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Introduction of Quantum Annealing and D-Wave Machines
Arithmer DX Div. Yuki Bando
1/36
2/36
Self Introduction
• Academic Background
Ph.D. (Science), Tokyo Institute of Technology
Theme: DFT and DFPT, Electron-Phonon Interactions
• Former Job
Postdoctoral Researcher at Nishimori Lab., Tokyo Tech.
Theme: Quantum Annealing
• Current Job
Applications of Machine Learning, Data Analysis
Yuki Bando (坂東優樹)
Outline
• About D-Wave Machines
• Theory of (Adiabatic) Quantum Annealing
• How to run D-Wave Machines
• Applications of D-Wave Machines
3/36
Reference
Theory Review
4/36
About D-Wave Machines
5/36
What is a ”quantum computer” ?
Quantum Computer
a high-performance computer taking advantage of quantum effects
Gate model (ゲート模型) Quantum Annealer (量子アニーリング方式)
From ibm.com From dwavejapan.com
6/36
What are ”quantum annealing” and “D-Wave machine?”
Quantum Annealing
Quantum algorithm to search the
global minimum (最適解) of a
combinatorial optimization problem
(組み合わせ最適化問題)
D-Wave Machine
Computer system implementing
quantum annealing on its hardware
From Wikipedia: Quantuma annealing
From Wikipedia: Quantuma annealing
• Analog computer
• Sampler (heuristic machine)
• Under a severe environment
7/36
D-Wave Machines
From dwavejapan.com
D-Wave machine is the first commercialized quantum computer.
From Wikipedia: Quantum annealing
Price: $10 Million Price: $15 Million
Human size
8/36
Theory of Quantum Annealing
9/36
Qubit
D-Wave Machine is composed of qubits (量子ビット)
quantum + bit → qubit
or
• qubit (quantum)
• bit (classical)
Superposition (重ね合わせ)
1 =
1
0
1
2
0 + 1 =
1
2
1
1
qubit
coupler
0 =
0
1
Measurement by 𝜎𝑖
𝑧
=
1 0
0 −1
10/36
Quantum Annealing: Framework
𝐻 𝑡 = 𝐴 𝑡 ∙ 𝐻𝐷 + 𝐵(𝑡) ∙ 𝐻𝑇
Hamiltonian
Schrödinger equation
(Quantum Mechanics)
𝑖ℏ
𝜕
𝜕𝑡
𝜓(𝑡) = 𝐻 𝑡 𝜓(𝑡)
𝐵 𝑡 =
𝑡
𝑡𝑎
𝐴 𝑡 = 1 −
𝑡
𝑡𝑎
𝑡𝑎
1
𝐎
𝐻𝐷: Driver term
𝐻𝑇: Target term
𝐴 𝑡 , 𝐵 𝑡 : Schedule
e.g.,
Schedule of
D-wave Advantage
𝐵 𝑠
𝑠 =
𝑡
𝑡𝑎
𝐴 𝑠
e.g., linear schedule
Solution 𝜓(𝑡𝑎) {𝜎𝑖
𝑧
} 𝜓(𝑡𝑎)
11/36
Quantum Annealing: Target Term
𝜎𝑖
𝑧
: Pauli matrix (𝑧) at 𝑖th site
(パウリ行列)
𝜎𝑖
𝑧
=
1 0
0 −1
e.g., Classical Ising model (古典イジング模型)
Many combinatorial optimization problems can be mapped onto the Ising model
𝐻𝑇 =
𝑖>𝑗
𝐽𝑖𝑗𝜎𝑖
𝑧
𝜎𝑗
𝑧
+
𝑖
ℎ𝑖𝜎𝑖
𝑧
magnetic field
Coupling strength
If ℎ𝑖 = 1 and the 𝑖th site’s state 1
0
,
−ℎ𝑖𝜎𝑖
𝑧
provide -1 energy gain
(lower energy is better)
12/36
Quantum Annealing: Combinatorial Optimization Problem
e.g.,
Traveling salesman problem
(find the shortest route)
A
E D
C
B A
E
D
C
B
[𝒒𝒂𝒊] A B C D E
1st 0 1 0 0 0
2nd 1 0 0 0 0
3rd 0 0 0 0 1
4th 0 0 0 1 0
5th 0 0 1 0 0
1st 0 1 0 0 0
𝑑𝛼𝛽
City: 𝛼, 𝛽
Turn:
𝑖
Total route length 𝐿 =
𝛼,𝛽 𝑖
𝑑𝛼𝛽𝑞𝑎𝑖𝑞𝛽𝑖+1
Target Hamiltonian (QUBO)
𝑞𝛼𝑖 = 0 or 1
𝐻𝑇 =
𝛼,𝛽 𝑖
𝑑𝛼𝛽𝑞𝑎𝑖𝑞𝛽𝑖+1 + 𝐜𝐨𝐧𝐬𝐭𝐫𝐚𝐢𝐧𝐭𝐬
𝑎
𝛼 𝑖
𝑞𝑎𝑖 − 1
2
+ 𝑏
𝑖 𝛼
𝑞𝑎𝑖 − 1
2
𝑞𝑎𝑖 → (𝐼 + 𝜎𝑖
𝑧
)/2
Binary
QUBO to Ising
13/36
Quantum Annealing: Driver Term
𝐻𝐷 = −
𝑖
𝜎𝑖
𝑥
𝜎𝑖
𝛼
𝛼 = 𝑥, 𝑦, 𝑧 : Pauli matrix
(パウリ行列)
𝜎𝑖
𝑥
=
0 1
1 0
Ground State (基底状態、GS) of Driver Term
Quantum fluctuation term
(量子揺らぎ項、横磁場項)
GS of 𝐻𝐷 =
1
2
𝑁
2
×
000 ⋯ 000 + 000 ⋯ 001 + 000 ⋯ 010 +
⋮
⋮
+ 101 ⋯ 111 + 011 ⋯ 111 + 111 ⋯ 111
superposition of all possible classical states
14/36
Quantum Annealing: GS of Driver Term
Ground state of −𝝈𝒊
𝒙
at the 𝒊𝐭𝐡 site
Grand state of 𝐻𝐷
GS of 𝐻𝐷 =
𝑖
GS 𝑖 =
𝑖
1
2
0 𝑖 + 1 𝑖
GS 𝑖 =
1
2
0 𝑖 + 1 𝑖
𝒊𝐭𝐡 site
1
2
0 𝑖 + 1 𝑖
GS of 𝐻𝐷 =
1
2
𝑁
2
×
000 ⋯ 000 + 000 ⋯ 001 + 000 ⋯ 010 +
⋮
⋮
+ 101 ⋯ 111 + 011 ⋯ 111 + 111 ⋯ 111
All (𝟐𝑵
) states are included in 𝐆𝐒 𝒐𝒇 𝑯𝑫 with same amplitude
∗∗ −𝜎𝑖
𝑥
𝐺𝑆 𝑖
= ±1 𝐺𝑆 𝑖
𝜎𝑖
𝑥
=
0 1
1 0
15/36
Quantum Annealing: Mechanism
𝑖ℏ
𝜕
𝜕𝑡
𝜓(𝑡) = 𝐻 𝑡 𝜓(𝑡)
𝐻 𝑡 = 𝐴 𝑡 ∙ 𝐻𝐷 + 𝐵(𝑡) ∙ 𝐻𝑇
• 𝐻 𝑡 = 0 = 𝐻𝐷
𝜓(𝑡 = 0) = GS of 𝐻𝐷
𝑡𝑎
𝑡
𝐵 𝑡
𝐴 𝑡
𝑡𝑎
1
• 𝐻 𝑡 = 𝑡𝑎 = 𝐻𝑇
𝜓(𝑡 = 𝑡𝑎) = GS of 𝐻𝑇 =
0 + 0 + 0 +
⋮
101 ⋯ 110
⋮
+0 + 0 + 0
• Start from GS of 𝐻𝐷
• Changing 𝐻 𝑡 by sufficiently large 𝑡𝑎
(Adiabatic Condition,)
• 𝜓(𝑡 = 𝑡𝑎) is optimal solution of 𝐻𝑇
Adiabatic Quantum Annealing
Energy
=
𝐸
𝑡
𝐻 𝑡 𝜙 = 𝐸 𝑡 𝜙
16/36
Quantum Annealing: Failure of Quantum Annealing
𝑖ℏ
𝜕
𝜕𝑡
𝜓(𝑡) = 𝐻 𝑡 𝜓(𝑡)
𝐻 𝑡 = 𝐴 𝑡 ∙ 𝐻𝐷 + 𝐵(𝑡) ∙ 𝐻𝑇
𝑡𝑎
𝑡
Energy
~𝐸
𝑡
Fast annealing cause state transition from the GS to other states
If 𝑡𝑎
′ ≪ 𝑡𝑎, 𝜓(𝑡) can not stay in the ground state (GS)
𝐵 𝑡
𝐴 𝑡
𝑡𝑎
1
𝑡𝑎
′
Not optimal solution
Transition
17/36
Quantum Annealing: Difference with Simulated Annealing(Classical)
Energy
Configuration
Optimal solution
Simulated Annealing
(熱焼きなまし法)
Quantum Annealing
(量子焼きなまし法)
Energy
Configuration
Optimal solution
Quantum annealing can pass through energy barrier
Using thermal fluctuations Using quantum fluctuations
18/36
Quantum Annealing: Failure of Simulated Annealing
Fast cooling trap the state in the local minimum
Energy
Configuration
Optimal solution
Energy Barrier
Simulated Annealing:
State 𝜙 𝑡 is updated using thermal fluctuation while cooling temperature 𝑇 𝑡
If 𝑡𝑎
′ ≪ 𝑡𝑎,
𝜙 𝑡 can not move anymore in the local state
𝑇 𝑡
𝑡𝑎
𝑡𝑎
′
Temperature
Large 𝑻 is necessary to get over the energy barrier.
19/36
Quantum Annealing: Theoretical Estimation
Worst Evaluation
(最悪評価)
0 < 𝜖 ≪ 1
log 𝜖 ≪ 𝜖−1
Simulated Annealing
Quantum Annealing
𝑡𝑎 ∝ 𝑒𝑎/𝜖𝑁
𝑡𝑎 ∝ 𝑒𝑏 log 𝜖 𝑁
𝑇(𝑡) → 𝜖
Temperature
Method Required time
𝐵(𝑡) → 𝜖
Quantum fluctuation
Required time to get optimal solution
(≡ Γ 𝑡 )
𝑁: System Size
20/36
Quantum Annealing: Details of Required Time
𝑡𝑎 ∝ 𝑁𝑎
𝑡𝑎 ∝ 𝑒𝑎𝑁
First order PT
(一次相転移)
Second order PT
(二次相転移)
𝑡𝑎
𝑡
Energy
Gap: ∆𝑚𝑖𝑛
PT
• Phase Transition (PT、相転移) occurs in QA
• Type of PT depends on the type of problem
• The required time 𝑡𝑎 depends on the type of PT
m
(磁化)
T
m
(磁化)
T
e.g., Magnetization
𝑁: System Size
21/36
How to run D-Wave Machines
22/36
From docs.dwavesys.com
D-Wave Cloud Service
D-Wave Inc. provides D-Wave cloud service called “Leap”
• One need to create a free account
• Leap provides free developer access, free time: one minute
(QPU usage at $2000/hour)
• D-Wave provides python SDK called “Ocean” to access to QPU,
https://cloud.dwavesys.com/leap
*QPU: quantum version of CPU
D-Wave Leap
23/36
View of D-Wave Leap (GUI part)
Service Information
Submission History
System Information
24/36
Available D-Wave Machines
D-Wave 2000Q Latest: D-Wave Advantage
Number of qubits: about 2000 Number of qubits: about 5000
25/36
D-Wave 2000Q D-Wave Advantage
Difference between 2000Q and Advantage
qubit
coupler
Chimera Graph Pegasus graph
6 different couplers per each qubit 15 different couplers per each qubit
26/36
Flowchart to run D-Wave Machine
From https://docs.ocean.dwavesys.com/
SPIN: -1 or 1
BINARY: 0 or 1
(binary quadratic model)
Coding part
Notebook part
27/36
Preparation to run D-Wave Machines by Python
pip install dimod
pip install dwave-system
get token
28/36
Coding Process
1. Define model
h = {0:-2.0, …, 7:1.0}, J = {(0,1):0.5, …, (6,7):-2.0}
model = BinaryQuadraticModel(h,J,”Binary” or SPIN”)
2. Set up D-Wave Sampler (QPU)
QPU = DWaveSampler(token=“token”, solver=“solver_name”)
3. Embed and sample Ising or QUBO model
sampler = EmbeddingComposite(QPU)
solutions = sampler.sample(model,**parameters)
4. Analyze solutions!
𝐻𝑇 =
𝑖>𝑗
𝐽𝑖𝑗𝜎𝑖
𝑧
𝜎𝑗
𝑧
+
𝑖
ℎ𝑖𝜎𝑖
𝑧
e.g., annealing time: 𝑡𝑎
29/36
Code Example
1 2
Result
30/36
Minor Embedding
𝐻𝑇 =
𝑖>𝑗
𝑁=4
𝐽𝑖𝑗𝜎𝑖
𝑧
𝜎𝑗
𝑧
Physical qubit
Logical qubit
(chain)
1
2
3
4
𝐽𝑖𝑗
𝐽′: Strong Ferro. Inter.
D-Wave machine can not implement a full-connected model directly。
”EmbeddingComposites” automatically do minor embedding
31/36
No Chain Breaking (simple case)
Chain Breaking
Chain Breaking (complicated case, 𝑱𝒊𝒋 and 𝒉𝒊 are random values)
32/36
Applications of D-Wave Machines: Industry
DENSO Volks Wagen
From dwavejapan.com
33/36
Applications of D-Wave machines: Academia
Observation of topological phenomena
in a programmable lattice of 1,800
qubits
https://arxiv.org/abs/1803.02047
International Journal of Theoretical Physics 21, 467–
488(1982)
Simulating physics with computers
Probing the Universality of Topological
Defect Formation in a Quantum Annealer:
Kibble-Zurek Mechanism and Beyond
https://arxiv.org/abs/2001.11637 etc…
34/36
Characteristic Properties and Future Prospects
Characteristic Properties
• There are hardware problems at each machine.
• Sometimes noise or bias affects results.
D-Wave 2000Q
at Burnaby
D-Wave 2000Q
at NASA
≠
Prospects
• There are several researches to speed up.
• Different scheduler may be added.
e.g., reverse annealing, pause, quench
• Larger scale, more couplers, more stable
35/36
Summary
• D-Wave machines work by quantum annealing algorithm
• Quantum annealing is theoretically faster than classical algorithms,
details depend on the type of problem
• There are several generations of D-Wave machines, and what they can
do is different.
• We can easily access and use D-Wave machines through Leap
36/36

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Introduction of Quantum Annealing and D-Wave Machines

  • 1. 2 0 2 1 / 0 4 / 2 2 Introduction of Quantum Annealing and D-Wave Machines Arithmer DX Div. Yuki Bando 1/36
  • 2. 2/36 Self Introduction • Academic Background Ph.D. (Science), Tokyo Institute of Technology Theme: DFT and DFPT, Electron-Phonon Interactions • Former Job Postdoctoral Researcher at Nishimori Lab., Tokyo Tech. Theme: Quantum Annealing • Current Job Applications of Machine Learning, Data Analysis Yuki Bando (坂東優樹)
  • 3. Outline • About D-Wave Machines • Theory of (Adiabatic) Quantum Annealing • How to run D-Wave Machines • Applications of D-Wave Machines 3/36
  • 6. What is a ”quantum computer” ? Quantum Computer a high-performance computer taking advantage of quantum effects Gate model (ゲート模型) Quantum Annealer (量子アニーリング方式) From ibm.com From dwavejapan.com 6/36
  • 7. What are ”quantum annealing” and “D-Wave machine?” Quantum Annealing Quantum algorithm to search the global minimum (最適解) of a combinatorial optimization problem (組み合わせ最適化問題) D-Wave Machine Computer system implementing quantum annealing on its hardware From Wikipedia: Quantuma annealing From Wikipedia: Quantuma annealing • Analog computer • Sampler (heuristic machine) • Under a severe environment 7/36
  • 8. D-Wave Machines From dwavejapan.com D-Wave machine is the first commercialized quantum computer. From Wikipedia: Quantum annealing Price: $10 Million Price: $15 Million Human size 8/36
  • 9. Theory of Quantum Annealing 9/36
  • 10. Qubit D-Wave Machine is composed of qubits (量子ビット) quantum + bit → qubit or • qubit (quantum) • bit (classical) Superposition (重ね合わせ) 1 = 1 0 1 2 0 + 1 = 1 2 1 1 qubit coupler 0 = 0 1 Measurement by 𝜎𝑖 𝑧 = 1 0 0 −1 10/36
  • 11. Quantum Annealing: Framework 𝐻 𝑡 = 𝐴 𝑡 ∙ 𝐻𝐷 + 𝐵(𝑡) ∙ 𝐻𝑇 Hamiltonian Schrödinger equation (Quantum Mechanics) 𝑖ℏ 𝜕 𝜕𝑡 𝜓(𝑡) = 𝐻 𝑡 𝜓(𝑡) 𝐵 𝑡 = 𝑡 𝑡𝑎 𝐴 𝑡 = 1 − 𝑡 𝑡𝑎 𝑡𝑎 1 𝐎 𝐻𝐷: Driver term 𝐻𝑇: Target term 𝐴 𝑡 , 𝐵 𝑡 : Schedule e.g., Schedule of D-wave Advantage 𝐵 𝑠 𝑠 = 𝑡 𝑡𝑎 𝐴 𝑠 e.g., linear schedule Solution 𝜓(𝑡𝑎) {𝜎𝑖 𝑧 } 𝜓(𝑡𝑎) 11/36
  • 12. Quantum Annealing: Target Term 𝜎𝑖 𝑧 : Pauli matrix (𝑧) at 𝑖th site (パウリ行列) 𝜎𝑖 𝑧 = 1 0 0 −1 e.g., Classical Ising model (古典イジング模型) Many combinatorial optimization problems can be mapped onto the Ising model 𝐻𝑇 = 𝑖>𝑗 𝐽𝑖𝑗𝜎𝑖 𝑧 𝜎𝑗 𝑧 + 𝑖 ℎ𝑖𝜎𝑖 𝑧 magnetic field Coupling strength If ℎ𝑖 = 1 and the 𝑖th site’s state 1 0 , −ℎ𝑖𝜎𝑖 𝑧 provide -1 energy gain (lower energy is better) 12/36
  • 13. Quantum Annealing: Combinatorial Optimization Problem e.g., Traveling salesman problem (find the shortest route) A E D C B A E D C B [𝒒𝒂𝒊] A B C D E 1st 0 1 0 0 0 2nd 1 0 0 0 0 3rd 0 0 0 0 1 4th 0 0 0 1 0 5th 0 0 1 0 0 1st 0 1 0 0 0 𝑑𝛼𝛽 City: 𝛼, 𝛽 Turn: 𝑖 Total route length 𝐿 = 𝛼,𝛽 𝑖 𝑑𝛼𝛽𝑞𝑎𝑖𝑞𝛽𝑖+1 Target Hamiltonian (QUBO) 𝑞𝛼𝑖 = 0 or 1 𝐻𝑇 = 𝛼,𝛽 𝑖 𝑑𝛼𝛽𝑞𝑎𝑖𝑞𝛽𝑖+1 + 𝐜𝐨𝐧𝐬𝐭𝐫𝐚𝐢𝐧𝐭𝐬 𝑎 𝛼 𝑖 𝑞𝑎𝑖 − 1 2 + 𝑏 𝑖 𝛼 𝑞𝑎𝑖 − 1 2 𝑞𝑎𝑖 → (𝐼 + 𝜎𝑖 𝑧 )/2 Binary QUBO to Ising 13/36
  • 14. Quantum Annealing: Driver Term 𝐻𝐷 = − 𝑖 𝜎𝑖 𝑥 𝜎𝑖 𝛼 𝛼 = 𝑥, 𝑦, 𝑧 : Pauli matrix (パウリ行列) 𝜎𝑖 𝑥 = 0 1 1 0 Ground State (基底状態、GS) of Driver Term Quantum fluctuation term (量子揺らぎ項、横磁場項) GS of 𝐻𝐷 = 1 2 𝑁 2 × 000 ⋯ 000 + 000 ⋯ 001 + 000 ⋯ 010 + ⋮ ⋮ + 101 ⋯ 111 + 011 ⋯ 111 + 111 ⋯ 111 superposition of all possible classical states 14/36
  • 15. Quantum Annealing: GS of Driver Term Ground state of −𝝈𝒊 𝒙 at the 𝒊𝐭𝐡 site Grand state of 𝐻𝐷 GS of 𝐻𝐷 = 𝑖 GS 𝑖 = 𝑖 1 2 0 𝑖 + 1 𝑖 GS 𝑖 = 1 2 0 𝑖 + 1 𝑖 𝒊𝐭𝐡 site 1 2 0 𝑖 + 1 𝑖 GS of 𝐻𝐷 = 1 2 𝑁 2 × 000 ⋯ 000 + 000 ⋯ 001 + 000 ⋯ 010 + ⋮ ⋮ + 101 ⋯ 111 + 011 ⋯ 111 + 111 ⋯ 111 All (𝟐𝑵 ) states are included in 𝐆𝐒 𝒐𝒇 𝑯𝑫 with same amplitude ∗∗ −𝜎𝑖 𝑥 𝐺𝑆 𝑖 = ±1 𝐺𝑆 𝑖 𝜎𝑖 𝑥 = 0 1 1 0 15/36
  • 16. Quantum Annealing: Mechanism 𝑖ℏ 𝜕 𝜕𝑡 𝜓(𝑡) = 𝐻 𝑡 𝜓(𝑡) 𝐻 𝑡 = 𝐴 𝑡 ∙ 𝐻𝐷 + 𝐵(𝑡) ∙ 𝐻𝑇 • 𝐻 𝑡 = 0 = 𝐻𝐷 𝜓(𝑡 = 0) = GS of 𝐻𝐷 𝑡𝑎 𝑡 𝐵 𝑡 𝐴 𝑡 𝑡𝑎 1 • 𝐻 𝑡 = 𝑡𝑎 = 𝐻𝑇 𝜓(𝑡 = 𝑡𝑎) = GS of 𝐻𝑇 = 0 + 0 + 0 + ⋮ 101 ⋯ 110 ⋮ +0 + 0 + 0 • Start from GS of 𝐻𝐷 • Changing 𝐻 𝑡 by sufficiently large 𝑡𝑎 (Adiabatic Condition,) • 𝜓(𝑡 = 𝑡𝑎) is optimal solution of 𝐻𝑇 Adiabatic Quantum Annealing Energy = 𝐸 𝑡 𝐻 𝑡 𝜙 = 𝐸 𝑡 𝜙 16/36
  • 17. Quantum Annealing: Failure of Quantum Annealing 𝑖ℏ 𝜕 𝜕𝑡 𝜓(𝑡) = 𝐻 𝑡 𝜓(𝑡) 𝐻 𝑡 = 𝐴 𝑡 ∙ 𝐻𝐷 + 𝐵(𝑡) ∙ 𝐻𝑇 𝑡𝑎 𝑡 Energy ~𝐸 𝑡 Fast annealing cause state transition from the GS to other states If 𝑡𝑎 ′ ≪ 𝑡𝑎, 𝜓(𝑡) can not stay in the ground state (GS) 𝐵 𝑡 𝐴 𝑡 𝑡𝑎 1 𝑡𝑎 ′ Not optimal solution Transition 17/36
  • 18. Quantum Annealing: Difference with Simulated Annealing(Classical) Energy Configuration Optimal solution Simulated Annealing (熱焼きなまし法) Quantum Annealing (量子焼きなまし法) Energy Configuration Optimal solution Quantum annealing can pass through energy barrier Using thermal fluctuations Using quantum fluctuations 18/36
  • 19. Quantum Annealing: Failure of Simulated Annealing Fast cooling trap the state in the local minimum Energy Configuration Optimal solution Energy Barrier Simulated Annealing: State 𝜙 𝑡 is updated using thermal fluctuation while cooling temperature 𝑇 𝑡 If 𝑡𝑎 ′ ≪ 𝑡𝑎, 𝜙 𝑡 can not move anymore in the local state 𝑇 𝑡 𝑡𝑎 𝑡𝑎 ′ Temperature Large 𝑻 is necessary to get over the energy barrier. 19/36
  • 20. Quantum Annealing: Theoretical Estimation Worst Evaluation (最悪評価) 0 < 𝜖 ≪ 1 log 𝜖 ≪ 𝜖−1 Simulated Annealing Quantum Annealing 𝑡𝑎 ∝ 𝑒𝑎/𝜖𝑁 𝑡𝑎 ∝ 𝑒𝑏 log 𝜖 𝑁 𝑇(𝑡) → 𝜖 Temperature Method Required time 𝐵(𝑡) → 𝜖 Quantum fluctuation Required time to get optimal solution (≡ Γ 𝑡 ) 𝑁: System Size 20/36
  • 21. Quantum Annealing: Details of Required Time 𝑡𝑎 ∝ 𝑁𝑎 𝑡𝑎 ∝ 𝑒𝑎𝑁 First order PT (一次相転移) Second order PT (二次相転移) 𝑡𝑎 𝑡 Energy Gap: ∆𝑚𝑖𝑛 PT • Phase Transition (PT、相転移) occurs in QA • Type of PT depends on the type of problem • The required time 𝑡𝑎 depends on the type of PT m (磁化) T m (磁化) T e.g., Magnetization 𝑁: System Size 21/36
  • 22. How to run D-Wave Machines 22/36
  • 23. From docs.dwavesys.com D-Wave Cloud Service D-Wave Inc. provides D-Wave cloud service called “Leap” • One need to create a free account • Leap provides free developer access, free time: one minute (QPU usage at $2000/hour) • D-Wave provides python SDK called “Ocean” to access to QPU, https://cloud.dwavesys.com/leap *QPU: quantum version of CPU D-Wave Leap 23/36
  • 24. View of D-Wave Leap (GUI part) Service Information Submission History System Information 24/36
  • 25. Available D-Wave Machines D-Wave 2000Q Latest: D-Wave Advantage Number of qubits: about 2000 Number of qubits: about 5000 25/36
  • 26. D-Wave 2000Q D-Wave Advantage Difference between 2000Q and Advantage qubit coupler Chimera Graph Pegasus graph 6 different couplers per each qubit 15 different couplers per each qubit 26/36
  • 27. Flowchart to run D-Wave Machine From https://docs.ocean.dwavesys.com/ SPIN: -1 or 1 BINARY: 0 or 1 (binary quadratic model) Coding part Notebook part 27/36
  • 28. Preparation to run D-Wave Machines by Python pip install dimod pip install dwave-system get token 28/36
  • 29. Coding Process 1. Define model h = {0:-2.0, …, 7:1.0}, J = {(0,1):0.5, …, (6,7):-2.0} model = BinaryQuadraticModel(h,J,”Binary” or SPIN”) 2. Set up D-Wave Sampler (QPU) QPU = DWaveSampler(token=“token”, solver=“solver_name”) 3. Embed and sample Ising or QUBO model sampler = EmbeddingComposite(QPU) solutions = sampler.sample(model,**parameters) 4. Analyze solutions! 𝐻𝑇 = 𝑖>𝑗 𝐽𝑖𝑗𝜎𝑖 𝑧 𝜎𝑗 𝑧 + 𝑖 ℎ𝑖𝜎𝑖 𝑧 e.g., annealing time: 𝑡𝑎 29/36
  • 31. Minor Embedding 𝐻𝑇 = 𝑖>𝑗 𝑁=4 𝐽𝑖𝑗𝜎𝑖 𝑧 𝜎𝑗 𝑧 Physical qubit Logical qubit (chain) 1 2 3 4 𝐽𝑖𝑗 𝐽′: Strong Ferro. Inter. D-Wave machine can not implement a full-connected model directly。 ”EmbeddingComposites” automatically do minor embedding 31/36
  • 32. No Chain Breaking (simple case) Chain Breaking Chain Breaking (complicated case, 𝑱𝒊𝒋 and 𝒉𝒊 are random values) 32/36
  • 33. Applications of D-Wave Machines: Industry DENSO Volks Wagen From dwavejapan.com 33/36
  • 34. Applications of D-Wave machines: Academia Observation of topological phenomena in a programmable lattice of 1,800 qubits https://arxiv.org/abs/1803.02047 International Journal of Theoretical Physics 21, 467– 488(1982) Simulating physics with computers Probing the Universality of Topological Defect Formation in a Quantum Annealer: Kibble-Zurek Mechanism and Beyond https://arxiv.org/abs/2001.11637 etc… 34/36
  • 35. Characteristic Properties and Future Prospects Characteristic Properties • There are hardware problems at each machine. • Sometimes noise or bias affects results. D-Wave 2000Q at Burnaby D-Wave 2000Q at NASA ≠ Prospects • There are several researches to speed up. • Different scheduler may be added. e.g., reverse annealing, pause, quench • Larger scale, more couplers, more stable 35/36
  • 36. Summary • D-Wave machines work by quantum annealing algorithm • Quantum annealing is theoretically faster than classical algorithms, details depend on the type of problem • There are several generations of D-Wave machines, and what they can do is different. • We can easily access and use D-Wave machines through Leap 36/36