high thresholds in two dimensions Robert Raussendorf, University of - - PowerPoint PPT Presentation

▶
high thresholds in two dimensions
SMART_READER_LITE
LIVE PREVIEW

high thresholds in two dimensions Robert Raussendorf, University of - - PowerPoint PPT Presentation

Oct 11, 2022 230 likes •745 views

Fault-tolerant quantum computation - high thresholds in two dimensions Robert Raussendorf, University of British Columbia QEC11, University of Southern California December 5, 2011 Outline Motivation Topological codes, Example 1:

slide-1
SLIDE 1

Fault-tolerant quantum computation -

high thresholds in two dimensions

Robert Raussendorf,

University of British Columbia

QEC11, University of Southern California December 5, 2011

slide-2
SLIDE 2

Outline

  • Motivation
  • Topological codes, Example 1: Surface codes
  • Twists, color, subsystems
  • Topological codes, Example 2: Subsystem color codes
  • Remarks
slide-3
SLIDE 3
slide-4
SLIDE 4

Introduction

Optical Lattice Greiner Lab Nist racetrack ion trap Monroe ion-photon link Superconducting qubit (Delft)

local global architecture

slide-5
SLIDE 5

Fault-tolerant quantum computation

  • Fault-tolerance is the art of maintaining the quantum speedup

in the presence of decoherence. Fault-tolerance theorem∗: If for a universal quantum computer the noise per elementary operation is below a constant non- zero error threshold ǫ then arbitrarily long quantum computa- tions can be performed efficiently with arbitrary accuracy. Remaining questions:

  • What is the value of the error threshold?
  • What is the operational cost of fault-tolerance?

*: Aharonov & Ben-Or (1996), Kitaev (1997), Knill & Laflamme & Zurek (1998), Aliferis & Gottesman & Preskill (2005)

slide-6
SLIDE 6

Our setting

superconducting฀ qubits quantum฀dots

  • ptical฀lattices

segmented฀ ion฀traps

  • 2D, nearest-neighbor translation-invariant interaction.
  • High fault-tolerance threshold
  • Moderate overhead scaling
slide-7
SLIDE 7

Part I:

Fault-tolerant quantum computation with the surface code

slide-8
SLIDE 8

Main ingredients

  • 1. Fault-tolerance from surface codes [Kitaev 97]:

Translation-invariant and short-range interaction.

  • 2. Topological quantum gates via time-dependent bound-

ary conditions.

holes in the code plane

macroscopic scale a log(# gates)

a

microscopic scale 2

qubit lattice, spacing a

microscopic view macroscopic view

slide-9
SLIDE 9

1.1 The surface code

harmless error syndrome at endpoint harmful error ZZ Z X X X X Z Z Z Z Z Z Z Z Z X X X X X X X X X Z = = plaquette stabilizer site stabilizer One qubit located on every edge Z X As Bp

|ψ = As|ψ = Bp|ψ, ∀|ψ ∈ HC, ∀s, p. (1)

  • Surface codes are stabilizer codes associated with 2D lattices.
  • Only the homology class of an error chain matters.
  • A. Kitaev,quant-ph/9707021 (1997).
slide-10
SLIDE 10

1.2 Topological error correction

  • Fault-tolerant data storage with planar code described by

Random plaquette Z2-gauge model (RPGM) [1].

[1] Dennis et al., quant-ph/0110143 (2001).

slide-11
SLIDE 11

1.4 How to encode qubits

  • Storage capacity of the code depends upon the topology of

the code surface.

slide-12
SLIDE 12

1.5 Surface code on plane with holes

site stabilizer not enforced XX X X plaquette stabilizer not enforced Z Z Z Z dual hole primal hole

  • There are two types of holes: primal and dual.
  • A pair of same-type holes constitutes a qubit.
slide-13
SLIDE 13

1.5 Surface code on plane with holes

Surface code with boundary:

dual hole dual hole rough฀boundary

Zd Xd

primal ฀hole primal ฀hole smooth฀boundary

Xp

p

Z p

primal qubit dual qubit

  • X-chain cannot end in primal hole, can end in dual hole.
  • Z-chain can end in primal hole, cannot end in dual hole.
slide-14
SLIDE 14

1.6 Encoded quantum gates

Now consider worldlines of holes.

slide-15
SLIDE 15

1.6 Encoded quantum gates

Topological quantum gates are encoded in the way worldlines of primal and dual holes are braided.

slide-16
SLIDE 16

1.6 A CNOT-gate

Xt X X

c c

dual hole dual hole rough฀boundary

Zd Xd

primal ฀hole primal ฀hole smooth฀boundary

Xp

p

Z p

primal qubit dual qubit

  • Propagation relation: Xc −

→ Xc ⊗ Xt.

  • Remaining prop rel Zc → Zc, Xt → Xt, Zt → Zc ⊗ Zt for

CNOT derived analogously.

slide-17
SLIDE 17

1.6 Topological quantum gates

slide-18
SLIDE 18

1.7 Universal gate set

  • Need one non-Clifford element:

fault-tolerant preparation of |A := X+Y

√ 2 |A.

Singular Qubit

  • FT prep. of |A provided through realization of magic state

distillation∗.

*: S. Bravyi and A. Kitaev, Phys. Rev. A 71, 022316 (2005).

slide-19
SLIDE 19

1.8 Error threshold

Perfect error correction: 11% With measurement error: 2.9% Error per gate: 0.75%[1] − 1.05%[2]

  • Comparison:

Highest known threshold [no geometric constraint] is 3% [3].

[1] R. Raussendorf and J. Harrington, Phys. Rev. Lett. 98, 190504 (2007). [2] D.S. Wang et al., Phys. Rev. A 83 020302(R) (2011). [3] E. Knill, Nature (London) 434, 39 (2005).

slide-20
SLIDE 20

1.8 Error Model

Error sources:

  • 1. |+-preparation:

Perfect preparation followed by 1-qubit partially depolarizing noise with probability pP.

  • 2. Λ(Z)-gates (space-like edges of L): Perfect gates followed by

2-qubit partially depolarizing noise with probability p2.

  • 3. Hadamard-gates (time-like edges of L): Perfect gates followed

by 1-qubit partially depolarizing noise with probability p1.

  • 4. Measurement: Perfect measurement preceeded by 1-qubit partially

depolarizing noise with probability pM .

  • No qubit is ever idle. (Additional memory error - same threshold)
  • For threshold set p1 = p2 = pP = pM =: p.
slide-21
SLIDE 21

Overhead

Operational overhead at 1/3 of error threshold.

slide-22
SLIDE 22

Overhead

Overhead in Knill’s scheme.

slide-23
SLIDE 23

Part II: Twists, Subsystems, and Color

slide-24
SLIDE 24

Twists

First consider twists in the surface code:

surface code

X X X X

Z Z Z Z

H H H H H H

X

45

  • Z

Z Z Z

X X X

  • H. Bombin, Phys. Rev. Lett. 105, 030403 (2010).
slide-25
SLIDE 25

Twists

X

Z Z Z Z

X X X X

Z Z Z Z

X X X

New lattice:

  • ld sites = white faces
  • ld faces= black faces
slide-26
SLIDE 26

Twists

X

Z Z Z Z

X X X

light cycle: measures plaquette stabilizers (included ``magnetic charge’’) dark cycle: measures site stabilizers (included ``electric charge’’)

slide-27
SLIDE 27

Twists

light cycle: measures plaquette stabilizers (included ``magnetic charge’’) dark cycle: measures site stabilizers (included ``electric charge’’)

slide-28
SLIDE 28

Twists

was dark face

slide-29
SLIDE 29

Twists

was dark face

slide-30
SLIDE 30

Twists

was dark face coloring inconsistent! twist

slide-31
SLIDE 31

Twists

twist Strings around twist do not close!

String changes color light <-> dark when passing the cut

slide-32
SLIDE 32

Twists

double loop closes

slide-33
SLIDE 33

Subsystem codes

Example: Bacon-Shor code

Z Z X Z Z Z Z Z Z Z Z X X X X X X X X X

Stabilizer generators: + translates + translates Encoded Pauli operators:

Z X Z Z Z X X X X

Encoded Z Encoded X

Z

slide-34
SLIDE 34

Subsystem codes

Example: Bacon-Shor code

Z Z X Z Z Z Z Z Z Z Z X X X X X X X X X

Stabilizer generators: + translates + translates Measured operators:

X Z Z X

+ translates + translates Encoded Pauli operators:

Z X Z Z Z X X X X

Encoded Z Encoded X

Z

* Do not mutually commute * Commute with stabilizer and encoded Pauli operators

slide-35
SLIDE 35

Subsystem codes

Z Z X Z Z Z Z Z Z Z Z X X X X X X X X X

Generated by the meaeasured operators

X Z Z X Z X Z Z Z X X X X Z

n-qubit Pauli group gauge group

(

)

stabilizer centralizer contains encoded Paulis

slide-36
SLIDE 36

Subsystem codes

Generated by the meaeasured operators

(

)

X Z Z X Z X Z Z Z X X X X Z

gauge group centralizer contains encoded Paulis

gauge qubits system qubits

acts here acts here

slide-37
SLIDE 37

Color codes

X Z Z Z Z Z Z X X X X X

  • Tri-valent graph
  • One qubit per site
slide-38
SLIDE 38

Color codes

X Z Z Z Z Z Z X X X X X

  • Two stabilizer generators per face, one X + one Z-type
  • Topology: #encoded qubits = 4 × #handles.
slide-39
SLIDE 39

Color

  • Closed strings represent encoded Pauli operators.
  • 3 strings can end in a comon vertex.
  • Must all have different color.

Image taken from: H. Bombin, MA. Delgado, PRL 97 (2006).

slide-40
SLIDE 40

Part III: Topological color subsystem codes

  • H. Bombin, New J. Phys. 13, 043005 (2011).
slide-41
SLIDE 41

TCSCs

lattice Λ

slide-42
SLIDE 42

TCSCs

Q1: What is the gauge group G?

decorated lattice Λ

Z1Z2, Y2X3 ∈ G.

slide-43
SLIDE 43

TCSCs

Q2: What is the centralizer Z(G)?

  • Elements of Z(G) are consistent shadings of Λ.
slide-44
SLIDE 44

TCSCs

Q3: What is the stabilizer S?

Z-type stabilizer X/Y-type stabilizer

  • Two stabilizer generators per (normal) face.
slide-45
SLIDE 45

TCSCs

Q4: What is a twist?

Z-type stabilizer X/Y-type stabilizer

?

Yes No

  • A twist is a face with odd number of edges.
  • Only one stabilizer generator per twist face.
slide-46
SLIDE 46

TCSCs

twist

  • A twist changes the color of an encirling string operator.
slide-47
SLIDE 47

TCSCs

X Z Encoding of a qubit in 4 twists

  • 1

Entangling gate: qubit 1 qubit 2

slide-48
SLIDE 48

TCSCs

X Z

  • 1

Entangling gate: qubit 1 qubit 2

What happens to Z1:

= =

Z1 − → Z1 ⊗ X2

  • Can implement entire Clifford group by braiding twists.
slide-49
SLIDE 49

More exotic codes out there

  • Tuarev-Viro codes: Universality without state distillation.
  • R. K¨
  • nig, G. Kuperberg and B.W. Reichardt, arXiv:1002.2816 (2010).
slide-50
SLIDE 50

Summary

Topological quantum codes are highly suitable for fault- tolerant quantum computation in 2D qubit lattices with nearest-neighbor interaction. Numbers:

  • Fault-tolerance threshold of up to 1% so far.
  • Moderate overhead scaling.

Suitable systems for realization:

  • Cold

atoms in

  • ptical

lattices, segmented ion traps, Josephson junction arrays, ...