The Surface Code in 2 and 3 Dimensions

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The Surface Code in 2 and 3 Dimensions

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


Outline


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 protecting logical qubits

from error by encoding 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.

Using this principle, we can define 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

Having different types of surface codes is useful as some are

better suited to certain 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 Codes (recap)

The 2D surface code uses three types of qubits:

Lattice qubits initialised to

Z-syndrome qubits initialised to

X-syndrome qubits initialised to

The operations performed between qubits are the CNOT

gate.

Error syndromes are determined by interactions 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 initialisation, 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


Implementing 2D codes

These error correction procedures require qubits to persist

after measurement.

Lattice qubits have to be capable of being 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 substituting entanglement for

persistence over time.

A sequence of entangled qubits is made to stand in for a

single qubit undergoing multiple measurements.

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

The measurement destroys the original 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 measuring 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

during the algorithm.

Clare Horsman The Surface Code in 2 and 3 Dimensions

Osaka University

23 February 2011


Building the lattice

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

throughout an algorithm before we start measuring it

No!

Entangling operations only occur between neighbouring

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 define a qubit by not enforcing a single (say) face

stabilizer.

In 3D this equates to removing a

qubit corresponding to a face

syndrome qubit. This is done by

measuring the qubit in the Z-basis. A

line of such qubits is measured out,

corresponding 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 measuring 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


Braiding

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

of the 2D code.

Logical gates, defined by qubit braiding, 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: chains become sheets; rings 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 defined 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 defining qubits with defects, we can turn an entire

surface (2D) or lattice volume (3D) into a qubit.

2D planar

code qubit:

Picture from A.

Fowler et al. Phys.

Rev. Lett. 104,

180503 (2010)

Merging 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

Having 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. They 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


Implementing the surface code (1)

2D code needs persisting qubits, and only requires nearest

neighbour entangling gates. Example: quantum dots.

Proposed quantum dot

lattice qubits.

Defining 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


Implementing 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 machine 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

There are different types of surface code.

The code space can be 2D or 3D.

Qubits can be defined by defects or surfaces.

Each type is useful for different implementations,

different computing 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

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