Simple Mitigation of Global Depolarizing Errors in Quantum Simulations
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Simple Mitigation of Global Depolarizing Errors in Quantum Simulations
Joseph Vovrosh*,1 Kiran E. Khosla,1 Sean Greenaway,1 Christopher Self,1 M. S. Kim,1 and Johannes Knolle2, 3, 1
1
Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom
2
Department of Physics TQM, Technische Universität München, James-Franck-Straße 1, D-85748 Garching, Germany
3
Munich Center for Quantum Science and Technology (MCQST), 80799 Munich, Germany
To get the best possible results from current quantum devices error mitigation is essential. In this work we
present a simple but effective error mitigation technique based on the assumption that noise in a deep quantum
circuit is well described by global depolarizing error channels. By measuring the errors directly on the device,
we use an error model ansatz to infer error-free results from noisy data. We highlight the effectiveness of our
mitigation via two examples of recent interest in quantum many-body physics: entanglement measurements and
real time dynamics of confinement in quantum spin chains. Our technique enables us to get quantitative results
from the IBM quantum computers showing signatures of confinement, i.e. we are able to extract the meson
arXiv:2101.01690v2 [quant-ph] 28 Apr 2021
masses of the confined excitations which were previously out of reach. Additionally, we show the applicability
of this mitigation protocol in a wider setting with numerical simulations of more general tasks using a realistic
error model. Our protocol is device-independent, simply implementable and leads to large improvements in
results if the global errors are well described by depolarization.
INTRODUCTION
Quantum computers are becoming large enough (∼ 50 −
100 qubits [1]) to, in principle, allow demonstrations of their
‘Quantum Advantage’ [2, 3]. However, the actual amount
of entanglement that can be generated in current devices is
constrained by noise and errors, limiting their ability to solve
complex problems such as quantum simulation. To address
this, various error mitigation strategies have recently been de-
veloped to counteract noise and boost the fidelity of experi-
mental results.
Error mitigation differs from fault tolerance. Fault tolerant
quantum computers will eventually be able to suppress errors
by encoding quantum information in a redundantly large num-
ber of qubits as error correcting codes [4, 5]. In this way they
will be able to execute arbitrarily deep circuits by the repeated
application of active error corrections. Unfortunately, these
encodings cannot be used in current devices as they require
smaller hardware errors and larger numbers of qubits than are
currently available. In contrast, error mitigation strategies are
applied to unencoded physical qubits. Rather than actively
correcting errors, they aim to estimate what the effect of the
error was and infer the error-free result. This current stage FIG. 1. Results from the IBM device Paris [6] of the second order
in the development of quantum computers has been dubbed Rényi entanglement entropy. Quench dynamics before and after
error mitigation of the second order Réyni entropy for the trans-
the noisy intermediate-scale quantum (NISQ) era [1] and is verse field Ising model with varying longitudinal field strengths.
expected to last for the foreseeable future. Here, J = 1, hx = 0.5 and n = 6. Note, 800 random unitaries
In the past few years, error mitigation has been a thriving were used to collect this data giving an uncertainty in the purity of
research direction as more and more quantum devices become δρ ∼ O(10−2 ). Clearly the mitigation allows results to go from
available. Error mitigation strategies typically address mea- qualitative agreement to quantitative agreement with the results ob-
surement errors [7–15] or the algorithms and gates employed tained numerically through exact diagonalization (ED).
for digital quantum simulation [16–23]. While some of these
techniques have proven useful for certain quantum algorithms,
their general success is hampered by the fact that they either these methods train classical computers to predict the error
rely on a high level of control of the quantum device itself, seen in a quantum device and use the results to infer the error-
or are specifically designed for a given quantum simulation free quantum simulations. While successful, these methods
problem, i.e. exploiting specific symmetries [24]. require a large increase in the classical computational over-
One promising direction is to employ machine learning al- head and are somewhat uncontrolled. The latter drawback is
gorithms [25–27] for error mitigation. Generally speaking, also true for popular protocols based on the idea of increas-2
ing errors in the device systematically and then extrapolating supplementary material). Last but not least, the large improve-
back to the zero error case [21, 28–34]. In general, how to ment in the quality of results on the IBM device motivate our
tune the error rates varies from device to device and reliable choice of depolarizing errors a posteriori.
fitting requires expert knowledge of the specific hardware. An n-single qubit depolarizing error channel can be mod-
Here, we propose a new protocol for gate error mitiga- elled via
tion combining a raft of desirable features: it is easily imple- X X n
p
mentable on any quantum device with little increase in work- E ⊗n (ρ) = (1 − p)n ρ + (1 − p)n−1 σαj ρσαj + ...
3
load, it is well rooted in a mathematical description of the α∈[x,y,z]j=1
errors, and it is suitable for any quantum algorithm of inter- (2)
est. Furthermore, as we show via specific examples, it can where E is the error channel, p is the probability of an error
lead to to large improvements in the performance of quantum occurring for each qubit (assumed to be equal for each qubit,
devices, for example see Fig. 1 for Rényi entropy results. and for each Pauli error) and ‘...’ indicates higher-order terms
The paper is organised as follows: First, we derive an ansatz corresponding to errors on multiple qubits [36]. One impor-
for the density matrix resulting from the action of a noisy tant feature of this mathematical formulation is that the second
quantum simulator with global depolarizing errors. Follow- term, which describes the depolarizing error, commutes with
ing this, we explain how this ansatz can be used for a general any unitary operator. Consequently, the error on the ith qubit
error mitigation protocol. We then demonstrate its effective- in a quantum circuit with purely depolarizing errors is
ness by studying real time dynamics of the transverse field Ii
Ising model (TFIM) with a longitudinal field. In that context, E i (ρ) = (1 − pi )ρ + p Tri [ρ] ⊗ (3)
2
some of the authors recently showed that signatures of con-
in which ρ is the density matrix, Ii /2 is the maximally mixed
finement and entanglement spreading can be observed on the
(i.e. completely depolarized) state for the ith qubit, Tri is the
IBM quantum computer [35]. With our new error mitigation
partial trace over the ith qubit and pi is the error on the ith
protocol we are able to measure the meson masses of confine-
qubit.
ment induced bound states directly on the IBM device and can
Instead of dealing with all combinations of single qubit er-
quantitatively measure entanglement spreading previously out
rors, we approximate the total error channel of Eq. (2), under
of reach. Furthermore, we corroborate these results with clas-
the assumption of symmetric depolarization Eq. (3), as an ef-
sical simulations of error models taken from an IBMQ device
fective depolarizing channel on the entire quantum state
in a more general setting to show the wide applicability of our
mitigation scheme. Finally, we close with a discussion and I ⊗n
ρ = (1 − ptot )ρexact + ptot (4)
outline future applications. 2n
where the effective total error probability
Q is ptot . In princi-
ple ptot is well approximated by i (1 − pi ), however, we
ERROR MITIGATION PROTOCOL do not make that identification here. We will soon show that
ptot can be measured directly on the device. The simplicity
Error channels can be conveniently modelled through of this ansatz allows it to be easily calculated. It has already
Kraus operators [36] which define a completely positive map been shown to be useful when mitigating measurement error
on the density matrix [37], however we later demonstrate that it is powerful tool
for mitigating global depolarizing errors which do not them-
Ki ρKi† such that Ki† Ki = 1.
X X
ρ→ (1) selves originate from local depolarizing errors. Furthermore,
i i we later numerically demonstrate that this ansatz does not rely
on the assumption of single- and two-qubit depolarizing er-
For a single qubit, choosing four Kraus operators to be pro- rors.
portional to the Pauli operators (identity included) defines a The many partial traces over single qubits, which would
depolarizing channel. The proportionality constants are re- have conserved some coherence in the remaining qubits, have
lated to the individual error probabilities and must satisfy the been replaced by the maximally mixed state I ⊗n /2n over the
identity constraint to preserve the trace of ρ. global quantum state, destroying all coherence. We stress that
Here, we concentrate on such depolarizing errors which are even though this may not be a good approximation for a sin-
inevitably present in any digital quantum simulator platform gle layer of qubit errors, it becomes a reasonable approxima-
and can be treated without much specific knowledge about the tion for the error channel of a many layer, many qubit circuit.
device performance (which also fluctuates over time). More- Eq. (4) is our basic ansatz for an effective error model after
over, our focus on depolarizing errors has the advantage of a many-layered unitary circuit and ρexact is the exact density
being treatable mathematically in a controlled way as detailed operator without noise (see Supplementary Material for more
below. In fact, given a deep enough quantum circuit (with details). With this ansatz one can analytically calculate the
sufficient qubits) for self-averaging of incoherent errors to oc- effect of errors on a measured observable, Ô, via
cur, this depolarizing error model is a good approximation to h i p
tot
the physical errors in a device (more details are given in the hÔi = (1 − ptot ) Tr Ôρexact + n Tr Ô . (5)
23
Our approach estimates ptot in order to apply error mitiga- our quantum circuit approximates the time evolution opera-
tion. We propose and test two approaches for finding ptot , the tor U (t) = exp{−iHt} where we wish to measure hO(t)i
first based on estimating the purity of the final state and the for a range of times. We can tune the circuit such that
second based on studying specific observables. To estimate t ∗ (Emax ) = 4
magnetization, which further simplifies with Tr[σ α ] = 0 in
Eq. (5) to
1.00
hσiα i = (1 − ptot )hσiα iexact . (8)
0.95
The time dependence can be calculated by applying a quan-
ED
tum circuit from a Trotterisation of the time evolution oper-
Mitigated Fit ator (for details of the implementation and quantum circuits
NT = 6
see Refs. [24, 35]). Increasing the number of trotter steps,
0.90 NT = 5
hσ z i
Mitigated Data Nt , leads to a deeper circuit and the ensuing increase in errors
results in a peculiar dampening of the magnetization dynam-
ics which can be removed via our error mitigation. In Fig.
0.85
2 we display the results of the local magnetization dynamics.
Our mitigation technique not only allows us to measure the
main oscillation frequency on the IBM device but in fact to do
0.80
so with quantitative agreement with the analytical results (the
latter is explained in Ref. [35]).
We note that an additional simplification can be used for the
0 1 2 3 local magnetization
tJ for added efficiency — no additional mea-
surement of Tr ρ2 was required because the result of hσ z i
is known at tJ = 0 and thus ptot can be inferred from the
FIG. 2. Results from the IBM machine before and after mitiga- corresponding measurement on the IBM device after running
tion. Quench dynamics of local magnetization from Trotterized time the time evolution circuit for tJ ≈ 0. This big simplifica-
evolution of the TFIM with longitudinal field before and after error
mitigation. Here, J = 1, hx = 0.5, hx = 0.75 and n = 7. Data
tion avoids the costly randomized measurement scheme and
is shown for, Nt , the number of trotter steps Nt = 5, 6. Results go should be generally applicable for observables whose value
from qualitative agreement to quantitative agreement. After fitting is known at tJ = 0. However, as this does not distinguish
a cosine function (dashed green) to the mitigated data (green dots) correlated noise from the uncorrelated noise assumed in our
the dominant frequency is clearly captured by the IBM device, for ansatz, it is possible for over-estimations in the mitigation to
details see Supplementary Material. Note, more details of the circuit occur, resulting in non-physical results. An example of this is
composition of the evolution operator can be found in [35] seen in Fig. 2 at time tJ = 0.1 in which the local magnetiza-
tion is measured to be greater than unity.
We have obtained a range of results with different number
The confining potential between fermions induces non-
of trotter steps. Under the assumption that (1 − ptot ) scales as
ergodic behaviour [61], which manifests itself through per-
(1 − pT )NT , where pT is the error in one trotter step, we can
sistent oscillations in the local magnetization, hσiα i (α ∈
extrapolate the results back to the error-free case, see green
{x, y, z}) and a slowing down of the entanglement spreading.
data points in Fig. 2. We are then in a position to suppress the
noise to a level which enables us to extract different meson
masses on the IBM device by a basic fit of the main oscilla-
Example 1: Measuring Meson Masses
tion frequency. In Fig. 3 we show the data for the first three
masses obtainable by starting from different initial states, see
The frequencies of the oscillating magnetization can be insets. Here, 7 spins are mapped onto 5 qubits resulting in a
mapped directly to the energy of the domain wall bound circuit with 5NT + 5 single qubit gates and 8NT CNOT gates
states [48], which have a large overlap with chosen initial with NT = 5, 6. Remarkably, we find quantitative agreement
states. These so called meson masses are defined as the energy between the mitigated results and the theoretical predictions
difference between the lowest excited states and the ground for the scaling of the meson masses with the transverse and
state. longitudinal fields [35].
Recent work using a trapped ion quantum simulator to sim-
ulate the long-range TFIM model showed how, by choosing
a variety of initial states, the meson masses can be measured Example 2: Entanglement Spreading
through the persistent oscillations of local magnetization [62].
Previous attempts to perform a similar measurement on a dig- As a second example of we study the suppression of half
ital quantum computer have failed for the short ranged TFIM, chain entanglement entropy spreading after the spin chain
Eq. (7), because the results are too noisy to resolve the smaller quench. For hz = 0 the entanglement entropy is expected to
amplitude of oscillations [35]. As our main result, we show increase linearly from the ballistic spreading of free fermionic
that our new error mitigation enables us to obtain the meson excitations [64]. However, with a non-zero longitudinal field
masses from the IBM device. this growth is suppressed in a characteristic fashion due to
We are mainly interested in the time dependence of the local confinement [47]. To observe this we have implemented the5
FIG. 3. Results from the IBM device Toronto [63] of meson masses. (a,b) The quench dynamics of the z-axis local magnetization is shown
for different initial states. Here, J = 1, hx = 0.75, hx = 0.75 and n = 7. (a) Results calculated via exact diagonalizion. (b) Mitigated
results from the IBM device. Here, clear dominant oscillations are extracted that quantitatively agree with the analytically derived values. (c)
An illustration of how the masses are extracted from the analytically derived energy levels [35]. (d) A comparison of the masses obtained
from the IBM device and the analytically derived values for varying hz showing this quantitative agreement. Note, more details of the circuit
composition of the evolution operator can be found in [35]
randomized measurement protocol of Ref. [40] on the IBM APPLICATIONS TO MORE GENERAL EXAMPLES
device for measuring the second order Réyni entanglement
entropy. In Ref. [35] two of us obtained qualitative agree-
ment for entanglement dynamics of six spins compared to an To demonstrate that our method works more generally, we
exact diagonalization (ED) calculation, but for a quantitative simulate example tasks of measuring the expectation value of
agreement a large shift of the results was needed. different operators. We choose a layered, brickwork circuit
consisting of a layer of single qubit rotations followed by a
By using the ansatz in Eq. (4) we can see that the effect
of layer of entangling CNOTs — the same circuit structure as
global depolarizing errors on measurements of Tr ρ2A , where for Trotterized evolution (see Supplementary material). How-
A is a subsystem in consideration, is ever, instead of time parameterized Trotter evolution, we sim-
ply choose random single qubit angles, corresponding to dif-
Tr ρ2A =(1 − ptot )2 Tr ρ2A,exact
ferent states, and therefore different expectation values.
ptot (1 − ptot ) ptot2 (9) In order to obtain a larger amount of data to support our
+ + nA .
2nA −1 2 mitigation technique, we numerically simulate the layered
circuits with noisy gates. Qiskit’s circuit simulator allows
one to simulate arbitrary gate-based noise models by speci-
If ptot is known, Tr ρ2A,exact can be extracted and the second
order Rényi entropy measurement is calculated via fying independent Kraus operators for each single- and two-
qubit gates. Using this noisy simulator, we can directly test
how gate-level non-depolarizing errors can result in an ef-
S (2) (ρexact ) = − log2 (Tr ρ2A,exact ).
(10) fective global circuit-level depolarizing error, and how well
our mitigation technique works in the presence of local non-
Fig. 1 shows how this mitigation protocol eliminates the er- depolarizing errors. To simulate a realistic noise model,
ror in the second order Rényi entropy results. With our error we take the Kraus operators directly from the ibmq santiago
mitigation protocol and Eq. (9) we now obtain quantitative backend [65]. This error model goes beyond the single gate
agreement with ED results for six spins and can account for depolarizing assumption, by including (asymmetric) thermal
the large shift of the results. relaxation.6
In order to test local, non-local, single, and many Pauli-
string operators, we have chosen the following operators: sin-
gle Z operator (local, single term), TFIM Hamiltonian (local,
many terms), Random Pauli string (non-local, single term),
and the molecular Hamiltonian of a H4 (non-local, many
terms). Here many ’terms refers’ to the number of non-
commuting Pauli strings that must be measured to construct
the expectation value, and locality refers to only containing
single and two-qubit Pauli strings. Note the four hydrogen
Hamiltonian (see Supplementary for details) is chosen as a
simple molecular test case with a large Hilbert space, and
where we can ignore complications from freezing out orbitals.
Figure 4 shows unmitigated and mitigated expectation val-
ues for each operator, and clearly demonstrates the added
value of purity measurements (see Supplementary for simu-
lation details). For expectation value mitigation (i.e. using a
single expectation value to calibrate ptot ), a single extra cir-
cuit is needed. For trace mitigation (using Eq. (9) to calibrate
ptot ) uses 500 extra circuits to find ptot ; these are split into FIG. 4. Error mitigation simulations with non-depolarizing er-
five groups to reduce the bias. Once ptot is calibrated for a rors. Unmitigated (black diamonds), trace mitigated (green circles)
given circuit depth, this single value is used for error mitiga- and single operator mitigated (red circles) expectation values for dif-
tion of the same circuit structure, but for different single qubit ferent operators, with the solid blue line showing perfect mitigation.
parameter angles. The numerical simulation uses the general error model taken from
ibmq santiago. Expectation values are evaluated with respect to pa-
As the main results we find that our global depolarising rameterized brick-like circuits with (CNOT) depth 18 (see supp for
assumption is an effective error channel for the whole circuit, details), and thirty random parameter points per plot. Operators are
and, crucially, it does not require each single- and two-qubit (a) Hamiltonian of 4 linear-equidistant hydrogen atoms (6 qubits),
errors to be themselves purely depolarising. (b) Single Z Pauli term (8 qubits), (c) TFIM Hamiltonian (6 qubits)
and (d) Single XY ZIXX string (6 qubits). These operators are
chosen to span local/global Pauli strings, and contain single/many
DISCUSSION non-commuting decompositions.
In this work we have proposed an error mitigation technique
which is simple to implement but which retains the mathe- sufficient. This should be true generally for quantum circuits
matical rigour of more complicated techniques. Our proto- which can be tuned to be close to the identity. However, this
col is directly applicable to any quantum simulation whose scalable simplification for obtaining ptot potentially faces the
measurements are basic expectation values, as exemplified by problem that it includes correlated errors which makes the ran-
our results for the time evolution of the local magnetization domised measurements preferable as long as it is feasible.
in spin chain dynamics. Though our error mitigation method The fact that our basic assumption of global depolarizing
may not replace full state tomography, we expect it to be use- errors leads to such large improvements in results is in itself
ful in measuring more complicated quantities beyond simple remarkable. The basic conclusion is that the total error on the
observables, as corroborated by our results for the entangle- IBM device is close to a global depolarizing error at least for
ment entropy. An interesting avenue of future research will our choice of problems. We note our error ansatz, Eq. (4) is
be an application to variational quantum eigensolver (VQE) an approximation even for single qubit depolarizing errors, let
problems [66] and other quantum circuits. alone the more complex channels that are no-doubt present in
The computational overhead of implementing our protocol physical devices. Nevertheless the ansatz works remarkably
is dominated by the cost of evaluating Tr ρ2 on the quan- well for correcting physical errors of large/deep circuits and
tum device. The number of corresponding measurements is is a no-lose addition to quantum simulation protocols. We
Nu Nm where Nu is the number of randomised unitaries and suggest this is because the depolarizing channel is a good ap-
Nm is the number of random measurements. Within the ran- proximation per gate in a many gate quantum circuit, even if
domised scheme of Ref. [39] it grows exponentially with the this approximation breaks down for single layers of gates. Of
system size [40], which poses a potential problem for large course, our mitigation falls short of accounting for large co-
quantum computers but is easily feasible for currently avail- herent or correlated errors and it will be a worthwhile endeav-
able NISQ devices. Alas, in some cases the costly random- our to think about a controlled extension of our basic ansatz
ized measurement scheme can be avoided entirely, i.e. in our Eq. (4) for the density matrix of a NISQ device. In particular,
benchmark example of spin chain dynamics a single measure- whether we can extend it to incorporate other aspects of error
ment of a local observable whose exact value is known was channels [33].7
As an application of our error mitigation we have presented (2019).
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We can construct the quantum channel for the full circuit.
[70] Assuming many qubits, but not necessarily all qubits have gates
applied. The non-unitary part of Eq. (11) commutes with itself over
[71] J. R. McClean, N. C. Rubin, K. J. Sung, I. D. Kivlichan, different layers meaning the a full circuit error channel is ap-
X. Bonet-Monroig, Y. Cao, C. Dai, E. S. Fried, C. Gidney, proximated as
B. Gimby, P. Gokhale, T. Häner, T. Hardikar, V. Havlı́ček, †
O. Higgott, C. Huang, J. Izaac, Z. Jiang, X. Liu, S. McArdle, Ecirc [ρ] = (1 − ptot )Ucirc ρUcirc + ptot I2⊗n /2n (12)
M. Neeley, T. O’Brien, B. O’Gorman, I. Ozfidan, M. D. Radin,
where (1 − ptot ) = l (1 − p0l ) (product over effective layer
Q
J. Romero, N. P. D. Sawaya, B. Senjean, K. Setia, S. Sim, D. S.
Steiger, M. Steudtner, Q. Sun, W. Sun, D. Wang, F. Zhang, and p0 ’s) holds dueQ to commuting non-unitary terms. While this is
R. Babbush, Quantum Sci. Technol. 5, 034014 (2020). not equal to g (1−pg ) (product over gate error probabilities),
[72] S. Bravyi, J. M. Gambetta, A. Mezzacapo, and K. Temme, the latter is a good zeroth-order estimate of this term.
arXiv (2017), 1701.08213. Finally we note the total map Eq. (12) unravels to an ef-
fective error model for single (i.e. one or two qubit) gates,
given by: Ei [ρ] = (1 − p00i )Ui ρUi† + p00i I2⊗n /2n . Note the
all-or-nothing description of the map: either the gate is imple-
mented perfectly, or the entire quantum state is destroyed, not
just part of the state as is the case for Tri [ρ] ⊗ Ii /2 etc. While9
this seems quite an unphysical model for a single gate, it is a to particle spin) are removed, reducing the number of qubits
good model of the effective gate error per gate in a multi-gate by two. For the TFIM, the Hamiltonian was reduced to two
circuit. non-commuting Pauli strings from which the total Hamilto-
nian can be evaluated. Finally the single Z, and XY ZIXX
(chosen randomly) operators were measured trivially. We do
Cosine Fitting not directly account for State-Preparation and Measurement
(SPAM) errors, but we note that we can (and do) include
In order to extract the frequencies from the local magneti- SPAM errors as contributing to ptot in our channel ansatz.
zation results presented in Fig. 2 and Fig. 3, a function that We numerically simulate circuits consisting of layers of pa-
contained a singular cosine was used, namely, rameterized u3 gates, generic single-qubit rotation gates with
3 Euler angles, followed by an entangling layer. The circuit
hσ z (t)i ∼ Ae−dt cos(ωt) + c1 t + c2 . (13) depth is defined by the number of entangling (CNOT) layers.
There is a final u3 layer before measurement. The parameter-
Here, A is the amplitude of the oscillations, D allows for de- ized circuit now creates a state ρ(θ), and varying θ changes the
cay of the amplitude, ω is the oscillation frequency and c1 and state and corresponding operator expectation values. For the
c2 give a time dependent shift. While this function can not numerical simulations we consider how our mitigation per-
capture long term behaviour of hσ z i, it is able to extract out forms for different depth circuits, and for the four tasks shown
the dominant oscillations for short time dynamics. in the main text. For each depth the calibration ptot must be
done independently, as ptot is an effective parameter for the
whole quantum channel and cannot simple be calculated a pri-
Statistical Errors Introduced by the Error Mitigation ori for different depths.
Calibration.— Two independent methods of calibrating
The error when calculating ptot from Tr ρ2 is given by ptot are considered. Firstly, via single expectation value: di-
rect simulation of a circuit with fixed parameters, which is
δ Tr ρ2 then compared to the error free expectation value to find ptot .
δptot = ±(1 − ptot ) , (14)
2(2n − Tr(ρ2 )) Secondly, via trace: from directly measuring Tr[ρ2 ] and esti-
mating ptot via Eq. (9).
where δptot is the errorin ptot and δ Tr ρ2 is the error in the For single expectation calibration, a single fixed parameter
measurement of Tr ρ2 . Note that δ Tr ρ2 can be calculated circuit was used to find hOi (for each operator O considered).
by resampling e.g. Jackknife or Bootstrap [35, 40]. Thus, This result was compared to the error free simulation of the
when using the proposed error mitigation on an observable Ô exact same circuit to estimate ptot . To evaluate these expecta-
this results additional uncertainty on top of the measurement tion values 40960 shots were used for each non-commuting
errors. More precisely, term in the decomposition of the given operator into Pauli
! strings. For trace mitigation, an extra 500 circuits are used
−n to estimate ptot using 8196 shots per circuit. This was broken
δmeas hÔi δptot hÔi − 2 Tr Ô
δhÔi = ± + , (15) up into five lots of 100 circuits to estimate ptot at five differ-
1 − ptot (1 − ptot )2 ent (random) circuit parameterizations. For each parameteri-
zation, the method of Ref. [39] is used to find five independent
where δhÔi is the error in the mitigated value of hÔi and estimates of ptot , which are then averaged to reduce the bias
δmeas hÔi is the error in the measurement of hÔi. Note, as the in the estimate of ptot for the circuit.
local magnetisation results presented in this paper bypassed Mitigation.— Once ptot is estimated either from the single
the measurement of T r(ρ2 ), the error contributing to δptot is value or trace measurement, it is used to mitigate further simu-
just shot noise. Thus, δhσiz i is of the on the order ∼ O(10−3 ) lated expectation values. New expectation values are obtained
and is not displayed in Fig. 2 or Fig. 3. by varying the parameters in the circuit, while keeping the cir-
cuit structure fixed. For each depth and each task, thirty dif-
ferent (parameterized) circuits are simulated, and mitigation
Numerical simulations applied. Figure 5 shows the mitigation results for different cir-
cuit depths. While the full trace-estimation is robust showing
All numerical simulations are performed using qiskit [7] a significant improvement in every case, the single expectation
with the santaiago error model (as of 12t h March 2021). value mitigation is more variable. This is due to the vanishing
For the hydrogen Hamiltonian, openfermion [71] was used expectation value of Pauli strings for random circuits, which
to compute the orbital integrals (for a row of four H-atoms in turn increases the effect of shot noise when inverting Eq. 8.
with 0.1 nm inter-atomic distance), and then map the prob- We note this shot noise problem is absent when calibrating a
lem to qubits using the symmetry conserving Bravyi-Kitaev Pauli string (or single operator) with a (noise free) unit expec-
transformation [72], finally the Z2 symmetries (corresponding tation value, as in the main text.10 FIG. 5. Complete mitigation simulations with non-depolarizing errors. Scaling of our mitigation schemes (circles) with circuit depth for four different operator expectation values. Single operator (red circles) show some improvement, while trace mitigated (green circles) show a robust improvement over the unmitigated results. As in the main text, numerical simulation uses the ibmq santiago error model, and circuits are layers u3 and CNOT, with the depth equal to the number of entangling layers. Note there are some combinations (e.g. TFIM at depth 14 etc) where the single expectation value mitigation found ptot = 1, and therefore did not improve results. Likewise there are other combinations (e.g. TFIM depth 22) where the single expectation value mitigation is omitted because it gave an unphysical estimate of ptot < 0. Inset numbers show ptot as estimated for the trace estimation.
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