This document proposes using all-photonic graph states generated from a few matter qubits to enable resource-efficient quantum communications over long distances. It finds that repeater-less transmission can be outperformed if the generation time of the repeater graph states is less than 6 x 10^4 seconds and other error thresholds are met. Specifically, the generation protocol must use a small number of matter qubits to deterministically create large photonic graph states that can encode logical qubits resistant to photon loss over repeater links of length 50 times the intrinsic loss length.

1. Resource-efficient quantum communications
using all-photonic graph state generated
from a few matter qubits.
Paul Hilaire, Edwin Barnes, and Sophia Economou

2. 2
Quantum communications
Photon loss 𝜂 𝑡 𝐿 = 𝑒
−
𝐿
𝐿 𝑎𝑡𝑡
(𝑡 𝑎𝑡𝑡= 𝐿 𝑎𝑡𝑡/𝑐 ≈ 0.1𝑚𝑠)
How to extend quantum communications to long distances?
𝐿 𝑎𝑡𝑡 ≈ 20𝑘𝑚

3. 3
𝐿0
Quantum Repeater
Memory-based + heralded entanglement
H. Briegel et al., Phys. Rev. Lett. (1998)
K. Azuma et al., Nat. Comm. (2015)
All-photonic repeater
M. Pant et al., Phys. Rev. A. (2017)
QR QRQR
Deterministic generation
using a few matter qubits
D. Buterakos et al., Phys. Rev. X (2017)

4. 4
𝐿0
Quantum Repeater
Memory-based + heralded entanglement
H. Briegel et al., Phys. Rev. Lett. (1998)
K. Azuma et al., Nat. Comm. (2015)
All-photonic repeater
M. Pant et al., Phys. Rev. A. (2017)
QR QRQR
Deterministic generation
using a few matter qubits
D. Buterakos et al., Phys. Rev. X (2017)
Rate per matter qubits
Which strategy?
𝑅(𝑄𝑀)
𝑁 𝑚
≤ 𝑅 𝑚𝑎𝑥
(𝑄𝑀)
=
𝑐
4𝐿
𝑅 𝑅𝐺𝑆
𝑁 𝑚
=
𝑃𝑅𝐺𝑆↔𝑅𝐺𝑆
𝐿
𝐿0
𝑁 𝑚 𝑇𝑅𝐺𝑆

5. 5
Repeater graph state
CZ gate
2𝑚 arms
K. Azuma et al., Nat. Comm. (2015)
Photonic qubit (|+⟩)
𝑚

6. 6
Measurement node
Repeater graph states
BSM

7. 7
Measurement node
Repeater graph states
???
???
Success!
???
Success!
Success!
𝑃𝑅𝐺𝑆↔𝑅𝐺𝑆 = (1 − 1 − 𝑃𝐵𝑒𝑙𝑙
𝑚
)
, 𝑃𝑝ℎ =𝑃𝐵𝑒𝑙𝑙 = 𝑃𝑝ℎ
2
/2
BSM
“Click”𝜂 𝑑𝜂 𝑐 𝜂 𝑡
𝐿0
2

8. 8
Measurement node
Repeater graph states
???
???
Success!
???
Success!
Success!
Z measurement
𝑃𝑅𝐺𝑆↔𝑅𝐺𝑆 = (1 − 1 − 𝑃𝐵𝑒𝑙𝑙
𝑚
)
× 𝑃 𝑍 2𝑚−2

9. 9
Measurement node
Generating a link over RGS
???
???
Success!
???
Success!
Success!
X measurement
Z measurement
𝑃𝑅𝐺𝑆↔𝑅𝐺𝑆 = 1 − 1 − 𝑃𝐵𝑒𝑙𝑙
𝑚
× 𝑃 𝑋 2
× 𝑃 𝑍 2𝑚−2

10. 10
Logical encoding
Missing ingredient → Loss tolerance
𝑃𝑅𝐺𝑆↔𝑅𝐺𝑆 ≤ 𝑃 𝑋 2
𝑃 𝑍 2𝑚−2
≤ 𝑃𝑝ℎ
2𝑚
≤ 𝜂 𝑡 𝐿0
𝑚
≤ 𝜂 𝑡(𝐿0)
No advantage over direct transmission…
=
M. Varnava et al., Phys. Rev. Lett. (2006)
Logical qubit
resistant against loss and error
𝑃𝑅𝐺𝑆↔𝑅𝐺𝑆 = 1 − 1 − 𝑃𝐵𝑒𝑙𝑙
𝑚
× 𝑃𝐿 𝑍 2𝑚−2
× 𝑃𝐿 𝑋 2
𝑃𝐿 𝑍 , 𝑃𝐿 𝑋
=𝑓(𝑡𝑟𝑒𝑒, 𝑃𝑝ℎ)
𝑃 𝑍 , 𝑃(𝑋)

11. 11
Generation time of a RGS
A. Russo et al., Phys. Rev. B (2018)
Deterministically with few solid-state ressources:
D. Buterakos et al., Phys. Rev. X (2017)
Spin-entangled
photon emission
CZ gateHadamard gate
𝑇𝑝ℎ 𝑇 𝐻 𝑇𝐶𝑍
𝑇𝐶𝑍 ≫ 𝑇 𝐻, 𝑇𝑝ℎ
≈ 0
𝑅(𝑅𝐺𝑆)
𝑁 𝑚
=
1
𝑇𝐶𝑍
𝑓(𝐿, 𝐿0, 𝑚, 𝑏0, 𝑏1)
𝜂 𝑐 𝜂 𝑑 = 1
= 𝑏1
𝑏0
𝑚
Maximize 𝑓 for fixed value of 𝐿
𝑇𝑅𝐺𝑆 = 𝑇𝐶𝑍 𝑁𝐶𝑍(𝑚, 𝑡𝑟𝑒𝑒)

12. 12
Who’s best? QM or RGS
𝑅 𝑚𝑎𝑥(𝐿, 𝑇𝐶𝑍)
Repeater-less wins
RGS wins (for sure)
QM can win
𝑇𝐶𝑍 ≤ 6 × 104
𝑡 𝑎𝑡𝑡 ≈ 60𝑛𝑠
RGS wins

13. 13
What does it look like?
𝑚 = 14
𝑏0 = 10
𝑏1 = 5
𝑁𝑝ℎ = 1708
Hard to get a perfect RGS.
Sensitivity to errors?

14. 14
Ideal case →More realistic case
Loss error
𝜂 𝑑𝜂 𝑐
𝑅 𝑠𝑘𝑟 = 𝑅 1 − 2ℎ 𝐹𝐴𝐵 𝜖 𝑚
𝜂 𝑐 𝜂 𝑑 ≠ 1
“Click”

1 − 𝜖 𝑚 𝜖 𝑚
Single qubit errors 𝜖 𝑚
𝜂 𝑐 𝜂 𝑑 ≥ 0.85
𝑇2 ≥ 2500 𝑇𝐶𝑍𝜖 𝑚 ≤ 2. 10−4
Purification can reduce 𝜖 𝑚 requirement
𝜖𝑚
𝜂𝑐𝜂𝑑
𝐿 = 50 𝐿 𝑎𝑡𝑡
𝐿 = 50 𝐿 𝑎𝑡𝑡

15. 15
Conclusion
𝑅
𝑁 𝑚
≥ 𝑐/4𝐿
How to use matter qubits for QR?
To generate RGS? Outperform repeater-less and memory-based
Protocols if
𝑐/4𝐿
Upper bound on resource-performance trade-off
𝑇𝐶𝑍 ≤ 6 × 10−4
𝑡 𝑎𝑡𝑡 ≈ 60𝑛𝑠
𝜂 𝑐 𝜂 𝑑 ≥ 0.85
𝑇2 ≥ 2500 𝑇𝐶𝑍
As quantum memory?
Full article: P. Hilaire, E. Barnes, and S. Economou (in preparation)
Thank you!