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 [15,23–25], 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 [26–29]. 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 Ak−1:0 of control op-
erations on the whole register, each represented mathe-
matically by completely positive (CP) maps: Ak−1:0 :=
{A0,A1,· · · ,Ak−1}, after which one obtains a final state
ρS
k(Ak−1:0) 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(Ak−1:0) = TrE[Uk:k−1Ak−1· · · U1:0 A0(ρSE
0)],(1)
where Uk:k−1(·) = uk:k−1(·)u†
k:k−1. 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 Ak−1:0
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(Ak−1:0) = Tr Υk:0 Πk⊗ˆ
Ak−1⊗ · · · 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:j−1. 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 ρ−log σ)] between a
process tensor Υk:0 and its closest Markov description,
the product of its marginals:
Υ(Markov)
k:0 =ˆ
Ek:k−1⊗ · · · ⊗ ˆ
E1:0 ⊗ρ0.(3)
We denote this generalised quantum mutual information
(QMI) for a given process by N(Υk:0). Classical shadow