14
FIG.7. ThisfigureshowcasesthenumericaloutcomesofTr{ρ2}andTr{ρ3}calculatedusingtheGSTmethodwhileaccounting
fornoise. Subfigure(a)presentsthenumericalresultsforTr{ρ2},andsubfigure(b)illustratesthenumericalresultsforTr{ρ3}.
The black lines denote the results obtained after the addition of random numbers, while the red lines represent the results
before the introduction of random gates.
Derivedfromthedatadepictedintheprovidedgraph, weareabletodistinctlydiscerntherepercussionsstemming
from the introduction or absence of noise. When noise is absent, the output data from the quantum circuit manifests
as consistently stable and precise. Nonetheless, the scenario takes a discernible turn upon the infusion of noise. The
introductionofnoiseprecipitatesvolatilityinthequantumcircuit’soutcomes,potentiallyculminatinginerrors. Noise
canengenderanunreliableexchangeofinformationamidstquantumbits,therebyinjectingsubstantialambiguityinto
thecomputationalprocess. Thesesourcesofnoisemightencompassphenomenasuchasdephasing,errorsinquantum
gate operations, imprecisions in measurements, or other external disturbances.
Inreal-worldscenarios,noisepresentsasignificanthurdlefortheadvancementofquantumcomputingandquantum
information processing. To counteract the disruptive influence of noise, researchers are tirelessly engaged in refining
quantum error correction and noise suppression techniques. The primary goal of these approaches is to bolster the
resilience of quantum circuits and heighten the precision of outcomes, ultimately striving for a more dependable
realm of quantum computing.Furthermore, the ramifications of noise can exhibit variability across diverse quantum
algorithms and tasks. As a result, practical applications necessitate a thorough assessment of algorithm robustness
alongside the prevailing noise levels. Achieving equilibrium between these considerations is crucial in determining the
most optimal course of action.
Appendix C: Subspace dimension
In this section, we study the dimensions of the invariant subspaces corresponding to nonlinear functions of dif-
ferent types of density matrices. Let’s start with the simplest case, Tr{ρm}. In our algorithm, we decompose
the computation of Tr{ρm} into the computation of Tr{G G ...G }, for k = 0,1,...,m. It is obvious that
q1 q2 qk
span(|ψ ⟩,|ψ ⟩,...,|ψ ⟩) forms a non-trivial invariant subspace of G G ...G . Therefore, the largest non-
q1 q2 qk q1 q2 qk
trivial invariant subspace involved in the calculation of Tr{ρm} is m-dimensional.
A more general nonlinear function is Tr{P ρ P ρ ...P ρ }. We can rewrite this function as Tr{P′ρ′ρ′ ...ρ′ },
1 1 2 2 m m 1 2 m
where ρ′ = (Πi P )ρ (Πi P )† and P′ = Πi P . Remark that although ρ′ ̸= ρ′ ̸= ... ̸= ρ′ , the algorithms
i l=1 l n l=1 l l=1 l 1 2 m
we proposed in this paper can also be used. During the calculation, we need to calculate Tr{P′G G ...G }, for
q1 q2 qk
k =0,1...,m, where P′ is a Pauli operator. Note that G acts on |ψ⟩ to get a superposition of |ψ ⟩ and |ψ⟩. Thus
qi qi
the non-trivial invariant subspace corresponding to PG G ...G is span(|ψ ⟩,...,|ψ ⟩,P|ψ ⟩,...,P|ψ ⟩), of
q1 q2 qk q1 qm q1 qm
which the largest possible dimension is 2m dimensions.