**The** **Surface** **Code** **in**

2 **and** 3 **Dimensions**

Clare Horsman

AQUA Group, Keio University

**The** **Surface** **Code** **in**

2 **and** 3 **Dimensions**

Clare Horsman

AQUA Group, Keio University

Outl**in**e

●

How many types of surface code are there

●

2D surface codes

●

3D surface codes

●

Planar vs. defect codes

●

Proposed implementations

●

Which one is best

●

Conclusions

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

**The** surface code **and** topology

**The** surface code is a method for protect**in**g logical qubits

from error by encod**in**g them **in** many physical qubits.

**The** error protection is topological: the overall geometry of

the physical qubits encodes the logical qubit.

Gates **in** the surface code are actions that change the

topology of a logical qubit **in** a set way.

Systems that look at first sight different can actually share

the same topological properties.

Us**in**g this pr**in**ciple, we can def**in**e different types of surface

code that all use the same topological properties for qubits.

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

**The** types of surface code

Hav**in**g different types of surface codes is useful as some are

better suited to certa**in** practical implementations than

others.

**The** three types of code that we will look at here are:

●

2D surface code

– best for solid state qubits

●

3D topological code

– best for optical qubits

●

Planar codes (2- **and** 3-D)

– best for small-scale applications (eg memory)

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

2D **Surface** **Code**s (recap)

**The** 2D surface code uses three types of qubits:

Lattice qubits **in**itialised to

Z-syndrome qubits **in**itialised to

X-syndrome qubits **in**itialised to

**The** operations performed between qubits are the CNOT

gate.

Error syndromes are determ**in**ed by **in**teractions between the

lattice qubits **and** the syndrome qubits.

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

Error syndromes **in** 2D

At the first timestep after **in**itialisation, Z-syndrome

measurements are performed

At the next step, X-syndrome measurements.

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

Implement**in**g 2D codes

**The**se error correction procedures require qubits to persist

after measurement.

Lattice qubits have to be capable of be**in**g measured

repeatedly by syndrome qubits **in** both Z- **and** X- bases.

Syndrome qubits need to repeatedly measure lattice qubits

**and** then be themselves classically measured to give the

error syndrome result (+1 or -1).

This is possible if the qubits are solid state (eg quantum

dots). But extremely unfeasible for photonic qubits **in** an

optical architecture.

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

3D “surface” codes

We can remove the requirement for qubits to undergo

multiple measurements by substitut**in**g entanglement for

persistence over time.

A sequence of entangled qubits is made to st**and** **in** for a

s**in**gle qubit undergo**in**g multiple measurements.

Each qubit is entangled with the next before it is measured.

**The** measurement destroys the orig**in**al qubit – very useful for

eg. photon detector measurements.

**The** “temporal” dimension of the code now becomes another

physical one: the code is a 3D lattice of entangled qubits.

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

**The** 3D lattice (1)

All qubits **in** the 3D lattice are prepared **in** , **and** measured

**in** the X basis. All operations between qubits are CPhase.

2D:

3D:

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

**The** 3D lattice (2)

Pictures from A. Fowler & K. Goyal, QIC 9,721 (2009)

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

**The** 3D lattice (3)

This lattice structure is a cluster state.

**The** X-measurements of each cluster qubit gives the

syndrome measurements.

Computation proceeds by measur**in**g out one face of the

cluster at each time step. This is equivalent to one round of

X- or Z- syndrome measurement **in** the 2D code.

Lattice qubits are destroyed by this measurement procedure.

**The** cluster state is first produced **and** then consumed

dur**in**g the algorithm.

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

Build**in**g the lattice

So do we need to build the entire cluster that we will use

throughout an algorithm before we start measur**in**g it

No!

Entangl**in**g operations only occur between neighbour**in**g

qubits – **in** all 3 dimensions. So only 3 cross-sections of the

lattice need to exist **and** be entangled at any one time.

**The** lattice can be created dynamically, **and** each new crosssection

entangled with the previous cross-section to create a

larger lattice.

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

Logical qubits **in** 3D

In 2D we def**in**e a qubit by not enforc**in**g a s**in**gle (say) face

stabilizer.

In 3D this equates to remov**in**g a

qubit correspond**in**g to a face

syndrome qubit. This is done by

measur**in**g the qubit **in** the Z-basis. A

l**in**e of such qubits is measured out,

correspond**in**g to the time that the

stabilizer is not enforced:

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

Lattice defects

As **in** the 2D model, larger defects can be created to support

a qubit with greater error correction.

In the 3D model, this is done by measur**in**g a tube of qubits

out **in** the Z-basis (which disentangles them from the lattice)

Picture from A. Fowler

& K. Goyal, QIC 9,721

(2009)

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

Braid**in**g

We can view the 3D model as a “spacetime representation”

of the 2D code.

Logical gates, def**in**ed by qubit braid**in**g, therefore become

3D loops between qubit tubes:

This can become quite complex...

Picture courtesy

of Simon Devitt,

NII

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

A 3D algorithm

Picture courtesy of Simon Devitt, NII

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

Logical operators

**The** logical operators X **and** Z for a qubit now become 2D

rather than 1D: cha**in**s become sheets; r**in**gs become tubes.

Picture from A. Fowler, K. Goyal, Quant.

Info. Comput. 9, 721-738 (2009)

Both sheets **and** tubes are of physical Z operators.

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

Lattice defects

Like the 2D code, the 3D one uses 2 defects per logical qubit.

This allows multiple qubits to be def**in**ed **in** the same lattice.

**The** code distance d is the distance around the perimeter of

one defect, **and** also the distance between defects.

Each qubit therefore needs a volume of the lattice with

cross-section (2d + d/4)*d.

We will need these types of codes for large-scale algorithms.

But can we build a distance d code with fewer qubits...

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

Planar codes

Instead of def**in****in**g qubits with defects, we can turn an entire

surface (2D) or lattice volume (3D) **in**to a qubit.

2D planar

code qubit:

Picture from A.

Fowler et al. Phys.

Rev. Lett. 104,

180503 (2010)

Merg**in**g two qubit surfaces/volumes forms a CNOT operation.

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

2D **and** 3D planar codes

Hav**in**g one qubit per surface/volume makes repeated logical

gate operations difficult. However, planar codes are useful

for communication **and** memory operations where gates are

not used. **The**y can be very small:

Smallest 2D planar code

Smallest 3D planar code

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

Implement**in**g the surface code (1)

2D code needs persist**in**g qubits, **and** only requires nearest

neighbour entangl**in**g gates. Example: quantum dots.

Proposed quantum dot

lattice qubits.

Def**in****in**g the logical

lattice on top of the

physical dot structure.

Pictures from R. Van Meter et

al. IJQI

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

Implement**in**g the surface code (2)

3D code can be used for photonic qubits. Examples: photonic

module lattice construction

Picture from S. Devitt

et al. ArXiv:1102.0370

Or “photon mach**in**e gun”

Picture from D.

Herrera-Marti et al.

ArXiv:10052915

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

Other implementations

…

…

…

You tell me!

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

Conclusions

**The**re are different types of surface code.

**The** code space can be 2D or 3D.

Qubits can be def**in**ed by defects or surfaces.

Each type is useful for different implementations,

different comput**in**g tasks, **and** for different levels

of technology.

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011

聞 いてくれてありがとう

Clare Horsman **The** **Surface** **Code** **in** 2 **and** 3 **Dimensions**

Osaka University

23 February 2011