate the utility of a sim- depolarizing noise channel E which acts on this state
ilar approach where the parameters of an assumed before a measurement of some observable of interest
noise model are learned using near-Clifford circuits X. The action of the channel can be described as
(pmCDR). follows
tr(X)
2.1 ZNE tr(E(ρ)X) = (1 − ) tr(ρX) + (3)
d
Zero-noise extrapolation is one of the most popular where d is the dimension of the system and ∈ (0, 1)
error mitigation strategies. It uses quantum circuit is a parameter characterizing the noise. Identifying
data collected at various hardware noise levels to esti- µ̂0 = tr(E(ρ)X) and
mate the value of a noise free observable. Intuitively,
by increasing the noise in a controlled manner and a1 = 1/(1 − ), a2 = − tr(X) (4)
d(1 − )
extrapolating to the zero-noise limit one can obtain
a more accurate estimate of an observable of inter- the noise-free expectation value tr(ρX) can be calcu-
est. Originally, this technique was presented within lated using Eq. (1). Therefore, in the case of a global
the context of stretching gate times to increase noise depolarizing channel CDR should perfectly mitigate
and using Richardson extrapolation to approach the the noise, assuming the Clifford circuit training set ac-
zero-noise limit [13]. More recently this has been curately captures its effect. This ansatz also perfectly
extended to hardware agnostic approaches through corrects certain types of measurement error [18].
2For observables X with tr(X) = 0 the linear term of blind extrapolation and is expected to outperform
in CDR appears to be redundant assuming the noise both ZNE and CDR in deep quantum circuits involv-
can be modelled with a global depolarising channel, ing many qubits. vnCDR makes use of Clifford cir-
shown to be an accurate description in some circum- cuits evaluated at several noise levels to train a more
stances [24]. However, we find including the constant general anstaz than that of CDR. Considering m near-
term in the anstaz allows for more flexible fitting of Clifford circuits and n + 1 noise levels cj ∈ C, a noisy
the training data, leading to a better mitigation in estimate of the observable expectation value is de-
general (e.g. for the data shown in Fig. 1 the abso- fined as xi,j . For each of the m circuits the corre-
lute error is improved by a factor of 1.2). An example sponding exact observable yi is computed classically.
of such a case can be seen in Appendix A. The training set T is taken as T = {(xi , yi )} where
xi = (xi,0 , . . . , xi,n ) is the vector of noisy expectation
2.3 Poor man’s CDR values produced by the ith circuit. This training data
is used to learn a function that takes a set of noisy
As previously mentioned, recently it has been shown estimates at the n + 1 different noise levels and out-
that a global depolarising channel (Eq. 3) appears to puts an estimate for the noise-free value. We use the
accurately describe the noise in a real device for small linear ansatz
system sizes [24, 29]. Indeed, this noise model pro-
vides the motivation for use of a linear anstaz in CDR. g(x; a, b) = a · x + b , (6)
Here, we implement a simplified version of CDR where where we have included a constant term b. Least-
short depth near-Clifford circuits are used to fit the squares regression is used on the dataset T to pick
parameter characterising a global depolarising noise optimal parameters a∗ , b∗ , i.e.,
channel: m
X 2
tr(X) (a∗ , b∗ ) = argmin (yi − (a · x + b)) . (7)
hXinoisy = (1 − ) hXiexact + . (5) a,b i=1
d
Therefore, g(x; a∗ , b∗ ) is expected to output a good
Training sets are constructed from the quantum cir- estimate for the noise-free expected value from a vec-
cuits of one and two Trotter steps. This data is used tor formed of the noisy expectation values at different
to determine 1 and 2 respectively. Due to the repet- controlled noise levels. This mitigation strategy is
itive structure of the circuit in a trotterised evolution also expected to perfectly mitigate for a global depo-
we assume the effect of the error on an observable can larising noise channel. Despite promising results the
be modelled using Eq. (5), with the parameters evolv- performance of vnCDR has not been extensively ex-
ing as (1 − )αNT , where NT is the number of Trotter plored on real quantum circuit data, motivating the
steps and α is some constant (see also [24]). Using analysis we present here.
1 and 2 , determined with the near-Clifford training Originally vnCDR was introduced with an ansatz
data, we can fit α and use this assumed model to cor- excluding the constant term b, appearing more similar
rect observables from circuits involving more Trotter to Richardson extrapolation [21]. For the observables
steps. We find α is close to one for the magnetisation we consider here including this parameter made for a
while for ∆ZZi (t) it is higher (e.g. the mean values of more accurate mitigation (e.g. for the data shown in
α are 1.01 and 1.30 for the magnetisation and ∆ZZ i (t) Fig. 1 the absolute error is improved by a factor of
results shown in Figs. 1 and. 3 respectively). 1.1).
The advantage of this approach is that it is only Overall, classically simulable near-Clifford circuits
necessary to produce two near-Clifford data sets. The can be used to inform the experimenter about the
parameters of the noise model are then learned and noise present in the device. CDR makes use of ex-
applied to observables from other circuits. This is tracting this data for every circuit to mitigate the re-
more convenient, having a reduced experimental and sults of an observable of interest from that particular
computational overhead. We note a similar technique circuit. Assuming a noise model, pmCDR makes use
was recently presented using estimation circuits con- of two Clifford data sets and uses this data to com-
sisting of only CNOT gates and combining this idea plete a mitigation on circuits with a repetitive struc-
with randomised compiling and zero-noise extrapola- ture at different depths. Data collected at various
tion [29]. artificial controlled noise rates also contains relevant
information to perform a mitigation as shown in ZNE.
2.4 vnCDR vnCDR conceptually unifies ZNE and CDR by collect-
ing near-Clifford circuit data at various noise levels.
Zero noise extrapolation and Clifford data regression
can be conceptually unified into one mitigation strat-
egy where Clifford circuit quantum data is used to 3 Model
inform the functional form of the extrapolation to
the zero-noise limit [21]. Intuitively, so called vari- Data-driven error mitigation is a promising approach
able noise Clifford data regression reduces the risk to reduce the impact of noise in near term quantum
3computers. One of the areas where quantum algo- Confinement suppresses the light cone spreading of
rithms are expected to show some advantage over clas- correlations [34]. This effect can be seen by measuring
sical methods is in simulating quantum many body the two point correlation function,
systems. A system which displays interesting many
ZZ
body dynamics is the TFIM with an additional lon- σi,j (t) = σ̂iZ σ̂jZ − σ̂iZ σ̂jZ . (12)
gitudinal field, providing a clear test bed for these
mitigation methods. In the presence of a longitudinal field and hX < 1
local observables after quenches exhibit oscillations
whose dominant frequencies are the energy gaps be-
3.1 Transverse-Longitudinal Ising model tween bound states [34] (see Appendix B). These en-
ergy gaps can be interpreted as meson masses. A suit-
The Hamiltonian of the quantum one-dimensional
able observable for measuring the meson masses is the
Ising model of length L with transverse and longi-
magnetisation, σiZ (t) = σ̂iZ . In order to avoid edge
tudinal fields is given by
effects we measure the magnetisation at the centre of
" L
X L
X L
X
# the chain for initial states without domain walls and
Z Z X Z at the outer edge of the domain wall for initial states
H = −J σ̂i σ̂i+1 + hX σ̂i + hZ σ̂i
i i=1 i=1 of two domain walls [23].
(8) In this work we explore the signatures of confine-
where J is an exchange coupling constant, which sets ment by measuring the probability distribution of
the microscopic energy scale and hX and hZ are the kinks ∆ZZi (t), the evolution of the two point corre-
transverse and longitudinal relative field strengths, re- ZZ
lation function σi,j (t) and the meson masses deter-
spectively. This model is integrable for hZ = 0 while mined by extracting the dominant frequency of the
for hZ 6= 0 it is only integrable in the continuum when oscillation of the magnetisation σiZ .
hX = 1 [30].
Setting hZ = 0, in the continuum limit, the diag-
3.2 Quantum simulation
onalisation of the Hamiltonian results in the descrip-
√ with mass m = 2J|1 − hX | and ve-
tion of a fermion We simulate the induced Hamiltonian dynamics using
locity v = 2J hX a where a is the chain spacing and a first order trotterised evolution of the initial state.
ka 1 [31]. At hX = 1, the system has a criti- We start by discretising the evolution operator in n
cal point and the low-energy behaviour of the system blocks such that
is described by a conformal field theory with central n
charge c = 1/2 [32]. For hX < 1, the system is in U (t) = e−iHt = e−iH∆t = U n (∆t) (13)
the ferromagnetic phase (with J > 0). This system
can be approximated by considering the low-energy with ∆t = t/n. Each evolution step operator is ap-
elementary particle excitations which are given by do- proximated using the first order Trotter expansion:
main walls between the two ground states of H with Y
hX = 0 [31], U (∆t) = e−iH∆t = e−ihk t/n + O((∆t)2 ) (14)
k
|ii = |↑ · · · ↑i−1 ↑i ↓i+1 ↓i+2 ↓i+3 · · · ↓i . (9)
where hk = −J σ̂kZ σ̂k+1
Z
+ hX σ̂kX + hZ σ̂kZ . To im-
These states are identified as fermions. plement e−ihk t/n on IBM devices, we decompose the
A longitudinal field hZ 6= 0 induces a confining po- quantum circuit to execute one Trotter step into
tential between pairs of domain walls, the native IBM gate set {RX (π/2), RZ (θ), X, CNOT}
(see Appendix D). This decomposition leads to a
|i, ni = |↑ . . . ↑i−1 ↓i . . . ↓i+n−1 ↑i+n . . . ↑i , (10) depth of 11 per Trotter step with 2(Q − 1) CNOT
gates for a system size Q > 2, where Q is the number
which increases linearly with the length of the do- of qubits. For a fixed time step δt one can evaluate
main, n. This leads to excitations formed from pairs the dynamics up to time t by repeated action of this
of domain walls Eq. (10), which are referred to as circuit NT times, where NT = t/δt.
mesons [33].
In order to show the temporal evolution of the po-
sition of fermions and mesons, we measure the prob- 4 Simulated Spin chain confinement
ability distribution of kinks,
In this section we display the results obtained after
1 applying the mitigation methods described above on
∆ZZ
i (t) = (1 − σ̂iZ σ̂i+1
Z
), (11) the trotterised evolution of a system of Q = 9 qubits.
2
We investigate three observables of interest: the mag-
from an initial state of 2 kinks (see Appendix B). This netisation, σiZ , to determine the masses of the mesons,
observable takes the value 0 when there are no kinks ∆ZZ
i , to visually demonstrate confinement and two-
ZZ
and is 1 when the i-th and (i+1)-th spins form a kink. point correlation function, σij , to explore how the
4Raw ZNE CDR vnCDR pmCDR
1.00
↑↑↑↑ ↑ ↑↑↑↑
0.75
0.50
0.25
0.00
0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4
1.00
↑↑↑ ↑ ↓↑↑↑↑
0.75
0.50
σz
0.25
0.00
0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5
1.00
↑↑↑ ↑ ↓↓↑↑↑
0.75
0.50
0.25
0.00
0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5
tJ
(a) (b) (c) (d) (e)
Figure 1: Temporal evolution of Z-axis local magnetisation with hX = 0.5 and hZ = 0.9 for three initial states (in each row)
with the observables mitigated using various techniques (columns). In all panels the exact diagonalised dynamics is shown as
a black-solid line. Raw observables are shown in the left most column (a) calculated at two noise levels C = {1, 3} (blue and
light blue points respectively) using the IBMQ Paris quantum computer. Black points and dashed lines show the trotterised
dynamics. Error bars show the distribution of observables calculated over 6 repeats of the circuits of interest, and central
points show the median. The raw observables at the first noise level were calculated to have a mean absolute error of 0.279.
ZNE reduced this error to 0.166, CDR to 0.094, vnCDR to 0.092 and pmCDR to 0.096.
mitigation techniques perform on a more complex, both other initialisations the Trotter step was 0.5J
non-local observable. Every circuit used, both in leading to a final circuit depth of 110 involving 160
training set construction for mitigation and in collect- CNOT gates. A different Trotter step is needed in
ing raw data, was evaluated with 8192 shots. We use these two cases to reproduce the smaller amplitude
the absolute error to quantitatively explore the per- and higher frequency oscillations observed with the
formance of the mitigation strategies implemented: initial state being all spins up.
hXimitigated − hXitrotterised The temporal evolution of the magnetisation is
error = , (15) shown in Fig. 1. The raw values for the magnetisa-
mean(hXitrotterised )
tion clearly decay towards the maximally mixed state
where the mean is taken over the time evolution. with circuit depth. With a higher noise level this can
be seen to occur more quickly, as expected. We mit-
igate the raw results using ZNE, CDR, vnCDR and
4.1 Local magnetisation evolution
pmCDR.
To determine the first meson masses we measure the From the time evolution of the magnetisation for
oscillations of the local magnetisation for three initial different values of hz (see Appendix C) we obtain the
states. We extract the dominant frequencies using a dominant frequencies shown in Fig. 2. In order to
single-frequency sinusoidal fit as in Ref. [24], calculate the frequencies it is not necessary to fit the
entire evolution of the magnetisation. We found more
σjZ = a1 e−a2 t cos(a3 t) + a4 t + a5 . (16)
accurate values are obtained by fitting times up to
We explore the evolution starting with the system around tJ = 3.
initialised as: all spins up, the central qubit in the All mitigation methods improve upon the raw data.
down state and all other qubits up and two central In particular CDR, vnCDR and pmCDR mitigate the
qubits in the down state with all others spin up. The effect of noise effectively in many cases, even for deep
system is evolved using a trotterised evolution of the circuits. In some cases, like those shown in Fig. 2,
Hamiltonian for a fixed time step. The circuit depth pmCDR performs as well as the other Clifford based
therefore grows linearly with the number of Trotter methods. This suggests due to the repetitive structure
steps. When the system is initialised to all spins up, of the quantum circuit the effect of noise on this ob-
the Trotter step was chosen to be 0.25J leading to a servable can be characterised more easily, using only
final circuit depth of 176 with 256 CNOT gates. In the first two Trotter steps. However, we find this is
5Trotterised Raw ZNE time. Confinement can be directly observed by com-
5.0 paring the evolution when hZ = 0 and hZ 6= 0. With
the presence of some transverse field we observe an
2.5
attraction between the kinks due to the confining po-
tential. These dynamics are more difficult to miti-
Frequency
(a) (b) (c)
gate in comparison with the local magnetisation as
CDR vnCDR pmCDR
to observe this effect the entire state of the system
5.0
is probed. This gives a good indication of average
2.5 performance of each method. In Fig. 3 we show the
evolution of ∆ZZ i with the various mitigation meth-
0.5 0.65 0.75 0.9 0.5 0.65 0.75 0.9 0.5 0.65 0.75 0.9 ods for hX = 0.5 and hZ = 0, 0.5, 0.9. We note that
hZ when hZ = 0 there are 9 less non-Clifford gates per
(d) (e) (f )
Trotter step.
Figure 2: Frequencies obtained at hX = 0.5 and various The oscillations that appear to be washed out in the
hZ values calculated from the exact diagonalised (dashed raw data are clearly recovered by CDR and vnCDR.
lines), trotterised (a) and the median raw (b) and median ZNE does partially recover the oscillations but not
mitigated results (c)-(f). Frequencies obtained for initial to the same accuracy. It is important to note that
states |↑↑↑↑↑↑↑↑↑i (dots), |↑↑↑↑↓↑↑↑↑i (diagonal cross) and pmCDR begins to fail at around tJ = 3 leading to an
|↑↑↑↓↓↑↑↑↑i (vertical cross) are plotted. increased absolute error in comparison with the raw
results.
In the pmCDR implementation since tr ∆ZZ 6= 0,
not as reliable as using a training set to learn the noise
the ansatz relating the mitigated and the noisy ob-
at each Trotter step, as done in CDR and vnCDR.
servables Eq. (5) has a linear term and a constant
This indicates the noise parameters can vary consid-
term dependent on the parameter . Therefore, to
erably beyond some circuit depth and also in short
obtain an error of the same magnitude as that of the
time frames, between runs. Still, it is quite remark-
magnetisation, the difference between the true pa-
able that a simple global noise model describes the
rameter and the one obtained from the fit must be
noise so accurately in some runs (see Appendix C for
smaller for ∆ZZi than for the magnetisation. If dur-
more examples).
ing operation the true noise model changes slightly
For more complicated observables treating the noise
this has a large impact on the results. In addition,
as purely depolarising breaks down more quickly and
because ∆ZZi is a two-qubit observable, the ansatz
pmCDR begins to perform worse. It should be noted
is perhaps too simple to fully characterise the noise.
that we do not implement measurement error mitiga-
The impact of measurement error is also detrimental.
tion. While we do not expect this to impact the per-
Furthermore, we note the results from pmCDR could
formance of CDR or vnCDR it should lead to worse
be improved by enforcing physical constraints on the
implementations of both pmCDR and ZNE. We also
mitigated values.
do not enforce any physical constraints on our miti-
Without mitigation the dynamics of the observables
gated observables, in order to asses the raw potential
is not significantly changed by the introduction of a
of the method. Therefore, occasionally pmCDR gives
transverse field. The separation velocity does appear
unphysical values, increasing the absolute error of the
to be reduced but no oscillatory dynamics is observed.
mitigation significantly.
However, with CDR and vnCDR these oscillations are
Overall, CDR and vnCDR show the most reliable
recovered and there is a striking visual contrast be-
mitigation of the magnetisation. They consistently
tween the dynamics with and without the presence
offer a quantitatively accurate mitigation for times
of a longitudinal field. We deduce from these results
up to tJ ∼ 5 and occasionally up to even longer
that CDR and vnCDR appear to be the more power-
times. Therefore, it can be concluded the computa-
ful mitigation strategies.
tional overhead necessary for these methods is useful
in mitigating the effect of noise. vnCDR does not offer
any significant visual advantage over CDR although 4.3 Correlation evolution
it does lead to the most accurate calculations of the
frequencies and often has a smaller absolute error on We also investigate the correlation with the central
average. ZNE and pmCDR perform consistently well qubit as the system evolves. This observable is non-
for shorter depth circuits. local and is formed by combining three observables
σ̂iZ σ̂5Z , σ̂iZ and σ̂5Z , which we mitigate sep-
arately before combining. In general, the correla-
4.2 Kink evolution
tion decreases with hZ while σ̂iZ σ̂5Z and σ̂5Z in-
Starting with the initial state |↑↑↑↓↓↑↑↑↑i, projecting crease. As the longitudinal field increases, it becomes
into the two kink subspace and measuring the ob- more complicated to mitigate the correlation since the
servable ∆ZZ
i shows the evolution of the kinks with difference between the values of the two correlation
6Trotterised Raw ZNE CDR vnCDR pmCDR
5
hz = 0 4 1.00
3
0.89
2
1 0.78
5 0.67
4
hz = 0.5
0.56
3
tJ
2
0.44
1
0.33
5
4 0.22
hz = 0.9
3
0.11
2
1 0.00
1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8
(a) (b) (c) sites (d) (e) (f )
Figure 3: The observable ∆ZZi projected into the 2-kink subspace, measured at various sites and Trotter steps when hX = 0.5,
hZ = 0 (upper row), hX = 0.5, hZ = 0.5 (middle row) and hX = 0.5, hZ = 0.9 (bottom row) . The initial state of the system
is |↑↑↑↑↓↑↑↑↑i. We mitigate the raw observables using ZNE (c), CDR (d), vnCDR (e) and pmCDR (f ). The raw observables
have a mean absolute error of 0.300. ZNE reduces this error to 0.197, CDR to 0.154, vnCDR to 0.147 and pmCDR increases
the absolute error to 0.410.
terms needs to be smaller. Therefore, the mitigation Trotterised Raw vnCDR
5
0.50
needs to perform very effectively on each term and the 4
0.44
0.38
correlation proves challenging to mitigate for general 3 0.31
values of hZ and hX . Thus, we focus on the case with
tJ
0.25
2 0.19
hZ < hX . This together with the edge effects due to 0.12
1 0.06
the finite size of the system results in values for the 0.00
correlation with hZ 6= 0 similar to values obtained 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
sites
with hZ = 0. (a) (b) (c)
However, CDR and vnCDR do provide an advan-
tageous mitigation with impressive visual results in Figure 4: Correlation of qubits at sites along the x axis with
some cases as shown in Fig. 4 where we exhibit the the central qubit for the TFIM with hZ = 0 and the system
correlation at hX = 0.9 and hZ = 0 with a Trot- initialised in the |↑↑↑↑↑↑↑↑↑i state. The mean absolute error
ter step of 0.25 which gives a final depth of 220 at relative to the trotterised dynamics for the raw observables
tJ = 5, involving 320 CNOT gates. The correlation is 0.716 , normalised by the mean value for the correlation
with hZ = 0.2 is shown in Appendix C. The dynamics across all times. vnCDR reduces this error to 0.457. In this
case vnCDR gave the best mitigated values for the correlation
which are almost entirely lost at late times are recov-
closely followed by CDR.
ered to qualitative accuracy by vnCDR. In this case
vnCDR mitigated results gave the lowest normalised
absolute error. CDR also performed well giving very
similar visual results. Showing that the dynamics of
a complex observable can be qualitatively recovered
for such deep circuits is a testament to the power of
CDR and vnCDR. structing the training set. More fine grained methods
Overall, we find that CDR and vnCDR lead to the such as random identity insertions [27] may be nec-
best mitigated results for the observables explored essary to obtain a clear contrast between CDR and
in this work in almost all cases. The advantage is vnCDR. Alternatively, the lack of advantage in using
particularly clear for more complex observables. In- multiple noise levels and near-Clifford training data
terestingly, although vnCDR does generally have the could be due to the number of shots used to evaluate
smallest absolute error the advantage over CDR is each circuit. More computational overhead might be
slight. This could be attributed to how noise is being necessary to obtain some improvement in the vnCDR
increased in the circuits of interest and when con- results in comparison with CDR.
75 Implementation details to the phase gate S n . Therefore, we replace some of
the RZ (θ) gates by the phase gate to some power n.
5.1 Scaling the noise Which gates (labelled i) to replace are chosen prob-
abilistically according to distribution,
We perform the noise amplification in our quantum
circuits using the so called fixed identity insertion 3
X
exp −||RZ (θi ) − S n ||2 /σ 2 ,
method [27]. We insert pairs of CNOT gates, which p(θi ) ∝ (17)
evaluate to the identity, after each CNOT implemen- n=0
tation in the original circuit. Assuming the vast ma- where ||.|| represents the Frobenius norm and sigma
jority of error is introduced by these entangling gates, is a constant parameter taken here as σ = 0.5. Addi-
this method amplifies the noise by the factor of CNOT tionally, which Clifford gate to replace a chosen RZ (θ)
gates introduced. In our experiments we found it op- rotation with is also chosen probabilistically,
timal to use noise levels C = {1, 3} when using ZNE
and vnCDR. Furthermore, a linear fit was used to p0 (n) ∝ exp −||RZ (θi ) − S n ||2 /σ 2 ,
(18)
extrapolate to the zero noise limit. We note that it
would be interesting to implement a more fine grained also with σ = 0.5. We find this choice of σ allows
noise amplification technique to explore if the results for construction of training sets which are diverse yet
obtained by ZNE and vnCDR could improve. Ad- biased to the circuit of interest.
ditionally, more complex functions could be used to In both CDR and vnCDR implementations 50 near
execute the extrapolation. Clifford circuits were constructed in this manner for
each circuit of interest. Half the non-Clifford gates
Method 1 Method 2 in each circuit were substituted, capped at 50 non-
1.0
Clifford gates.
Two approaches were compared: replacing gates
0.5 throughout the entire circuit (method 1) and restrict-
Exact
ing the replacements to appear beyond a certain depth
0.0 (method 2). We found that method 2 produces more
similar observables to the circuit of interest, while still
-0.5 being sufficiently diverse. An example of a two CDR
-0.2 0 0.2 0.4 0.6 0.8 -0.2 0 0.2 0.4 0.6 0.8
training sets constructed with both methods is shown
Noisy in Fig. 5. This example reflects the general trend ob-
served, with training circuits being more similar to
Figure 5: Distribution of exact and noisy magnetisation pro- the circuit of interest when restricting Clifford sub-
duced by the near-Clifford training circuits constructed us- stitutions to a fixed portion of the circuit. This kind
ing two different methods for time t = 4 (8 Trotter steps) of training set leads to a better mitigation for deeper
with hX = 0.5, hZ = 0.9 and the initial state |↑↑↑↑↓↓↑↑↑i. circuits (see Fig. 6).
Method 1 refers to using probabilistic replacements through- This can be motivated by visualising a Clifford re-
out the entire circuit. Method 2 refers to when every non-
placement as a unitary transformation on the original
Clifford gate is fixed to appear after some circuit depth. The
blue star shows the noisy and exact result for the observable
circuit. To minimise the action of this unitary one can
from the circuit of interest. imagine naively maximising the section of the circuit
left unchanged, so forcing the Clifford substitution
to appear as late as possible. We replace all non-
Clifford gates in the second half of the circuit up to
5.2 Near-Clifford circuit training set 50 non-Clifford gates. Beyond 50 non-Clifford gates
we restrict all Clifford substitutions to appear at the
Constructing the set of circuits that make up the greatest possible circuit depth. Fig. 6(c) shows the
training set is a key feature of both CDR and vnCDR. dispersion of each training set constructed by both
Intuitively, one desires a set of circuits close, in some methods at various circuit depths. We use a measure
sense, to the circuit of interest while also being di- of dispersion to indicate the closeness of the training
verse enough to accurately train the ansatz. In order circuits to the circuit of interest, defined as:
to construct such a set of circuits we follow the proto-
m
col presented in Refs. [18, 21]. In this work we restrict 1 X
q
(xi − hXinoisy )2 + (yi − hXiexact )2 , (19)
substitutions to a portion of the circuit beyond some m i
depth.
First the circuit of interest is decomposed into where m is the number of training circuits and xi , yi
the native gate set of the IBM quantum computers are the noisy, exact expectation values for the observ-
{RX (π/2), RZ (θ), X, CNOT}. These gates are Clif- able of interest for each of the training circuits and
ford with the exception of RZ (θ) which is only Clifford hXinoisy , hXiexact are the noisy, exact expectation
when θ = nπ/2, where n = 0, 1, 2, 3 and correspond values for the circuit of interest.
8Raw CDR
1.0 1.0 0.6
Dispersion
0.8
0.8 0.4
σ4z
0.6
0.6 0.2
0.4 Method 1
Method 2
0.4 0.0
0.0 2.5 5.0 7.5 0.0 2.5 5.0 7.5 0.0 2.5 5.0 7.5
tJ
(a) (b) (c)
Figure 6: Evolution of the |↑↑↑↓↓↑↑↑↑i state with hX = 0.5 and hZ = 0.9. Exact diagonalised (black-solid), Trotterisied
(black-dashed) and raw results shown in (a). Data mitigated using CDR with two training set construction methods is shown
in (b), where brown crosses show the results from mitigating with a training set constructed with method 1 and orange points
show results using method 2. The final circuit depth is 176 with 216 CNOT gates. Error bars show distribution of six repeats
of the circuit of interest. The dispersion of each training set constructed by both methods at various circuit depths is shown
in (c).
In Fig. 6(b) an example of CDR is shown success- as presenting a simplified implementation of CDR,
fully mitigating noise in deep circuits, with this figure so-called pmCDR inspired by Ref. [24]. Using these
showing the dynamics of the magnetisation being re- techniques we have shown it is possible to calculate
covered up to the final circuit depth of 176. Oscilla- the first meson masses with quantitative accuracy for
tions in the magnetisation are recovered after they systems of 9 qubits, the largest system explored in a
all but vanish from the raw data. The dispersion study of this type. Clifford based mitigation methods
increases less quickly with circuit depth for method show the best performance overall. We have demon-
2 than for method 1, shown in Fig. 6(c), suggesting strated quantitative accuracy can be obtained using
method 2 makes for more reliable training sets. This CDR and vnCDR from observables produced by cir-
is reflected in the more accurate mitigation results cuits with depths of up to 176 involving hundreds of
obtained. CNOT gates. Furthermore, we have shown CDR and
In the case of pmCDR the training sets for the first vnCDR enable the recovery of dynamics which appear
two Trotter steps were used to train the model as completely washed out due to noise, highlighted in our
outlined in Section 2.3. measurements of the observable ∆ZZ i and the two-site
Once the circuits in the training set are executed (at correlation. pmCDR does work well consistently for
two noise levels for each circuit of interest) this data shorter depth circuits, but begins to struggle as depth
is used to train the CDR and vnCDR ansatzes. We increases. A similar trend is observed for ZNE. Com-
found for the majority of the observables investigated bining pmCDR with other mitigation strategies such
here the mitigation improved by repeatedly training as measurement error mitigation [35], random compi-
the given anstaz on a randomly selected subset of the lation and ZNE for the estimation of the noise param-
total training data. We used 200 subsets with data eters could improve its performance [29]. In general
from 5 circuits each, taking the final mitigation as CDR and vnCDR are advantageous due to the more
the median mitigated observables produced from each general ansatzes fitted with training data which re-
subset. We leave systematic investigation of this boot- flect the noise acting on the circuit of interest more
strapped training method for a later work. All the ob- accurately.
servables of interest here can be calculated from the We have shown that making Clifford substitutions
counts measured in the Z basis. Therefore, data from in a fixed region of the circuit of interest, beyond
the same training set from each circuit could be used some depth, makes for a more accurate mitigation.
to mitigate the noise on all the observables of interest. The best training set construction method to use in
general is still an open question. Clifford circuits are
clearly useful mitigation strategies, but their perfor-
6 Conclusion mance could be enhanced with the development of
well studied methods to construct a faithful training
In this work we have simulated the dynamics of a set. Furthermore, the exploration of more complex
quantum quench on the TFIM using a trotterised ansatzes is sure to provide promise in mitigating noise,
evolution on a quantum computer. We applied sev- as well as using training data suited to specific prob-
eral data-driven error mitigation techniques, as well lems [20]. Finally, the combination of these methods
9with more recent error mitigation advances such as ter J. Love, Alán Aspuru-Guzik, and Jeremy L.
virtual distillation appears to be a promising research O’Brien. A variational eigenvalue solver on a
direction [12]. It would also be interesting to explore photonic quantum processor. Nature Commu-
recent variational algorithms [36] in conjunction with nications, 5(1):4213, Jul 2014. ISSN 2041-1723.
Clifford circuit based error mitigation to obtain some DOI: 10.1038/ncomms5213.
computationally non-trivial results. [5] M Cerezo, Andrew Arrasmith, Ryan Babbush,
Overall, improvement in quality of available miti- Simon C Benjamin, Suguru Endo, Keisuke Fu-
gation techniques and quantum hardware becoming jii, Jarrod R McClean, Kosuke Mitarai, Xiao
more widely accessible opens the possibility of near Yuan, Lukasz Cincio, and Patrick J Coles. Vari-
term useful quantum advantage. Near-Clifford circuit ational quantum algorithms. arXiv preprint
based mitigation methods are demonstrating their po- arXiv:2012.09265, 2020. URL https://arxiv.
tential to become the staple error mitigation tech- org/abs/2012.09265.
nique. [6] Kunal Sharma, Sumeet Khatri, Marco Cerezo,
and Patrick J Coles. Noise resilience of
variational quantum compiling. New Jour-
Acknowledgments nal of Physics, 22(4):043006, 2020. URL
We thank Piotr Czarnik for useful discussions. We https://iopscience.iop.org/article/10.
also thank the IBM Quantum team for making de- 1088/1367-2630/ab784c.
vices available via the IBM Quantum Experience. [7] P. J. J. O’Malley, R. Babbush, I. D. Kivlichan,
The access to the IBM Quantum Experience has been J. Romero, J. R. McClean, R. Barends, J. Kelly,
provided by the CSIC IBM Q Hub. A.S.G is sup- P. Roushan, A. Tranter, N. Ding, B. Campbell,
ported by the Spanish Ministry of Science and In- Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth,
novation under grant number SEV-2016-0597-19-4. A. G. Fowler, E. Jeffrey, E. Lucero, A. Megrant,
M.H.G is supported by “la Caixa” Foundation (ID J. Y. Mutus, M. Neeley, C. Neill, C. Quin-
100010434), Grant No. LCF/BQ/DI19/11730056. tana, D. Sank, A. Vainsencher, J. Wenner, T. C.
This work has also been financed by the Span- White, P. V. Coveney, P. J. Love, H. Neven,
ish grants PGC2018-095862-B-C21, QUITEMAD+ A. Aspuru-Guzik, and J. M. Martinis. Scalable
S2013/ICE-2801, SEV-2016-0597 of the ”Centro de quantum simulation of molecular energies. Phys.
Excelencia Severo Ochoa” Programme and the CSIC Rev. X, 6:031007, Jul 2016. DOI: 10.1103/Phys-
Research Platform on Quantum Technologies PTI- RevX.6.031007.
001. [8] Cristina Cirstoiu, Zoe Holmes, Joseph Io-
sue, Lukasz Cincio, Patrick J Coles, and An-
drew Sornborger. Variational fast forward-
References ing for quantum simulation beyond the coher-
ence time. npj Quantum Information, 6(1):
[1] Prakash Murali, Jonathan M. Baker, Ali Javadi- 1–10, 2020. URL https://www.nature.com/
Abhari, Frederic T. Chong, and Margaret articles/s41534-020-00302-0.
Martonosi. Noise-adaptive compiler mappings
[9] Thomas E O’Brien, Stefano Polla, Nicholas C
for noisy intermediate-scale quantum comput-
Rubin, William J Huggins, Sam McArdle, Ser-
ers. In Proceedings of the Twenty-Fourth In-
gio Boixo, Jarrod R McClean, and Ryan Bab-
ternational Conference on Architectural Sup-
bush. Error mitigation via verified phase esti-
port for Programming Languages and Operating
mation. arXiv preprint arXiv:2010.02538, 2020.
Systems, ASPLOS ’19, page 1015–1029, New
URL https://arxiv.org/abs/2010.02538.
York, NY, USA, 2019. Association for Comput-
ing Machinery. ISBN 9781450362405. DOI: [10] Bálint Koczor. Exponential error suppression
10.1145/3297858.3304075. for near-term quantum devices. arXiv preprint
[2] Lukasz Cincio, Yiğit Subaşı, Andrew T Sorn- arXiv:2011.05942, 2020. URL https://arxiv.
borger, and Patrick J Coles. Learning the quan- org/abs/2011.05942.
tum algorithm for state overlap. New Jour- [11] William J Huggins, Sam McArdle, Thomas E
nal of Physics, 20(11):113022, nov 2018. DOI: O’Brien, Joonho Lee, Nicholas C Rubin, Ser-
10.1088/1367-2630/aae94a. gio Boixo, K Birgitta Whaley, Ryan Babbush,
[3] Lukasz Cincio, Kenneth Rudinger, Mohan and Jarrod R McClean. Virtual distillation
Sarovar, and Patrick J Coles. Machine learn- for quantum error mitigation. arXiv preprint
ing of noise-resilient quantum circuits. arXiv arXiv:2011.07064, 2020. URL https://arxiv.
preprint arXiv:2007.01210, 2020. URL https: org/abs/2011.07064.
//arxiv.org/abs/2007.01210. [12] Piotr Czarnik, Andrew Arrasmith, Lukasz Cin-
[4] Alberto Peruzzo, Jarrod McClean, Peter Shad- cio, and Patrick J. Coles. Qubit-efficient ex-
bolt, Man-Hong Yung, Xiao-Qi Zhou, Pe- ponential suppression of errors. arXiv preprint
10arXiv:2102.06056, 2021. URL https://arxiv. tion of domain wall confinement and dynam-
org/abs/2102.06056. ics in a quantum simulator. arXiv preprint
[13] Kristan Temme, Sergey Bravyi, and Jay M. arXiv:1912.11117, 2020. URL https://arxiv.
Gambetta. Error mitigation for short-depth org/abs/1912.11117.
quantum circuits. Phys. Rev. Lett., 119: [24] Joseph Vovrosh, Kiran E. Khosla, Sean Green-
180509, Nov 2017. DOI: 10.1103/Phys- away, Christopher Self, Myungshik Kim, and
RevLett.119.180509. Johannes Knolle. Efficient mitigation of depo-
[14] Matthew Otten and Stephen K Gray. Recover- larizing errors in quantum simulations. arXiv
ing noise-free quantum observables. Physical Re- preprint arXiv:2101.01690, 2021. URL https:
view A, 99(1):012338, 2019. DOI: 10.1103/Phys- //arxiv.org/abs/2101.01690.
RevA.99.012338. [25] T. Giurgica-Tiron, Y. Hindy, R. LaRose,
[15] E. F. Dumitrescu, A. J. McCaskey, G. Hagen, A. Mari, and W. J. Zeng. Digital zero noise
G. R. Jansen, T. D. Morris, T. Papenbrock, extrapolation for quantum error mitigation. In
R. C. Pooser, D. J. Dean, and P. Lougovski. IEEE International Conference on Quantum
Cloud quantum computing of an atomic nucleus. Computing and Engineering (QCE), pages 306–
Phys. Rev. Lett., 120:210501, May 2018. DOI: 316, 2020. DOI: 10.1109/QCE49297.2020.00045.
10.1103/PhysRevLett.120.210501. [26] Ryan LaRose, Andrea Mari, Peter J. Karalekas,
[16] Abhinav Kandala, Kristan Temme, Antonio D. Nathan Shammah, and William J. Zeng. Mi-
Córcoles, Antonio Mezzacapo, Jerry M. Chow, tiq: A software package for error mitigation
and Jay M. Gambetta. Error mitigation extends on noisy quantum computers. arXiv preprint
the computational reach of a noisy quantum pro- arXiv:2009.04417, 2020. URL https://arxiv.
cessor. Nature, 567(7749):491–495, Mar 2019. org/abs/2009.04417.
ISSN 1476-4687. DOI: 10.1038/s41586-019-1040- [27] Andre He, Benjamin Nachman, Wibe A. de Jong,
7. and Christian W. Bauer. Zero-noise extrapola-
[17] Zhenyu Cai. Multi-exponential error extrap- tion for quantum-gate error mitigation with iden-
olation and combining error mitigation tech- tity insertions. Phys. Rev. A, 102:012426, Jul
niques for nisq applications. arXiv preprint 2020. DOI: 10.1103/PhysRevA.102.012426.
arXiv:2007.01265, 2020. URL https://arxiv. [28] Suguru Endo, Simon C. Benjamin, and Ying
org/pdf/2007.01265.pdf. Li. Practical quantum error mitigation for near-
[18] Piotr Czarnik, Andrew Arrasmith, Patrick J future applications. Phys. Rev. X, 8:031027, Jul
Coles, and Lukasz Cincio. Error mitigation with 2018. DOI: 10.1103/PhysRevX.8.031027.
clifford quantum-circuit data. arXiv preprint [29] Miroslav Urbanek, Benjamin Nachman, Vin-
arXiv:2005.10189, 2020. URL https://arxiv. cent R. Pascuzzi, Andre He, Christian W.
org/abs/2005.10189. Bauer, and Wibe A. de Jong. Mitigat-
[19] Armands Strikis, Dayue Qin, Yanzhu Chen, Si- ing depolarizing noise on quantum computers
mon C Benjamin, and Ying Li. Learning- with noise-estimation circuits. arXiv preprint
based quantum error mitigation. arXiv preprint arXiv:2103.08591, 2021. URL https://arxiv.
arXiv:2005.07601, 2020. URL https://arxiv. org/abs/2103.08591.
org/abs/2005.07601. [30] A. B. Zamolodchikov. Integrals of motion
[20] Ashley Montanaro and Stasja Stanisic. Error and s matrix of the (scaled) t=t(c) ising
mitigation by training with fermionic linear op- model with magnetic field. International Jour-
tics. arXiv preprint arXiv:2102.02120, 2021. nal of Modern Physics A, 04(16):4235–4248,
URL https://arxiv.org/abs/2102.02120. October 1989. ISSN 0217-751X. DOI:
[21] Angus Lowe, Max Hunter Gordon, Piotr 10.1142/S0217751X8900176X.
Czarnik, Andrew Arrasmith, Patrick J. Coles, [31] Subir Sachdev. Quantum phase transitions.
and Lukasz Cincio. Unified approach to data- Cambridge University Press, Cambridge ; New
driven quantum error mitigation. arXiv preprint York, second edition edition, 2011. ISBN
arXiv:2011.01157, 2020. URL https://arxiv. 9780521514682.
org/pdf/2011.01157.pdf. [32] G. Mussardo. Statistical field theory: an in-
[22] Adam Smith, M. S. Kim, Frank Pollmann, and troduction to exactly solved models in statistical
Johannes Knolle. Simulating quantum many- physics, chapter 10, page 368. Oxford gradu-
body dynamics on a current digital quantum ate texts. Oxford University Press, Oxford ; New
computer. npj Quantum Information, 5(1), nov York, 2010. ISBN 9780199547586.
2019. DOI: 10.1038/s41534-019-0217-0. [33] Barry M. McCoy. The Connection between sta-
[23] W. L. Tan, P. Becker, F. Liu, G. Pagano, K. S. tistical mechanics and quantum field theory. In
Collins, A. De, L. Feng, H. B. Kaplan, A. Kypri- 7th Physics Summer School on Statistical Me-
anidis, R. Lundgren, W. Morong, S. Whitsitt, chanics and Field Theory, pages 26–128, 3 1994.
A. V. Gorshkov, and C. Monroe. Observa- [34] Marton Kormos, Mario Collura, Gabor Takács,
11and Pasquale Calabrese. Real-time confine- 1.5
ment following a quantum quench to a non-
integrable model. Nature Physics, 13(3):246–249,
March 2017. ISSN 1745-2473, 1745-2481. DOI: 1.0
Exact
10.1038/nphys3934.
[35] Lena Funcke, Tobias Hartung, Karl Jansen, Ste-
fan Kühn, Paolo Stornati, and Xiaoyang Wang. 0.5
Measurement error mitigation in quantum com-
puters through classical bit-flip correction. arXiv
preprint arXiv:2007.03663, 2020. URL https: 0.0
//arxiv.org/abs/2007.03663. 0.0 0.1 0.2 0.3
[36] Joe Gibbs, Kaitlin Gili, Zoë Holmes, Ben- Noisy
jamin Commeau, Andrew Arrasmith, Lukasz
Cincio, Patrick J. Coles, and Andrew Sorn- Figure 7: Distribution of exact and noisy magnetisation pro-
duced by the near-Clifford training circuits constructed for
borger. Long-time simulations with high fi-
time t = 3.75 (15 Trotter steps) with hx = 0.5, hz = 0.5
delity on quantum hardware. arXiv preprint and the initial state |↑↑↑↑↑↑↑↑↑i. The blue star shows the
arXiv:2102.04313, 2021. URL https://arxiv. noisy and exact result for the observable from the circuit of
org/abs/2102.04313. interest. The continuous and dashed lines correspond to the
[37] Pasquale Calabrese, Fabian H. L. Essler, and ansatz Eq. (1) with a2 6= 0 and a2 = 0, respectively. We see
Maurizio Fagotti. Quantum Quench in the that the constant term a2 provides a clear advantage since
Transverse-Field Ising Chain. Physical Re- the blue point is contained in the black line.
view Letters, 106(22):227203, June 2011. DOI:
10.1103/PhysRevLett.106.227203.
to the separation between the domain walls and the
[38] Joseph Vovrosh and Johannes Knolle. Con-
second allows nearest neighbour interactions due to
finement and Entanglement Dynamics on a
hopping. Therefore, a pair of kinks will experience
Digital Quantum Computer. arXiv preprint
an oscillatory motion due to the confining potential
arXiv:2001.03044, 2020. URL https://arxiv.
resulting in a meson.
org/abs/2001.03044.
In the case with hZ = 0, σ Z (t) decays to zero expo-
[39] Cheng-Ju Lin and Olexei I. Motrunich. Quasi-
nentially for any quench with hX < 1 [37]. However,
particle explanation of the weak-thermalization
when a longitudinal field is introduced, the dynam-
regime under quench in a nonintegrable quantum
ics changes and an oscillatory behaviour is observed
spin chain. Phys. Rev. A, 95:023621, Feb 2017.
with various frequencies from which the masses of
DOI: 10.1103/PhysRevA.95.023621.
the two kinks bound states can be extracted. For
this purpose, we consider the states |i, ni indicated
in Eq. (10) which are eigenstates of the Hamiltonian
A CDR training set example with hX = 0 and we perform a quench up to a certain
value hX < 1. To obtain the eigenstates of the system
We show an example of a Clifford training set which after the quench we use the 2-kink model introduced
provides a much more faithful mitigation with the before as it is a good approximation of the low en-
ansatz including a constant term (see Fig. 7). Using ergy behaviour of the system even for large values of
an ansatz which contains the constant clearly allows hX [23]. To diagonalise the Hamiltonian Eq. (20) we
for more flexible fitting of the training data. start by changing to the momentum space,
L−(n+1)
1 X n
|k, ni = p e−ikj−ik 2 |j, ni ,
B Meson masses L − (n + 1) j
(21)
In order to understand the phenomenon of confine-
so that the Hamiltonian becomes
ment it is useful to project the Hamiltonian Eq. (8)
into the two kink subspace with basis {|i, ni}: X
k
H= V (n) |k, ni hk, n| + 2hX cos
2
X k,n
H= [V (n) |i, ni − JhX (|i − 1, n + 1i
i,n
(|k, ni hk, n + 1| + |k, ni hk, n − 1|)] . (22)
+ |i + 1, n − 1i + |i, n − 1i + |i, n + 1i)] hi, n| This Hamiltonian is diagonal in the basis of states
X
(20) |k, αi = Cα Jn−νk,α (xk ) |k, ni (23)
n
where V (n) = 2JhZ n. The first term of this Hamil- where J is the Bessel function
of the first kind, νk,α =
tonian represents a confining potential proportional Ek,α /2hX , xk = 2hZ cos k2 /hx and Cα is a coefficient
12to normalise the state [38]. Therefore, |k, αi are the than that presented in the main text. This could be
eigenstates of the Hamiltonian with hX 6= 0 and hZ 6= attributed to the number of non-Clifford gates per
0 and we can write the state of the system at time t Trotter step being greater when hZ 6= 0. This means
as for the same circuit depth more gates need to be sub-
X stituted when forming the training circuits, making
|Ψ(t)i = hk, α|Ψ(0)i e−iEk,α t |k, αi (24) for a less reliable dataset in general.
k,α
where |Ψ(0)i is the initial state. Using this expression, 5
Trotterised Raw vnCDR
0.50
the expected value of a certain observable O is 4
0.44
0.38
X 3 0.31
hΨ(t)| O |Ψ(t)i = hΨ(0)|k, αi hq, β|Ψ(0)i ·
tJ
0.25
2 0.19
k,α
0.12
q,β 1 0.06
−i(Eq,β −Ek,α )t 0.00
· hk, α| O |q, βi e 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9
(25) sites
(a) (b) (c)
where we see an oscillatory behaviour with frequen-
Figure 8: Correlation of qubits at sites along the x axis with
cies equal to energy differences between eigenstates.
the central qubit for the TFIM with hZ = 0.2 and the system
A method of obtaining the masses corresponding to initialised in the |↑↑↑↑↑↑↑↑↑i state. The mean absolute error
different excited states, mα = E0,α − E0,0 , is to use relative to the trotterised dynamics for the raw observables
initial states whose dominant oscillation frequency is is 0.521. vnCDR reduces this error to 0.402. In this case
ωα = E0,α+1 − E P0,α because then, the masses mα are vnCDR gave the best mitigated values for the correlation
α
given by mα = β=0 ωβ . closely followed by CDR.
With the parameters hX and hZ that we are us-
ing, if we consider the initial states |i = 4, n = 1i =
|↑↑↑↑↓↑↑↑↑i and |i = 4, n = 2i = |↑↑↑↑↓↓↑↑↑i, the
dominant frequencies are ω1 and ω2 , respectively. The Trotterised Raw ZNE
highest coefficients,
7.5
Cα Jn−νk,α 5.0
ck,α,n = p , (26)
L − (n + 1) 2.5
(a) (b) (c)
Mass
in the expansion Eq. (25) written in the basis |j, ni are CDR vnCDR pmCDR
those corresponding to the states |k = 0, α = 2i and
7.5
|k = 0, α = 1i for the initial state |i = 4, n = 1i and
5.0
|k = 0, α = 3i and |k = 0, α = 2i for the initial state
|i = 4, n = 2i. 2.5
0.5 0.65 0.75 0.9 0.5 0.65 0.75 0.9 0.5 0.65 0.75 0.9
It should be noted that we cannot use the two-kink hZ
approximation to obtain the energy of states with n = (d) (e) (f )
0, such as the state |↑↑↑↑↑↑↑↑↑i. However, a dominant
oscillation frequency is also observed in the temporal Figure 9: Masses obtained at hX = 0.5 and different lon-
evolution of the magnetisation using this initial state. gitudinal fields from the exact diagonalised (dashed lines),
simulated (a), raw (b) and mitigated data (c)-(f). Masses
This oscillation frequency is to be understood as ω0 =
obtained for initial states |↑↑↑↑↑↑↑↑↑i (dots), |↑↑↑↑↓↑↑↑↑i
E0,1 −E0,0 since it corresponds to the energy required
(diagonal cross) and |↑↑↑↓↓↑↑↑↑i (vertical cross) are plotted.
to create a particle with momentum zero [39].
Following this prescription, we show in Fig. 9 the
masses corresponding to the frequencies of Fig. 2.
C Additional data
D Quantum circuit for trotterised evo-
Here we present the results for the time evolution of
the local magnetisation with different values of hZ
lution
used to obtain the frequencies shown in Fig. 2. (see
Fig. 10). We also show a colour plot of the magnetisa- The quantum circuit to evolve the system by one Trot-
tion for each qubit in the system for hX = 0.5, hZ = ter step for a 5 spin system is shown in Fig. 12. In
0.9 (see Fig. 11). Furthermore, we show the additional general for a system with an odd number of qubits
results for the correlation when hZ = 0.2 (see Fig. 8). Q each Trotter step has a depth of 11 with 2(Q − 1)
The mitigation of the correlation here is a little worse entangling gates and 3Q − 1 non-Clifford gates.
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