Optimal quantum spatial search on random temporal networks

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by  Shantanav Chakraborty

Optimal quantum spatial search on random temporal networks
Shantanav Chakraborty, Leonardo Novo, Serena Di Giorgio, and Yasser Omar
Instituto de Telecomunica¸co˜es, Physics of Information and Quantum Technologies Group, Lisbon, Portugal and
Instituto Superior T´ecnico, Universidade de Lisboa, Portugal
(Dated: January 16, 2017)
Toinvestigatetheperformanceofquantuminformationtasksonnetworkswhosetopologychanges
in time, we study the spatial search algorithm by continuous time quantum walk to find a marked
nodeonarandomtemporalnetwork. Weconsideranetworkofnnodesconstitutedbyatime-ordered
sequenceofErdo¨s-R´enyirandomgraphsG(n,p),wherepistheprobabilitythatanytwogivennodes
areconnected: aftereverytimeintervalτ,anewgraphG(n,p)replacesthepreviousone. Weprove
analyticallythatforanygivenp,thereisalwaysarangeofvaluesofτ forwhichtherunningtimeof
√
thealgorithmisoptimal,i.e.O( n),evenwhensearchontheindividualstaticgraphsconstituting
thetemporalnetworkissub-optimal. Ontheotherhand,thereareregimesofτ wherethealgorithm
7
issub-optimalevenwheneachoftheunderlyingstaticgraphsaresufficientlyconnectedtoperform
1
optimal search on them. From this first study of quantum spatial search on a time-dependent
0
network, it emerges that the non-trivial interplay between temporality and connectivity is key to
2
the algorithmic performance. Moreover, our work can be extended to establish high-fidelity qubit

n transfer between any two nodes of the network. Overall, our findings show that one can exploit
a temporality to achieve optimal quantum information tasks on dynamical random networks.
J

2
2 Temporal networks are ubiquitous: natural, techno- These networks are obtained as a sequence of Erd¨os-
logical and social networks typically have time-varying R´enyi random graphs G(n,p): after every time interval
]
h topologies. Recently, such networks have been exten- τ, a new graph G(n,p) replaces the previous one. This
p sivelystudiedattheclassicallevel[1–5]. However,quan- problem can also be viewed as spatial search on a com-
- tum dynamics on temporal networks has largely been plete graph with dynamical structural defects, i.e. where
t
n unexplored. Intuitively, one could expect that the un- links can randomly vanish and reappear over time, as in
a
controlled dynamical loss and emergence of links would a dynamical percolation problem.
u
q hindertheperformanceofquantuminformationtasksre- We define the temporality of a network as the frequency
[ alised on networks, namely for communication, compu- withwhichagivennetworkchangesitstopologyascom-

tation and sensing. But could this temporal character pared to the relevant energy scale of the Hamiltonian
2
actually yield any advantages for such tasks? In this representing the network, and thus 1/τ is a measure of
v
2 work, we consider the spatial search algorithm by con- temporality. Naturally, the introduction of this new fea-
9 tinuous time quantum walk [6] to find a marked node on ture leads to a much richer behaviour in the algorithmic
3 atemporalnetwork,andestablishanalyticallythatthere dynamics, as compared to the static scenario. In fact,
4
are regimes where its performance is optimal. nowtheoptimalityofthealgorithmdependscruciallyon
0
This algorithm was first introduced in Ref. [6] and has the interplay between τ and p.
.
1 been extensively studied on particular static graphs [7– In our work, we find a new threshold of p, namely
0 √
10]. Furthermore the analog analogue of Grover’s algo- p = log(n)/ n, such that for p ≥ p the algo-
7 temp temp
1 rithm [11] can be perceived as spatial search by quan- rithm is optimal irrespective of the temporality of the
: tum walk on the complete graph [6]. Recently, the algo- network. On the other hand, we show that sufficiently
v
rithm was proven to be optimal for Erd¨os-R´enyi random high temporality ensures that the algorithm retains its
i
X graphs, i.e. graphs of n nodes with each link existing optimality for arbitrarily low values of p. This holds
r between any two nodes with probability p [12, 13], as even when the underlying random graphs are comprised
a
long as p ≥ p = log3/2(n)/n [14]. Moreover, as a ofmostlyisolatednodesandsmalltreeswhicharegraphs
static
random graph can also be obtained by randomly remov- where, inthestaticcase, quantumsearchwouldnotpro-
ing links from a complete graph, these results can be vide any speed-up.
seen as an analysis of the robustness of quantum search Interestingly there also exists an intermediate regime
on the complete graph to random loss of links. Note p ≤ p < p where the spatial search algorithm
static temp
that quantum dynamics on static Erd¨os-R´enyi random is optimal on the underlying random graphs, whereas
graphsandothercomplexnetworkshavebeenstudiedin for a certain interval of τ, this is no longer the case.
Refs. [15, 16]. Also, some properties of the evolution of We find that when the temporality of the network coin-
quantum walks on dynamical percolation graphs such as cideswiththeenergyscaleoftheHamiltonianrepresent-
the mixing time, return probabilities and spreading were ing the network, the algorithmic running time is peaked.
studied in Refs. [17–19]. By gradually lowering or increasing the temporality, the
In this paper, we study how the quantum spatial running time of the algorithm decreases, and after a cer-
searchalgorithmperformsonrandomtemporalnetworks. tain threshold of temporality, becomes optimal – a be-
2
haviour also observed in Ref. [20] for the analog version almost surely the initial condition of the algorithm |s(cid:105),
of Grover’s algorithm albeit in a different context. Our as was shown to be the case in Lemma 2 of Ref. [14].
results show that quantum information processing tasks To obtain the regime where the optimality of the algo-
can be performed optimally on dynamically disordered rithm is maintained as a function of τ and p, we use
structures. time-dependentperturbationtheory. However,thisprop-
Quantum spatial search on random temporal networks.— ertyaboutthespectrumofadjacencymatricesofrandom
A temporal network is a dynamically evolving network graphsdoesnotholdwhenpisbelowtheaforementioned
of n vertices that alters its topology after a given time threshold. So, forsuchregimes, weconstructalinearsu-
interval. Asaresultlinksappearanddisappearafterev- peroperator that describes the average dynamics of the
ery time interval. If initially the network is represented algorithm on random temporal networks. We present
by a graph G , then after a time interval τ, the topol- each of these approaches separately.
1
ogy of the network changes and we obtain a new graph Quantum spatial search on random temporal networks
G , and so on. Thus, within a time t, a temporal net- having p ≥ p .— As long as p ≥ log3/2(n)/n, the
2 static
work may be represented by a sequence of static graphs eigenstate corresponding to the maximum eigenvalue of
G ={G ,G ,...,G }, where t=mτ and m∈N. theadjacencymatrixofanErd¨os-R´enyirandomgraphis
temp 1 2 m
Naturally, a random temporal network is represented by almost surely the state |s(cid:105) with eigenvalue np [14]. Thus
a network that is a sequence of random graphs. Let us the adjacency matrix of each of the random graphs ap-
consider Erd¨os-R´enyi random graphs G(n,p). A random pearinginG (n,p,τ)wouldsatisfythisproperty. Let
temp
temporal network G (n,p,τ) is a temporal sequence A denotetheadjacencymatrixoftherandomgraphap-
temp j
of Erd¨os-R´enyi random graphs such that after a time pearingatthejthtimeinstance(i.e.afteratimet=jτ).
t = mτ, the network will be defined as G (n,p,τ) = Then,eachoff-diagonalentryofA is1withprobabilityp
temp j
{G (n,p),G (n,p),...,G (n,p)}, where G (n,p) repre- and0withprobability1−p. LetB =A −np|s(cid:105)(cid:104)s|+pI
1 2 m j j j
sents the random graph at the jth time interval. We where B is a random matrix with each off-diagonal en-
j
shall focus on the optimality of the spatial search algo- try having mean 0 and variance p, with the diagonal en-
rithm by continuous time quantum walks on these net- tries being zero, and I is the identity matrix. We define
works and thus first introduce the algorithm briefly. the search Hamiltonian for G (n,p,τ) as in Eq. (1)
temp
Let G represent a graph of n vertices V = {1,...,n}. by choosing γ = 1/(np). By expressing each of the
We consider the Hilbert space spanned by the local- adjacency matrices appearing in G (n,p,τ) as men-
temp
ized quantum states at the vertices of the graph H = tioned previously, we obtain the following search Hamil-
span{|1(cid:105),|2(cid:105),...,|n(cid:105)}. The search Hamiltonian corre- tonian:
sponding to G is given by
m
(cid:88)
H (t)=−|w(cid:105)(cid:104)w|−|s(cid:105)(cid:104)s|− γB f (t,τ), (2)
H =−E|w(cid:105)(cid:104)w|−γA , (1) search j j
search G (cid:124) (cid:123)(cid:122) (cid:125)
j=1
where |w(cid:105) corresponds to the solution node of the search H0 (cid:124) (cid:123)(cid:122) (cid:125)
V(t)
problem marked by the local site energy E, γ is a real
number and A is the adjacency matrix of the graph G where f (t,τ)=Θ(t−(j−1)τ)−Θ(t−jτ), where Θ(x)
G j
[21]. Notethattypically||γA ||=E andhencetheorder is the Heaviside function, and m = T/τ is the number
G
oftheeigenvalues(energyscale)ofH isdetermined of instances of random graphs appearing throughout the
search √
by E. Generally this energy is chosen to be 1, as this evolution time of T = O( n). Here H induces a ro-
0
would imply that the quantum simulation of |w(cid:105)(cid:104)w| for tation in the two dimensional subspace spanned by |w(cid:105)
time t would correspond to O(t) queries to the standard and |s(cid:105), whereas V(t) will induce a coupling between
Grover oracle [6]. Henceforth, we shall choose E = 1. thissubspaceandthen−2degenerateeigenspaceofH .
0
The initial state of the algorithm is usually chosen to Also, H is the search Hamiltonian corresponding to the
0
be the equal superposition of all vertices, i.e. the state quantum walk on a complete graph and has been proven
√
|s(cid:105) = (cid:80)n |i(cid:105)/ n. The quantum search algorithm is to be optimal for search [6, 11]. In this case, we treat
i=1
said to be optimal on graph G if there exists a value of V(t) as a perturbation to H and use time-dependent
√ 0
γ such that after a time T = O( n), the probability of perturbation theory. Let |ψ(t)(cid:105) be the wavefunction of
obtaining the solution upon a measurement in the basis thequantumwalkobtainedbyevolvingunderH (t).
search
of the vertices is |(cid:104)w|e−iHGT|s(cid:105)|2 =O(1) [6]. Theerrorprobabilityinducedbytheperturbationisthus
√
In order to analyze this algorithm on G (n,p,τ), we (cid:15)=1−|(cid:104)w|ψ(T)(cid:105)|2, where T =O( n).
temp
use two separate approaches to prove our results for dif- We are interested in calculating when the average er-
ferent ranges of p. For p ≥ p = log3/2(n)/n, we ror probability (cid:104)(cid:15)(cid:105) is bounded as a function of τ and
static
use the fact that the the maximum eigenvalue of the ad- p. Whenever (cid:104)(cid:15)(cid:105) ∼ o(1), the algorithm outputs the
√
jacency matrix of each of the random graphs appearing solution state |w(cid:105) with probability 1 − o(1) in O( n)
during the time evolution is separated from the bulk of time. Without loss of generality, we intend to bound
the spectrum, and the eigenstate corresponding to it is (cid:104)(cid:15)(cid:105) = O(1/log(n)). We prove that the average error
3
searchHamiltonianintheformofarandommatrix,with
eachentrybeingatimedependentrandomvariablewith
a predefined autocorrelation function and with a certain
cut-offfrequency. Theauthorsfindthatwhenthecut-off
frequency of noise scales much faster or slower than the
energy scale of the Hamiltonian, the algorithm retains
its optimality. On the other hand, when they scale sim-
ilarly (i.e. when the cut-off frequency of noise is O(1)),
the average error is bounded by a constant only when
the ratio of the norm of the perturbation Hamiltonian
andthatoftheunperturbedsearchHamiltonianscalesas
O(n−1/4). Analogously, we find that for networks with
constant temporality, the average error is constant when
√
FIG. 1: Average running time of the quantum spatial p∼1/ n, in which case the aforementioned ratio is also
search algorithm as a function of τ for Gtemp(200,0.06,τ) ||V(t)||/||H || = ||B || = O(n−1/4), where we have used
(in blue dots) and G (200,0.1,τ)(in red squares). Each 0 j √
temp the fact that ||B || = O( np) [22, 23]. This shows that
point is averaged over 100 realizations. As predicted, the j
theglobalfeaturesoftheresponseofthisalgorithmwith
average running time peaks at τ ∼1, when the temporality
respecttothetypicalnoisetimescalesforthesetwomod-
coincides with the energy scale of the search Hamiltonian.
Away from this peak, the average running time decreases els is quite similar.
gradually towards the optimal running time (indicated by Note that we also recover the scenario of the spatial
the solid line). search algorithm on a static random network by choos-
√
ing τ = O( n). In this case the average error is always
bounded for p≥p , thereby recovering the results of
static
probability is given by (for the derivation see Sec. I of
Ref. [14].
the Supplemental Material):
Quantum spatial search on random temporal networks
 (cid:18) τ (cid:19) having p < pstatic.— Here we shall prove that for ran-
O p√n if τ <O(1) dom temporal networks with sufficiently high temporal-
(cid:104)(cid:15)(cid:105)= (cid:18) (cid:19) . (3) ity,thespatialsearchalgorithmisoptimalforarbitrarily
1
O pτ√n if τ ≥O(1) low values of p. For this regime of p, the results ob-
tained previously no longer hold, as |s(cid:105) is not an eigen-
Firstly, we are interested in finding the regime of p for state of the adjacency matrix (and np is no longer the
which the algorithm is robust to temporality. From maximum eigenvalue) of an Erd¨os-R´enyi random graph.
√
Eq. (3) we find that as long as p ≥ p = log(n)/ n, For p < log(n)/n, the underlying random graphs are no
temp
the average error is bounded irrespective of any 0<τ ≤ longer connected [24]. Moreover p << 1/n, the static
√
O( n). For lower values of p, temporality becomes cru- random graphs appearing during the time evolution of
cialtotheoptimalityofthealgorithmandinfactforthe the algorithm are extremely sparse and are mostly com-
rangeofpbetweenp andp thereexisttwosepa- prisedofisolatednodesandtrees. Inparticular,weshall
static temp
rateregimesoftemporalitythatdeterminetheoptimality focus on finding a regime of optimality of the search
of the algorithm: a fast temporality regime and a slow algorithm for p ≤ 1/n, while we refer the readers to
temporality regime such that if the topology of the net- Sec. II of the Supplemental Material for results when
√
work alters faster than τfas√t =O(p n/log(n)) or slower 1/n<p<pstatic.
than τ = O(log(n)/(p n)), the algorithm remains Inthisregimewefollowadifferentapproach: weconsider
slow
optimal. The behavior of the algorithm in the interme- theevolutionofthequantumstateaveragedoverallpos-
diateregimeofτ <τ <τ isalsointeresting, albeit sible realizations of a random graph using the density
fast slow
suboptimal. As the temporality of the network increases matrix formalism. The number of possible realizations
fromτ , thealgorithmicrunningtimeincreaseswithit of G(n,p) is |G| = 2N, where N = (cid:0)n(cid:1). The average
fast 2
peaking at τ = O(1), after which it gradually decreases dynamics of the algorithm on a random temporal net-
until τ = τ . To confirm this, we plot in Fig. 1 the work after one time step τ is described by the following
slow
average running time of G (200,0.06,τ) (blue dots) superoperator:
temp
and G (200,0.1,τ) (red squares) as a function of τ.
temp
|G|
As predicted, the average running time peaks when the (cid:88)
Φ(ρ)= p e−iHrτρeiHrτ =(cid:104)e−iHrτρeiHrτ(cid:105), (4)
temporality1/τ ≈1andapproachestheoptimalrunning r
r=1
time (solid line) away from the peak.
AsimilarbehaviourhasalsobeenobservedinRef.[20]for where p is the probability of the rth realization and
r
theanalogversionofGrover’salgorithm,forthefollowing H =|w(cid:105)(cid:104)w|+γA with A being the adjacency ma-
r Gr Gr
noise model: the authors consider a perturbation to the trix corresponding to the rth realization of G(n,p). Let
4
(cid:104)X(cid:105) represent the expected value of X. The evolution of refer to Sec. II of Supplemental Material)
√
the algorithm after m=O( n/τ) time steps is given by
Φm(ρ). The first order expansion of the superoperator  1
 if 1/n2 ≤p≤1/n
yields: n5/2log(n)
τ ≤ p . (9)
√ if p<1/n2
nlog(n)
Φ(ρ)=(cid:104)ρ−iτ[H ,ρ](cid:105)+δ (5)
r
=ρ−iτ[|w(cid:105)(cid:104)w|+|s(cid:105)(cid:104)s|,ρ]+δ (6) In general our results imply that although p is well
below the percolation threshold, and in fact the tem-
=Φ (ρ)+δ, (7)
0
poral network consist of graphs that do not have giant
components and are mostly composed of isolated nodes
wherethesecondstepfollowsbecausetheexpectedvalue
and trees of O(1) nodes, sufficiently high temporality
of each entry of A is p and so (cid:104)A (cid:105) = np|s(cid:105)(cid:104)s|.
Gr Gr can still lead to optimal search. This cannot be achieved
Thus (cid:104)H (cid:105) = |w(cid:105)(cid:104)w|+|s(cid:105)(cid:104)s|, which is the same as H
r 0 byperformingaquantumwalkonanyofthesestructures
defined in Eq. (2), and is optimal for quantum spatial
appearing as a static network. This has been confirmed
search. Here δ is the error induced by truncating the
in Fig. 2, wherein we plot (in blue points) the average
superoperator Φ after the first order and is given by
δ ≤ (cid:80)∞ (τk/k!)(cid:10)||Hk||(cid:11). Note that the superopera- runningtimeofthequantumspatialsearchalgorithmon
k=2 r random temporal networks G (50,0.0008,τ) with a
temp
valueofpiswaybelowpercolationthreshold(p=2/n2).
As expected, for sufficiently low values of τ, the running
time of the algorithm is close to the optimal running
√
time of T = π n/2 (solid line) and increases as τ is
increased. We summarize the regimes of τ and p where
the algorithm is optimal in Fig. 3. For derivations refer
to the Supplemental Material.
Discussion.— We have proven analytically that for any
givenp,thereisalwaysarangeofvaluesofτ forwhichthe
running time of the spatial search algorithm by continu-
ous time quantum walk on a random temporal network
√
G (n,p,τ) is optimal, i.e. O( n). Indeed, we find
temp
that the non-trivial interplay between p and the tempo-
FIG. 2: Average running time of the quantum spatial
rality of the network is key to the algorithm’s perfor-
search algorithm on G (50,0.0008,τ) (dots) as a function
temp
mance (see Fig. 3).
of τ. Each point is averaged over 50 realizations. Note that √
p=2/n2 and even then, for small enough τ, the algorithm We obtain a threshold p = log(n)/ n above which
temp
runs in optimal time (solid line). As τ is increased, the the algorithm is optimal irrespectively of τ, i.e. of how
algorithmic running time increases. fast or slowly the links appear and disappear in the dy-
namical network.
tor Φm0 describes approximately the standard evolution Wealsofindthat,forsufficientlylowvaluesofτ,thealgo-
of the algorithm under the Hamiltonian (cid:104)Hr(cid:105), and thus rithmisoptimalforanyvalueofp. Thismeansthathigh
weintendtoboundtheerrorobtainedbyusingthesuper- temporality allows optimal performance even when p is
operator Φ instead of the superoperator Φ0 to describe well below the static percolation threshold, i.e. when the
the dynamics for each of the m timesteps. This is given underlyingstaticgraphsarecomprisedmostlyofisolated
by : nodes and trees of constant depth.
Interestingly, for p < p < p , the algorithm is
static temp
(cid:15)=||Φm(ρ)−Φm(ρ)||≤mδ. (8) optimal on each static random graph, but not always on
0
the temporal network composed by the sequence of such
Thus to bound (cid:15) we need to bound (cid:104)||A ||(cid:105). Since graphs. In the sub-optimal regime, the algorithmic run-
Gr
p≤1/n, the underlying random networks are extremely ningtimeispeakedwhenthetemporalityofthenetwork
sparse, contain isolated nodes and very few links. Thus coincides with the energy scale of the search Hamilto-
||A || is bounded by the sum of the individual links nian. We can move away from this regime by decreasing
Gr
of the random graphs. As p decreases further (i.e. orincreasingthetemporality: therunningtimeoftheal-
p << 1/n), the aforementioned bound is better as the gorithm will then decrease accordingly, reaching the op-
underlying networks have fewer and fewer links. For a timal performance at τslow or τfast respectively.
given range of p we find the bound for τ where (cid:15) ≤ Note that our results on spatial search can also be ex-
O(1/log(n)). In fact we obtain that (for the derivations tended to perform high-fidelity state transfer of a qubit
5
FIG. 3: Summary of analytical results: thresholds of τ above or below which the quantum spatial search algorithm on a
random temporal network of n nodes is optimal for a given range of p.
between any two nodes of a random temporal network Acknowledgments — The authors thank the
[14, 25, 26]. support from Fundac¸˜ao para a Ciˆencia e a
Finally, our findings can also be interpreted as an Tecnologia (Portugal), namely through pro-
analysis of the robustness of the quantum spatial search grammes PTDC/POPH/POCH and projects
algorithm and the state transfer protocol on a complete UID/EEA/50008/2013, IT/QuSim, IT/QuNet, Pro-
graph with dynamical structural defects. Furthermore, QuNet, partially funded by EU FEDER, from the
they pave the way to study quantum dynamics on EU FP7 project PAPETS (GA 323901), and from the
non-Markovian temporal networks [27], as well as to JTF project The Nature of Quantum Networks (ID
exploit temporality as a control mechanism to improve 60478). Furthermore SC, LN and SDG acknowledge the
or protect the effectiveness and efficiency of quantum support from the DP-PMI and FCT (Portugal) through
information tasks on dynamical networks. SFRH/BD/52246/2013, SFRH/BD/52241/2013 and
PD/BD/114332/2016 respectively.
Supplemental Material
I. Optimality of quantum search on random temporal networks when p ≥ p — Let A denote the
static j
adjacency matrix of the random graph appearing at the jth time instance (i.e. after a time t = jτ). Then each
off-diagonal entry of A is 1 with probability p and 0 with probability 1−p. Let
j
B =A −np|s(cid:105)(cid:104)s|+pI. (S1)
j j
The matrix B is a random matrix with each off-diagonal entry having mean 0 and variance p with the diagonal
j
entries being zero, and I is the identity matrix. As long as p ≥ log3/2(n)/n, the eigenstate corresponding to the
maximum eigenvalue of the A is almost surely the state |s(cid:105) with eigenvalue np. Thus, the adjacency matrix of each
j
of the random graphs appearing in G (n,p,τ) would satisfy this property. So we have that, for all j
temp
B |s(cid:105)=0. (S2)
j
We shall use this property in our derivation at a later stage. We have that
m
(cid:88)
H (t)=−|w(cid:105)(cid:104)w|−|s(cid:105)(cid:104)s|− γB f (t,τ), (S3)
search j j
j=1
where f (t,τ)=Θ(t−(j−1)τ)−Θ(t−jτ) with Θ(x) being the Heaviside function. Let H =−|w(cid:105)(cid:104)w|−|s(cid:105)(cid:104)s| and
j 0
V(t)=(cid:80)m γB f (t,τ). Let |s (cid:105) represent thestate thatis anequal superposition ofall nodes ofthe network other
j=1 j j w¯ √ √
thanthesolution. ThegroundstateandfirstexcitedstateofH are|λ (cid:105)=(|w(cid:105)+|s (cid:105))/ 2and|λ (cid:105)=(|w(cid:105)−|s (cid:105))/ 2
0 √ 1 w¯ √ 2 w¯
respectively with the corresponding energies being λ = −1−1/ n and λ = −1+1/ n. The remaining n−2
1 2
eigenstates form a degenerate subspace of energy 0. Without loss of generality let the solution node |w(cid:105) = |1(cid:105) and
then the remaining eigenstates are represented as
n
1 (cid:88)
|λ (cid:105)= √ ω(k−2)j|j(cid:105), 3≤k ≤n, (S4)
k
n−1
j=2
where ω =ei2π/(n−1).
6
We treat V(t) as a perturbation to H and use first order time-dependent perturbation theory. Let the wavefunction
0
corresponding to the evolution under H after a time t be
0
n
|φ(t)(cid:105)=(cid:88)c(0)|λ (cid:105) (S5)
k k
k=1
√
As H is optimal for quantum spatial search, after a time T = O( n), the wavefunction is localized at the solution
0
node |w(cid:105), i.e.
|λ (cid:105)+|λ (cid:105)
|φ(T)(cid:105)=|w(cid:105)= 1 √ 2 . (S6)
2
Now, in the presence of the time-dependent perturbation term V(t), after the time t, assume that we would obtain
the wavefunction |ψ(t)(cid:105). The error probability induced by the perturbation is thus
(cid:15)=1−|(cid:104)w|ψ(T)(cid:105)|2. (S7)
We are interested in calculating when the average error is bounded as a function of τ and p.
Using time-dependent perturbation theory, we obtain that
n
(cid:88)
|ψ(t)(cid:105)= ck(t)e−iλkt|λk(cid:105). (S8)
k=1
Here
ck(t)=c(k0)−i(cid:88)c(x0)(cid:90) teωkxt1vkx(t)dt−(cid:88)c(y0)(cid:90) t(cid:90) t1ei(ωkxt1+ωxyt2)vkx(t1)vxy(t2)dt1dt2, (S9)
x 0 x,y 0 0
where ω =λ −λ and
kx k x
m
(cid:88)
v (t)= (cid:104)λ |γB |λ (cid:105)f (t,τ). (S10)
kx k j x j
j=1
Observe that as each entry of B has mean 0,
j
E[v (t)]=0, (S11)
kx
where E[X] represents the expected value of random variable X. Also from Eq. (S6) we find that the solution state
|w(cid:105) has non-zero overlap in only the ground and first excited states of H . This yields
0
1(cid:104) (cid:105) (cid:104) (cid:105)
|(cid:104)w|ψ(t)(cid:105)|2 = |c (t)|2+|c (t)|2 +Re c∗(t)c (t)e−iω12t . (S12)
2 1 2 1 2
In general in calculating Eq. (S12), we obtain terms such as
(cid:90) t (cid:90) t
c∗l(t)ck(t)=c(k0)c∗l(0)−ic∗l(0)(cid:88)c(y0) eiωlyt1vly(t1)dt1+ic(k0)(cid:88)c∗y(0) e−iωkyt1vk∗y(t1)dt1
y 0 y 0
(cid:18)(cid:90) t (cid:19)(cid:18)(cid:90) t (cid:19)
(cid:88)
+ c∗x(0)c(y0) e−iωkxt1vk∗x(t1)dt1 eiωkyt1vky(t1)dt1
x,y 0 0
(S13)
−(cid:88)c(k0)c∗y(0)(cid:90) t(cid:90) t1e−i(ωlxt1+ωxyt2)vl∗x(t1)vx∗y(t2)dt1dt2
x,y 0 0
−(cid:88)c∗l(0)c(y0)(cid:90) t(cid:90) t1ei(ωkxt1+ωxyt2)vkx(t1)vxy(t2)dt1dt2.
x,y 0 0
As we are interested in calculating the average error (cid:104)(cid:15)(cid:105), the first order terms of Eq. (S13) are going to be zero owing
to Eq.(S11). Moreover the second order terms require calculating correlation functions of the following form
m
(cid:88)
(cid:104)v (t )v∗ (t )(cid:105)= E[(cid:104)λ |γB |λ (cid:105)(cid:104)λ |γB |λ (cid:105)]f (t ,τ)f (t ,τ)δ . (S14)
ab 1 ac 2 a x b c y a x 1 y 2 xy
x,y=1
7
When a,b,c∈{1,2}, we use Eq. (S2)and the fact that (cid:104)w|B |w(cid:105)=0 andobtain that thecorrelation function is zero.
j
Subsequently the contribution of the corresponding integrals to (cid:104)(cid:15)(cid:105) is zero. In fact the terms that would contribute
are the terms that couple the (n−2) fold degenerate subspace with the ground and first excited states. Without loss
of generality, assume in Eq. (S14) that a=1 and b,c>2. We have
m
(cid:88)
(cid:104)v (t )v∗ (t )(cid:105)= E[(cid:104)λ |γB |λ (cid:105)(cid:104)λ |γB |λ (cid:105)]f (t ,τ)f (t ,τ)
1b 1 1c 2 1 x b c y 1 x 1 x 2
x=1
m (S15)
(cid:88)
= E[(cid:104)w|γB |λ (cid:105)(cid:104)λ |γB |w(cid:105)]f (t ,τ)f (t ,τ). [Using(S2)]
x b c y x 1 x 2
x=1(cid:124) (cid:123)(cid:122) (cid:125)
Ex [...]
abc
Now as independent entries of the random matrix B are uncorrelated, we have
j
n
1 (cid:88)
Ex [...]= E[ (cid:104)w|γB |α(cid:105)ω(b−2)α(cid:104)β|γB |w(cid:105)ω−(c−2)βδ ] (S16)
abc 2(n−1) x x αβ
α,β=2
n
1 (cid:88) 1
= γ2p= . (S17)
2(n−1) 2n2p
α=2
Thus,
m
1 (cid:88)
(cid:104)v (t )v∗ (t )(cid:105)= f (t ,τ)f (t ,τ). (S18)
1b 1 1c 2 2n2p x 1 x 2
x=1
Note that the result in Eq. (S18) is independent of b and c. For any b,c > 2 the same result would be obtained.
Moreover we find that the autocorrelation function obtained in Eq. (S18) has the same form even when a,b∈{1,2}
and c > 2. The term where all of a,b,c > 2 is never encountered. Thus the coupling of the ground and first excited
statetoeachofthen−2degenerateeigenstatesyieldthesamevalueofthecorrelationfunctions,i.e.(cid:104)v (t )v∗ (t )(cid:105)=
pq 1 pr 2
(cid:104)v (t )v∗ (t )(cid:105) for all p∈{1,2} and q,r >2.
px 1 py 2 √
Subsequently, we calculate the average error after a time T = O( n). This involves calculating integrals of the
following forms
1(cid:88)m (cid:90) jτ (cid:90) t1
I± = dt dt e±i(ωxzt1−ωyzt2)(cid:104)v∗ (t )v (t )(cid:105), (S19)
xyz 2 1 2 xz 1 yz 2
j=1 (j−1)τ (j−1)τ
(S20)
wherex,y ∈{1,2}andz >2. AlsothecorrelationfunctionsinI± arethesameasthoseofI± foranyz >2. Thus
xy3 xyz
it suffices to replace z by 3 and we have that
1(cid:88)m (cid:18)e±i(ωx3−ωy3)(j−1)τ(e±i(ωx3−ωy3)−1)τ e±i(ωx3−ωy3)(j−1)τ −e±i(ωx3jτ−ωy3(j−1)τ)(cid:19)
I± = + . (S21)
xy3 2 ω (ω −ω ) ω ω
y3 x3 y3 x3 y3
j=1
From Eq. (S21), we have three cases, namely, when x=y, x>y and x<y. The average probability of error is given
by
(cid:32) 2 (cid:33)
n−2 (cid:88)
(cid:104)(cid:15)(cid:105)= (I+ +I− )+2Re[e−iω12T(I− +I+ +I− +I+ )] (S22)
2 xy3 xy3 113 223 123 213
x,y=1
The first set of integrals satisfy:
(cid:88)2 (I+ +I− )=O(cid:32)γ2p√n(cid:88)2 sin2(ωx3τ)(cid:33). (S23)
xx3 xx3 τ ω2
x=1 x=1 x3
On the other hand, the remaining integrals satisfy:
√
(cid:88)2 (cid:18)γ2p n (cid:19)
(I+ +I− )=O (1−cos(τ)) . (S24)
xy3 xy3 τ
x=1
8
Thusweobtaintwodifferentcasesnamely,whenτ ≥O(1)andwhenτ <O(1). Inthelattercase,wecanapproximate
sin2(τ)≈τ2 and cos2(τ)≈τ2. Finally we have that
 (cid:18) (cid:19)
τ
O p√n if τ <O(1)
(cid:104)(cid:15)(cid:105)= (cid:18) (cid:19) . (S25)
1
O pτ√n if τ ≥O(1)
√ √
• When p≥p =log(n)/ n: The algorithm would is optimal for any 0≤τ ≤O( n).
temp
√
• When p = log3/2(n)/n ≤ p ≤ p : There exists two regimes of temporality, τ = O(log(n)/(p n)
static √ temp slow
and τ = O(p n/log(n)) such that if the topology of the network changes faster than τ or slower than τ ,
fast fast slow
the algorithm is optimal.
II. Optimality of quantum search on random temporal networks when p<p — Here we demonstrate
static
that the spatial search algorithm is optimal for random temporal networks even when each of the underlying static
networks are below the percolation threshold. We consider the evolution of the quantum state averaged over all
possible realizations of a random graph. The number of possible realizations of G(n,p) is |G|=2N, where N =(cid:0)n(cid:1).
2
The average dynamics of the algorithm on a random temporal network after one time step τ is given by the following
superoperator
|G|
(cid:88)
Φ(ρ)= p e−iHrτρeiHrτ, (S26)
r
r=1
where p is the probability of the rth realization and H = |w(cid:105)(cid:104)w|+γA with A being the adjacency matrix
r r Gr Gr √
corresponding to the rth realization of G(n,p). Thus the evolution of the algorithm after m = n/τ time steps is
given by Φm(ρ). The first order expansion of the superoperator yields
Φ(ρ)=(cid:104)ρ−iτ[H ,ρ](cid:105)+δ (S27)
r
=ρ−iτ[|w(cid:105)(cid:104)w|+|s(cid:105)(cid:104)s|,ρ]+δ (S28)
=Φ (ρ)+δ, (S29)
0
where the second step follows because expectation value of each entry of A is p and so (cid:104)A (cid:105) = np|s(cid:105)(cid:104)s|. Thus
Gr Gr
(cid:104)H (cid:105)=|w(cid:105)(cid:104)w|+|s(cid:105)(cid:104)s| and the error (cid:15) induced by this truncation is the sum of all higher order terms given by
r
δ ≤(cid:88)∞ τk (cid:10)||Hk||(cid:11). (S30)
k! r
k=2
Let us consider the kth order term of δ. This is
δ = τk (cid:10)||Hk||(cid:11) (S31)
k k! r
≤ τk (cid:10)||H ||k(cid:11)= τk (cid:10)|||w(cid:105)(cid:104)w|+γA ||k(cid:11) (S32)
k! r k! Gr
≤ τk (cid:10)||1+γA ||k(cid:11). (S33)
k! Gr
Also notice that the total error after mτ time steps is
(cid:15)=||Φm(ρ)−Φm(ρ)||≤mδ. (S34)
0
Bounding the error δ (and subsequently (cid:15)) boils hinges upon bounding ||A ||. We will consider different regimes of
Gr
p and obtain bounds on τ for which the (cid:15)=O(1/log(n)).
• When log(n)/n ≤ p < log3/2(n)/n : The maximum degree of each node (d ) of the random graphs are
max
no far from the average degree np. In fact it is known that d /np = O(1). Also since for any graph G,
max
9
||A || ≤ d , we have that in this regime of p, δ ≤ O(τk/k!). This gives us that δ ≤ O(τ2). Subsequently,
G √ max k √
(cid:15)≤O(τ n) which implies that for (cid:15) to be bounded, τ <1/( nlog(n)).
• When log(n)/n < p ≤ c/n such that c > 1 : The underlying random graphs are no longer connected. The
maximum degree of each node is no longer close to the average degree. In this regime thus we use the trivial bound
√
that d ≤ n−1 ≈ n and obtain that δ ≤ O(τ2/p2) and hence (cid:15) ≤ O( nτ/p2). In fact τ < 1/(n5/2log(n)) is
max
sufficient for the error to be bounded. Using a better bound for ||A || would improve the bound on τ.
Gr
• When p ≤ 1/n : In this regime d << n − 1 and hence we use a different bound for ||A ||. As there
max Gr
are very few links in the underlying random graphs, we use the fact that for any graph G, ||A || ≤ |E(G)|, where
G
|E(G)| is the total number of edges of graph G. For a random graph G(n,p) the probability of having l edges follows
a binomial distribution, i.e. (cid:0)N(cid:1)pl(1−p)N−l. Thus Eq. (S31) can be written as
l
τk (cid:88)N (cid:18)N(cid:19)
δ ≤ pl(1−p)N−l(1+γl)k (S35)
k k! l
l=0
As p≤1/n, we have γ =1/(np)≥1. So
(2γτ)k (cid:88)N (cid:18)N(cid:19) (2γτ)k
δ ≤ lkpl(1−p)N−l = B (N,p), (S36)
k k! l k! k
l=0
where B (N,p)representsthe kthmomentofa Binomialdistribution. Now thereare twodistinct case. Firstly, when
k
p < 1/n2, B (N,p) = O(Np) = O(n2p/2) and secondly for 1/n2 ≤ p ≤ 1/n, B (N,p) = O(Nkpk). We deal with
k k
these cases separately. So when p<1/n2 we have
∞
(cid:88)
δ ≤ (cid:15) (S37)
k
k=2
n2p(cid:88)∞ (2γτ)k n2p
≤ = (e2γτ −1−2γτ) (S38)
2 k! 2
k=2
≤O(2n2pγ2τ2). (S39)
So when p<1/n2 we have that the total error is
√
(cid:18) (cid:19)
(cid:15)≤O(cid:0)m2n2pγ2τ2(cid:1)=O nτ . (S40)
p
√
So the error (cid:15) = O(1/log(n)) as long as τ < p/( nlog(n)). Now for the case where 1/n2 ≤ p ≤ 1/n we have
B (N,p) = O(Nkpk) and a similar derivation yields that ∆ is bounded as long as 1/(n5/2log(n)). Thus we find
k
that the dynamics of the algorithm is well approximated by the superoperator Φ (ρ) and the dynamics is restricted
0
to the two dimensional space spanned by the target state |w(cid:105) and the (almost) initial state |s (cid:105). Thus after a time
√ w¯
T =mτ =O( n), the probability of finding the solution state |w(cid:105), P (T)≈1.
w
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