Resource-efficient simulation of noisy quantum circuits and application to network-enabled QRAM optimization Luís Bugalho1 2 3 4Emmanuel Zambrini Cruzeiro5Kevin C.

2025-04-29 0 0 5.57MB 28 页 10玖币

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Resource-eﬃcient simulation of noisy quantum circuits and application to

network-enabled QRAM optimization

Luís Bugalho,1, 2, 3, 4 Emmanuel Zambrini Cruzeiro,5Kevin C.

Chen,6, 7 Wenhan Dai,7, 8 Dirk Englund,6, 7 and Yasser Omar1, 2, 3

1Instituto Superior Técnico, Universidade de Lisboa, Portugal

2Physics of Information and Quantum Technologies Group,

Centro de Física e Engenharia de Materiais Avançados (CeFEMA), Portugal

3PQI – Portuguese Quantum Institute, Portugal

4Sorbonne Université, CNRS, LIP6, 4 Place Jussieu, Paris F-75005, France

5Instituto de Telecomunicações, Lisbon, 1049-001, Portugal

6Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

7Department of Electrical Engineering and Computer Science,

Massachusetts Institute of Technology, Cambridge, MA 02139, USA

8Department of Computer Science, University of Massachusetts, Amherst, Massachusetts 01003, USA

Giovannetti, Lloyd, and Maccone (2008) proposed a quantum random access memory (QRAM)

architecture to retrieve arbitrary superpositions of N(quantum) memory cells via O(log(N)) quan-

tum switches and O(log(N)) address qubits. Towards physical QRAM implementations, Chen et

al. (2021) recently showed that QRAM maps natively onto optically connected quantum networks

with O(log(N)) overhead and built-in error detection. However, modeling QRAM on large networks

has been stymied by exponentially rising classical compute requirements. Here, we address this

bottleneck by: (i) introducing a resource-eﬃcient method for simulating large-scale noisy entangle-

ment, allowing us to evaluate hundreds and even thousands of qubits under various noise channels;

and (ii) analyzing Chen et al.’s network-based QRAM as an application at the scale of quantum

data centers or near-term quantum internet; and (iii) introducing a modiﬁed network-based QRAM

architecture to improve quantum ﬁdelity and access rate. We conclude that network-based QRAM

could be built with existing or near-term technologies leveraging photonic integrated circuits and

atomic or atom-like quantum memories.

Introduction

A quantum random access memory (QRAM) is an es-

sential computational primitive for many quantum algo-

rithms. The ability to perform a QRAM query in log(N)

time steps, where N= 2nis the number of memory

cells, implies polynomial speed-ups for applications such

as quantum machine learning [1], matrix inversion [2],

quantum imaging [3], and quantum searching [4]. De-

spite its clear importance to quantum information pro-

cessing, a QRAM has yet to be realized experimentally.

Hence, ﬁnding a suitable architecture that can be real-

ized in the near-future remains an active research subject

in the theoretical and experimental domains.

In this article, we present a method to simulate large-

scale entanglement accounting for various sources of

noise. We are able to eﬃciently simulate circuits with

thousands of qubits under dephasing, amplitude damp-

ing, and CNOT errors. Based on our simulation model,

we present a QRAM architecture for photonic network-

based QRAM based on Ref. [5]. The feasibility assess-

ment is based on realistic parameters extracted from re-

cent experiments, which we will refer to throughout the

article.

A classical RAM [6] consists of a binary tree leading

to a ﬁnal layer of memory cells, each corresponding to an

unique address. The address is represented as a series of

bits, with each bit corresponding to a layer of the binary

tree. Each bit of an address describes how the bus signal

propagates in the layer: to the right or to the left child

node. Hence, the nodes of the binary tree act as switches

for the address. When provided with a n-bit address, the

RAM returns a bit string fkassociated to the memory

cell labeled k. This is called the fan-out scheme [7].

A QRAM is the quantum analog of the RAM, similarly

consisting of addresses, quantum switches, and memory

cells in the form of qubits. In particular, with a quantum

address state, over the set of address qubits a, given by

|ψ′

in⟩=Pn

j=1 αj|j⟩a, one can retrieve data from a super-

position of memory cells. A QRAM query is deﬁned via

the following transformation,

|ψin⟩=|ψ′

in⟩|∅⟩b−→ |ψout⟩=

j=1

αj|j⟩a|Dj⟩b(1)

where |∅⟩ represents an ancillary state, over the bus qubit

b, which transforms into the retrieved data state after

querying. In this article, we will restrict our investi-

gations to classical data, i.e. |Dj⟩are separable bits.

A direct conversion of classical fan-out protocol to the

quantum realm is ineﬃcient since it requires maintain-

ing quantum coherence over an exponential number of

connections [7].

Three main schemes have been investigated to date:

the fan-out scheme that was already described, the

bucket brigade model, and the teleportation-based

scheme. Important ﬁgures of merit for the QRAM are

the ﬁdelity of the above transformation and the query

time. For a detailed study and comparison of the ﬁrst

two schemes, please refer to Ref. [8].

arXiv:2210.13494v2 [quant-ph] 4 Dec 2023

In the bucket brigade (BB) model [7, 9], the number

of qubits of the device scales as O(2n), as does the num-

ber of gates. Moreover, the original protocol [7] includes

an additional third state in each node, called the “wait”

state in order to prevent the exponential scaling of the

amount of decoherence, with respect to the memory size.

However, Hann et al. [8] have shown that the origin of

the noise resilience of the BB model is the amount of en-

tanglement among the memory’s components and not the

presence of the “wait” state, as one can devise a BB model

without the “wait” state that still achieves a polynomial

scaling of the decoherence with respect to the number of

memory addresses n.

More recently, Chen et al. presented a photonic

network-based QRAM scheme [5] that makes use of quan-

tum teleportation of addresses from a quantum computer

to the QRAM binary tree. Such a scheme greatly in-

creases the protocol’s eﬃciency by teleporting the reg-

isters to the layers (initially prepared in GHZ states) in

parallel as opposed to in series, thereby circumventing the

event of a single qubit loss collapsing the entire tree state.

Additionally, the proposed QRAM maps onto quantum

networks, leading to potential applications in distributed

quantum computing and sensing.

However, Chen et al. left as an open challenge the sim-

ulation of the scheme on large-scale networks since the

computational complexity scales exponentially with the

number of qubits. In this work, we bypass this problem

resorting to more eﬃcient ways of modeling the noise in

stabilizer states. Moreover, this method generalizes to

other quantum networking tasks with similar construc-

tions, such as protocols for distributed quantum compu-

tation.

This comes in line with the fact that distributing en-

tanglement is central in quantum information processing

schemes ranging from quantum computing to sensing to

communications [10–12]. Simulation of distributed en-

tanglement in a network setting, be it a long-distance net-

work such as a possible future quantum internet [13], or

small-distance quantum local area network (QLAN) [14],

is important to assessing the limitations imposed by near-

term quantum technologies. The architecture of the

QRAM considered in this paper, building on photonic

network-based QRAM proposed in Ref. [5], involves a se-

ries of exponentially growing GHZ states, with the largest

having as many qubits as there are memory cells. Each

GHZ state spans across a physical layer in the QRAM

architecture, and the number of nodes per layer grows

exponentially with the number of memory cells 2n≡N

to be addressed, as shown in Fig. 1.

Computer simulations of noisy quantum processes in

such a system quickly becomes computationally inten-

sive [15–17] due to the density matrices growing exponen-

tially in size with the number of qubits. Even though the

entire QRAM protocol deﬁnition, i.e the retrieval of data

given an input address (see Eq. 1), requires more than

just Cliﬀord operations, creating the routing state over

the QRAM architecture only uses Cliﬀord gates. These

operations are the ones used to create these GHZ states

and the teleporting the address state onto the QRAM ac-

cess layers. Moreover, the operations required to access

the QRAM after the routing state is distributed over the

routing nodes only grows with the logarithm of the num-

ber of qubits of the QRAM, in comparison to the linear

amount of operations required to create the routing state.

This is the reason why noise in the system mostly comes

from the GHZ states before access. In particular, this

set of operations to create the routing state can be clas-

sically simulated eﬃciently [16]. This approach enables

an explicit and eﬃcient description of all the intermedi-

ary states, up to local unitary corrections. In this article,

we develop eﬃcient methods to simulate large-scale noisy

entanglement by characterizing the impact of noise at all

intermediate steps, and apply these tools to simulate a

noisy QRAM.

Quantum

Computer

...

Quantum RAM

Access Tree

Memory Cells

Layer 1

Layer 2

Layer 3

D1D2D3D4D5D6D7

FIG. 1. Overview of a teleportation-based QRAM ar-

chitecture. A quantum RAM in the form of a binary tree

comprises GHZ states for each physical layer. The left-most

node of each layer iis entangled with an ancillary qubit in

a remote quantum computer, which hosts the query address

qubits (blue). Bell state measurement in the quantum com-

puter then teleports the address state onto the access tree.

The elementary operations to constructing GHZ states in a

photonic integrated circuit (PIC) QRAM are identical to the

ones over [5].

There are several architectures for a QRAM. Here, we

focus on the optically mediated quantum network-based

QRAM architecture introduced in Ref. [5], as it oﬀers sev-

eral key beneﬁts: implementation in quantum networks

compatible with envisioned quantum internet architec-

ture and quantum data centers, and faster query times

and possibility of executing in a non-local manner by

means of teleportation. Hence, this scheme works un-

der any network-like architecture, be it locally (e.g. on a

chip) or across large distances (e.g. over a quantum in-

ternet). Without loss of generality, we characterize each

node of the architecture as one of a spin-photon network

that could be implemented in photonic integrated circuits

(PIC).

The architecture of the QRAM is similar to previous

models, such as BB and the fan-out models. The main

e1 e2 e1 e2 e1 e2

Electron Photon Interaction

FIG. 2. Probabilistic CNOT. Execution of a CNOT gate

between two electrons, e1 and e2, mediated by a photon.

diﬀerence concerns the execution of the protocol and the

resources available at each node. In this architecture, one

considers two agents: the quantum computer, which pre-

pares the addresses, and the QRAM or quantum access

tree (see Fig. 1). The quantum computer must provide

an address state with n= log2Nqubits, where Nis the

total number of memories (for simplicity assume n∈N).

The QRAM has a binary tree architecture, with nphys-

ical layers, where the kth layer (k∈ {1, ..., n −1}) has

2k−1quantum nodes. As we describe next, in each phys-

ical layer, all the nodes share a GHZ state, which is used

to teleport the address state onto the QRAM itself, al-

lowing for an ancilla qubit to access the memories in the

correct superposition.

As for the type of physical implementation chosen,

and without loss of generality, we focus on a QRAM im-

plementation involving solid-state spin qubits integrated

into PICs, an approach that is promising in terms of scal-

ability. In particular, we consider diamond nanophotonic

cavities coupled with silicon-vacancy centers [18, 19] as

each QRAM tree node. Each emitter contains an elec-

tronic spin that directly interacts with the photonic ad-

dress register qubits and an accompanying nuclear spin

acting as a long-lived memory. By entangling the elec-

tronic spin with the photon via cavity reﬂection, consec-

utive reﬂection of a photon oﬀ two neighboring nodes

and subsequent heralding achieves spin-spin entangle-

ment. This remote entangling strategy is repeatedly used

to generate a GHZ state across each layer. Such opera-

tions are probabilistic (see Fig. 2): the photon has a non-

zero probability of being lost to the environment before

reﬂecting oﬀ two cavities and arriving at the detector.

On the other hand, it is possible to perform close to de-

terministic two-qubit gates between the electronic and

nuclear spin qubits, albeit with a larger error [20, 21].

For this reason, we term this architecture teleportation-

based deterministic QRAM, or TD-QRAM.

In these types of systems, the main contributors to er-

rors are (i) spin phase errors (at rate 1/T2), (ii) spin ﬂip

errors (at rate 1/T1), and (iii) errors in hyperﬁne gates

between electron and nuclear spins (see Supplementary

Table I). We leave out photon-electron interactions, as

one could conceive trading-oﬀ the eﬃciency ηfor arbi-

trarily high ﬁdelity in the cavity-reﬂection based scheme

proposed in Ref. [22] in the high-cooperativity and over-

coupling regime.

Hence, we explore diﬀerent values for T1,T2of both

electronic and nuclear spin qubits, and peand pnfor

the probabilities of error in electronic and nuclear spin

CNOTs. For the remaining of this article, we set Tn

100 Te

1≡100 T1and Tn

2= 100 Te

2≡100 T2. Nuclear

spins have a higher coherence time as they are much less

coupled to the noisy spin-bath compared to electronic

spins. Reported values of characteristic times go, ex-

perimentally, up to Te

1∼1 s,Te

2∼10 ms [23], and

there are theoretical predictions of being able to reach

pe, pn= 10−2∼10−4[24, 25]. Moreover, we detail other

important physical parameters of this type of system,

used for the simulations, in Supplementary Table I.

Results

Simulating the eﬀects of decoherence for a TD-

QRAM. To simulate the QRAM initialization protocol,

we use NetSquid [26] under the stabiliser formalism and

extract all the parameters of the noise channels before

implementing them in simulations, for instances: tim-

ing parameters for every qubit used throughout the sim-

ulation, all the noisy CNOTs with corresponding error

probabilities, and to which qubits and at which step it

is applied. From here, we compute the ﬁdelity of the ﬁ-

nal QRAM state by substituting all these values into the

expressions presented in the methods.

We start by presenting the simulation of a 212-qubit

QRAM in Fig. 3. Here, we detail individually the ﬁdeli-

ties of the GHZ state distributed at each physical layer

of the QRAM. The ﬁdelity of the full state of the QRAM

is given by:

F(QRAM) =

n−1

i=1

FLayeri,|GHZ⟩2i−1+1,

where |GHZ⟩q=1

√2|0⟩⊗q+|1⟩⊗q

(2)

i.e., the ﬁdelity of the entire tree (or the QRAM) is de-

ﬁned as the product of the ﬁdelities of each physical layer

(see Supplementary Methods for more details). We dis-

tinguish access ﬁdelity from tree ﬁdelity, where the for-

mer refers to the ﬁdelity of the state retrieved after ac-

cessing the memory cells (|ψout⟩in Eq. 1), and the latter

refers to the multipartite state ﬁdelity of the binary tree

constituting the QRAM. Only the access ﬁdelity depends

on the address and bus qubits.

One observes an exponential decrease of the ﬁdelity

with the number of the layer (notice the logarithmic scal-

ing on the y-axis corresponding to the ﬁdelity). This

agrees with the GHZ state size increasing exponentially

with the number of layers, i.e. scaling 2k. When one

qubit in this multipartite state suﬀers an error, the en-

tire state is aﬀected.

One critical ﬁgure of merit that we extract from the

NetSquid simulations is the query time. As demonstrated

in Ref. [5], the query eﬃciency scales logarithmically with

the number of qubits. Extracting from multiple queries

of the QRAM, we obtain the query times (apart from

10110 2103

N um ber of M em or y Qubits

0.0

0.2

0.4

0.6

0.8

1.0

F i de lity

0.0

0.0091

0.6204

0.9525

0.999

CNOT = 10 2

CNOT = 10 3

CNOT = 10 4

CNOT = 10 5

CNOT = 0

10110 2103

N um ber of M em or y Qubits

1 10 2

1 10 3

1 10 4

1 10 5

1 10 6

1 10 7

F i de lity

0.9 692

0.9 967

= 0.5

= 0.6

= 0.7

= 0.8

= 0.9

= 0.5; T2=

1 2 3 4 5 6 7 8 9 10 11 12

Layer N um ber

1 10 2

1 10 3

1 10 4

F i de lity

Layer 2

Layer 3

Layer 4

Layer 5

Layer 6

Layer 7

Layer 8

Layer 9

Layer 1 0

Layer 1 1

Layer 1 2

= 0.5

= 0.6

= 0.7

= 0.8

= 0.9

10110 2103

N um ber of M em or y Qubi ts

105

Quer y t ime ( ns)

= 0.5

= 0.6

= 0.7

= 0.8

= 0.9

128

~0.87

c d

FIG. 3. TD-QRAM Simulations. (a) TD-QRAM access protocol for 12 layers, with the eﬃciency of generating a Bell

pair swept from η= 50% to η= 90%. The noise analysis considers only dephasing and damping errors. The ﬁnal ﬁdelity is

calculated according to Eq. 2, with T1= 20 ms,T2= 10 ms, and ϵCNOT = 0 for each layer. (b) Query times with varying

sizes from 2 layers to 12 layers, and sweeping the eﬃciency of generating a Bell pair from η= 50% to η= 90%. There is an

expected logarithmic scaling of the query time with the number of qubits. (c) TD-QRAM noise analysis with dephasing errors,

T2= 10 ms (ﬁlled lines) and T2= 100 ms (traced lines), with ﬁxed amplitude-damping error T1= 2 s. We consider diﬀerent

QRAM sizes from 2 layers to 12 layers as well as various eﬃciencies of generating a Bell pair from η= 50% to η= 90%. (d)

TD-QRAM noise analysis with noisy CNOTs, pe=pn∈ {0,10−5,10−4,10−3,10−2}, for a QRAM with the number of layers

ranging from 2 to 12. The dephasing time is ﬁxed at T2= 100 ms, and the amplitude-damping time is ﬁxed at T1= 2 s.

The eﬃciency of generating a Bell pair is ﬁxed at η= 90%. The ﬁnal ﬁdelity mainly depends on the number of noisy CNOTs

performed throughout the protocol and has little dependence on the eﬃciency. All the error bars over the data correspond to

the error of the average value over 100 simulations of the protocol.

a logarithmic factor derived from making the bus qubit

traverse the binary tree) in Fig. 3.

Dephasing and Damping Errors for TD-QRAM.

Considering only the eﬀects of dephasing and amplitude-

damping errors in the spin qubits, we take T2=∞ms for

amplitude-damping errors only, and then T2= 10 ms and

T2= 100 ms with a ﬁxed T1= 2 s [23], see Supplementary

Table I. We also set the CNOT error rate to 0. We present

the simulation results for the TD-QRAM scheme under

memory dephasing for increasing QRAM size, as shown

in Fig. 3.

Looking closely at Figure 3(c), one can observe that

the eﬀect of amplitude-damping shows an identical be-

havior to the one of dephasing and amplitude-damping

combined, i.e., with the same type of scaling. However,

it is residual comparatively to the eﬀect of dephasing.

This is easily explainable by the time-scales of the co-

herence times of the corresponding noises (T1and T2)

in the memory diﬀer by orders of magnitude, with the

ﬁrst, T1, being usually much longer than the latter, T2,

i.e. T1∼1 s [23]. For this reason, its impact can be

neglected relative to other sources of error.

Dephasing, Damping and Noisy CNOTs for TD-

QRAM. The only type of error missing in the analysis

is the error derived from the use of noisy CNOTs. Illus-

trated in Fig. 3, the dephasing and damping errors min-

imally contribute to inﬁdelity. We now analyse the case

for noisy CNOTs on top of ﬁxed T1= 2 s and T2= 100 ms

(note we now switch to linear scale in the y-axis for the

ﬁdelity, due to the set of values present for the diﬀerent

simulations). For simplicity, we consider equal CNOT

error probability, ϵCNOT, for both electronic and nuclear

CNOTs, and vary ϵCNOT from 10−5to 10−2as shown in

Fig. 3:

These simulations show that the CNOT gates domi-

nate the overall error in the QRAM state ﬁdelity in the

TD-QRAM. For instance, to access a 128-qubit QRAM,

one needs ﬁdelities of the CNOT gates to be somewhere

near 99.9% to obtain an access ﬁdelity exceeding 90%. In

this architecture, while the query times do not increase

linearly with the size of the memory, the errors do. Ex-

pectedly, applying an error to a single qubit of a GHZ

state contributes in the same order for the entire state.

The price to pay for performing CNOTs with such large

error rates deterministically could be circumvented by

near-perfect yet probabilistic CNOTs [22, 27] via cavity-

based electron spin-photon interactions, as opposed to

deterministic yet error-prone nuclear-electron spin cou-

pling. In light of this, we explore a hybrid teleportation-

based QRAM architecture in the following section.

Teleportation-based Stochastic QRAM . In the

TD-QRAM protocol, the entanglement generation and

swap (Fig. 8) operation are still probabilistic given the

ﬁnite chance of photon loss. Hence, these probabilistic

CNOTs are done in parallel throughout each physical

layer to improve eﬃciency. After an EPR pair is created

between two electron spins, however, transferring entan-

glement onto the nuclear spins is a deterministic proce-

dure. Thereby, the query time grows sub-linearly. As

noted before addressing the TD-QRAM scheme, this de-

terministic CNOT based on nuclear-electron spin inter-

action mainly dominates the inﬁdelity of the GHZ state,

motivating us to contemplate an alternative solution.

Since the decoherence errors from T1and T2contribute

much less to the inﬁdelity relative to electron-nuclear spin

CNOT, replacing some of the noisy deterministic CNOTs

with probabilistic CNOTs helps improve the ﬁdelity de-

spite reducing eﬃciency. As we will show, this leads

to higher QRAM tree state ﬁdelities, albeit with longer

query times. We call this architecture ‘teleportation-

based stochastic QRAM’, or TS-QRAM.

Relying solely on probabilistic CNOTs in every step of

the protocol would be very ineﬃcient since the probabil-

ity of generating a GHZ state diminishes exponentially

with the number of nodes. In other words, if one entan-

glement attempt fails during construction of a GHZ state,

the entire state collapses. Since each linking process is

heralded, there are ways to circumvent this by choosing a

speciﬁed order to perform the CNOTs, similar to entan-

glement swapping in a repeater chain [28, 29]. Here, the

probabilistic swapping operations are equivalent to the

probabilistic CNOTs, and measuring the middle node is

analogous to joining smaller GHZ states to form a larger

GHZ state. Abstractly, they describe the same problem,

which allows us to use the solutions provided by Ref. [29].

Next, we present an in-depth analysis of the trade-oﬀ be-

tween ﬁdelity and query rate as a function of error rates

and physical implements.

Increasing T1and T2.To decrease the number of

employed deterministic CNOTs, and taking into account

that these always happen when the electronic spins in-

teract with the nuclear spins, it is natural to consider

dropping the nuclear spins altogether. This is motivated

by the fact that we can perform CNOTs, albeit proba-

bilistically, between the electron spins. The downside is

that electron spins suﬀer from having shorter coherence

times than their nuclear counterparts. Still, it is advanta-

geous to consider such schemes to avoid the use of noisier

deterministic CNOTs.

To minimize the consequently increased decoherence,

one could conceive schemes for increasing the T1and T2

times for the electrons, since these are the ones now caus-

ing the ﬁdelity bottleneck, together with the required

time to query the memory.

Presently, the SiV’s electronic spin’s T1time is shown

to be longer than 1 s [23], thereby posing no concern

over depolarisation. On the other hand, its T2coherence

time is limited to tens of milliseconds [23] even under dy-

namical decoupling. The main dephasing mechanism is

attributed to the surrounding nuclear spin bath, which is

weakly coupled to the electronic spin of interest via hy-

perﬁne interaction [30]. A potential avenue to improving

the electronic spin’s T2is therefore to “purify” its environ-

ment by materials engineering [31]. By producing SiV in

a carbon-13 free matrix, for example, the coherence time

may be further extended.

Nevertheless, our numerical analyses of the hybrid

scheme show ﬁdelities still exceeding 60% for a reason-

able CNOT error rate of 10−3and 1024 memory cells,

using a T2of 100 ms. For such a result, a probability of

success of about 70% for the CNOT is required.

The Teleportation-based Stochastic QRAM

Protocol. In the TD-QRAM protocol, there are

two steps occurring in parallel across each layer in the

QRAM: one for generating EPR pairs across every other

node and another for linking all the states into a larger

GHZ state, via sharing EPR pairs in-between nodes hold-

ing the previously shared EPR pairs (see Fig. 8). This

could be made in parallel because the linking operations

are deterministic.

In the TS-QRAM protocol, however, we must now con-

sider an order for the linking step that depends on the

node’s position, similar to the quantum repeater chain

problem [26, 29, 32]. If a linking process fails, the sub-

set of qubits that would have become entangled must

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Resource-efficientsimulationofnoisyquantumcircuitsandapplicationtonetwork-enabledQRAMoptimizationLuísBugalho,1,2,3,4EmmanuelZambriniCruzeiro,5KevinC.Chen,6,7WenhanDai,7,8DirkEnglund,6,7andYasserOmar1,2,31InstitutoSuperiorTécnico,UniversidadedeLisboa,Portugal2PhysicsofInformationandQuantumTechnologie...

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Resource-efficient simulation of noisy quantum circuits and application to network-enabled QRAM optimization Luís Bugalho1 2 3 4Emmanuel Zambrini Cruzeiro5Kevin C.

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