Optimized Telecloning Circuits Theory and Practice of Nine NISQ Clones Elijah Pelofskey Andreas B artschiy Stephan Eidenbenzy

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Optimized Telecloning Circuits: Theory and

Practice of Nine NISQ Clones

Elijah Pelofske∗†, Andreas B¨

artschi∗†, Stephan Eidenbenz†

†CCS-3 Information Sciences, Los Alamos National Laboratory, Los Alamos, NM 87544, USA

∗Corresponding author: epelofske@lanl.gov

Abstract—Although perfect copying of an unknown quantum

state is not possible, approximate cloning is possible in quantum

mechanics. Quantum telecloning is a variant of approximate

quantum cloning which uses quantum teleportation to allow

for the use of classical communication to create physically

separate clones of a quantum state. We present results of a

of 1→9universal, symmetric, optimal quantum telecloning

implementation on a cloud accessible quantum computer - the

Quantinuum H1-1 device. The H1-1 device allows direct creation

of the telecloning protocol due to real time classical if-statements

that are conditional on the mid-circuit measurement outcome of

a Bell measurement. In this implementation, we also provide an

improvement over previous work for the circuit model description

of quantum telecloning, which reduces the required gate depth

and gate count for an all-to-all connectivity. The demonstration

of creating 9approximate clones on a quantum processor is the

largest number of clones that has been generated, telecloning or

otherwise.

Index Terms—NISQ computing, Quantum telecloning, quan-

tum cloning, Bell state, single qubit state tomography

I. INTRODUCTION

Due to the no cloning theorem, perfect copies of an

unknown quantum state can not be made [35]. However,

approximate copies of an unknown state can be made [7]—

this process is referred to as quantum cloning. There are

a large number of variants of quantum cloning algorithms,

and therefore it is helpful to classify these algorithms using

different characteristics. Symmetric quantum cloning means

that all generated clones are identical and therefore have the

same ﬁdelity, whereas asymmetric quantum cloning is where

the clones can be different. Universal quantum cloning is state

independent, i.e., the clone quality is not dependent on the state

being cloned. In state dependent versions of quantum cloning,

on the other hand, the clone quality is dependent on the state

which is cloned [13].

The optimal theoretical approximate clone ﬁdelity limit for

symmetric universal quantum cloning can be exactly computed

[23], [29], and is given in Eq. (1). A ﬁdelity of 1indicates

LA-UR-22-30899; this work was supported by the U.S. Department of

Energy through the Los Alamos National Laboratory. Los Alamos National

Laboratory is operated by Triad National Security, LLC, for the National

Nuclear Security Administration of U.S. Department of Energy (Contract No.

89233218CNA000001). We acknowledge the use of IBM Quantum services

for this work. The views expressed are those of the authors, and do not reﬂect

the ofﬁcial policy or position of IBM or the IBM Quantum team. This research

used resources of the Oak Ridge Leadership Computing Facility, which is a

DOE Ofﬁce of Science User Facility supported under Contract DE-AC05-

00OR22725.

that the two quantum states being compared are identical,

whereas a ﬁdelity of 0indicates that the two quantum states are

orthogonal, while a maximally mixed (e.g. completely noisy)

1-qubit state results in a ﬁdelity of 0.5.

FN→M=MN +M+N

M(N+ 2) (1)

A quantum cloning process which produces clones that can

achieve this bound is referred to as optimal [15], [34]. Quan-

tum telecloning is a combination of quantum teleportation

[6] and optimal quantum cloning [23]. Quantum telecloning

allows the distribution of quantum information, speciﬁcally

approximate clones of an unknown quantum state, to be

distributed to different parties [14]. In particular, the usage

of classical communication of the measurement of a Bell state

allows different parties (which could be spatially separated) to

conditionally apply quantum operations to their qubits in order

to generate optimal clones of the unknown quantum state.

Algorithm 1details the telecloning process. Quantum cloning

is of particular interest for quantum information processing

and quantum networking [5], [9], [24], [29].

Designing quantum telecloning circuits to run on Noise

Intermediate Scale Quantum (NISQ) [27] equipment is an

open challenge because of the need for optimized, low-depth

circuits due to noise and decoherence on NISQ devices. Using

Dicke state preparation improvements [1]–[4], [12], [16], the

authors of [25] build an explicit circuit model description for

implementing telecloning on NISQ devices. Our contributions

are a signiﬁcantly improved telecloning circuit over previous

work that we show to be useful on a NISQ device by

generating 9 clones. More precisely, we:

1) Provide an optimized circuit model algorithm of 1→M

optimal, universal, symmetric telecloning for an all-to-

all connectivity.

2) Report experimental results of 1→9quantum tele-

cloning on the Quantinuum H1-1 device, which is the

largest experimental demonstration (in terms of the

number of clones) of optimal universal symmetric clones

generated on a NISQ computer.

We ﬁnd that (i) our 1→9quantum telecloning circuit has

211 two-qubit gates; (ii) the ﬁdelities we achieve vary across

the nine individual clones and also across the four different

cloned states that we tested, but generally range from 0.55 to

0.67 (theoretical optimum is 19

27 ≈0.7037, see (1)) with an

average of 0.59 across all experiments, which is comparable

arXiv:2210.10164v2 [quant-ph] 30 Nov 2022

Algorithm 1 Quantum 1→MTelecloning Protocol

State Preparation:

1: A message qubit qmis prepared by a sender

2: A quantum telecloning state T C is constructed with

-(M−1) ancilla qubits A,1Port qubit P, and

-Mclone qubits C(sent to the receivers).

Teleportation:

3: A bell measurement is made between qmand P, and the

results are communicated classically to the clone holders.

4: The clone holders use the result of the bell measurement

to decide whether to apply X- and/or Z-gates to the clone

qubits in order to construct the approximate clones:

-Φ+: apply nothing - Φ−: apply Z-gate

-Ψ+: apply X-gate - Ψ−: apply X- then Z-gate

Result:

5: Mapproximate clones of qmhave been generated with

theoretical maximal ﬁdelity described by Eq. (1).

to the ﬁdelities achieved by e.g., the IBM Q devices for

much smaller 1→3telecloning circuits with ancilla [25];

(iii) beyond the standard use cases for telecloning, we ﬁnd

telecloning to be a good benchmark to test NISQ devices as

a more detailed analysis reveals asymmetries in the results

that are due to hard- and control-software details of the H1-1

device.

Earlier works have looked at telecloning with either two

or three clones and mostly on custom-built devices, whereas

our results include nine clones achieved on a fairly standard

general-purpose, cloud-accessible quantum computing NISQ

device: [22] reports ﬁdelities of 0.58 for two teleclones; [33]

describes a device for 3 teleclones achieving ﬁdelity 0.64 for

two and 0.49 for the third clone; [10] provides a high-ﬁdelity

implementation of three teleclones on a custom-built photonics

device; ﬁnally, [25] runs a large number of two and three

telecloning experiments on IBM Q quantum computers as

well as on Quantinuum’s H1-1 device and reporting ﬁdelities

of 0.82 and 0.76 for two and three teleclones, respectively,

on Quantinuum H1-2 and signiﬁcantly lower ﬁdelities for

the IBM Q devices with additional parameter studies of

circuits with and without ancilla qubits and error mitigation

techniques. See [14] for a review on telecloning and [29] for

a review on quantum cloning.

Figures are generated using a combination of Qiskit [32],

Matplotlib [8], [17], QuTiP [18], [19] and mayavi [28]. All

data, code, and extra ﬁgures are available on a public Github

repository1.

II. METHODS

A. Algorithm improvements

Figure 2, illustrated with quantikz [21], details the all-to-all

telecloning circuit implementation. This circuit construction

reduces the number of two qubit gates (CNOTs) required by

≈33% compared to previous work [25], while keeping the

1https://github.com/lanl/Quantum-Telecloning

Fig. 1. Two different views of Bloch sphere representations of the four

Tetrahedral basis states which we want to create approximate teleclones of.

circuit depth the same. Our circuit construction works for any

M; we illustrate it for M= 9, the largest such circuit that

can be executed on current Quantinuum H1 devices.

For 1→Mquantum telecloning, one generates a message

qubit qmand a telecloning state A(M−1)P CMon M−1

ancilla, 1 port, and Mclone qubits of the form [23]

|AM−1P CMi=1

√M+1 XM

i=0 |DM

iiAM−1P|DM

iiCM,(2)

where |DM

iidenotes the uniform superposition over all M-

qubit states of Hamming weight iwith real amplitudes, e.g.,

|D3

1i= [|001i+|010i+|100i]/√3. After creation of this

state, the ancilla qubits are discarded.

It is known [25] that such a telecloning state can be created

using Dicke state unitaries DSU(M), based themselves on

Split & Cyclic Shift unitaries SCS(m)[3], deﬁned by:

DSU(M): |1i0M−ii 7→ |DM

ii,

SCS(m): |1i0m−ii 7→ √m−i

m|1i0m−ii+√i

m|1i−10m−i1i.

We make two observations, cf. Figure 2:(i) the tele-

cloning state can be created by applying two Dicke state

unitaries DSU(M)in parallel to the (easily creatable)

input state 1

√M+1 PM

i=0 |1i0M−iiAM−1P|1i0M−iiCM, and

(ii) Dicke state unitaries are made up of Split & Cyclic

Shift unitaries: DSU(M) = QM

m=1 SCS(m)⊗IdM−m. We

make use of these observations combined with the fact that

ancilla qubits are discarded:When considering the application

of DSU(M)to the ancilla and port qubits, only the ﬁrst

SCS(M)unitary involves the port qubit, the following SCS(m)

unitaries comprising a DSU(M−1) unitary act only on

the to be discarded ancilla qubits. These unitaries can thus

be removed without consequence to the telecloning protocol,

resulting in a reduction from 330 to 211 CNOT gates for

1→9telecloning.

Finally, after distribution of the clones to the receivers,

the standard teleportation protocol [6] is executed. In this,

the sender measures its |message,portiqubits in the Bell

basis. The result is communicated classically to the receivers,

which – depending on the measurement – adjust their clones

with/without local Pauli-Xand/or Zgates, see Algorithm 1.

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OptimizedTelecloningCircuits:TheoryandPracticeofNineNISQClonesElijahPelofskey,AndreasB¨artschiy,StephanEidenbenzyyCCS-3InformationSciences,LosAlamosNationalLaboratory,LosAlamos,NM87544,USACorrespondingauthor:epelofske@lanl.govAbstractAlthoughperfectcopyingofanunknownquantumstateisnotpossible,app...

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Optimized Telecloning Circuits Theory and Practice of Nine NISQ Clones Elijah Pelofskey Andreas B artschiy Stephan Eidenbenzy

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