Filtering crosstalk from bath non-Markovianity via spacetime classical shadows G. A. L. White1 2K. Modi3 4yand C. D. Hill1 5 6z 1School of Physics University of Melbourne Parkville VIC 3010 Australia

2025-05-06 0 0 740.37KB 9 页 10玖币
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Filtering crosstalk from bath non-Markovianity via spacetime classical shadows
G. A. L. White,1, 2, K. Modi,3, 4, and C. D. Hill1, 5, 6,
1School of Physics, University of Melbourne, Parkville, VIC 3010, Australia
2School of Physics and Astronomy, Monash University, Clayton, Victoria 3800, Australia
3School of Physics and Astronomy, Monash University, Clayton, VIC 3800, Australia
4Centre for Quantum Technology, Transport for New South Wales, Sydney, NSW 2000, Australia
5School of Mathematics and Statistics, University of Melbourne, Parkville, VIC, 3010, Australia
6Silicon Quantum Computing, The University of New South Wales, Sydney, New South Wales 2052, Australia
From an open system perspective non-Markovian effects due to a nearby bath or neighbouring
qubits are dynamically equivalent. However, there is a conceptual distinction to account for: neigh-
bouring qubits may be controlled. We combine recent advances in non-Markovian quantum process
tomography with the framework of classical shadows to characterise spatiotemporal quantum cor-
relations. Observables here constitute operations applied to the system, where the free operation
is the maximally depolarising channel. Using this as a causal break, we systematically erase causal
pathways to narrow down the progenitors of temporal correlations. We show that one application
of this is to filter out the effects of crosstalk and probe only non-Markovianity from an inaccessible
bath. It also provides a lens on spatiotemporally spreading correlated noise throughout a lattice
from common environments. We demonstrate both examples on synthetic data. Owing to the scal-
ing of classical shadows, we can erase arbitrarily many neighbouring qubits at no extra cost. Our
procedure is thus efficient and amenable to systems even with all-to-all interactions.
INTRODUCTION
In the race to fault tolerant quantum computing, mag-
nified sensitivity to complex dynamics in open quan-
tum systems requires increasingly tailored characterisa-
tion and spectroscopic techniques [18]. Correlated dy-
namics are one particularly pernicious class of noise, and
can be generated from a variety of sources, including
inhomogeneous magnetic fields, coherent bath defects,
and nearby qubits, see Figure 1a [9,10]. Concerningly,
these effects are often omitted from quantum error cor-
rection noise models despite being ubiquitous in noisy
intermediate-scale quantum (NISQ) hardware [5,6,11
14].
Temporal – or non-Markovian – correlations are el-
ements of error that are correlated between different
points in time, as mediated by interactions with an exter-
nal system [6,15]. A process is said to be non-Markovian
if the total dynamics do not factorise into a product of
dynamical maps [16], a stronger condition than the well-
known completely-positive divisibility of dynamics [17].
The specific mediator of these effects is both conceptually
and experimentally relevant device information. Is it con-
trollable, or is it part of the inaccessible bath? Relatedly,
if the dynamics of two nearby qubits do not spatially fac-
torise, this is known as crosstalk. If one qubit is traced
out, then entangling crosstalk – such as the always-on
ZZ interactions in transmon qubits [18] – can generate
temporal correlations for the second qubit. Whether the
dynamics look non-Markovian depends on whether it is
feasible or not to dilate the characterisation to multiple
qubits and account for the variables responsible for these
correlations. Typically, it is not. Since crosstalk and
bath non-Markovianity can easily be conflated, it is cru-
a
b
Crosstalk
Bath non-
Markovianity
FIG. 1. System of interacting qubits and an inaccessible non-
Markovian environment (E). aA target qubit qnmay interact
via crosstalk mechanisms with other qubits, {q1, q2, q3,· · · },
in a quantum device, as well as defects in the bath and fluc-
tuating classical fields B+δB.bThe non-Markovian corre-
lations for that system may be separated into different causal
pathways by which the correlations are mediated. Causal
breaks (depicted in grey) erase any temporal correlations from
a given pathway, allowing one to infer the various contribu-
tions to total non-Markovianity from nearby qubits and envi-
ronment.
cial to find robust methods that can not only account for
their behaviour, but distinguish them.
In this Letter, we establish a systematic, concrete, and
efficient approach to the two pragmatic questions: (1)
arXiv:2210.15333v2 [quant-ph] 26 Apr 2023
2
if non-Markovian dynamics are detected across different
timescales for a qubit, do they come from neighbouring
qubits or a nearby bath? And (2) how can we determine
when two qubits are coupled to a shared bath generat-
ing common cause non-Markovian effects. The solutions
here have highly practical implications. Namely, whether
curbing the correlated effects is achievable through con-
trol or fabrication methods [19,20]. Process tensor to-
mography (PTT) is a recently developed generalisation
to quantum process tomography, and can guarantee an
answer to these questions and more, but the number of
experiments required grows as O(d2kn+N) to find corre-
lations across ksteps over Nqudits [6].
The basic premise of our work is to apply the method
of classical shadows [21] to PTT, resolving these prob-
lems. The classical shadow philosophy implements ran-
domised single-shot measurements to learn properties of
a state, granting access to an exponentially larger pool of
observables at fixed locality. Employing this, instead of
reconstructing the whole multi-time process for an entire
quantum register, we can estimate and analyse each of
the fixed-weight process marginals. Marginalising over
a measurement is equivalent to measuring and throwing
the outcome away. To marginalise over a process input is
equivalent to inputting a maximally mixed state. Hence,
these are maximal depolarising channels at no extra cost,
which act as causal breaks on controllable systems.
When suitably placed, these operations eliminate tem-
poral correlations as mediated on the chosen Hilbert
spaces, thus allowing non-Markovian sources to be
causally tested. We illustrate this idea in Figure 1b. The
end result is the simultaneous determination of the bath-
mediated non-Markovianity on all qubits. Our approach
hence only depends on the individual system size (in this
work, qubits), and is a physics-independent way for us to
test the relevant hypotheses. We are also able to simulta-
neously compute all spacetime marginals, extending the
randomised measurement toolkit to the spatiotemporal
domain [22].
SPATIOTEMPORAL CLASSICAL SHADOWS
By virtue of the state-process equivalence for multi-
time processes [15,2325], quantum operations on dif-
ferent parts of a system at different times constitute ob-
servables on a many-body quantum state. This allows
state-of-the-art characterisation techniques to be applied
to quantum stochastic processes. Classical shadow to-
mography [21,22] is one such technique, and already
has many generalisations and applications [2629]. Mea-
suring classical shadows allows for exponentially greater
observables to be determined about a state, provided
sufficiently low weight. But this restriction means the
technique has limitations for the study of temporal cor-
relations (which are high weight) in contrast to spatial
ones, as discussed in Ref. [12]. Our work expands on this
to the multi-qubit-multi-time case, and identifies other
desirable applications of classical shadows to multi-time
processes.
Definitions and Notation.– Consider a quantum device
with a register of qudits Q:= {q1, q2,· · · , qN}across a
series of times Tk:= {t0, t1,· · · , tk}. We take the whole
quantum device to define the system: HS:= NN
j=1 Hqj.
The device interacts with an external, inaccessible envi-
ronment whose space we denote HE. The k-step open
process is driven by a sequence Ak1:0 of control op-
erations on the whole register, each represented mathe-
matically by completely positive (CP) maps: Ak1:0 :=
{A0,A1,· · · ,Ak1}, after which one obtains a final state
ρS
k(Ak1:0) conditioned on this choice of interventions.
Note that where we label an object with time informa-
tion only, that object is assumed to concern the entire
register. These controlled dynamics have the form:
ρS
k(Ak1:0) = TrE[Uk:k1Ak1· · · U1:0 A0(ρSE
0)],(1)
where Uk:k1(·) = uk:k1(·)u
k:k1. Now let the Choi
representations of each Ajbe denoted by a caret, i.e.
ˆ
Aj=Aj⊗I[|Φ+ihΦ+|] = Pnm Aj[|nihm|]⊗|nihm|. Then,
the driven process in Equation (1) for arbitrary Ak1:0
uniquely defines a multi-linear mapping across the regis-
ter Q– called a process tensor, Υk:0 – via a generalised
Born rule [15,24]:
pS
k(Ak1:0) = Tr Υk:0 Πkˆ
Ak1 · · · ˆ
A0T,(2)
At each time tj, the process has an output index oj
(which is measured), and input index ij+1 (which feeds
back into the process). The details of process tensors can
be found in the appendix, but are not crucial to under-
standing this work. The two important properties that
we stress are: (i) a sequence of operations constitutes an
observable on the process tensor via Equation (2), gen-
erating the connection to classical shadows, and (ii) a
process tensor forms a collection of possibly correlated
completely positive, trace-preserving (CPTP) maps, and
hence may be marginalised in both time and space to
yield the jth CPTP map describing the dynamics of the
ith qubit ˆ
E(qi)
j:j1. A process is said to be Markovian if and
only if its process tensor is a product state across time.
The measure of non-Markovianity we use throughout this
work is that described in Ref. [16]. Specifically, it is the
relative entropy S[ρkσ] = Tr [ρ(log ρlog σ)] between a
process tensor Υk:0 and its closest Markov description,
the product of its marginals:
Υ(Markov)
k:0 =ˆ
Ek:k1 · · · ˆ
E1:0 ρ0.(3)
We denote this generalised quantum mutual information
(QMI) for a given process by Nk:0). Classical shadow
摘要:

Filteringcrosstalkfrombathnon-MarkovianityviaspacetimeclassicalshadowsG.A.L.White,1,2,K.Modi,3,4,yandC.D.Hill1,5,6,z1SchoolofPhysics,UniversityofMelbourne,Parkville,VIC3010,Australia2SchoolofPhysicsandAstronomy,MonashUniversity,Clayton,Victoria3800,Australia3SchoolofPhysicsandAstronomy,MonashUniver...

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