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)