Quantum Error Mitigation via
Quantum-Noise-Effect Circuit Groups
Yusuke Hama（Quemix Inc.）
Hirofumi Nishi（Tokyo Institute of Technology & Quemix Inc.）APS March Meeting 2023
March 7th 2023
Reference: Y. Hama and H. Nishi, arXiv:2205.13907.

Introduction 1: Research and Development of Quantum Computers
Research and Development of Quantum Computers•

Various types of quantum hardware

e.g., superconducting circuits, trapped ions, etc.
M. Kjaergaad et al., Appl. Phys. Rev. 6, 021318 (2019);
Ann. Rev. Conmat. Phys. 11, 369 (2020).
C. D. Bruzewicz et al., Appl. Phys. Rev. 6, 021314 (2019).
Superconducting circuits
F. Yoshihara et al., Phys. Rev. Lett. 120,
183601 (2018).

• Quantum algorithms for material science, quantum
chemistry, combinatorial optimization problems

Trapped Ions
C. D. Bruzewicz et al.,Appl. Phys. Rev. 6, 021314 (2019).

e.g., Variational Quantum Eigensolver (VQE)
Quantum Approximate Optimization Algorithm (QAOA)
A. Peruzzo, et al., Nat. Commun. 5, 4213 (2014); J. R. McClean et al.,New. J. Phys. 18 023023 (2016).
E. Farhi et al., arXiv:1411.4028.

•

Demonstration for quantum supremacy withsuperconducting circuit devices
F. Arute et al., Nature 574, 505 (2019).

Schematic for VQE implementation
B. Bauer et al., Chem. Rev. 2020, 120, 12685.

Introduction 2: Quantum Error Mitigation

Drawback: Fragile against quantum noise
Tradional error correcting schemes•

Quantum error correcting (QEC) codes
S. J. Devitt et al., Rep. Prog. Phys. 76, 021318 (2013).

M. Martonosi and M.
Roetteler,
arXiv:1903.10541.

problem: QEC codes are not implemented in near-term quantum computers (NISQ devices)→ necessity for searching alternative error correcting (mitigating) schemes

Quantum error mitigation (QEM) 1. zero-noise extrapolation

K. Temme et al., Phys. Rev. Lett. 119, 180509 (2017).
Y. Li and S. C. Benjamin, Phys. Rev. X 7, 021050 (2017);
S. Endo et al., Phys. Rev. X 8, 031027 (2018).
B. Koczor, Phys. Rev. X 11, 031057 (2021);
New. J. Phys. 23, (2021) 123047.
W. J. Huggins Phys. Rev. X 11, 041036 (2021).
Y. Hama and H. Nishi, arXiv:2205.13907.

2. probabilistic error cancellation
3. Exponential suppression derangement/virtual distillation
4. Quantum-Noise-Effect circuit groups

Modeling Noisy Quantum Computation
This work: QEM for quantum computational errors which occur during gate
operations, e.g., amplitude damping (AD).
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Goal: Establish Quantum-Error Mitigation (QEM) schemes via softwareIdeas and strategy for quantum error mitigation (QEM)

Ideas:⇣
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⇢

real

=⇢

ideal

<latexit sha1_base64="omq8mP20zzlcUTMZynr8N4AE4Ns=">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</latexit>

⇢

+⌧ ·

real

AD ideal
2
(⇢
)
+
O(⌧
)
1

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AD ideal real) =1 ⇢AD ideal
AD
⇢
+
⌧
·
1
1 (

AD ideal
AD ideal real
ideal
2
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·
)⇡⌧·
1 (⇢
⇢
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=
⇢
+
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).
1
first-order QEM<latexit sha1_base64="6s7voSqOpBA/xl41z2JR6aOjlds=">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</latexit>⌧ ·(AD ideal
2
⇢
)
+
O(⌧
).
1
AD ideal)
1 (⇢Applications of our QEM method to QAOA
1X
HC =ZQri ⌦ ZQrj
2
<latexit sha1_base64="0ipm2Pwi73c1Dz+O0zKCELzOQZg=">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</latexit>• QAOA Hamiltonian

1

(i,j)

•

QEM measure: |expnoQEM - expideal|/| expQEM - expideal |RTQEM>1 implies
the validity of our QEM.
<latexit sha1_base64="3+sMEJeACni89nZMyDFQElIEObw=">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</latexit>⌧=log cos2

✓

#⌧
2

◆

quantum circuit

solid curves: simulation results with our original code
open circles: simulation results with Qiskit code

Comparison with PEC
M=181
NPECsamp=100
PEC simulation is
performed by
Cirq and Mitiq

2. Variances

1. Ratio<latexit sha1_base64="dWvsKwSFojDhpDw4AZDxCY/t2D4=">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</latexit>

RTPEC/QEM (M, #⌧ , l) =|hÔiPEC (M, #⌧ , l)|hÔiQEM (#⌧ )NPEC/QEM: # of PEC data points
satisfying RTPEC/QEM >1

hÔiideal |hÔiideal |

.

PEC
Nsamp

<latexit sha1_base64="AIPWK8Ho+9auIYsj5pGiWxGluQE=">AAAIEXicrVO9b9NAFL82QIr5aAsLEotFlSqRQrhkAISoVFFVsBT63Uq9Jro4F8eq7Rj7Elpc/wPdEAMDEyAGxF/AhmABMQFi6MKOGIvEghDvLk5D0nwV4ZPtd+/e73fv/e5d3jENj2O8OzAYOXL0WHTouHLi5KnTwyOjZ5a9csXV2JJWNsvuap56zDRstsQNbrJVx2XUyptsJb8xJdZXqsz1jLK9yLcctm5R3TaKhkY5uHKjkesxkme6YfvUNHSbFQIlRjjb5P78YpDza+bs9NSluemZIIjPJEmVurzEOM0RTitJM6FOqKToUs3fJia1dZOppES5fydQVeJKR26fpT3BxR5Ao8CoGajbgb+t9ggVWcabd0j0SQ/8Kaid2YV9KVSlVRviGbpFs5muJd3ONlbDQI9aTpBQx0Err2LlfHMiHWT9ToFiYylpOugSA1qwIo//s+q9tAx1Ia6hl3gim0mqxC5zqhPSKkMb1f+DCIrahwh9adAuv0Mo0JAgpTT3R25kDKewfNSDRjo0xiZv7kz9Jg8+zpZHB74iggqojDRUQRZiyEYcbBNR5MFYQ2mEkQO+deSDzwXLkOsMBUgBbAWiGERQ8G7AV4fZWui1YS44PYnWYBcTXheQKorhL/gF3sPv8Ev8Df/qyOVLDpHLFvzzNSxzcsM75xZ+9kRZ8Oeo1EB1zZmjIroqczUgd0d6RBVaDV+9/2hv4dp8zB/HT/F3yP8J3sVvoQK7+kN7PsfmH3dhXwQlBXMRFLCbeMXoVUcd5e0jDoNJtqhRn+elqskWhZPA4oLPgUhx1pvgEdnW+sOVHAx8ZmhthsgkYMW3IrtDWJa0Rce4kMs9iTEgmkt9dZlZSXIKvruwl4gUnWhATKPS/kdnTcSetV7s3LetParIu8FASSJzLEAECWvmf8X64WrQcjp1bB7iKpK/Pbq+HjTVrMAr8rSlGqY8A12e3wT0kgVDgfuebr3dB43lTCp9OZWZg4t/A9WeIXQeXUBx4LmCJtEtNIuWkBZ5Fnkf+RT5HH0YfRV9HX1TCx0cCDFnUdMT/fAHHGs1zg==</latexit>

2
PEC
PEC (M, #⌧ , Nsamp ) =

X
l=1

PEC
Nsamp

2
PEC
QEM (#⌧ , Nsamp ) =

X
l=1

1
PEC
Nsamp

1
PEC
Nsamp

⇣
⇣

hÔiPEC (M, #⌧ , l)hÔiQEM (#⌧ , l)Result: Our method outperforms PEC.

hÔiidealhÔiideal

⌘2

⌘2
.

,

QEM under NISQ parameters
superconducting circuit

trapped ion

Superconducting circuits
F. Yoshihara et al., Phys. Rev. Lett. 120, 183601 (2018).

Trapped Ions
C. D. Bruzewicz et al., Appl. Phys. Rev. 6, 021314 (2019).

quantum device

Two-qubit gate time

Decay rate

Perturbation ordersuperconducting
circuits

100 nsec

5 kHz (1/2T1)

1sttrapped ions

100 µsec

0.05 Hz (1/2T2)

1stAdvantages of our quantum error mitigation (QEM)
scheme•It can be implemented with any type of quantum device•

It can be applied to any type of quantum algorithm under various types ofquantum noise channels with an arbitrary strength.

•

Computational cost of our QEM is polynomial with respect to the product of thedepth of a quantum algorithm and the number of qubits.

•

Both the errors associated with gate operations and those occurred due to theadditional operations for QEM are self-consistently mitigated in our scheme.

•

It can be applied to other quantum technologies,
e.g., quantum metrology: arXiv:2303.01820.