Timing Regimes in Quantum Networks and their Physical Underpinnings
draft-hajdusek-qirg-timing-physics-00
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| Document | Type | Active Internet-Draft (individual) | |
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| Last updated | 2026-03-02 | ||
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draft-hajdusek-qirg-timing-physics-00
Quantum Internet Research Group M. Hajdušek
Internet-Draft R. Van Meter
Intended status: Informational Keio University
Expires: 3 September 2026 2 March 2026
Timing Regimes in Quantum Networks and their Physical Underpinnings
draft-hajdusek-qirg-timing-physics-00
Abstract
Entangling quantum networks build on new physical mechanisms to
distribute quantum entanglement among a set of nodes over a set of
links. To design a complete network protocol stack with proper
division of responsibilities into layers, hardware and protocol
engineers must share an understanding of those physical mechanisms
and use a common vocabulary. This document bridges the abstract
concepts described in RFC 9340 and the underlying physics to
engineering concerns such as timing constraints on arrival of photons
and exchange of supporting classical messages. The equations
presented here will serve as reference points for architectural
decisions in future documents, allowing future documents to deal
directly in code without complex mathematics. Application-layer
developers will not need the low-level physics presented here.
About This Document
This note is to be removed before publishing as an RFC.
The latest revision of this draft can be found at https://moonshot-
nagayama-pj.github.io/draft-hajdusek-qirg-timing-physics/draft-
hajdusek-qirg-timing-physics.html. Status information for this
document may be found at https://datatracker.ietf.org/doc/draft-
hajdusek-qirg-timing-physics/.
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Source for this draft and an issue tracker can be found at
https://github.com/moonshot-nagayama-pj/draft-hajdusek-qirg-timing-
physics.
Status of This Memo
This Internet-Draft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
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Table of Contents
1. Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1. Goals . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2. Non-Goals . . . . . . . . . . . . . . . . . . . . . . . . 5
3. Interferometric Stabilization . . . . . . . . . . . . . . . . 5
3.1. Hong-Ou-Mandel interference . . . . . . . . . . . . . . . 6
3.2. Polarization . . . . . . . . . . . . . . . . . . . . . . 8
3.2.1. Interference of photons from two independent EPPS . . 10
3.3. Spectral distinguishability . . . . . . . . . . . . . . . 11
3.3.1. Pure states . . . . . . . . . . . . . . . . . . . . . 11
3.3.2. Mixed states . . . . . . . . . . . . . . . . . . . . 12
3.3.3. Example 1: Gaussian wave packets . . . . . . . . . . 13
3.4. Wave Packet Overlap . . . . . . . . . . . . . . . . . . . 14
3.4.1. Example: Gaussian wave packets . . . . . . . . . . . 15
4. Detector Timing Windows . . . . . . . . . . . . . . . . . . . 16
4.1. Detector basics . . . . . . . . . . . . . . . . . . . . . 16
4.2. Acceptance window . . . . . . . . . . . . . . . . . . . . 18
4.3. Separation in a train of wavepackets . . . . . . . . . . 18
5. Measurement basis selection . . . . . . . . . . . . . . . . . 20
5.1. Measurement basics . . . . . . . . . . . . . . . . . . . 20
5.1.1. Single-qubit measurements . . . . . . . . . . . . . . 20
5.1.2. Two-qubit measurements . . . . . . . . . . . . . . . 21
5.2. Measurements on quantum memories . . . . . . . . . . . . 21
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5.2.1. Trapped ions . . . . . . . . . . . . . . . . . . . . 22
5.3. Measurements on photonic qubits . . . . . . . . . . . . . 23
6. Optical Switch Control . . . . . . . . . . . . . . . . . . . 24
7. Pre-configured Event-driven Tasks . . . . . . . . . . . . . . 26
8. Urgent but Not Synchronization-critical Tasks . . . . . . . . 27
9. Host-side Application-level Tasks . . . . . . . . . . . . . . 27
10. Background Tasks . . . . . . . . . . . . . . . . . . . . . . 29
11. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 30
Appendix A. References . . . . . . . . . . . . . . . . . . . . . 30
A.1. Normative . . . . . . . . . . . . . . . . . . . . . . . . 30
A.2. Informative . . . . . . . . . . . . . . . . . . . . . . . 30
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 30
1. Prologue
In 1982, Digital Equipment Corporation, Intel, and Xerox published
*The Ethernet: A Local Area Network Data Link Layer and Physical
Layer Specifications*. This 120-page document specifies pretty much
everything: diameter of the coaxial cable, its impedance, dispersion,
maximum cable length, voltages and currents, signal rise times, etc.
The types of physical connectors allowed. How a bit is encoded in
the signal. How a frame is demarcated. How collisions are detected.
The format of messages. Addressing. Multicasting. Polynomials for
error correction. It's ALL there.
Equally importantly, it specifies *timing requirements*. For
example, the rise time for a signal on the coaxial cable shall be 25
± 5 nanoseconds. The total worst-case round-trip delay is calculated
in a table to be 46.38 microseconds. How the entries in that table
are combined to produce that number is fairly obvious; however, the
numerical entries themselves are mostly unjustified in the
specification itself, only stated. One exception is the statement,
"Rise and fall times meet 10,000 series ECL requirements," referring
to a specific series of well-known digital emitter-coupled logic
parts, and hence incorporating a great deal of prior knowledge and
work by reference.
In the quantum world, we are starting from first principles. Hence,
we must begin at the beginning. We want to have specifications like
Ethernet's, but first we must describe how the entries in e.g. the
physical propagation delay budget are determined. The role of this
document is to provide the underpinnings that give a shared
understanding of how the basic numbers are determined and how they
can be combined in a particular system design.
Thanks, DIX Ethernet creators, for showing the way!
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2. Introduction
Quantum networks that distribute end-to-end entanglement involve a
number of tasks with varying demands on timing precision and jitter.
The design of a quantum network will involve a layered protocol
architecture where different layers take responsibility for meeting
these differing constraints. This document describes the various
timing regimes, from most to least stringent, in order to assist the
process of making key design decisions.
The range of time scales of interest extends from ensuring the sub-
wavelength stability of optical paths up to batch monitoring of the
operation of the network itself. Light with a wavelength of 1.5
micrometers (common in communications, including quantum
communications) has a frequency of approximately 200 THz (2E+14 Hz),
for a cycle time of 5E-15 seconds. Ranging from sub-wavelength
stabilization through background operations such as routing,
therefore, covers some 16 or more decimal orders of magnitude. Add
in a 24-hour thermal drift that must be compensated for in many
cases, and we reach twenty decimal orders of magnitude from the
bottom to the top. Naturally, meeting this range of demands requires
the use of a variety of mechanisms. This document avoids specifying
solutions to the problems, and instead presents the functions and how
their requirements are calculated (or measured). Thus, each
individual network design should apply the methods introduced here
and present a numerical summary of the resulting values, after which
corresponding solutions can be proposed and implemented.
Summary of timing regimes:
* *Interferometric stabilization:* photon wavepacket overlap,
technology dependent, roughly nanoseconds.
* *Detector timing windoes:* opening and closing of detector timing
windows, detector recovery time: nanoseconds to microseconds.
* *Measurement basis selection (if required in BSA):* performance
will constrain entanglement attempt rate.
* *Optical switch control:* switching of trains of wave packets.
* *Pre-configured event-driven tasks:* timing-triggered or
measurement-triggered execution of quantum circuits, microseconds
* *Urgent but not synchronization-critical tasks:* execution of
classical code that processes RuleSet messages and selects or
creates new quantum circuits for execution, milliseconds
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* *Host-side application-level tasks:* post-measurement operations,
milliseconds
* *Background tasks:* link tomography calculations, routing table
updates, seconds to minutes
Some of these can only be achieved using high-quality hardware, while
others are software tasks. Detailed analysis of these regimes will
affect core software design in each network node type.
2.1. Goals
* Identify and provide introduction to the physical principles
related to timing regimes in quantum networks.
* Provide justification behind specific design choices discussed in
our other documents.
* Serve as a reference for other quantum network specifications.
2.2. Non-Goals
* Detailed physical derivations.
* Exhaustive coverage of all existing quantum platforms and
technologies.
* New research results.
3. Interferometric Stabilization
Entanglement distribution in quantum networks is performed by
entanglement swapping (ES) on photonic qubits. Central to photonic
ES is the Hong-Ou-Mandel (HOM) interference
(https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.59.2044),
regardless of the photonic qubit encoding or of the particular
protocol implementing photonic ES. We begin by introducing the
notation used, giving a brief overview of the effect, as well as
discussing how to quantify the effect. We then continue with a
discussion of the requirements that must be satisfied in order to
observe the effect. This section follows quite closely this
excellent tutorial (https://arxiv.org/abs/1711.00080).
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3.1. Hong-Ou-Mandel interference
Consider two photons incident on a beamsplitter (BS) with
reflectivity r. We label the input modes by a and b, and the output
modes by c and d. We are interested in the observed behavior at the
output modes of the BS. There are four possible cases that may
occur:
* Case A: photon in mode a is reflected, while photon in mode b is
transmitted.
* Case B: both photons are transmitted.
* Case C: both photons are reflected.
* Case D: photon in mode a is transmitted, while photon in mode b is
reflected.
The input state can be expressed as
† †
|ψ⟩ = a b | 0 ⟩
ab j k ab
where the daggered operators represent bosonic creation operators,
which create a single photon in the corresponding input port of the
BS. The indices j and k represent other properties of the photons
that determine how distinguishable the photons are. For example, j
and k could represent
* polarizations (for polarization-encoded qubits),
* spectral modes,
* temporal modes (for time-bin encoded qubits),
* arrival time,
* transverse spatial mode.
Action of the BS on the input modes is the following:
† ┌───┐ † ┌─┐ † † ┌─┐ † ┌───┐ †
a -> ╲│1-r c + ╲│r d , b -> ╲│r c - ╲│1-r d
The output state of the two photons is
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┌──────┐ † † † † † † ┌──────┐ † †
|ψ⟩ = ( ╲│r(1-r) c c + r c d - (1-r) c d - ╲│r(1-r) d d ) |0⟩
cd j k k j j k j k cd
For a 50:50 BS when r=1/2:
1 ⎛ † † † † † † † † ⎞
|ψ⟩ = ─ ⎜ c c + c d - c d - d d ⎟ |0⟩
cd 2 ⎝ j k k j j k j k ⎠ cd
From this expression, we can see that when j=k, in other words when
the input photons are indistinguishable, the output state has the
following form,
1
|ψ⟩ = ──── ( |2⟩ - |2⟩ )
cd ┌─┐ c d
╲│2
The probability amplitudes for the cases where both input photons are
transmitted or both reflected (Cases B and C in the figure above)
interfere destructively. Perfectly indistinguishable input photons
always exit the BS in the same ouput mode. It is this interference
effect that is at the heart of quantum networking.
In order to quantify the effect that distinguishability has on HOM
interference, we consider the *probability of a coincidence
detection*, p_c, where one photon is detected in the BS output mode
c, and the other photon in output mode d. This probability is
defined as
p = ⟨ψ| P ⊗P | ψ⟩
c cd c d cd
where P_i, for i=c,d, are the projection operators representing a
detection of a single photon in output mode i of the BS. For
completely indistinguishable input photons that undergo the full HOM
interference, we have p_c=0. On the other hand, for fully
distinguishable photons, the probability of a coincidence detection
attains its maximum value p_c=1/2.
An often-used measure that quantifies the degree of HOM interference
is the *visibility* V, defined via the probability of a coincidence
detection,
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max min
p - p
c c min
V = ──────────────── = 1 - 2 p
max c
p
c
where we used the fact that the maximum probability of a coincidence
detection is 1/2. We observe that the visibility varies from V=0 for
fully distinguishable input photons to V=1 for perfectly
indistinguishable ones.
Visibility V plays a useful role when modelling the effects of
imperfect HOM interference in the context of entanglement swapping.
Consider the case when the input photons a, b are entangled with
auxiliary systems s_1 and s_2, respectively. The BSA performs ES by
measuring the input photons, entangling systems s_1 and s_2 in the
process. Fidelity of the new entangled pair is directly proportional
to the visibility V of the HOM interference. Non-ideal HOM
interference can be modelled as a two-qubit dephasing
(https://journals.aps.org/prl/abstract/10.1103/
PhysRevLett.130.050803),
no-deph deph
ρ = V ×ρ + (1 - V) ×ρ
s s s s s s
1 2 1 2 1 2
where superscript no-deph denotes a density matrix resulting from an
ideal ES at the BSA with unit visibility of the HOM interference, and
superscript deph denotes a fully dephased state obtained by setting
all off-diagonal elements of the density matrix to zero.
In the following subsections, we address and quantify how
distinguishable photons affect the visibility of the HOM
interference.
3.2. Polarization
We now consider the case when the input photons differ in their
polarization degree of photons. The maximum probability of a
coincidence detection is obtained for orthogonally polarized photons,
for example when j=H and k=V. Here, H denotes horizontal
polarization and V denotes vertical polarization. The output state
of the two photons is
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1
| ψ⟩ = ─ ⎛ |1;H⟩ |1;V⟩ + |1;V⟩ |1;H⟩ - |1;H⟩ |1;V⟩ - |1;H⟩ |1;V⟩ ⎞
cd 2 ⎝ c c c d c d d d ⎠
We can observe that maximum probbility of coincidence is 1/2.
In general, the two input photons will have polarizations given by
two unit vectors, j=ε and k=ε'. The output state can be written as
1
|ψ⟩ = ─ ⎛ |1;ϵ⟩ |1;ϵ′⟩ + |1;ϵ′⟩ |1;ϵ⟩ - |1;ϵ⟩ |1;ϵ′⟩ - |1;ϵ⟩ |1;ϵ′⟩ ⎞
cd 2 ⎝ c c c d c d d d ⎠
The projection operators corresponding to a detection even at
detector i ($i=a,b$) are given by
P = |1;ϵ⟩ ⟨1;ϵ| + |1;ϵ′⟩ ⟨1;ϵ′|
i i i i i
Either an ε-polarized or an ε'-polarized photon is detected in the
output mode i. The probability of coincidence is then
1 ⎛ 2 ⎞ 1 2
p = ⟨ψ| P ⊗P |ψ⟩ = ─ ⎝ 1 - ⎢ ⟨ϵ′|ϵ⟩⎢ ⎠ = ─ sin θ
co cd c d cd 2 2
where the overlap between the polarization unit vectors is
parametrized by θ, and can be written as
⟨ϵ′|ϵ⟩= cosθ
We can define the corresponding visibility as a function of the angle
between the two polarization vectors,
2
V(θ) = 1 - 2 p = cos θ
c
When the photons have identical polarization, the visibility reaches
its maximum of 1. On the other hand, when the photons are fully
distinguishable and their polarization vectors are orthogonal,
visibility vanishes.
Ensuring that the two input photons are indistinguishable in their
polarization degree of freedom is critical for proper operation of
the BSA. Care must be therefore taken to characterize the photons
just before they are incident onto the BS, as it is possible for the
polarization of a photon to *drift* during its transmission and
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change its state from the one that the photon possessed immediately
after emission. This is often the case in fiber-based quantum
networks, where polarization of photons is particularly sensitive to
mechanical stresses and temperature gradients affecting the fiber.
This issue may be sidestepped by using polarization-maintaining
fibers that are designed to suppress coupling between linearly-
polarized orthogonal states of light. However, these incur
prohibitive costs for long-distance quantum communication, and may
actually introduce unwanted coupling between linearly and circularly
polarized light.
3.2.1. Interference of photons from two independent EPPS
The preceding discussion was concerned with two independent pure
photons of different polarization. In the context of quantum
networking, a much more common scenario is that of two entangled
pairs of photons originating from two independent EPPS nodes, where
two qubits, one from each pair, are incident onto a BS and undergo
HOM interference. The two pairs are in the following initial state,
⎛ iθ ⎞ ⎛ iθ ⎞
1 ⎜ 1 ⎟ 1 ⎜ 2 ⎟
|ψ⟩ = ────⎜ |HV⟩ + e |VH⟩ ⎟, |ψ⟩ = ────⎜ |HV⟩ + e |VH⟩ ⎟
a b ┌─┐⎜ a a a a ⎟ b b ┌─┐⎜ b b b b ⎟
1 2 ╲│2 ⎝ 1 2 1 2 ⎠ 1 2 ╲│2 ⎝ 1 2 1 2 ⎠
where θ_1 and θ_2 represent the polarization drift induced in the
single-mode fiber. Photons a_2 and b_1 are incident onto a BS, where
they undergo HOM interference. Following the same calculation as
above, it can be shown that the probability of a coincidence event is
p_{c} = 1/4, regardless of the polarization drift. This suggests
that the visibility is insensitive to the polarization drift.
However, the polarization drift must be tracked regardless because it
affects the fidelity of the post-ES state of photons a_1 and b_2. It
is therefore important to characterize the polarization drift at the
BSA at regular intervals and compensate for it. This can be done at
the nodes generating the photon pairs at the cost of the BSA having
to communicate polarization drift results to the ends nodes. Or it
can be compensated for directly at the BSA using waveplates at the
cost of increased complexity of the BSA. In Krutyanskiy et.al.
(https://journals.aps.org/prl/abstract/10.1103/
PhysRevLett.130.050803), polarization drift characterization and
compensation at the BSA takes a few minutes and is performed every 20
minutes.
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3.3. Spectral distinguishability
Another important source of distinguishability in HOM interference is
the spectral property of the input photons. The photon wave packet
of a photon can be represented by its *spectral amplitude function*
ϕ(ω) that satifies the normalization condition:
⌠ 2
⎮ dω|ϕ(ω)| =1
⌡
Two input photons become distinguishable if their respective spectral
amplitude functions are not equal. We restrict our discussion to
Gaussian spectral amplitude functions but the same methods generalize
to arbitrary photons. The two photons may have different central
frequencies or different standard deviations.
In this subsection, we analyze the requirements in terms of the
photonic spectral amplitude function that lead to high visibility of
the HOM interference.
3.3.1. Pure states
We begin the discussion by focusing on pure states of the input
photons first. Single-photon state with a spectral amplitude
function ϕ(ω) is a superposition written as
⌠ †
|1;ϕ⟩ = ⎮ dωϕ(ω) a (ω) |0⟩
a ⌡ a
where creation operator creates a photon in the BS input mode a with
frequency ω. Two input photons with arbitrary spectral functions ϕ
and φ are described by
⌠ † ⌠ †
|ψ⟩ = |1;ϕ⟩ |1;𝜑⟩ = ⎮ dω ϕ(ω ) a (ω ) ⎮ dω 𝜑(ω ) b (ω ) |0 ⟩
ab a b ⌡ 1 1 1 ⌡ 2 2 2 ab
We assume that the BS acts on the different frequency modes
independently, and that the reflectivity is frequency-independent.
Applying the same transformation rules for the creation operators,
the output state of the two photons is
1 ⌠ ⌠ ⎡ † † † † † † † † ⎤
|ψ⟩ = ─ ⎮ dω ϕ(ω ) ⎮ dω 𝜑(ω ) ⎢ c (ω ) c (ω ) + c (ω ) d (ω ) - c (ω ) d (ω ) - d (ω ) d (ω ) ⎥ |0⟩
cd 2 ⌡ 1 1 ⌡ 2 2 ⎣ 1 2 2 1 1 2 1 2 ⎦ cd
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The projection operators corresponding to a detection event in output
mode c and output mode d are given by
⌠ † ⌠ †
P = ⎮ dωc (ω) |0⟩ ⟨0| c (ω), P = ⎮ dωd (ω) |0⟩ ⟨0| d(ω)
c ⌡ c c d ⌡ d d
The probability of a coincidence detection is then
1 1⌠ ∗ ⌠ ∗
p = ─ - ─⎮ dω ϕ (ω )𝜑(ω ) ⎮ dω 𝜑 (ω ) ϕ(ω )
c 2 2⌡ 1 1 1 ⌡ 2 2 2
The form of this expression is the same as the one in subsection on
polarization above, where the probability of a coincidence detection
depended on the overlap between the polarization vectors ε and ε'.
Now, p_c depends on the overlap between the spectral amplitude
functions. If the input photons are fully distinguishable, their
respective spectral amplitude functions ϕ(ω) and φ(ω) are orthogonal
and the integrals vanish, meaning p_c=1/2. On the other hand, for
completely indistinguishable input photons we have ϕ(ω)=φ(ω), and due
to the normalization condition we obtain p_c=0.
3.3.2. Mixed states
Previous discussion of pure states can be extended to include mixed
states of the input photons. Such states will inevitably arise due
to imperfections in the preparation procedure and due to the input
photons being entangled with other degrees of freedom. These can
include other photons or quantum memories.
The mixed state of an input photon is described by the following
density matrix:
⎲ ⎲
ρ = ⎳ u |1;ϕ ⟩ ⟨1;ϕ | , ⎳ u =1
a k k k a k a k k
where the state of the photon is a mixture of pure single-photon
states with spectral amplitude function ϕ_k(ω), weighted by
probability u_k. The two-photon input state can be written as
in ⎲
ρ = ⎳ u v |1;ϕ ⟩ |1;𝜑 ⟩ ⟨1;ϕ | ⟨1;𝜑 |
ab kk′ k k′ k a k′ b k a k′ b
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It is not necessary to repeat the entire calculation we did for pure
states. Due to linearity of quantum mechanics, we can immediately
write the expression for the probability of coincidence as a sum of
pure-state coincidence probabilities weighted by u_k and v'_k:
1 1 ⎲ ⌠ ∗ ⌠ ∗
p = ─ - ─ ⎳ u v ⎮ dω ϕ (ω )𝜑 (ω ) ⎮ dω 𝜑 (ω ) ϕ (ω )
c 2 2 kk′ k k′ ⌡ 1 k 1 k′ 1 ⌡ 2 k′ 2 k 2
3.3.3. Example 1: Gaussian wave packets
In this example, we consider input photons with Gaussian spectral
amplitude functions. The spectral amplitude functions are given by
2
(ω-ω̅ )
i
-───────
2
2σ
1 i
ϕ (ω) = ────────── e , for i=a,b
i 1/4 ┌──┐
π ╲ │σ
╲│ i
The probability of a coincidence detection is then
2
(ω̅ -ω̅ )
a b
-────────
2 2
σ σ σ +σ
1 a b a b
p = ─ -─────── e
c 2 2 2
σ + σ
a b
*Case A (different central frequencies)*
We assume that the two spectral amplitude functions have the same
standard deviation, which simplifies the expression for the
probability of a coincidence detection to
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⎛ 2 ⎞
⎜ (ω̅ -ω̅ ) ⎟
⎜ a b ⎟
⎜ -──────── ⎟
⎜ 2 ⎟
1 ⎜ 2σ ⎟
p = ─ ⎝ 1 - e ⎠
c 2
We observe that for identical photons, the probability of a
coincidence detection vanishes. For fully distinguishable wave
packets, when the difference between central frequencies diverges,
the probability approaches 1/2. The visibility as a function of the
difference between the central frequencies is
2
(ω̅ -ω̅ )
a b
-────────
2
2σ
V(ω̅ -ω̅ ) = e
a b
*Case B (different standard deviations)*
The spectral amplitude functions have the same central frequencies,
which gives the following expression for the probability of
coincidence and visibility,
σ /σ 2σ /σ
1 b a b a
p = ─ - ────────────, V(σ /σ ) = ────────────
c 2 2 b a 2
1 + (σ /σ ) 1 + (σ /σ )
b a b a
3.4. Wave Packet Overlap
So far we have assumed that the two input photons arrive at the BS at
exactly the same time. In this subsection, we address this
unrealistic assumption and quantify how temporal distinguishability
affects the visibility of HOM interference. Even for photons with
identical spectral amplitude functions, different arrival times
result in decreased overlap between the photons' wave packets,
diminishing the visibility of the HOM interference.
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Without loss of generality we assume that photon b is delayed by a
time τ, which transforms its creation operator,
† † -iωτ
b (ω) →b (ω) e
Two input photons with arbitrary spectral functions ϕ and φ, with
photon b arriving late, are described by
-iω τ
⌠ † ⌠ † 2
|ψ⟩ = |1;ϕ⟩ |1;𝜑⟩ = ⎮ dω ϕ(ω ) a (ω ) ⎮ dω 𝜑(ω ) b (ω ) e |0⟩
ab a b ⌡ 1 1 1 ⌡ 2 2 2 ab
We assume that the BS acts on the different frequency modes
independently, and that the reflectivity is also frequency-
independent. Applying the same transformation rules for the input
creation operators, the output state of the two photons is
-iω τ
1 ⌠ ⌠ 2 ⎡ † † † † † † † † ⎤
|ψ⟩ = ─ ⎮ dω ϕ(ω ) ⎮ dω 𝜑(ω ) e ⎢ c (ω ) c (ω ) + c (ω ) d (ω ) - c (ω ) d (ω ) - d (ω ) d (ω ) ⎥ |0⟩
cd 2 ⌡ 1 1 ⌡ 2 2 ⎣ 1 2 2 1 1 2 1 2 ⎦ cd
For pure input states, the probability of a coincidence detection is
-iω τ iω τ
1 1⌠ ∗ 1 ⌠ ∗ 2
p = ─ - ─⎮ dω ϕ (ω )𝜑(ω )e ⎮ dω 𝜑 (ω ) ϕ(ω ) e
c 2 2⌡ 1 1 1 ⌡ 2 2 2
while for mixed states is can be generalized to the following form,
-iω τ iω τ
1 1 ⎲ ⌠ ∗ 1 ⌠ ∗ 2
p = ─ - ─ ⎳ u v ⎮ dω ϕ (ω )𝜑 (ω )e ⎮ dω 𝜑 (ω ) ϕ (ω ) e
c 2 2 kk′ k k′ ⌡ 1 k 1 k′ 1 ⌡ 2 k′ 2 k 2
3.4.1. Example: Gaussian wave packets
Consider two identical pure Gaussian wavepackets that arrive at the
BS with a time difference given by τ. The probability of coincidence
and the corresponding visibility are given by
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⎛ 1 2 2 ⎞ 1 2 2
⎜ -─σ τ ⎟ -─σ τ
1 ⎜ 2 ⎟ 2
p = ─ ⎝ 1 - e ⎠, V(τ) = e
c 2
4. Detector Timing Windows
In this section, we discuss how properties of single-photon detectors
(SPDs) affect the timing regimes in quantum networks. An ideal SPD
generates an electrical signal after absorbing a photon, and
generates no signal in the absence of a photon. This is not always
true for real-world SPDs (https://www.nature.com/articles/
nphoton.2009.230).
4.1. Detector basics
Performance of SPDs can be quantified by the following
characteristics,
* *Spectral range:* SPDs are sensitive over a limited range of
wavelengths. This range depends on the materials used in the
fabrication of the detector. Typical spectral ranges are in the
near-infrared, around 1550nm, where commercial optical fibers
perform best in terms of photon loss rates.
* *Detection efficiency:* The overall probability that an incoming
photon registers a count, denoted by η. This efficiency can be
further broken down. Probability of losing the photon before it
reaches the detector is described by the _coupling efficiency_,
η_coupling. The type of material and geometry of the detector
determine the photon _absorption efficiency_, η_absorption.
Finally, the probability that an electric signal is generated upon
successful absorption of a photon is described by the _registering
efficiency_, η_registering. The overall _system detection
efficiency_ is given by the product of these three,
η = η ×η ×η
sde coupling absorption registering
The _device detection efficiency_ is given by
η = η ×η
dde absorption registering
Detection efficiency affects the rate at which entanglement can be
distributed.
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* *Recovery time:* Denoted by τ_recovery and also known as 'dead'
time. It is the time duration following an absorption of a photon
during which the detector is unable to reliably detect another
photon. Recovery time affects the maximum detection rate. If the
source of photons has low efficiency, the clock rate does not need
to be limited by the recovery time, as majority of the trials will
not produce a photon. This could also be the case if the
probability of losing the photon is high (either due to loss in
fiber or due to low system detection efficiency η_sde). On the
other hand, if the photon source is highly efficient, it is
important to ensure that the separation between the wavepackets is
longer than τ_recovery to ensure effcient use of the generated
photons.
* *Dark count rate:* SPDs have a finite chance to produce an output
electric signal even in the absence of a photon. This may be
caused by materials properties of the detector, biasing
conditions, or external noise. It is usually given in Hz (counts
per second). Dark counts decrease the fidelity of the distributed
entangled states.
* *Timing jitter:* Denoted by J_timing. Describes the variation in
time between the photon being absorbed and the output electric
signal being generated.
The table below shows the above characteristics for a SNSPD
(https://singlequantum.com/wp-content/uploads/2022/12/SQ-General-
Brochure.pdf).
+=============================+=========+=========+
| Wavelength | 800 nm | 1550 nm |
+=============================+=========+=========+
| System detection efficiency | > 90% | > 90% |
+-----------------------------+---------+---------+
| Recovery time | 10 ns | 20 ns |
+-----------------------------+---------+---------+
| Dark count rate | < 1 Hz | < 1 Hz |
+-----------------------------+---------+---------+
| Timing jitter | < 15 ps | < 15 ps |
+-----------------------------+---------+---------+
Table 1
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4.2. Acceptance window
In the previous Section, we used τ to denote the difference between
the arrival time of the photons at the BSA. However, due to emission
jitter it is impossible to know the precise time of arrival of a
photon. The only information that is available comes from the
electric output signal of a detector. Time of detection is in
general different from the time of photon arrival due to finite time
needed to generate the output signal described by the timing jitter
J_timing. Therefore, we will use τ to denote the difference in
detection time of the two photons.
Measurement at the BSA is successful when the correct pattern of
detector clicks is observed, and the difference in detection times τ
is smaller than a given detection *acceptance window*, T_window. The
size of this window affects both the fidelity and the generation rate
of the entangled pairs that the link produces. Large acceptance
windows produce high rates but low fidelity, while small acceptance
windows result in low rates and high fidelity. The appropriate size
of the acceptance window must be chosen in order to satisfy the
demands of the application requesting the entangled states. Reaching
the requested fidelity should take priority over high generation
rate.
4.3. Separation in a train of wavepackets
Current experiments on quantum repeaters use single quantum memory
per QNIC. As quantum technologies improve, it is likely that QNICs
will be equipped with multiple quantum memories. This will allow for
generation of link-level entanglement in a multiplexed manner, where
trains of photons, each originating from a different memory inside
the same QNIC, are sent to the BSA. The photons making up a train
must be well separated such that upon a successful BSM, the BSA can
uniquely identify which two photons were measured. We refer to the
minimum separation between the photons as the *separation time*
T_separation.
The size of the separation time depends on the following:
* *Wave packet shape:* Individual photons cannot have overlapping
spatial wavepackets, which may lead to incorrect assignement of
entangled qubits following a successful meadurement at the BSA.
We will use T_photon to denote the length of a wavepacket in
seconds.
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* *Detector recovery time:* Spacing the wavepackets too close to
each other may result in some of the photons being lost due to the
detector recovering following a detection event, leading to
inefficient use of initially generated entangled pairs (either
memory-photon or photon-photon).
* *Memory emission jitter:* The separation between the wavepackets
must take into account the probabilistic nature of photon emission
from a quantum memory in order to prevent wavepacket overlap.
* *Detector timing jitter:* Generation of the electric signal
following absorption of a photon varies in duration, leading to a
variance in timing of the detection event. This may lead to the
BSA mislabelling which photons were part of a successful
measurement if their wavepackets are spaced too closely.
General (conservative) separation time should therefore be set to
T ≥T + J + J
separation photon emission timing
The above discussion assumes that the photons can be generated nearly
on-demand. This is a fair assumption in the case of quantum memories
based on trapped ions (https://ora.ox.ac.uk/objects/
uuid:604c53b9-8df8-4e45-8103-10fd81eb3366). Here, the memory must be
first initialized by cooling it to its ground state, a process which
takes <1ms. The memory is then excited by a laser pulse of
approximately 50 microseconds that generates a photon.
In the case of memory-less link architectures, the picture is
slightly different. Here, EPPS nodes utilizing the principle of
spontaneous parametric down-conversion (SPDC) generate entangled
photon pairs. Each photon is sent to a different BSA, where they are
measured with a photon originating from a different EPPS node. SPDC
is an inefficient process with success probability of around 10^{−6}
per pump photon. In system design, the intensity of the pump laser
is adjusted so that the average number of photons is appropriate;
generally this must be set below one photon per time window in order
to avoid polluting the signal with two-photon states. This means
that most of the time windows given by the separation time will not
contain a photon. However, the separation time should be maintained
in order to correcly identify the photons that were part of a
successful measurement at the BSA. The separation time governs the
maximum rate at which EPPS attempts to generate the entangled photon
pairs, which is given by 1/T_separation.
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5. Measurement basis selection
We have encountered Bell-state measurements performed by the BSA on
photonic qubits that are needed for entanglement swapping. These
measurements were static in the sense that we did not need to change
the measurement basis. Observed detection pattern determined whether
the post-measurement state of the remote quantum systems (either
quantum memories or other photons) was |Ψ^+〉 or |Ψ^−〉. Projection
onto two of the four Bell-basis states was achieved probabilistically
without actively applying any transformations on the photonic qubits.
We will see in this Section that the photonic BSA is a very special
case in this regard, and that changing the basis of the measurement
is an indispensable part of quantum networking. Entanglement
swapping on stationary qubits stored in quantum memories is not
possible without applying appropriate unitaries first that change the
basis of the measurement. There are also cases, where change of
basis is required even when dealing with only photonic qubits. An
example of this are the so-called all-photonic quantum repeaters,
where measurement basis is conditioned on the outcomes of previous
measurements, leading to the requirement of very fast basis
switching.
5.1. Measurement basics
We will first discuss quantum measurements in general before
discussing concrete implementations and their timing requirements
based on their physical implementations.
5.1.1. Single-qubit measurements
For simplicity, we begin with measurements on a single qubit before
generalizing to two qubit measurements. Consider a general state of
the qubit, |ψ〉 = α |0〉 + β |1〉. Measurement in an arbitrary basis M
projects |ψ〉 onto one of the eigenvectors of M. Probabilities of the
two possible measurement outcomes are given by the squared modula of
the overlaps between the initial state |ψ〉 and the eigenvectors of
the observable M.
2 ⟂ ⟂ 2
Pr(|ϕ⟩;|ψ⟩)=|⟨ϕ|ψ⟩| , Pr(|ϕ ⟩;|ψ⟩)=|⟨ϕ |ψ⟩|
It is often difficult to directly measure the qubit in an arbitrary
basis when it comes to real-world implementation. In such a case,
the qubit needs to be pre-rotated by an appropriate unitary
operation, and then measured in the Z basis, which can usually be
implemented in a straightforward way. This approach greatly
simplifies the implementation of arbitrary measurements.
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Consider that the observable M is related to the Pauli Z by unitary
U:
†
M = U Z U
This means the unitary U relates the eigenvectors of the two
observables,
⟂
|ϕ⟩= U |0⟩, and |ϕ ⟩= U |1⟩
We can perform measurement in the M basis by applying adjoint of U to
the initial state |ψ〉, and then measuring it in the Pauli Z basis.
This can be easily verified by rewriting the above probabilities
corresponding to the two measurement outcomes,
2 † 2 †
Pr(|ϕ⟩;|ψ⟩) = |⟨ϕ|ψ⟩| = |⟨0| U |ψ⟩| = Pr(|0⟩; U |ψ⟩)
and
⟂ ⟂ 2 † 2 †
Pr(|ϕ ⟩;|ψ⟩) = |⟨ϕ |ψ⟩| = |⟨1| U |ψ⟩| = Pr(|1⟩; U |ψ⟩)
5.1.2. Two-qubit measurements
The same principle of changing the measurement basis can be
generalized to two qubits. This time state |ψ〉 represents a general
two-qubit state, unitary U ^{†} acts on both qubits, which are both
finally measured in Pauli Z basis. In the majority of cases, we are
interested in performing measurements in the Bell basis. Required
unitary is the Hermitian conjugate of the unitary that creates a Bell
pair when the qubits are both initialized in |0〉.
5.2. Measurements on quantum memories
In this Section, we discuss various methods of implementing
measurements of quantum memories. These methods vary based on the
quantum technology used as the quantum memory, and even within the
same technology there are usually variations. We are mainly
concerned with giving an overview of the different measurement
methods, and their respective timing regimes.
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5.2.1. Trapped ions
Trapped ions possess two degrees of freedom
(https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.75.281).
The first one is the motional degree of freedom, resulting from the
ion oscillating around its equilibrium position in the trap. The
second one is the internal degree of freedom, represented by the
ground state |g〉 and the excited state |e〉. It is the latter degree
of freedom which is used to encode a qubit and hence acts as a
quantum memory.
Measurement in the *Pauli Z* basis is performed by *electron
shelving* via the use of a third atomic level |r〉, with much shorter
life time than the excited state (https://www.amazon.co.jp/Quantum-
World-Ultra-Cold-Atoms-Light/dp/1783266163) |e〉, τ_e ≫ τ_r. The ion
is illuminated by light tuned to resonate with the transition |g〉
<−>|r〉, represented by the red straight arrow in the Figure above.
If fluorescence is immediately observed, this corresponds to
measuring the ion in the ground state |g〉. If no fluorescence is
observed, the ion is measured in the excited state |e〉.
Hypothetically a single fluorescent photon would be sufficient,
however, the fluorescent photons are only rarely captured into the
measurement apparatus (typically involving lenses and a camera) and
observed, and stray photons are also often captured, so a relatively
long *integration time* is used to confirm the fluorescence with high
probability. (Solid-state systems such as quantum dots and
superconducting qubits also need relatively long integration times in
their measurement processes.) Combined with laser pulses that apply
a single-qubit rotation, measurement of a *single ion in an arbitrary
basis* can be performed in 1-2 ms
(https://journals.aps.org/prl/abstract/10.1103/
PhysRevLett.130.050803).
The *CNOT gate* can be applied in two different ways. The original
proposal is due to Cirac and Zoller
(https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.74.4091),
where the ions needed to be cooled to their collective motional
ground state first. This approach was demonstrated experimentally
using calcium ions (https://www.nature.com/articles/nature01494).
Execution of the gate took around 600 microseconds, with the achieved
fidelity being less than 0.8. The second approach is due to Molmer
and Sorensen (https://journals.aps.org/prl/abstract/10.1103/
PhysRevLett.82.1835), and is more robust against motional excitation.
This led to high-fidelity demonstrations of >0.99, and gate times of
around 50 microseconds.
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5.3. Measurements on photonic qubits
Measurement of polarization-encoded photonic qubits can be performed
with the aid of a *polarizing beam splitter* (PBS), a *half
waveplate* (HWP), a *quarter waveplate* (QWP), and two detectors
(https://link.springer.com/chapter/10.1007/978-3-540-44481-7_4) (one
detector is enough in fact but less efficient). The idea is the same
as in the case of measurements performed on stationary qubits
discussed above. Setting the HWP and QWP at particular angles
applies the unitary U ^{†} that picks the basis of the measurement,
while the PBS filters out vertical and horizontal polarizations that
then get detected by the detectors placed in the output paths of the
PBS. Horizontal polarization gets transmitted through the PBS, while
vertical polarization gets reflected.
General pure state of a polarization-encoded qubit can be written as
⎛θ⎞ iϕ ⎛θ⎞
|ψ⟩= cos⎜─⎟|H⟩+ e sin⎜─⎟|V⟩
⎝2⎠ ⎝2⎠
This is directly equivalent to expressing the qubit state in the
computational basis, and can be visualized with the help of the
*Poincaré sphere*. Polarization of light is manipulated by
waveplates. Waveplate rotated by an angle α (zero is aligned with
the horizontal axis) rotates the polarization state around an axis,
located at an angle of 2α with the horizontal state |H〉 in the
horizontal plane. Half waveplate rotates the polarization state by
an angle π, while a quarter waveplate rotates by an angle π/2 in the
Poincaré sphere. The action of the half waveplate is captured by the
corresponding unitary operations in linear polarization basis:
⎡ cos2α sin2α⎤
U (α) = ⎣ sin2α -cos2α⎦
HWP
Unitary matrix representing the action of a quarter waveplate in
linear polarization basis:
⎡ 2 2 ⎤
⎢ cos α+ isin α (1-i)cosαsinα⎥
U (α) = ⎢ 2 2 ⎥
QWP ⎣ (1-i)cosαsinα sin α+icos α ⎦
The idea behind measurements in arbitrary basis
⟂
{|ψ⟩, |ψ ⟩}
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is to choose the angles for the waveplates such that the following
transformation is achieved:
⟂
U U |ψ⟩→|H⟩, U U |ψ ⟩→|V⟩
HWP QWP HWP QWP
Settings for the three Pauli bases are summarized in the table below.
+===================+======+======+
| Measurement basis | HWP | QWP |
+===================+======+======+
| linear (Z) | 0 | 0 |
+-------------------+------+------+
| diagonal (X) | Pi/8 | Pi/4 |
+-------------------+------+------+
| circular (Y) | 0 | Pi/4 |
+-------------------+------+------+
Table 2
Changing the basis of measurement requires mechanical rotation of the
waveplates and coordination with the detectors. The waveplates can
be rotated by a motorized rotator device, which can be adjusted at a
rate of around 1 degree per 100ms. Therefore, for a rotation of 45
degrees, the motor requires areound 4.5s. During the rotation
interval, any results obtained from the detectors must be discarded
as they correspond to measurements in an undesired basis.
6. Optical Switch Control
Optical switches play an essential role in distributed computing and
communication systems. Their job is to guide light from a given
input to the desired output. Optical switches have a number of
important characteristics such as _insertion loss_, _crosstalk_, and
_size_. In the context of timing regimes, we will focus on the
following characteristics in this section,
* *Switching time:* time required to reconfigure the switch.
* *Propagation time delay:* time required for the photon to travel
across the switch.
Two approaches to switching are of relevance to our discussion. The
first approach is the _crossbar switch_ with all-to-all connectivity.
Such an N× N switch can be reconfigured to accomodate all possible N!
permutations of input-output pairs. One usual implementation of a
crossbar switch is using *microelectromechanical systems (MEMS)*
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relying on small movable parts such as popup micromirrors, rotating
prisms or spinning holographic disks
(https://onlinelibrary.wiley.com/doi/book/10.1002/0471213748). MEMS
have usually low insertion loss and crosstalk, however due to their
mechanical nature they suffer from slow switching times, which range
from 10 microseconds to 10 miliseconds.
Crossbar switches are important in classical switching networks and
are use in classical control systems in some quantum technologies.
In the context of quantum networks, it is often not necessary for the
switch to be able satisfy all possible N! input-output permutations.
For example, the switch can be placed behind a pool of entangled
photon pair sources (EPPS) in order to route entangled photons
towards end nodes requesting a connection
(https://opg.optica.org/jocn/fulltext.cfm?uri=jocn-
8-5-331&id=340335). Or the switch can be placed before a pool of
Bell State Analyzers (BSA) and route input pairs of photons to the
desired BSA, where they undergo measurement in the Bell basis
(https://ieeexplore.ieee.org/document/10821447).
Both of these designs consider a 2× 2 switch as the basic building
block, which is implemented with *integrated photonics* and
controlled electro-optically. Applied electric fields are used to
alter the refractive index of the material (such as lithium niobate)
to change the state of the switch from a BAR state to a CROSS state.
Switching times for electro-optical switches are much faster, varying
from 10 nanoseconds to 10 microseconds.
The optical switch introduces a *propagation time delay*. For some
MEMS switches, this delay can be as low as 25 nanoseconds
(https://www.viavisolutions.com/en-us/literature/polatis-series-6000-
osm-network-switch-module-data-sheets-en.pdf). In general, this
delay time varies with the choice of input-output ports. This
variation is probably insignificant in most classical contexts, but
any delay between the arrival times of photon pairs at the same BSA
may result in decreased visibility further lowering the fidelity of
the post-measurement state. The issue of arrival time delay arises
in the case of integrated switches used in paired-egress BSA pools.
The propagation delay introduced by the switching fabric depends on
the design of the switch.
An example where time delays arise is the triangular switch design,
introduced by Koyama et.al. (https://ieeexplore.ieee.org/
document/10821447). Photons entering the switch from different ports
need to traverse vastly different number of switching points. The
significance of this time delay ultimately depends on the type of
photons used. Photons with longer envelopes, such as those emitted
from trapped ions, may be more robust to the propagation time delays
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introduced by the optical switch. Photons with very short envelopes,
such as the ones originating from an SPDC source, are expected to be
very susceptible to any propagation time delays.
In order to compensate for the propagation time delay, and ensure
acceptable visibility at the BSAs, it is neccesary to adjust the path
length of the photons with *optical delay lines* (ODLs). For small
enough optical switches, it may be possible to characterize the
propagation time delays for given photon pairs assigned to a
particular BSA prior to the opration of the switch. This would allow
the ODLs to be set to precomputed configurations based on the
connection request patterns. This approach will most likely not
scale, at least in its general form, to larger optical switches.
Further complication that arises during the operation of the switch
is also related to maintaining indistinguishability of the photons.
As the photons traverse the switching points, their *polarization*
changes leading to a decrease in the visibility of HOM interference
at the BSA. This polarization drift must be characterized and
compensated if acceptable levels of visibility are to be maintained.
Polarization drift characterization and compensation is a regular
step in modern experiments in quantum communications. For example,
in the Innsbruck demonstration of remote-entanglement generation over
230m (https://journals.aps.org/prl/abstract/10.1103/
PhysRevLett.130.050803), data acquisition was stopped every 20
minutes in order to correct for the polarizaiton drift. This process
took *several minutes*. In the worst case scenario, this process
needs to take place after every reconfiguration of the optical switch
leading to severely limited multiplexing capabilities.
Finally, given a set of connection requests, the optical switch must
compute the state of all switch points to *route* the photons
correctly. The reconfigurably non-blocking designs proposed in [15]
come with efficient routing algorithms that achieve this. Given the
need for path-length adjustment with ODLs and polarization drift
correction, it is expected that computing the configuration of all
switching points will not be the bottleneck during operation of the
optical switch.
7. Pre-configured Event-driven Tasks
In this section, we discuss synchronization-critical tasks that must
be conducted when an event occurs. Most stationary qubits are under
the control of a classical analog circuit that includes a local
oscillator (LO) coupled to the corresponding frequency of the qubit
itself. Avoiding drift between the _understood_ phase of the qubit
and the _actual_ phase of the LO is a key part of hardware design for
a qubit, but is beyond the scope of this document.
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Events can be _local_ to a quantum computer or repeater node, or
_remote_, generally implying reception and processing of a classical
message.
In some cases, when a memory is used to emit a photon, the ultimate
disposition of the qubit in memory might be measurement immediately
after the emission of the photon (as in QKD). Alternatively, in
systems involving QEC, immediately after emission of the photon, the
memory qubit may be encoded into a logical qubit. In general, such
events can trigger execution of a local quantum circuit.
For links using HOM-based entanglement generation, inevitably there
is a delay between the BSA operation completing successfully and the
generation, transmission and reception of the confirmation message.
Over distances of a few kilometers, this can require a few
microseconds.
8. Urgent but Not Synchronization-critical Tasks
Some events trigger a computation, or series of computations, that
are too complex to be compiled directly into a form for execution by
an ASIC or FPGA. For example, hybrid or adaptive algorithms such as
VQE, if executed in a distributed fashion, might require a
substantial statistical computation to adjust the parameters used in
the creation of the ansatz.
9. Host-side Application-level Tasks
The service provided by a quantum network is entangled states, which
may be either delivered to applications on quantum computers, held in
limited-capability quantum memories for later release and use, or
directly measured as creation is completed, corresponding to the
capabilities of COMP, STOR and MEAS end node types, respectively.
An application that uses the services of a quantum network passes
through several phases:
* *Planning*: selecting application-level tasks and communication
partners, including defining application quantum circuits (for
distributed computation) or measurement bases and characteristics
(for QKD or sensing applications), required distributed quantum
states (at the moment, presumed to be Bell pairs), distributed
entanglement fidelity, and number or rate of entangled states
needed.
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* *Computational resource preparation*: for distributed quantum
computation, allocation of quantum computing resources attached to
the quantum network, compilation of application circuits for
processing and consumption of the entangled states delivered by
the network service.
* *Connection setup*: the classical process of establishing
communication between nodes. Depending on the network
architecture, this may include allocation of resources at repeater
nodes.
* *Real-time receipt and disposition of entangled states*: as the
network delivers entangled states to the end nodes, they will be
consumed by applications, ultimately resulting in classical
information which may determine further quantum actions at the
level of immediate, result-dependent actions.
* *Near-real time computation*: many quantum algorithms are hybrid
classical/quantum computation and may require larger-scale
adaptation or recompilation of application quantum circuits.
* *Post-quantum processing*: the classical data generated by the
quantum circuits or measurements can be delivered to larger
classical computation and communication systems, e.g. for use as
cryptographic keys or as part of a much larger computation. At
this point, processing is completely decoupled from the quantum
network.
* *Connection teardown*: After completion of the quantum network
service requested by the application or larger IT service (e.g.,
encrypted classical connection), resources along the communication
path can be recovered; partially complete entangled states are
discarded or repurposed, and physical resources reallocated to
other connections.
* *Computational resource release*: reserved quantum computational
resources are released.
* *Completion*: the end-to-end hybrid quantum+classical service is
terminated.
All of the above except those marked real-time and near-real time are
almost entirely insensitive to timing issues, except as necessary for
the end-to-end service to meet the users' needs. If allocated
resources sit unused for extensive periods of time, the service
delivery of the network as a whole may be negatively impacted;
introduction of proper pricing or admission control may be needed to
resolve such issues.
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10. Background Tasks
Network operations include a number of tasks that monitor and
maintain the integrity and performance of the network. In the case
of a quantum network, uses of the quantum portion of the network can
often be deferred until the network is idle or pre-scheduled time
slots arrive, in order to minimize the impact on application
requests. Once the quantum operations are begun, of course, they are
subject to all of the constraints listed above, but the accompanying
classical calculation and inter-node reconciliation can proceed in
the background.
Such tasks include:
* *Link monitoring*: Each link must be monitored continuously in
order to inform routing (below) and RuleSet creation during
connection setup. Reconstruction of the link density matrix and
entanglement success rates involve classical information sharing
between the two nodes at opposite ends of the link. This
information must be shared reliably but does not have hard real-
time constraints, as so is well suited to transmission over a
reliable protocol such as TCP without concern for delays. The
required classical information is the outcomes of measurements of
the quantum portion of the link. That data can be collected from
entangled states specifically assigned to the link monitoring
task. It can also be collected from application-targeted uses of
the link, provided that appropriate coordination can be achieved
and connection privacy maintained.
* *Routing*: Creation and update of routing tables at each node is
an ordinary, distributed classical task that shares the
information collected about links as above. The expected
completion time of this tasks should be quick enough that the
network converges to provide seamless service upon topology
changes. Unless nodes are mobile, propagation and recalculation
of such changes at the level of seconds should be acceptable.
* *Malicious use monitoring*: It is known that a hijacked or
malfunctioning repeater can be used to impede the overall service
of the network or even to partition the network. It is also known
that QKD-derived monitoring of the network using randomly selected
measurement bases on a portion of the network capacity can serve
as a detection mechanism for this malicious behavior.
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11. Acknowledgments
The authors would like to thank Andrew Todd for crucial help in
building the document, and Shota Nagayama, Akihito Soeda and Monet
Tokuyama Friedrich for useful early discussions on the direction of
the document. This work was supported by the JST Moonshot R&D
program under Grant Number JPMJMS226C.
Appendix A. References
A.1. Normative
A.2. Informative
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* A. M. Branczyk, Hong-Ou-Mandel Interference, [_arXiv:1711.00080_ (2017)](https://arxiv.org/abs/1711.00080).
* V. Krutyanskiy _et al._, Entanglement of Trapped-Ion Qubits Separated by 230 Meters, [_Phys. Rev. Lett._ **130**, 050803](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.130.050803) (2023).
* R. H. Hadfield, Single-photon detectors for optical quantum information applications, [_Nature Photonics_ **3**, 696](https://www.nature.com/articles/nphoton.2009.230) (2009).
* Single Quantum, [Datasheet](https://singlequantum.com/wp-content/uploads/2022/12/SQ-General-Brochure.pdf).
* D. Nadlinger, Device-independent key distribution between trapped-ion quantum network nodes, DPhil Thesis, [Oxford University (2022)](https://ora.ox.ac.uk/objects/uuid:604c53b9-8df8-4e45-8103-10fd81eb3366).
* D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Quantum dynamics of single trapped ions, [_Rev. Mod. Phys._ **75**, 281 (2003)](https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.75.281).
* C. Gardiner, and P. Zoller, The Quantum World of Ultra-Cold Atoms and Light: The Physics of Quantum-Optical Devices, [Imperial College Press, (2015)](https://www.amazon.co.jp/Quantum-World-Ultra-Cold-Atoms-Light/dp/1783266163).
* J.I. Cirac, and P. Zoller, Quantum Computations with Cold Trapped Ions, [_Phys. Rev. Lett._ **74**, 4091 (1995)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.74.4091).
* F. Schmidt-Kaler _et. al._, Realization of the Cirac–Zoller controlled-NOT quantum gate, [_Nature_ **422**, 408 (2003)](https://www.nature.com/articles/nature01494).
* K. Molmer, and A. Sorensen, Multiparticle Entanglement of Hot Trapped Ions, [_Phys. Rev. Lett._ **82**, 1835 (1999)](https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.82.1835).
* J. Benhelm _et al._, Towards fault-tolerant quantum computing with trapped ions, [_Nature Physics_ **4** 463 (2008)](https://www.nature.com/articles/nphys961).
* J. Altepeter, D.F.V. James, and P.G. Kwiat, Qubit quantum state tomography, [_Lecture Notes in Physics_ **649**, 113 (2004)](https://link.springer.com/chapter/10.1007/978-3-540-44481-7_4).
* B. E. A. Saleh, and M. C. Teich, Fundamentals of Photonics, [John Wiley & Sons, (2019)](https://onlinelibrary.wiley.com/doi/book/10.1002/0471213748).
* R. J. Drost, T. J. Moore, and M. Brodsky, Switching Networks for Pairwise-Entanglement Distribution, [_Journal of Optical Communications and Networking_, **8**, 331 (2016)](https://opg.optica.org/jocn/fulltext.cfm?uri=jocn-8-5-331&id=340335).
* M. Koyama, C. Yun, A. Taherkhani, N. Benchasattabuse, B. O. Sane, M. Hajdušek, S. Nagayama, R. Van Meter, Optimal Switching Networks for Paired-Egress Bell State Analyzer Pools, [_arXiv:2405.09860_ (2024)](https://arxiv.org/abs/2405.09860).
* Polatis Series 6000i Instrument Optical Matrix Switch, [https://www.viavisolutions.com/en-us/literature/polatis-series-6000-osm-network-switch-module-data-sheets-en.pdf](https://www.viavisolutions.com/en-us/literature/polatis-series-6000-osm-network-switch-module-data-sheets-en.pdf).
Authors' Addresses
Michal Hajdušek
Keio University
Email: michal@sfc.wide.ad.jp
Rodney Van Meter
Keio University
Email: rdv@sfc.wide.ad.jp
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