Quantum Error Mitigation using Symmetry Expansion
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Quantum Error Mitigation using Symmetry Expansion
Zhenyu Cai1,2
1 Department of Materials, University of Oxford, Oxford, OX1 3PH, United Kingdom
2 Quantum Motion Technologies Ltd, Nexus, Discovery Way, Leeds, LS2 3AA, United Kingdom
Even with the recent rapid developments in computational tasks to be performed that are well be-
quantum hardware, noise remains the biggest yond classical capabilities [1, 2], prompting the ques-
challenge for the practical applications of any tion of whether it is possible to perform any prac-
near-term quantum devices. Full quantum er- tically useful tasks with the noisy intermediate-scale
ror correction cannot be implemented in these quantum (NISQ) devices that we will soon have.
devices due to their limited scale. There- One of the biggest roadblocks for such a goal is
fore instead of relying on engineered code how to alleviate the damage done by the noise in
arXiv:2101.03151v3 [quant-ph] 14 Sep 2021
symmetry, symmetry verification was devel- these NISQ devices with the limited quantum resource
oped which uses the inherent symmetry within we have. This brings us to quantum error mitiga-
the physical problem we try to solve. In tion, which unlike quantum error correction, mostly
this article, we develop a general framework relies on performing additional measurements instead
named symmetry expansion which provides a of employing additional qubits to mitigate the dam-
wide spectrum of symmetry-based error miti- ages caused by errors [3–6]. Quantum error mitiga-
gation schemes beyond symmetry verification, tion has been successfully implemented in various ex-
enabling us to achieve different balances be- perimental settings [7–11]. Symmetry verification is
tween the estimation bias and the sampling one such error-mitigation technique that projects the
cost of the scheme. We show that certain noisy output quantum state back into the symmetry
symmetry expansion schemes can achieve a subspace defined by the physical problem we try to
smaller estimation bias than symmetry veri- solve [12, 13]. Besides measuring all symmetries in
fication through cancellation between the bi- every circuit run and discarding the runs that fail any
ases due to the detectable and undetectable symmetry test, symmetry verification can also be car-
noise components. A practical way to search ried out in a post-processing way with a much simpler
for such a small-bias scheme is introduced. measurement scheme inspired by quantum subspace
By numerically simulating the Fermi-Hubbard expansion [12, 14]. Since stabiliser code decoding can
model for energy estimation, the small-bias be viewed as symmetry verification with engineered
symmetry expansion we found can achieve an symmetry, we can also apply similar post-processing
estimation bias 6 to 9 times below what is techniques to reduce the practical challenges in the
achievable by symmetry verification when the stabiliser code implementation [15].
average number of circuit errors is between 1 More recently, two teams have independently de-
to 2. The corresponding sampling cost for ran- veloped a quantum error mitigation scheme that uses
dom shot noise reduction is just 2 to 6 times multiple copies of the noisy quantum device to sup-
higher than symmetry verification. Beyond press errors exponentially as the number of copies in-
symmetries inherent to the physical problem, creases [16, 17]. The method was named virtual dis-
our formalism is also applicable to engineered tillation by one of the teams. Its core idea relies on
symmetries. For example, the recent scheme the permutation symmetry among the copies, but the
for exponential error suppression using multi- way it is implemented does not involve projecting the
ple noisy copies of the quantum device is just a noisy state into the correct symmetry subspace, and
special case of symmetry expansion using the thus does not fall within the symmetry verification
permutation symmetry among the copies. framework. In fact, the corresponding permutation
symmetry verification [18–20] can only suppress er-
rors linearly with the increase of the number of copies
1 Introduction instead of exponentially.
The ultimate goal of implementing a fully fault- In this article, we will provide a general frame-
tolerant quantum error correction scheme for quan- work named symmetry expansion that encompasses a
tum algorithms with provable speed-up may still be much wider range of symmetry-based error mitigation
years away. However, the recent rapid advance in schemes beyond symmetry verification. We will dis-
quantum computing hardware has enabled certain cuss ways to search within this wide range of symme-
try expansion schemes and identify one that can out-
Zhenyu Cai: cai.zhenyu.physics@gmail.com
perform symmetry verification, just as virtual distilla-
Accepted in Quantum 2021-09-14, click title to verify. Published under CC-BY 4.0. 1tion arises from the permutation symmetry group and Similarly for biases, we often care more about the
vastly outperform the permutation symmetry verifica- fractional deviation from the ideal value, which we
tion. will call the relative bias and denoted as:
We will start by introducing the general concepts of hO0 i − hOem i Bias [Oem ]
quantum error mitigation in Section 2 and the frame- em (O) = =− . (2)
hO0 i hO0 i
work of symmetry expansion in Section 3. Then in
Section 4 and Section 5, we will discuss the estima- The additional minus sign relative to the bias is for a
tion bias and the implementation costs for symmetry simpler connection to quantum state infidelity later,
expansion. Moving onto Section 6, we will outline a but the essential idea does not change.
practical way to identify a good symmetry expansion When trying to compare two different error mitiga-
scheme. In the end, we will finish with an explicit tion schemes, if one of the schemes has both a smaller
numerical example for the application of symmetry bias and a smaller variance (sampling cost), then that
expansion in Section 7, along with the conclusion and would simply be the better scheme. However, the
outlook in Section 8. more common and more interesting case is when one
scheme has a smaller bias while the other has a smaller
variance, which means we need to compare the bias-
2 Quantum Error Mitigation variance trade-off between the two schemes more care-
fully. As the number of samples increase, the shot
In quantum error correction, we encodes the quan-
noise contribution from the variance will decrease and
tum information into a code space by employing addi-
the error due to the bias will become more and more
tional qubits, thus we can recover the noiseless state
dominant, and as a result the small-bias scheme would
ρ0 by project our noisy state ρ back into the code
become more and more favourable. In Appendix A,
space. On the other hand, in quantum error mitiga-
we have shown that in practice the minimum number
tion, instead of trying to recover the ideal state, we
of samples required for a small-bias estimator A to
are trying to recover the ideal measurement statistics
outperform a small-variance estimator B for a Pauli
for some observable. For a given observable O, we
observable O is:
will slightly abuse the notation and denote the ran-
dom variable corresponding to its measurement re- CA − CB
N∗ . . (3)
sults of the unmitigated noisy state ρ as simply O, B (O)2 − A (O)2
i.e. hOi = Tr(Oρ), while the random variable cor-
As we will see later, the general framework of sym-
responding to the measurement results of the noise-
metry expansion will enable us to predict the bias and
less state ρ0 is denoted as O0 , i.e. hO0 i = Tr(Oρ0 ).
variance of different symmetry-based error-mitigation
Using the noisy circuit to estimate the noiseless ex-
schemes, and thus enable us to choose a suitable
pectation value can lead to a large estimation bias
scheme for our given scenario (or at least rule out
|Bias [O]| = |hOi − hO0 i| when there is a large amount
some impractical schemes).
of noise present. Hence, we can instead try to con-
struct an error-mitigated estimator Oem using the
noisy data obtained from different noisy circuit con- 3 Symmetry Expansion
figurations, which would hopefully lead to a smaller
bias in the estimates: We are given a circuit that would ideally generate the
noiseless state |ψ0 i which is stabilised by a group of
|Bias [Oem ]| ≤ |Bias [O]|. symmetry operations G:
Note that throughout this article, we often refer to the G |ψ0 i = |ψ0 i ∀G ∈ G.
magnitude of bias simply as “bias” just like above, the
exact meaning should be clear from the context. The group of symmetry is usually identified through
An error-mitigated estimator that reduces the bias our knowledge of the structure of the circuit and/or
will usually come at a cost of a larger variance the physical meaning of the state (see Appendix B).
The symmetry subspace that |ψ0 i lives in, within
Var [Oem ] ≥ Var [O] ∼ Var [O0 ] which all G ∈ G will act trivially, is defined by the
projector [21]:
which means that a larger number of samples are
needed to reduce the shot noise in the sample esti- 1 X
ΠG = G. (4)
mate. Very often we care more about the factor of in- |G|
G∈G
crease of variance for a given error mitigation scheme
compared to the unmitigated scheme, which will be By definition we have
called its sampling cost: Π2G = ΠG , ΠG |ψ0 i = |ψ0 i . (5)
Var [Oem ] Note that since G is a unitary group, we know that
Cem = . (1)
Var [O] for any G ∈ G we also have G† ∈ G. Therefore ΠG
Accepted in Quantum 2021-09-14, click title to verify. Published under CC-BY 4.0. 2being an uniform average of elements in G will be where Γw~ is the symmetry expansion operator :
Hermitian: Π†G = ΠG . P
In reality, the output of the circuit is a noisy state Γw~ = PG∈G
wG G
(12)
ρ that might have components outside the symme- G∈G wG
try subspace. Hence, we will project ρ back into the
symmetry subspace to mitigate the damage of the and ρw~ is the symmetry-expanded ‘density operator’:
noise and we will call this process symmetry verifica-
tion [12, 13]. The effective density matrix we obtain Γw~ ρ Γw~ ρ
ρw~ = = . (13)
after symmetry verification is simply: Tr(Γw~ ρ) hΓw~ i
ΠG ρΠG ΠG ρΠG The symmetry expansion operation acting on ρ here
ρsym = = . (6) is not necessarily a positive map. Hence, though ρw~
Tr(ΠG ρ) hΠG i
has unit trace as expected, it is not necessarily pos-
Throughout this article, we will use hOi = Tr(Oρ) itive semi-definite and thus is not necessarily a valid
to denote the expectation value of some observable O density operator in the usual sense. Note that Γw~ is
from the noisy circuit. Using Eq. (6), the expectation also a symmetry operator:
value of some observable O after symmetry verifica-
tion will be: Γw~ |ψ0 i = |ψ0 i , Γw~ ΠG = ΠG . (14)
Tr(OΠG ρΠG ) hΠG OΠG i
Tr(Oρsym ) = = . (7) An important class of symmetry expansion would
hΠG i hΠG i
be a uniform expansion using a subset of symmetry
In direct symmetry verification, we will measure the operators F ⊆ G (note that F is not necessarily a
observable O and the symmetry verification result ΠG group), which would be our main focus later on. We
in the same circuit run, and discard the circuit runs ~ = ~F
will denote the corresponding weight vector as w
that has ΠG measured to be 0. This would require and the symmetry expansion operator is simply
[O, ΠG ] = 0 (8) 1 X
Γ~F = F.
|F|
and thus we have: F ∈F
Tr(OΠG ρ) hOΠG i When F contains only one symmetry operator F , we
Tr(Oρsym ) = = . (9)
hΠG i hΠG i will slightly abuse the notation and write F = F and
thus ΓF~ ≡ F .
Alternatively, Eq. (9) will also hold if we have The unmitigated noisy state is simply a special case
of symmetry expansion using just the identity opera-
[ρ, ΠG ] = 0. (10)
tor:
How this may arise will be further discussed in Sec-
tion 6.4. Note that the normalisation factor hΠG i is ΓI~ = I, ρI~ ≡ ρ.
essentially the fraction of circuit runs that pass the
symmetry verification (i.e. those runs with ΠG mea- Symmetry verification, when used for the purpose of
sured to be 1). expectation value estimation in Eq. (9), is equivalent
We can also apply symmetry verification in a post- to a uniform symmetry expansion over the full set of
processing way. Using Eq. (4), we see that we can symmetry operators G:
actually obtain hΠG i by uniformly sampling random
~ = ΠG ,
ΓG ~ ≡ ρsym .
ρG (15)
G ∈ G. Similarly, we can also obtain hOΠG i by uni-
formly sampling random OG for G ∈ G. In each
Hence, from here on, our discussion of general sym-
circuit run, if we measure O along with a random
metry expansion schemes will also be applicable to
G ∈ G, we can then compose them to obtain both
G and OG and thus effectively obtaining samples for • the unmitigated case: w ~
~ = I,
hΠG i and hOΠG i, which in turn can give us the esti-
mate of Tr(Oρsym ) via Eq. (9). • the symmetry-verified case: w ~
~ = G.
Instead of sampling uniformly from G, we can sam-
ple each symmetry G with a positive weight wG , and As discussed in Refs. [12], symmetry verification is
we will call this more general sampling scheme sym- just a special case of quantum subspace expansion [14]
metry expansion. By denoting the sampling weight and the idea of changing the weight of the symme-
distribution as w,
~ the resultant observable following try in the projection operator has been discussed be-
our new sampling scheme will be: fore under the quantum subspace expansion frame-
work [15]. Assuming all symmetry operators G ∈ G
Tr(OΓw~ ρ) hOΓw~ i commute with the observable O or the noisy state ρ (a
Tr(Oρw~ ) = = (11)
Tr(Γw~ ρ) hΓw~ i special case of Eq. (8) and Eq. (10)), the expectation
Accepted in Quantum 2021-09-14, click title to verify. Published under CC-BY 4.0. 3value after performing quantum subspace expansion via both theoretical analysis and numerical simula-
using the expansion operator Γ~v is given by tions [16, 17].
D E
Tr OΓ~v ρΓ~†v OΓ~†v Γ~v
= D E . (16) 4 Estimation Bias of Symmetry Ex-
Tr Γ~v ρΓ~†v Γ~†v Γ~v
pansion
Such quantum subspace expansion using symmetry
operators will simply be called symmetry subspace 4.1 Absolute Infidelity
expansion (not to be confused with symmetry expan- Using Eq. (2), the relative bias for a given observable
sion). Compared to Eq. (11), we see that symme- O when symmetry expanded with weight w ~ is simply:
try subspace expansion is simply the special case of
symmetry expansion in which the expansion opera- w~ (O) = 1 −
Tr(Oρw~ )
. (18)
tor Γw~ is the square of some other expansion oper- Tr(Oρ0 )
ator: Γw~ = Γ~†v Γ~v , i.e. Γw~ is positive semi-definite.
Here ρ0 = |ψ0 i hψ0 | and ρw~ are the ideal state and
In Appendix C, we have shown that the symmetry
symmetry-expanded state, respectively. Recall that
subspace expansion scheme that would maximise the
the unmitigated case is simply w ~ = I~ while the
fidelity against the ideal state is simply the symme- ~
symmetry-verified case is w
~ = G.
try verification scheme. Non-uniform weight distribu-
To show that symmetry verification can produce
tion for symmetry subspace expansion will only bring
a smaller estimation bias than the unmitigated case,
advantages over symmetry verification when we are
one possible way is compare their output state fidelity
restricted to a truncated set of symmetry operators
against the ideal state:
that do not form a group [15]. Hence, in this article
we will only focus on the comparison between sym- Unmitigated: FI~ = Tr(ρρ0 ) = hρ0 i (19)
metry verification and symmetry expansion since we hρ0 i
are interested in the case in which we have access to Symmetry-verified: FG
~ = Tr ρG
~ ρ0 = . (20)
ΓG
the full set of symmetries. ~
To perform direct symmetry verification for a gen- Here we have used Eq. (13) and Eq. (14). We see that
eral symmetry group, we can have an ancilla register the symmetry-verified fidelity is just the unmitigated
with its different basis states corresponding to dif- noisy fidelity hρ0 i boosted by a factor of ΓG
~
−1
.
ferent elements in the group, then perform control- We might be tempted to generalise the above argu-
symmetry operations from the ancilla to the data ments and try to predict the performance of symmetry
register and measure out the ancilla as outlined in expansion using the symmetry-expanded fidelity :
Ref. [19]. For Pauli symmetry groups (stabiliser
groups), their direct symmetry verification can be car- hρ0 i
Fw~ = Tr(ρ0 ρw~ ) = , (21)
ried out by simply measuring the group generators hΓw~ i
since they are abelian. Similarly, for post-processing
which is just the unmitigated noisy fidelity hρ0 i
symmetry verification, Pauli symmetries are much −1
boosted by a factor of hΓw~ i . However, unlike the
easier to measure compared to general symmetries
unmitigated and symmetry-verified cases, Tr(ρ0 ρw~ ) is
since they are Hermitian. Hence, Pauli symmetries
not upper-bounded by 1 anymore since ρw~ might not
are most relevant to the practical implementations of
be positive semi-definite as mentioned in Section 3.
symmetry-based error mitigation techniques and will
Hence, instead we construct a metric that measure
be our main focus in this article. As a result, the
how far Fw~ = Tr(ρ0 ρw~ ) deviate from the ideal value
symmetry expansion operator Γw~ we look at will be
of 1, which is equivalent to the magnitude of the bias
Hermitian, and since all the symmetries are unitary,
in Eq. (18) with the observable being the ideal state
we have
ρ0 :
−1 ≤ hΓw~ i ≤ 1. (17) hρ0 i
|w~ (ρ0 )| = |1 − Fw~ | = 1 − . (22)
Though our discussion mainly focuses on Pauli sym- hΓw~ i
metries, many of the arguments hereafter are still ap- For the unmitigated and symmetry-verified cases we
plicable to general symmetries. An example for the have:
application of symmetry expansion beyond Pauli sym-
metry is the virtual distillation scheme mentioned in I~(ρ0 ) = I~(ρ0 ) = 1 − FI~ = 1 − hρ0 i (23)
Section 1, which makes use of the permutation sym- hρ0 i
metries among identical copies of the quantum sys- G ~ (ρ0 ) = 1 − FG
~ (ρ0 ) = G ~ =1− . (24)
ΓG
~
tem. The details of how it relates to symmetry ex-
pansion are outlined in Appendix H. Being a symme- which are simply the output state infidelities in the
try expansion scheme, it was shown to vastly outper- corresponding cases. Hence, we will call this metric
form the corresponding symmetry verification scheme the absolute infidelity.
Accepted in Quantum 2021-09-14, click title to verify. Published under CC-BY 4.0. 4In this article, we are going to use the absolute in- 2
Absolute Infidelity and its Components
fidelity, which is the magnitude of the bias with ob- Absolute Infidelity: | w( 0)|
Undetectable Bias: Bu, w
servable ρ0 , as our metric for the magnitude of the Detectable Bias: Bd, w
Unmitigated Infidelity
bias for a given error mitigation scheme. We have Symmetry-verified Infidelity
further discussed why this is a reasonable metric in 1
Appendix D, and the validity of this metric will also
I( 0)
be further verified in our numerical simulation later ( 0)
in Section 7. 0
4.2 Mechanism of Symmetry Expansion
The erroneous component of the symmetry-expanded -1
0 0 1
state ρw~ is defined to be the components that are Symmetry Expansion Expecation Value, w
orthogonal to the ideal state ρ0 , which can be ob-
tained by applying the projector 1 − ρ0 . Within these
Figure 1: The change in the absolute infidelity along with its
erroneous components, there are components living detectable and undetectable components against the change
within the symmetry subspace defined by the projec- in the noisy expectation value of the symmetry expansion
tor ΓG
~ , which are undetectable via symmetry verifica- operator. We always have hρ0 i ≤ Γ~G since the symmetry-
tion. The coefficient of these undetectable erroneous verified fidelity in Eq. (20) is always smaller than 1. Note that
components in ρw~ is simply: the absolute infidelity is simply the magnitude of the sum of
the detectable bias and the undetectable bias: |w~ (ρ0 )| =
~ − hρ0 i
ΓG
~ (1 − ρ0 )ρw
Bu,w~ = Tr ΓG ~ = . (25) |Bu,w~ + Bd,w~ |.
hΓw~ i
On the other hand, the erroneous components living
outside the symmetry subspace would be detectable, which is plotted in Fig. 1. Let us now go through
whose coefficient in ρw~ is: these different regions of symmetry expansion.
hΓw~ i − ΓG ~
Standard behaviour: 1 ≥ hΓw~ i ≥ ΓG
Bd,w~ = Tr 1 − ΓG
~ (1 − ρ0 )ρw
~ = . 4.2.1 ~
hΓw~ i
(26) Here both the detectable bias Bd,w~ and the unde-
tectable bias Bu,w~ are positive and thus the absolute
From Eq. (22), we see that the absolute infidelity is
infidelity is simply
simply the magnitude of the sum of the detectable
and undetectable error coefficients: |w~ (ρ0 )| = Bu,w~ + Bd,w~ .
|w~ (ρ0 )| = |Bu,w~ + Bd,w~ |. (27)
At the start of this region we have hΓw~ i = hIi = 1,
Hence, Bd,w~ and Bu,w~ are simply the bias of the which is simply the noisy unmitigated case. As we
symmetry-expanded estimation of the observable ρ0 modify our symmetry expansion scheme to decrease
due to the detectable and undetectable errors, respec- hΓw~ i, the undetectable bias Bu,w~ will increase while
tively. We will simply call them detectable and unde- the detectable bias Bd,w~ will decrease. In this region,
tectable bias. In the special case of w ~ = I, ~ i.e. the the absolute infidelity |w~ (ρ0 )| will decrease overall as
unmitigated case, Bu,I~ and Bd,I~ are simply the unde- a result of decrease in hΓw~ i.
tectable and detectable error probability, respectively. When we reach the point of hΓw~ i = ΓG ~ , we have
From Eq. (22), we can see that symmetry expan- arrived at symmetry verification, in which we have
sion with negative hΓw~ i will lead to |w~ (ρ0 )| ≥ 1 ≥ Bd,G~ = 0. Hence, symmetry verification have re-
I~(ρ0 ) , i.e. the corresponding symmetry expan- moved all the detectable error components and we
sion scheme will performs worse than the unmitigated have |w~ (ρ0 )| = Bu,w~ .
case, thus there is no point of applying symmetry ex-
pansion in this regime. Therefore, we will only con- 4.2.2 ~ > hΓw
Bias cancellation: ΓG ~ i ≥ hρ0 i
sider symmetry expansion operators with positive ex-
pectation values: 0 < hΓw~ i ≤ 1. In this regime, Bu,w~ In this region the detectable bias Bd,w~ turns to nega-
will increase as hΓw~ i decrease and will always be pos- tive and thus the overall error becomes
itive, while Bd,w~ will decrease as hΓw~ i decrease and
|w~ (ρ0 )| = Bu,w~ − |Bd,w~ |
will become negative once hΓw~ i < ΓG ~ . Hence, the
expression of the absolute infidelity can be split into We see that since the detectable bias Bd,w~ and the un-
the following regions: detectable bias Bu,w~ have different signs, the absolute
infidelity become the difference between the magni-
Bu,w~ + Bd,w~
1 ≥ hΓw~ i ≥ ΓG ~
|w~ (ρ0 )| = Bu,w~ − |Bd,w~ | ΓG tude of detectable and undetectable biases, allowing
~ > hΓw ~ i ≥ hρ0 i
cancellation between the detectable and undetectable
|Bd,w~ | − Bu,w~ hρ0 i > hΓw~ i > 0
bias and thus reducing the overall bias.
Accepted in Quantum 2021-09-14, click title to verify. Published under CC-BY 4.0. 5In this region, the absolute infidelity |w~ (ρ0 )| will We see that I~(ρ0 ) is proportional to the sum of Pu
still decrease overall as a result of decrease in hΓw~ i. and Pd since none of the errors are removed. We also
As we arrive at the point of hΓw~ i = hρ0 i, we have have G
~ (ρ0 ) being proportional to just Pu since the de-
Bu,w~ = |Bd,w~ |, which means perfect cancellation be- tectable errors are removed by symmetry verification.
tween the detectable and undetectable bias, and thus On the other hand, G ~ (ρ0 ) is proportional to the dif-
the absolute infidelity is reduce to |w~ (ρ0 )| = 0. ference between Pu and Pd , allowing cancellation be-
tween detectable and undetectable error probability,
4.2.3 Negative infidelity: 0 ≤ hΓw~ i ≤ hρ0 i which is why symmetry expanding with w ~ = G ~ can
achieve a smaller bias than symmetry verification.
In this region, we still have negative detectable bias
Bd,w~ , and thus there will still be cancellation between
the detectable and undetectable biases. However, now 5 Implementation Costs of Symmetry
the detectable bias has grown to a larger magnitude
than the undetectable bias and thus in this region we Expansion
have w~ (ρ0 ) being negative, which means:
5.1 Implementation
|w~ (ρ0 )| = |Bd,w~ | − Bu,w~ .
In direct symmetry verification, we need to measure
The absolute infidelity |w~ (ρ0 )| will increase as the whole set of symmetry generators and the observ-
we decrease hΓw~ i. When hΓw~ i decrease beyond able in the same circuit run, and discard runs that fail
−1 −1 any of the symmetry tests. By using additional circuit
−1
2 hρ0 i − ΓG
~ , we will have |w~ (ρ0 )| >
structures to simultaneously diagonalise all the ob-
~ (ρ0 ) , i.e. symmetry expansion will have a larger
G servables and symmetries we want to measure [22, 23],
bias than symmetry verification. When hΓw~ i decrease we would only need to perform local measurements
−1
beyond 2 hρ0 i − 1
−1
, we will have |w~ (ρ0 )| > directly on the data register without needing any an-
cilla. The additional circuit structure required can
I~(ρ0 ) , i.e. symmetry expansion will have a larger
be vastly simplified (or even removed) by using post-
bias than the unmitigated state.
processing symmetry verification and more generally
symmetry expansion, for which we only need to mea-
4.2.4 Simple example of bias cancellation
sure one symmetry operator along with the observ-
As explained above, some symmetry expansion ables in each circuit run. The Fermi-Hubbard simu-
schemes can achieve smaller absolute infidelity than lation that we will study in our numerical simulation
symmetry verification due to the cancellation between later in Section 7 is an example of no additional cir-
the detectable and undetectable bias. We will further cuit structure required to perform our measurements.
illustrate this through the simple example in which The detailed measurement scheme for this case can
there is just one single non-trivial symmetry opera- be found in Ref. [24].
tor: G = {I, G}. Since we are discarding circuit runs that fail the
We will compare the performance of three schemes: symmetry test in direct symmetry verification, we
are essentially changing the form of the effective er-
• Unmitigated w ~ hΓw~ i = 1
~ = I:
ror channels in the circuit. Therefore combining di-
• Symmetry-verified w ~ hΓw~ i = Γ~ =
~ = G: 1+hGi rect symmetry verification with other error mitiga-
G 2
tion techniques like quasi-probability and error ex-
• Symmetry-expanded w ~ hΓw~ i = hGi
~ = G: trapolation would require elaborate schemes [5] since
As mentioned, the undetectable/detectable biases they all rely on our knowledge about the error chan-
for the unmitigated case are simply the probability nels. On the other hand, there is no discarding circuit
that some errors undetectable/detectable by the sym- runs in symmetry expansion, and thus it can be com-
metry G have occurred in a given circuit run, which bined with other error mitigation techniques straight-
we will denote as Pu /Pd : forwardly, e.g. by simply performing symmetry ex-
pansion following Eq. (11) using hOΓw~ i and hΓw~ i that
~ − hρ0 i
Pu = Bu,I~ = ΓG are error-mitigated using the other error mitigation
Pd = Bd,I~ = 1 − ΓG techniques.
~
Hence, we can write the absolute infidelities of various
5.2 Sampling Cost
schemes in terms of Pu and Pd using Eq. (22):
Now looking beyond the circuit implementation, we
Unmitigated: I~(ρ0 ) = Pu + Pd
would want to know how many circuit runs are needed
Pu for symmetry expansion. For a given error mitigation
Symmetry-verified: G
~ (ρ0 ) =
1 − Pd (28) scheme we have defined the corresponding sampling
Pu − Pd cost in Eq. (1) as the factor of increase in the number
Symmetry-expanded: G ~ (ρ0 ) = .
1 − 2Pd of circuit runs required compared to the unmitigated
Accepted in Quantum 2021-09-14, click title to verify. Published under CC-BY 4.0. 6case. As shown in Appendix G, the sampling cost for Hence, the probability that there is no errors in the
implementing direct symmetry verification and sym- circuit will be:
metry expansion are
P0 = e−µ .
−1 −2
Cdir ∼ ΓG
~ , Cw~ ∼ hΓw~ i . (29)
Let us use f to denote the effective fraction of errors
As can be seen from Section 4.1, in the region where that would move the ideal state |ψ0 i out of the 1D
hΓw~ i ≥ hρ0 i (i.e. w~ (ρ0 ) is positive), the symmetry- subspace it spanned. Then the average number of
expanded fidelity behaves like regular fidelity and thus effective errors occurring on the ideal state is simply
can be use as a metric for the estimation bias of sym- f µ, and the fidelity hρ0 i is just the probability that
metry expansion. For all symmetry expansions in this zero effective errors happens:
region, including post-processing symmetry verifica-
tion, if we are allowed C sampling cost for our error hρ0 i ∼ P0 = e−f µ . (30)
mitigation technique,
√ symmetry expansion can give us
a factor of C boost in the fidelity based on Eq. (21) In practice, we usually assume f = 1 for crude esti-
and Eq. (29). Hence, we see that symmetry expansion mation of hρ0 i, which gives hρ0 i = e−µ .
in this regime is as ‘cost-effective’ as post-processing Moving on, using Eq. (12), hΓw~ i is simply
symmetry verification. Direct symmetry verification, P
G∈G wG hGi
on the other hand, can achieve a factor of C boost hΓw~ i = P . (31)
in the fidelity and thus is more ‘cost-effective’ but is G∈G wG
more difficult to implement due to the measurement Hence, we can estimate hΓw~ i by simply measuring G
requirement and harder to combine with other error with sampling weight wG at the end of the circuit.
mitigation techniques as discussed in Section 5.1.
In the region where hΓw~ i < hρ0 i, the symmetry-
expanded fidelity is not well-defined anymore and thus 6.2 Search for a Small-bias Symmetry Expan-
our metric for estimation bias becomes the absolute sion
infidelity as discussed in Section 4.1. In this region we In many of the practical application, due to the large
can achieve the same bias as the symmetry expansion amount of circuit noise present, very often the main
in the hΓw~ i ≥ hρ0 i region, but at a smaller hΓw~ i and bottleneck of quantum error mitigation is still trying
thus at a larger sampling cost Cw~ . Hence, symmetry to achieve a small enough estimation bias. Therefore,
expansion in this region is less ‘cost effective’ than we hope to use symmetry expansion to further reduce
post-processing symmetry verification. the estimation bias of symmetry-based error mitiga-
tion method beyond symmetry verification. Here we
6 Searching for Suitable Symmetry Ex- will discuss how to search for a symmetry expansion
scheme that has a smaller absolute infidelity and thus
pansion Schemes a smaller bias than symmetry verification.
The absolute infidelity |w~ (ρ0 )| of a given symmetry
6.1 Estimation of Fidelity and Symmetry Ex- expansion can be estimated using hρ0 i and hΓw~ i us-
pectation Value ing Eq. (22). However, due to the approximation we
made in estimating hρ0 i and hΓw~ i, it is usually hard
As discussed in Section 4 and Section 5, we can esti-
to fine-tune our expansion weights w ~ such that the
mate the bias and the sampling cost of a given symme-
absolute infidelity |w~ (ρ0 )| reach the exact value we
try expansion scheme using the expectation value of
want. On the other hand, the precision of our |w~ (ρ0 )|
its symmetry expansion operator hΓw~ i and the unmit-
estimates is often enough for comparing among the
igated fidelity hρ0 i. Hence, in order to compare among
restricted class of symmetry expansion scheme men-
different symmetry expansion schemes, we need to
tioned in Section 3, in which we only consider a uni-
first perform estimations of hρ0 i and hΓw~ i.
form expansion using a subset of symmetry operators
We will call the expected number of errors occurring
F ⊆ G. For such symmetry expansion schemes, we
in each circuit run the mean circuit error count and
have:
denote it using µ. In practical NISQ applications, we
would need to have a circuit that is large enough such Γ~F =
1 X
hF i .
that the number of error locations is much greater |F|
F ∈F
than 1 and is not too noisy such that the mean circuit
error count is of order unity: µ ∼ 1. In this limit, In order for the symmetry expansion scheme with
using the Le Cam’s theorem, the probability that n ~ = ~F to achieve an absolute infidelity ~F (ρ0 )
w
errors occurring in a given circuit run will follow the smaller than some threshold δ, using Eq. (22) we need
Poisson distribution provided the noise is Markovian: to have:
µn hρ0 i hρ0 i
Pn = e−µ . ≤ Γ~F ≤ . (32)
n! 1+δ 1−δ
Accepted in Quantum 2021-09-14, click title to verify. Published under CC-BY 4.0. 7Finding symmetry expansion schemes that has a Note that fI = 0. Using Eq. (31), we then have:
smaller bias than symmetry verification is simply the
wG e−2fG µ
P
case of δ = G~ (ρ0 ).
hΓw~ i ∼ G∈G . (37)
One way to construct symmetry expansions that
P
G∈G wG
fulfil this requirement is by first identifying the set
of symmetry operator R ⊆ G consists of all the sym- We can use Eq. (30) and Eq. (36) to rewrite Eq. (33)
metry operators whose noisy expectation value falling at the circuit error rate µ as:
within this range:
e−f µ e−f µ
≤ e−2fR µ ≤ ∀R ∈ R.
hρ0 i hρ0 i 1+δ 1−δ
≤ hRi ≤ ∀R ∈ R. (33)
1+δ 1−δ
In the limit of small δ, the first order approximation
In such a way, any subset F ⊆ R will always satisfy of the inequality above gives:
Eq. (32).
Using Eq. (32) we see that the scheme that has −δ ≤ − (2fR − f ) µ ≤ δ ∀R ∈ R.
a smaller Γ~F − hρ0 i can achieve a smaller estima-
i.e.
tion bias. Following Eq. (30) and assuming f ≈ 1, we
have hρ0 i ≈ e−µ . Hence, we should be able to achieve 2fR µ = f µ + O(δ) ∀R ∈ R.
a small estimation bias using the set of symmetry op-
erators that minimise Γ~F − e−µ , which is denoted In this small δ limit, the relative bias of symmetry ex-
as panding using any F ⊆ R can be obtain using Eq. (22):
G∗ = arg min Γ~F − e−µ . (34) |F|e−f µ 1 X
F⊆R ~F (ρ0 ) = 1 − P −2f µ
≈ (2fF − f ) µ,
F ∈F e
F |F|
F ∈F
We need to be careful about the case in which we (38)
have two schemes whose Γ~F are of similar distance
to e−µ , one from above and one from below. In this which is of O(δ) as expected. The corresponding sam-
case, the scheme with larger Γ~F is expected to lead pling cost can be obtained using Eq. (29):
to a smaller bias as it will have Γ~F closer to the ac-
tual hρ0 i (which is lower-bounded by e−µ ), and it is 2
|F| −2f µ
also expected to have a smaller sampling cost as dis- C~F = P 2 ∼ e . (39)
−2fF µ
F ∈F e
cussed in Section 5.2. Hence, the scheme with larger
Γ~F is generally preferred in such a case. We will We see that the sampling cost grows exponentially
call the symmetry expansion scheme found in the way with the mean circuit error count µ, similar to other
outlined above the small-bias expansion scheme. mainstream quantum error mitigation techniques [6].
In Appendix D and later in the numerical simu- Using Eq. (38) with f ≈ 1, the search for the small-
lation (Section 7), we can see that we usually have bias expansion scheme in Eq. (34) can now be written
w~ (O) ∼ w~ (ρ0 ). Hence, using Eq. (3), the minimum as
number of samples required for the small-bias expan-
sion scheme (w ~ =G ~ ∗ ) to outperform symmetry veri- 1 X
~ G∗ = arg min (2fF − 1) . (40)
fication (w~ = G) for a Pauli observable is simply: F⊆R |F|
F ∈F
CG ~ ∗ − CG Similarly to our discussion
P before, when we have two
~
N∗ . . (35)
2
~ (ρ0 ) − G
G ~ ∗ (ρ0 )
2
schemes with similar |F|
1
F ∈F (2fF − 1) , the scheme
with smaller |F| F ∈F fF will have larger hΓw~ i and
1
P
6.3 Alternative Search Strategy using Knowl- thus is preferred.
edge of Noise
When we have detailed knowledge about the noise 6.4 Reducing Search Space using Equivalent
channels, we can try to estimate hΓw~ i analytically Symmetry
instead. For a given Pauli symmetry G, if we can
If a given symmetry operator S commutes with the
approximately decomposed all noise channels within
noisy state ρ, which also implies S −1 commuting with
each error location into components that are de-
ρ, we then have:
tectable by G and components that are undetectable,
with fG being the fraction of errors detectable by G hGi = Tr S −1 SGρ = Tr SGS −1 ρ = SGS −1 .
averaged over all error locations, then as shown in (41)
Appendix E, we have:
One possible way that such a symmetry S can arise is
hGi ∼ e−2fG µ . (36) when it corresponds to some qubit permutations. In
Accepted in Quantum 2021-09-14, click title to verify. Published under CC-BY 4.0. 8such a case, we would naturally design a state prepa- parametrised. This circuit can be used in for example
ration circuit with a layout that respects this symme- variational eigen-solvers. We would perform simula-
try. Since we usually assume the nature of the noise tions for the 2 × 2 and 2 × 3 half-filled Fermi-Hubbard
of a given type of circuit components is the same for model, which corresponds to a 8-qubit circuit with 144
all of its instances in the circuit, we can expect the two-qubit gates and a 12-qubit circuit with 336 two-
resultant noisy state ρ coming out of this circuit will qubit gates, respectively. All two-qubit gates are as-
also follow the same symmetry: SρS −1 = ρ, i.e. ρ sumed to be affected by two-qubit depolarising noise
commutes with S. of the same strength, which is of the form:
Eq. (41) implies we can obtain the same value for p X
hΓw~ i even if we sample SGS −1 in place of G. Hence, D(ρ) = (1 − p)ρ + P ρP
|P2 |
G and SGS −1 are equivalent when we are consid- P ∈P2
ering hΓw~ i, which in turn means they are equiva-
where P2 is the full set of two-qubit Pauli opera-
lent when trying to construct a symmetry expansion
tors. Similar simulations are also performed for bit-
scheme with a given absolute infidelity and a given
flip noise in Appendix J, which yield similar results.
sampling cost.
We will look at a range of circuits with randomly
We will denote the set of symmetry operators that
generated parameters and thus has random ideal out-
commute with ρ as S, which is a subgroup of G. We
put states. Using tv,w = 1 and Uv = 2 for our Hamil-
say that two elements G and G0 are equivalent if they
tonian, we can measure the energy of the ideal output
give the same noisy expectation value. Alternatively,
state and compare it to the energies of the noisy out-
based on our arguments above, we can write this as
put state and the error-mitigated output states. In
G ∼ G0 iff ∃S ∈ S SGS −1 = G0 . our simulation, we will focus on circuits whose ideal
output energies magnitude are larger than 0.5 since in
We can prove that this is an equivalence relation, and practice we are usually interested in states with non-
thus will partition the whole symmetry group into negligible energy magnitude (otherwise the sampling
different equivalence classes. The symmetry opera- cost for high energy precision will be very large). The
tor within each equivalence class will have the same simulation code used is available online [25].
effect when sampled in symmetry expansion for calcu-
lating the fidelity against the ideal states. Hence, to 7.2 Different Symmetry Expansion Schemes
find the suitable symmetry expansion, we only need
to pick one element from each equivalence class and The ideal circuit will conserve the number of fermions
sample over them instead of sampling over the whole within each spin subspace. Hence, the symmetry op-
symmetry group. This vastly reduces the search space erator we can enforce on the output state would be
for the suitable expansion weight distribution. In the the spin-up/down number parity symmetry, denoted
extreme case that ρ commutes with all symmetry op- as G↑/↓ . In the Jordan-Wigner encoding, G↑/↓ is just
erators G ∈ G, then the equivalence classes are just the tensor product of the Z operators acting on the
the conjugacy classes of G. qubits corresponding to the spin-up/down orbitals.
Together they can generate the number parity sym-
metry group
7 Numerical Simulation
G = {I, G↑ , G↓ , Gtot }, (43)
7.1 Problem Setup where Gtot = G↑ G↓ is the total fermion number parity
The physical problem that we will be looking at is symmetry. As discussed in Section 6.2, our main focus
the 2D Fermi-Hubbard model with nearest-neighbour will be on the symmetry expansion schemes that are
interaction, with the Hamiltonian given as: uniform expansion of subsets of symmetry operators
X X F ⊆ G.
H=− tv,w a†v,σ aw,σ + a†w,σ av,σ + Uv nv,↑ nv,↓ . The Hamiltonian in Eq. (42) is invariant under ex-
σ,hv,wi v change of spin up and spin down. This spin-exchange
symmetry is denoted as S, and will naturally lead
| {z } | {z }
nearest-neighbour hopping onsite repulsion
(42) to a circuit construction that respects S. Hence, as
shown in Section 6.4, we can prove that G↑ and the
Here a†v,σ /av,σ are the creation/annihilation opera- corresponding conjugated symmetry SG↑ S −1 = G↓
tors of site v with spin σ and nv,↑/↓ are the number would be equivalent when used for symmetry expan-
operators. The exact circuit we used is outlined in sion. Note that for our practical implementation of
Ref. [24], which has the same form as a first-order the circuit, the trotter step implemented actually does
trotterisation circuit. We use the Jordan-Wigner en- not exactly commute with S, thus the performance of
coding to map the individual interaction terms and G↑ and G↓ for symmetry expansion are only approxi-
fermionic swaps to two-qubit gates in the circuit, with mately the same in our case. However, they are close
the gates that correspond to the interaction terms enough such that the symmetry expansion schemes
Accepted in Quantum 2021-09-14, click title to verify. Published under CC-BY 4.0. 90.8 0.8
Relative Bias in Energy Estimate, | (H)|
Relative Bias in Energy Estimate, | (H)|
0.7 0.7
0.6 0.6
0.5 0.5
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0.0 0.0
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Circuit Mean Error Count, Circuit Mean Error Count,
(a) 8-Qubit (b) 12-Qubit
1.0 1.0
0.8 0.8
Absolute Infidelity, | ( 0)|
Absolute Infidelity, | ( 0)|
0.6 0.6
0.4 0.4
0.2 0.2
0.0 0.0
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Circuit Mean Error Count, Circuit Mean Error Count,
(c) 8-Qubit (d) 12-Qubit
100
{I} Unmitigated
41 {I, G }
Sampling Cost Factor, C
{I, Gtot}
Full Verification
{G }
10 {Gtot}
6 7 {G , Gtot} - Small-bias Expansion
3
1
0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
Circuit Mean Error Count,
(e)
Figure 2: Performance metrics for different symmetry expansion schemes against the increase of mean circuit error count
under depolarising noise: the relative bias in the energy estimate for (a) 8-qubit and (b) 12-qubit simulations; the absolute
infidelity for (c) 8-qubit and (d) 12-qubit simulations; (e) the sampling cost for all simulations. Data are averaged over
circuit configurations with randomly generated parameters. The shaded areas indicate the spread of the lines over different
parameters. The symmetry expansions we consider are all uniform expansion of a given set of symmetry operators. The legend
indicates the set of symmetry operators used in the corresponding symmetry expansion. When the set of symmetry operators
forms a group, the corresponding symmetry expansion will be symmetry verification which is labelled using solid lines. All
the other symmetry expansion are labelled using dashed lines. Three of the most representative methods: without mitigation
(black), symmetry verification using the full symmetry group G = {I, G↑ , G↓ , Gtot } (green) and the small-bias symmetry
expansion using G∗ = {G↓ , Gtot } (brown) are labelled with thicken lines.
Accepted in Quantum 2021-09-14, click title to verify. Published under CC-BY 4.0. 10that have G↓ in place of G↑ or vice versa are consid- 8-Qubit 12-Qubit
ered to be identical to simplify our comparison be- Cost
tween different expansions. |(H)| |(ρ0 )| |(H)| |(ρ0 )|
As shown in Appendix I, the fraction of errors de- Unmitigated 0.493 0.601 0.390 0.610 1.0
tectable by Gtot is estimated to be fGtot ∼ 15
8
and that
by G↑/↓ is estimated to be fG↑/↓ ∼ 5 . Using these
2 Verified 0.229 0.279 0.175 0.299 3.2
and following Eq. (40) along the discussions there, Expanded 0.027 0.027 0.029 0.009 6.5
the small-bias expansion scheme that we will find is
simply uniform expansion with the set of symmetry
operators:
G∗ = {G↑/↓ , Gtot }. (44) (a) µ = 1
8-Qubit 12-Qubit
7.3 Results Cost
|(H)| |(ρ0 )| |(H)| |(ρ0 )|
The absolute relative bias in the energy estimate,
denoted as |(H)| over different mean circuit error Unmitigated 0.739 0.838 0.618 0.848 1.0
counts µ for the 8-qubit and 12-qubit simulations are Verified 0.504 0.567 0.413 0.595 7.1
shown in Fig. 2 (a) and (b). The corresponding ab-
solute infidelity |(ρ0 )| are shown in Fig. 2 (c) and Expanded 0.051 0.063 0.051 0.025 41.4
(d). From the close resemblance between the shapes
of the curves for |(H)| and |(ρ0 )|, we see that |(ρ0 )|
is indeed a good metric for comparing the estima-
tion bias for different symmetry expansion methods (b) µ = 2
as discussed in Section 4. This further validates our
approach of searching for good symmetry expansions Table 1: The relative bias in energy estimate |(H)|, absolute
based on |(ρ0 )| in Section 6 and our assumption of infidelity |(ρ0 )|, and the sampling costs for three of the most
w~ (O) ∼ w~ (ρ0 ) in Eq. (35). representative symmetry expansions in our simulation: the
The symmetry expansion schemes that we look at unmitigated noisy state (Unmitigated), the symmetry verifi-
cation using the full symmetry group G (Verified) and the
are uniform expansions of different subsets of sym-
small-bias symmetry expansion using G∗ = {G↓ , Gtot } (Ex-
metry operators F ⊆ G as discussed in Section 7.2.
panded) at the mean circuit error rate (a) µ = 1 and (b)
When F forms a group, then symmetry expanding µ = 2. Note that the cost for symmetry verification here are
with F is simply performing symmetry verification for post-processing symmetry verification.
using the symmetry group F. We see that in our
examples in Fig. 2, out of all symmetry verification
schemes (solid lines), the symmetry verification using using G∗ = {G↓ , Gtot }, whose data are summarised
the full symmetry group G can achieve the lowest bias in Table 1 (a). We see that at µ = 1, the energy es-
as expected. When we look at all symmetry expan- timate bias |(H)| for the unmitigated noisy output
sion schemes including symmetry verifications, we see sits at ∼ 0.5 for the 8-qubit example and ∼ 0.4 for
that the small-bias scheme we found in Section 7.2 the 12-qubit example. Symmetry verification using G
can achieve the lowest bias. can reduce the bias to ∼ 0.2, more than halving the
The sampling costs are shown in Fig. 2 (e), which bias, at the sampling cost of ∼ 3. The bias in the en-
is essentially the plot of the function Cw~ = Γ~F
−2 ergy estimate can be further reduced to ∼ 0.03 when
in Eq. (29) for different F. Since the estimated de- we apply the small-bias expansion scheme in Eq. (44),
tectable error fraction fG↑/↓ and fGtot are indepen- which is ∼ 15 times reduction compared to the unmit-
dent of the number of qubits and the circuit param- igated noisy state and ∼ 6 times reduction compared
eters, it means that hΓw~ i is mostly independent of to symmetry verification. The sampling cost for the
the number of qubits and the circuit parameters using symmetry expansion is CG ~ ∗ ∼ 6, which is only twice
Eq. (37), and thus our cost plot is practically the same the cost of symmetry verification. We see a similar,
for both the 8-qubit and 12-qubit cases and for circuits if not larger, improvement of the absolute infidelity
with any parameters. Note that the costs for the sym- |(ρ0 )| by using the small-bias symmetry expansion
metry verifications here are for post-processing veri- scheme over the unmitigated noisy state and symme-
fication. If we go for direct verification instead, then try verification. Using Eq. (35), we have
we need to take the square root of the cost. CG ~ ∗ − CG ~
Let us zoom in at the mean circuit error count µ = 1 N∗ . 2 2
∼ 40
~ (ρ0 ) − G
G ~ ∗ (ρ0 )
and focusing on the three most representative symme-
try expansions (thicken lines): the unmitigated noisy i.e. the minimum number of samples needed for the
state, the symmetry verification using the full symme- small-bias expansion to outperform symmetry verifi-
try group G and the small-bias symmetry expansion cation for a Pauli observable is around 40, which is a
Accepted in Quantum 2021-09-14, click title to verify. Published under CC-BY 4.0. 11sampling number we can easily reach in practice and between the biases due to the detectable and unde-
thus the small-bias scheme should be much preferred tectable noise components. For any given symmetry
in practice. expansion scheme, we have shown ways to estimate its
Now we will move to µ = 2 with the data sum- bias and sampling cost using the overall circuit error
marised in Table 1 (b). The bias in energy estimate rate and measurements of the symmetry operators.
|(H)| for the unmitigated noisy output now rise to Such performance prediction can be further simplified
∼ 0.74 for the 8-qubit example and ∼ 0.62 for the 12- if we have knowledge about the error channels in the
qubit example. Symmetry verification can reduce the circuit. Using the bias prediction, we can search for
relative bias to ∼ 0.5 and ∼ 0.4 for the 8-qubit and 12- the uniform symmetry expansion over some subset of
qubit case, respectively, with a sampling cost of ∼ 7. symmetry operators that has the smallest predicted
The small-bias symmetry expansion on the other hand bias. The scheme we found is simply called the small-
can still achieve a low relative bias of ∼ 0.05, which bias expansion scheme.
is around 13 times reduction compared to the unmit- We then applied our methods to the energy estima-
igated state and around 9 times reduction compared tion of Fermi-Hubbard model simulation with random
to symmetry verification. Similar to the µ = 1 case, circuit parameters for both the 8-qubit and 12-qubit
again we see a similar if not larger improvements in cases. From our simulation, we see that indeed abso-
the absolute infidelity |(ρ0 )| compared to |(H)| when lute infidelity is a good metric for predicting the bias
we use the small-bias symmetry expansion. However, in our energy estimate. When there is on average 1
due to the high circuit error rate µ = 2, the small-bias error per circuit run, the relative energy estimation
symmetry expansion would have a high sampling cost biases can be reduced from 0.4 ∼ 0.5 for the unmiti-
of CG~ ∗ ∼ 41. Using Eq. (35), we have gated noisy state and ∼ 0.2 for symmetry verification
to ∼ 0.03 for the small-bias symmetry expansion. The
CG ~ ∗ − CG
N∗ .
~
∼ 100 sampling cost for symmetry verification in this case is
2 2
~ (ρ0 ) − G
G ~ ∗ (ρ0 ) 3, i.e. we need 3 times more circuit runs for shot noise
reduction, while the sampling cost for the small-bias
i.e. the minimum number of samples needed for the
symmetry expansion is around 6. Hence, symmetry
small-bias expansion to outperform symmetry verifi-
expansion can reach an estimation accuracy 6 times
cation for a Pauli observable is around 100. This is
beyond what is achievable by symmetry verification
larger than the µ = 1 case, but still reachable in prac-
at a sampling cost that is only 2 times higher. When
tice.
there is on average 2 errors per circuit run, we found
Alternatively, at µ = 2, we can opt for symmetry-
that the factor of bias reduction brought by the small-
expanding with only G↓ , which can also achieve
bias symmetry expansion further increases, but the
a much lower bias than symmetry verification at
sampling cost also rise to 41 due to the high circuit er-
G~ ↓ (ρ0 ) ≈ 0.2 as seen from Fig. 2, but at a lower sam- ror rate. The small-bias scheme will still outperform
pling cost than the small-bias scheme at CG ~ ↓ ∼ 27. symmetry verification given a reasonable amount of
Hence, the minimum number of samples needed to samples allowed, but we can also turn to another ex-
outperform symmetry verification for a Pauli observ- pansion schemes with a slightly higher bias (still lower
able is around N ∗ . 70 in this case, which is smaller than symmetry verification) at a lower sampling cost.
than the small-bias scheme.
In practice, choosing the error-mitigation scheme
with the right bias-variance trade-off would largely
8 Conclusion depends on the precision we wish to reach and/or the
number circuit runs allowed for our experiment. Sym-
In this article, we have constructed a general frame- metry expansion is able to provide us with a wider
work named symmetry expansion for symmetry-based range of symmetry-based schemes beyond symmetry
error mitigation techniques. Different symmetry verification to fit our various practical requirements
expansions correspond to different sampling weight on the bias-variance trade-off. The estimation biases
distributions among the symmetry operators, with of the symmetry expansion can be further improved if
symmetry verification corresponding to the uniform we could develop effective ways to search for the rel-
weight distribution over the full symmetry group. The evant symmetry expansion schemes, which could be
effective ‘density operator’ after symmetry expansion done by for example constructing a better metric be-
is not positive semi-definite, thus instead of using the yond the absolute infidelity or using Clifford approx-
infidelity against the ideal state to predict the esti- imation learning-based method outlined in Ref. [26].
mation bias of a given symmetry expansion, we intro- In this article, we have only considered positive
duced the metric absolute infidelity against the ideal sampling weights for symmetry expansion and it
state to predict the estimation bias. Using the abso- would be natural to extend this to negative sam-
lute infidelity as the metric, we have shown that some pling weights like in the quasi-probability error mit-
symmetry expansion schemes can achieve a smaller igation [3, 4]. On the other hand, it would also be
bias than symmetry verification through cancellation interesting to see if any method can be developed
Accepted in Quantum 2021-09-14, click title to verify. Published under CC-BY 4.0. 12q
to modify the sampling distribution within quasi- The root mean square error MSE Oem will indi-
probability like how we go from symmetry verifica- cate the estimation precision we can reach in prac-
tion to symmetry expansion. Part of this has been tice given N samples. We see that the contribution
explored in Ref. [26]. from the variance term can be reduced by increasing
We have provided an explicit example of the ap- the number of samples N , but the overall precision is
plication of symmetry expansion for Pauli symmetry lower-bounded by the bias contribution |Bias [Oem ]|.
group and for virtual distillation (Appendix H) in this Suppose we are required
article. One could look into the more detailed ap- q to obtain an answer whose
root mean square error MSE Oem is below a cer-
plication of symmetry expansion in the other cases,
e.g. non-abelian symmetry group. Note that non- tain error bound δ, we need to first make sure our
abelian symmetry expansion is much easier to im- estimation bias is below the error bound:
plement than direct non-abelian symmetry verifica- |Bias [Oem ]| < δ. (45)
tion, as non-abelian symmetry verification cannot be
done by simply measuring their generators. However If both schemes have their biases below this thresh-
in this case, the expectation values of the symmetry old, then we will compare the number of sam-
operators might not be real any more, and thus we ples
q required for them to reach this error threshold
need to redevelop our methods for finding a suitable
MSE Oem = δ:
symmetry expansion, possibly involving using com-
plex weights. We might use methods in Ref. [27] to Var [Oem ]
Nδ = 2
measure such complex expectation values. δ2 − Bias [Oem ]
All of our symmetry expansion arguments are di- The scheme with a smaller Nδ will be our go-to
rectly applicable to the stabiliser symmetry group in scheme.
the stabiliser codes. It would be interesting to look at In this article, we do not have a fixed precision re-
an explicit example of how symmetry expansion might quirement for our estimators, thus we turn to another
perform in the context of stabiliser code like what has way to compare between different error mitigation
been done for quantum subspace expansion [15]. To schemes.
find the optimal symmetry expansion in this context, If we are given a fixed number of samples allowed
one might want to draw inspiration from Ref. [28] instead of being given the required precision, and the
about ways to reduce the search space. In addition, two error mitigation schemes are labelled 1 and 2,
it would also be interesting to see how symmetry ex- 2 2
one with a smaller bias Bias [O1 ] < Bias [O2 ] while
pansion can be applied to the symmetry present in the other with a smaller variance Var [O1 ] > Var [O2 ],
continuous variable systems like bosonic codes [29– then the two schemes will achieve the same mean
31]. square error when the number of samples is:
Var [O1 ] − Var [O2 ]
N∗ = 2.
Acknowledgements 2
Bias [O2 ] − Bias [O1 ]
When the number of allowed samples is more than this
The numerical simulations are performed using
threshold N > N ∗ , then the error due to bias domi-
QuESTlink [32], which is the Mathematica interface
nates and thus scheme 1 with smaller bias is preferred,
of the high-performance quantum computation sim-
on the other hand when N < N ∗ , then variance is the
ulation package QuEST [33]. The author is grateful
larger error source and thus scheme 2 with smaller
to those who have contributed to both these valuable
variance is preferred.
tools.
Using the expression of relative bias in Eq. (2) and
The author would like to thank Simon Benjamin
sampling cost in Eq. (1), we have:
and Balint Koczor for valuable discussions and Tyson
Jones for his help on the usage of QuESTlink. C1 − C2 Var [O]
N∗ =
The author is supported by the Junior Research Fel- 2 (O) − 1 (O) hO0 i2
2 2
lowship from St John’s College, Oxford and acknowl-
For a Pauli observable O, we have Var [O] = 1 −
edges support from the QCS Hub (EP/T001062/1). 2
hOi . We usually only care about Pauli observable
with expectation values hOi and hO0 i that have non-
A Comparison between Error Mitiga- negligible magnitude, because otherwise a large num-
ber of samples is needed to reach high accuracy and
tion Techniques the contribution of the Pauli observable to the actual
physical properties we care about (which would be a
Using Oem to denote the sample mean of Oem after N
linear sum of Pauli observables) is small. Hence, we
samples, the mean square error of the error-mitigated 2
sample mean estimator Oem is simply: have Var [O] . hO0 i and thus
C1 − C2
2 2 Var[Oem ] N∗ . . (46)
MSE Oem = Oem − O0 = Bias [Oem ] + N . 2 (O)2 − 1 (O)2
Accepted in Quantum 2021-09-14, click title to verify. Published under CC-BY 4.0. 13You can also read
