Quantum information meets quantum phases:From entanglement to topological quantum computation

Bowen Shi

The Ohio State University

April 06 2018

Quantum information Seminar Series

Outline of the talk

• Qubits and entanglement:• 1 qubit, 2 qubits and many qubits

• hide quantum information among entanglement

• long range entanglement

• Topological orders:• the ground states on a torus and open surfaces

• excitations: anyons (fusion and braiding)

• long range entangled and good for store quantum information

• Topological quantum computation:• Ising anyons (non-universal)

• Fibonacci anyons (universal quantum computation)

• Current progress of quantum computing (Majorana)

A qubit vs a classical bit:

quantum superposition(coherence)

pure state

pure statedensity matrix

𝑆2

A qubit vs a classical bit:

quantum superposition(coherence)

pure state

pure statedensity matrix

classical superposition(noncoherence)

𝑆2

A qubit vs a classical bit:

quantum superposition(coherence)

pure state

pure statedensity matrix

mixed state

classical superposition(noncoherence)

𝐷3

2 qubits and quantum entanglement:

A B

• factorizable if

• entangled if

𝑆2 𝑆2𝑆7

subsystems

⊗

Many qubits:

𝑨𝟏

• want states to look the same on relatively large subsystems

𝑨𝟐 𝑨𝟑 𝑨𝒏… …

?

?

Many qubits:

𝑨𝟏

• want states to look the same on relatively large subsystems

𝑨𝟐 𝑨𝟑 𝑨𝒏… …

?

?𝐴

𝐵lose a qubit 𝐴

coherence is lost (bad)

Many qubits:

𝑨𝟏

• want states to look the same on relatively large subsystems

𝑨𝟐 𝑨𝟑 𝑨𝒏… …

highly entangled states

Many qubits:

𝑨𝟏

• want states to look the same on relatively large subsystems

𝑨𝟐 𝑨𝟑 𝑨𝒏… …

highly entangled states

The information (𝑐0 and 𝑐1) can be recovered on any 3 sites. (Good!)

5 qubits quantum error correction code.

Why we would like to hide information in entanglement?

Many qubits:

• be able to correct errors (losing qubits)

5 qubits quantum error correction code

1 logical qubit in 5 physical qubits

𝐴

Why we would like to hide information in entanglement?

Many qubits:

• be able to correct errors (losing qubits)

• immune to decoherence

5 qubits quantum error correction code

1 logical qubit in 5 physical qubits

photon(environment)

Why we would like to hide information in entanglement?

Many qubits:

• be able to correct errors (losing qubits)

• immune to decoherence

5 qubits quantum error correction code

photon(environment)

1 logical qubit in 5 physical qubits

a quantum book

Long range entanglement is needed (for quantum books)

Quantum circuit: (generate entanglement from a product state)

• 4 qubits example:

𝑢12 𝑢34

𝑢23

Long range entanglement is needed (for quantum books)

Quantum circuit: (generate entanglement from a product state)

• General case: (n qubits)

𝑈2

𝑈1

Long range entanglement is needed (for quantum books)

Short range entangled states:

related to a product state by a depth 𝑙 quantum circuit

Long range entangled states:

depth 𝑙 grows with system size

Example: 1D SPT ground state

Example: topological order

Theorem: To make information unaccessible for all subsystems smaller than length scale 𝐿, need depth 𝑙 > 𝐿. (need long range entanglement to hide information among entanglement)

Topological orders in 2D (long range entangled)

• Ground state degeneracy (topological dependent)

14

16

# come from toric code

• Ground state degeneracy (topological dependent)

14

16

topologically a disk(see no difference)

Topological orders in 2D (long range entangled)

# come from toric code

• Ground state degeneracy (topological dependent)

14

16

topologically an annulus(see some difference)

𝑆7

# come from toric code

Topological orders in 2D (long range entangled)

• Ground state degeneracy (topological dependent)

14

16

topologically 𝑇2\𝐷2

(see everything)

𝑆7

Topological orders in 2D (long range entangled)

# come from toric code

𝑆7

On open surface

• Ground state degeneracy (topological dependent)

With two types of boundaries

2 ground states (for the toric code model)

a topological qubit experimentally easier than a torus

?

have not been achievedexperimentally

𝑎

ത𝑎

Anyons – low energy excitations of a topological order

General properties:• superselection sectors:

• braiding:

• fusion:

nontrivial braiding if

ത𝑎 is the antiparticle of 𝑎 for Abelian anyon models, always a unique 𝑐

Ising anyon model:• Superselection sectors:

• Fusion rules:

Degeneracy from non-Abelian anyons:

• 4 𝜎 anyons with total charge 1:

• claim:

Non-Abelian Anyons (carry degeneracy)

(𝜎 is non-Abelian)

is a topological qubit

Fibonacci anyon model:• Superselection sectors:

• Fusion rules:

Degeneracy from non-Abelian anyons:

• 4 𝑎 anyons with total charge 1:

• claim:

Non-Abelian Anyons (carry degeneracy)

(𝑎 is non-Abelian)

is a topological qubit

Summary:

• Both Ising anyon and Fibonacci anyon could store quantum information immune to decoherence.

• How about quantum computation?

Non-Abelian Anyons (carry degeneracy)

6 qubits

to store information is not enough• initialize• quantum gates• readout

Quantum computation:

Initialize + apply quantum gates + readout

quantum gates (unitary)

Universal quantum gate set:• could realize all unitary transformations.

(powerful)• give you a universal quantum computer

Theorem: any generic 2-qubit gate give you a universal quantum gate set. (assume it could apply to any 2 nearby qubits)

• apply quantum gates before decoherence

• readout in a specific basis

Topological quantum computation:

Initialize + apply quantum gates + readout

quantum gates (unitary)are braiding operations

a 1 qubit gate a 2 qubits gate

Topological quantum computation:

Initialize + apply quantum gates + readout

quantum gates (unitary)are braiding operations

a 1 qubit gate a 2 qubits gate

Quantum gates for Ising anyon:

Single qubit gates:

Topological quantum computation:

Initialize + apply quantum gates + readout

quantum gates (unitary)are braiding operations

a 1 qubit gate a 2 qubits gate

Quantum gates for Ising anyon:

2-qubit gates:

Single qubit gates:

Topological quantum computation:

Initialize + apply quantum gates + readout

quantum gates (unitary)are braiding operations

a 1 qubit gate a 2 qubits gate

Quantum gates for Ising anyon:

not universal gate set

2-qubit gates:

Single qubit gates:

Topological quantum computation:

Initialize + apply quantum gates + readout

quantum gates (unitary)are braiding operations

a 1 qubit gate a 2 qubits gate

Quantum gates for Fibonacci anyon:

Single qubit gates:

Theorem: any SU(2) elements can be constructed with arbitrary accuracy by finite # of braiding of 4 Fibonacci anyons.

2-qubit gates:

Theorem: any SU(13) elements can be constructed with arbitrary accuracy by finite # of braiding of 8 Fibonacci anyons.

universal

Quantum computation current progress:

• D-Wave, 128 qubits, May 2011, quantum annealing

• IBM, 16 qubits, May 2017, universal

• Google, 72 qubit, March 2018, universal

• Microsoft, 1 Majorana qubit, April 1 2018

Quantum computation current progress:

• D-Wave, 128 qubits, May 2011, quantum annealing

• IBM, 16 qubits, May 2017, universal

• Google, 72 qubit, March 2018, universal

• Microsoft, 1 Majorana qubit, April 1 2018

not universal

• expected to get longer coherence time

• how to braiding?• make it universal?

limited # of error rate of each gate (coherence time is not an important issue here)

References:

5-qubit error correction code:

Pastawski, Yoshida, Harlow, Preskill 2015 (arxiv: 1503.06237)

Yoshida’s talk at PI: Decoding a black hole. http://pirsa.org/displayFlash.php?id=17040026.

The quantum book analogy:

John Preskill’s talk: Quantum is different. https://www.youtube.com/watch?v=31NswlprSKk.

Quantum Circuit, short range entanglement and long range entanglement:

Chen, Gu, Wen 2010 (arxiv: 1004.3835)

Haah 2014 (arxiv: 1407.2926)

Locally indistinguishable states of topological orders and information in subsystems:

Shi, Lu 2018 (arxiv: 1801.01519)

Quantum computation and universal quantum computation:

John Preskill’s lecture notes, Chapter 6: http://www.theory.caltech.edu/people/preskill/ph229/notes/chap6.pdf

Anyons and topological quantum computation:

John Preskill’s lecture notes, Chapter 9: http://www.theory.caltech.edu/~preskill/ph219/topological.pdf

About Ising anyon quantum gates: Fan, Garis 2010 (arxiv: 1003.1253)

Fibonacci anyon: Bonesteel, Hormozi, Zikos, Simon 2005 (arxiv: quant-ph/0505065)

References:

Current progress of quantum computation:

Wikipedia, List of quantum processors: https://en.wikipedia.org/wiki/List_of_quantum_processors#cite_note-Lant-1

IBM Q devices information: https://quantumexperience.ng.bluemix.net/qx/devices.

Majorana:

News April 1 2018: https://www.neowin.net/news/microsoft-quantum-computing-scientists-have-captured-a-majorana-quasiparticle

Quantized Majorana conductance: http://www.nature.com/articles/nature26142.

Theory paper about braiding of Majorana: Alicea, Oreg, Refael, Oppen, Fisher 2010 (arxiv: 1006.4395)