Embedding Logical Qubits into the D-Wave Hardware Graph
Keith Britt keithbritt@utk.edu
February 26 th , 2014

Agenda
• A Little Bit About Quantum Computing
• A Little Bit About D-Wave
• The D-Wave Hardware
• Transforming the Hardware into a Graph
• Embedding Literature “Review”
• Embedding Limits
• Example Embedding(s)
• Homework Problems

Quantum Computing
• In the Quantum World, classical physics give way to 2 phenomena:
– Superposition
– Entanglement
• We can use these phenomena to massively parallelize computation and also do some crazy stuff that is otherwise impossible (truly random numbers, data teleportation, etc.)

Quantum Computing
• Parallelism Scales Exponentially with the Number of Qubits ( 2 𝑛
)
• Several Models of Quantum Computation
– Gate/Circuit Model
– Adiabatic Model
– Topological model (and more…)
• The First Commercially Available Quantum
Computer (D-Wave) Uses the Adiabatic Model

D-Wave
• Canadian Company Founded by Physicists and
Material Scientists
• Created a Quantum Chip Using Adiabatic
Model Running at Extremely Cold
Temperatures and Minimal Interference
• Very Controversial… Is the Computer Really
Quantum? Is It Worth It?
– Simulations Beat It
– Classical Adiabatic Computer Matched It

Ising Model
𝐸 𝑠 = 𝑖,𝑗 ∈ 𝑛𝑒𝑖𝑔ℎ
𝐽 𝑖,𝑗 𝑠 𝑖 𝑠 𝑗
+ ℎ 𝑖 𝑠 𝑖 𝑖 ∈𝑉 𝑠 𝑖
: State (Spin/Value) of the Qubit ( −1, 1  0, 1 )
𝑉 : Set of Qubits ℎ 𝑖
: Strength of the Qubit Value
𝐽 𝑖,𝑗
: Strength of the Qubit-Qubit Interaction
𝐸 𝑠 : The Total Energy of the System http://dwave.files.wordpress.com/2010/12/weightedmaxsat_v2.pdf

D-Wave Hardware http://dwave.wordpress.com/2011/12/01/vesuvius-a-closer-look-512-qubit-processor-gallery/

So is Hardware Just a Line Graph?
Not Valid

Unit Cell Graph Representation
• Each Qubit Becomes a Vertex
• Each Coupler Becomes an Edge Between
Vertices
Valid

Ising Model
𝐸 𝑠 = 𝑖,𝑗 ∈ 𝑛𝑒𝑖𝑔ℎ
𝐽 𝑖,𝑗 𝑠 𝑖 𝑠 𝑗
+ ℎ 𝑖 𝑠 𝑖 𝑖 ∈𝑉
Vertex Weights
Edge Weights 𝑠 𝑖
: State (Spin/Value) of the Qubit ( −1, 1  0, 1 )
𝑉 : Set of Qubits ℎ 𝑖
: Strength of the Qubit Value
𝐽 𝑖,𝑗
: Strength of the Qubit-Qubit Interaction
𝐸 𝑠 : The Total Energy of the System http://dwave.files.wordpress.com/2010/12/weightedmaxsat_v2.pdf

Linking Unit Cells

Linking Unit Cells
Not Valid

Linking Unit Cells
Valid

• 512 Vertices (Qubits)
• 16 Intra Unit Cell Edges per Unit Cell
• 8 – 16 Inter Unit Cell
Edges per Unit Cell

Literature Review
• Choi, Vicky. Minor-Embedding in Adiabatic Quantum
Computation: I. The Parameter Setting Problem. Quantum
Information Processing, 7, pp 193 – 209,
2008. arXiv:0804.4884v1 [quant-ph].
• Choi, Vicky. Minor-embedding in adiabatic quantum
computation: II. Minor-universal graph design. Quantum
Information Processing: Volume 10, Issue 3 (2011), Page
343. arXiv:1001.3116v2 [quant-ph].
• Klymko, C., Sullivan, B., Humble, T. Adiabatic quantum
programming: minor embedding with hard faults.
Quantum Information Processing: Volume 13, Issue 3
(2014), pp 709 – 729. arXiv:1210.8395v2 [quant-ph].

D-Wave Embedding Limits
• Logical Qubit is Not the Same as a Physical
Qubit
• Can Always Embed N Logical Qubits onto 𝑁 2
Physical Qubits
• This is Not a Tight Upper-bound, Even for
Complete Graphs
• Choi’s Triad Technique Able to Embed 𝐾
17
128-Qubit Processor ( 17 > 128 ) on

Logical to Physical
• A Logical Qubit can be Spread Over Multiple
Physical Qubits as Long as There is a Common
Path Between All The Physical Qubits
VALID NOT VALID

Something We’re Not Going Over
𝐸 𝑠 = 𝑖,𝑗 ∈ 𝑛𝑒𝑖𝑔ℎ
𝐽 𝑖,𝑗 𝑠 𝑖 𝑠 𝑗
+ ℎ 𝑖 𝑠 𝑖 𝑖 ∈𝑉
Vertex Weights
Edge Weights
𝐽 𝑖,𝑗 and ℎ 𝑖 values determine a lot about how aligned qubits are with one another. This gets a bit too outside of graph theory, but there’s an easy to read explanation at: http://www.dwavesys.com/sites/default/files/Map%20Coloring%20WP2.pdf

An Example
• Embed 𝑊
4 into a Unit Cell

http://arxiv.org/pdf/1001.3116v2.pdf

http://arxiv.org/pdf/1001.3116v2.pdf

Embedding 𝐾
8
• Start by Embedding
𝐾
4 in a Unit Cell

Embedding 𝐾
8
• Next Embed the
Other 𝐾
4
Cell in a Unit

Embedding 𝐾
8
• Next Embed 𝐾
4,4
Unit Cell in a

Embedding 𝐾
8
• Tada! 𝐾
8 is embedded in the D-Wave
Hardware

Homework Questions
• Embed 𝐾
12 into the D-Wave Hardware Graph in 6 or Fewer Unit Cells.

Homework Questions
• Embed the Peterson Graph into the D-Wave
Hardware Graph (Diamond Isomorphism) in 6 or Fewer Unit Cells.