QKD in the NISQ era enhancing secure key rates via quantum error correction Shashank Kumar Ranu12 Anil Prabhakar1 Prabha

2025-05-02 0 0 830.46KB 31 页 10玖币

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QKD in the NISQ era: enhancing secure key rates

via quantum error correction

Shashank Kumar Ranu1,2, Anil Prabhakar1, Prabha

Mandayam2

1Department of Electrical Engineering, Indian Institute of Technology Madras,

Chennai, India

2Department of Physics, Indian Institute of Technology Madras, Chennai, India

E-mail: ee16s300@ee.iitm.ac.in,anilpr@ee.iitm.ac.in,

prabhamd@physics.iitm.ac.in

12 October 2022

Abstract. Error mitigation is one of the key challenges in realising the full potential

of quantum cryptographic protocols. Consequently, there is a lot of interest in adapting

techniques from quantum error correction (QEC) to improve the robustness of quantum

cryptographic protocols. In this work, we benchmark the performance of diﬀerent QKD

protocols on noisy quantum devices, with and without error correction. We obtain the

secure key rates of BB84, B92 and BBM92 QKD protocols over a quantum channel

that is subject to amplitude-damping noise. We demonstrate, theoretically and via

implementations on the IBM quantum processors, that B92 is the optimal protocol

under amplitude-damping and generalized amplitude-damping noise. We then show

that the security of the noisy BBM92 protocol crucially depends on the type and the

mode of distribution of an entangled pair. Finally, we implement an error-corrected

BB84 protocol using dual-rail encoding on a noisy quantum processor, and show that

the dual-rail BB84 implementation outperforms the conventional BB84 in the presence

of noise. Our secure key rate calculation also takes into account the eﬀects of cnot

imperfections on the error rates of the protocols.

1. Introduction

Quantum key distribution (QKD) oﬀers the promise of secure communication between

two parties, Alice and Bob, in the presence of an eavesdropper, Eve. When Eve

attempts to steal information from the quantum channel, she also inevitably introduces

disturbances in the channel and reveals herself. Since the ﬁrst proposal by Bennett

and Brassard in 1984 [1], there have been many advances, both theoretically and

experimentally [2, 3, 4]. So far, QKD protocols have been implemented over thousands of

kilometers over free-space channels [5], hundreds of kilometers over ﬁber-optical channels

[3, 4], and tens of meters over underwater quantum communication channels [6]. QKD

key rates are aﬀected by scattering, absorption, damping, and other noise models

prevalent in these quantum channels. Amplitude-Damping (AD) and Generalized

arXiv:2210.05297v1 [quant-ph] 11 Oct 2022

QKD in the NISQ era: enhancing secure key rates via quantum error correction 2

Amplitude-Damping (GAD) are two such noise models that degrade the performance

of a QKD system, especially in turbulent media such as air and seawater.

In parallel, the area of quantum computing has also progressed dramatically,

culminating in the demonstration of quantum supremacy by various groups [7, 8, 9].

Established technology giants (IBM, Microsoft, Google, Honeywell, etc.) and startups

(Rigetti, IonQ, Xanadu, etc.) are in the race to build a fully fault-tolerant and scalable

quantum computer. However, the current generation of quantum computers are Noisy

Intermediate Scale Quantum (NISQ) devices with noisy gates, qubit readout errors,

and small coherence times [10]. Benchmarking these NISQ devices forms a critical step

towards coming up with the practical applications of these quantum processors.

Eﬀects of AD noise on the BB84 QKD protocol have been studied in the asymptotic

as well as ﬁnite-key regime [11, 12]. However, the performance of the BB84 protocol in

the presence of GAD noise has not been studied. Entanglement-based protocols such

as the BBM92 protocol form a vital subclass of QKD protocols. Eﬀects of AD noise

on the Bell states have been extensively studied in the context of teleportation ﬁdelity

between two parties [13], but its eﬀects on the bit and the phase error rates of an

entanglement-based QKD protocol are yet to be quantiﬁed.

In this work, we merge ideas from two diﬀerent areas of quantum technologies,

namely, QKD and quantum computing. We employ a quantum processor to mimic

the amplitude-damping channel and implement QKD protocols on it. Such an

implementation on a quantum processor helps us observe the eﬀects of AD noise on

the performance of QKD protocols without having to physically implement the protocol

over a noisy, long-distance quantum channel. Our results will also help us design eﬃcient

QKD protocols over turbulent channels such as seawater and free-space. The insights

we gain from this study will also help in the eventual design and characterization of

quantum memories to be used as quantum repeaters, in the presence of AD noise.

We invoke techniques from quantum error correction to mitigate the eﬀects of

amplitude-damping noise on the secure key rates of QKD protocols. Recently, there have

been preliminary works studying the performance of encoded quantum repeater-based

QKD protocols, where the well-known three-qubit repetition code has been employed

for encoding quantum information [14, 15]. Here, in a deviation from the standard

approach of using stabilizer codes to correct for arbitrary noise, we rely on noise-adapted

quantum error correction [16] to improve the secure key rates of QKD protocols over an

amplitude-damping channel.

One of the simplest error-detecting codes tailored to protect against amplitude-

damping noise is the dual-rail code, involving encoding a logical qubit in just a pair

of physical qubits [17]. In previous work, dual-rail encoding has been used to correct

readout asymmetry in a BB84 implementation on the IBM quantum processors [18],

where the swap gate was used to realise a quantum channel between Alice and Bob.

The swap gates are implemented using cnot gates, thereby making depolarizing error

the dominant noise in this BB84 implementation of [18]. Moreover, the dual-rail encoder

and decoder used in [18] consist of 4 cnot gates, leading to a high intrinsic error rate.

QKD in the NISQ era: enhancing secure key rates via quantum error correction 3

Furthermore, the results in [18] do not address the eﬀects of state preparation errors,

gate imperfections, channel noise, and qubit readout errors of the NISQ device on the

phase and bit error rates of their dual-rail-based protocol.

In our work, we present two variants of the dual-rail-encoded BB84 protocol and

introduce a theoretical framework to describe the eﬀects of cnot imperfections as well

as damping noise on the performance of our dual-rail BB84 implementations. First,

we use an error detection scheme presented in [19] to reduce the number of cnots in

the dual-rail BB84 implementation. We compare the performance of this error-detected

scheme against the conventional BB84 protocol under AD noise. Our study also helps

to identify the optimal prepare-and-measure QKD protocol under AD noise.

Furthermore, we also address some practical questions in the context of the BBM92

protocol using our theoretical framework and implementations. In particular, we identify

the Bell-EPR states that are most suited for the BBM92 protocol in the presence of AD

noise. We also determine an optimal way to share an entangled pair between Alice and

Bob for the BBM92 protocol. Finally, we estimate the eﬀects of channel asymmetry,

that is, diﬀerent damping probabilities of Alice and Bob’s channel, on the performance

of the BBM92 protocol.

The rest of this paper is organized as follows. Sec. 2 of this work gives an overview

of AD and GAD noise. We also discuss the robustness of the dual-rail encoding against

both noise models. Sec. 3 describes the eﬀects of AD and GAD noise on the secure

key rates of diﬀerent QKD protocols. We also consider various practical imperfections

that aﬀect QKD protocols under AD noise. We next study the performance of dual-rail

encoded BB84 against AD noise in Sec. 4. Assuming non-ideal quantum circuits, we

also estimate the eﬀects of gate imperfections on the performance of the dual-rail BB84

protocol. Sec. 5 of our paper describes the implementation of diﬀerent QKD protocols

on IBM quantum processors. Finally, we summarize and discuss future outlook in Sec. 6.

2. Preliminaries

2.1. Amplitude-damping channel

The amplitude-damping (AD) channel models the decay of an excited state of a two-

level atom due to the spontaneous emission of a photon. Concretely, the interaction

between a two-level atom and the electromagnetic ﬁeld is described using the Jaynes-

Cummings Hamiltonian (HJC). Time evolution of the joint system is described by the

unitary operator U=e−iHJC t. Fixing the detuning parameter of the Jaynes-Cummings

interaction as 0 and tracing over the ﬁeld leads to a decay probability for the two-level

atom that has an exponential dependence on time [17].

Let |0iAdenote the ground state of a two-level atom and |1iArepresent the

excited state of the atom. We assume the “environment” described by the modes of

the electromagnetic ﬁeld to be in the vacuum state |0iE. Let γ(t) denote the decay

probability, with 0 ≤γ≤1. After a time t, the excited state decays to the ground state

QKD in the NISQ era: enhancing secure key rates via quantum error correction 4

with a probability γ(t), and the atom emits a photon. Hence, the environment makes

a transition from the state |0iE(“no photon”) to the state |1iE(“one photon”). This

evolution is described as [17],

|0iA⊗ |0iE→ |0iA⊗ |0iE,

|1iA⊗ |0iE→p1−γ(t)|1iA⊗ |0iE+pγ(t)|0iA⊗ |1iE.(1)

This disspiative behavior is captured by a single-qubit operator of the form,

AAD

1=pγ(t)|0ih1|.(2)

Note that we will drop the dependence of γon time henceforward for notational

simplicity. AAD

1annihilates the ground state and causes the excited state to decay

to the ground state. However, the Kraus operator AAD

1alone does not specify a physical

map, since AAD†

1AAD

1=γ|1ih1|. The Kraus operators of any channel should satisfy the

condition PiA†

iAi=I. We satisfy this completeness condition by choosing another

operator AAD

0such that,

AAD†

0AAD

0=I−AAD†

1AAD

1=|0ih0|+p1−γ|1ih1|.(3)

Thus, the operators AAD

0and AAD

1are valid Kraus operators for the AD channel, and

are represented in matrix form as,

AAD

0= 1 0

0√1−γ!, AAD

1= 0√γ

0 0 !.(4)

2.2. Generalized amplitude-damping channel

The GAD channel models the dissipation eﬀects due to the interaction of a system with

a purely thermal bath at ﬁnite temperature. GAD noise is one of the primary sources

of noise in superconducting quantum processors and in linear optical systems with low-

temperature background noise [20, 21]. The Kraus representation of the GAD channel

is,

Λ(ρ) =

i=3

i=0

AGAD

iρA†GAD

i,(5)

where the 2 ×2 matrix representation of the Kraus operators of GAD channel are

AGAD

0=√p 1 0

0√1−γ!, AGAD

1=√p 0√γ

0 0 !,

AGAD

2=p1−p √1−γ0

0 1!,

AGAD

3=p1−p 0 0

√γ0!.(6)

QKD in the NISQ era: enhancing secure key rates via quantum error correction 5

Here, γis the damping parameter and 0 ≤p≤1. The action of Kraus operators of the

GAD channel on the computational basis vectors {|0i,|1i} can be expressed as,

AGAD

0|0i=√p|0i, AGAD

0|0i=√pp1−γ|1i,

AGAD

1|0i= 0, AGAD

1|1i=√p√γ|0i,

AGAD

2|0i=p1−pp1−γ|0i, AGAD

2|1i=p1−p|1i,

AGAD

3|0i=p1−p√γ|1i, AGAD

3|1i= 0.(7)

Note that we can obtain the Kraus operators of the amplitude-damping channel from

Eq. (6) for p= 1. As the name suggests, the GAD channel generalizes the AD channel

for a bath at a ﬁnite temperature. Hence, GAD noise leads to transitions from |0i→|1i

as well as |1i→|0i. Such a transformation may happen in practical QKD systems due

to polarization ﬂuctuations.

2.3. Robustness of dual-rail encoding against amplitude-damping noise

We now brieﬂy review the dual-rail encoding and how it acts as an error-detecting code

against single-qubit amplitude-damping noise [17]. Dual-rail encoding maps a qubit

onto a two-qubit subspace as shown below.

|0i→|01i≡|0iL,|1i→|10i≡|1iL.(8)

The circuit shown in Fig. 1 implements such an encoding. Using Eq. (4) we can write

Figure 1: Circuit for dual-rail qubit. Here, q0can be in any arbitrary state.

the Kraus operators for the dual-rail qubits as,

M0=AAD

0⊗AAD

M1=AAD

0⊗AAD

M2=AAD

1⊗AAD

M3=AAD

1⊗AAD

1(9)

And the action of these Kraus operators on the logical qubits can be described as shown

below.

M0|01i=p1−γ|01i,

M1|01i= 0,

M2|01i=√γ|00i,

M3|01i= 0,

M0|10i=p1−γ|10i,

M1|10i=√γ|00i,

M2|10i= 0,

M3|10i= 0.

(10)

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QKDintheNISQera:enhancingsecurekeyratesviaquantumerrorcorrectionShashankKumarRanu1;2,AnilPrabhakar1,PrabhaMandayam21DepartmentofElectricalEngineering,IndianInstituteofTechnologyMadras,Chennai,India2DepartmentofPhysics,IndianInstituteofTechnologyMadras,Chennai,IndiaE-mail:ee16s300@ee.iitm.ac.in,anilp...

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QKD in the NISQ era enhancing secure key rates via quantum error correction Shashank Kumar Ranu12 Anil Prabhakar1 Prabha

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