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How to show magic with random unitaries and RMPS

Liyuan Chen1

​

1Applied Mathematics,

School of Engineering and Applied Science

Harvard University

​

March 25, 2023

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Outline

  • Introduction
  • Random matrices
  • Randomness in recent physics (QI/HEP/CMP)
  • Quantum resource (Entanglement, coherence, Magic)
  • Magic of RMPS

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Outline

  • Introduction
  • Random matrices
  • Randomness in recent physics (QI/HEP/CMP)
  • Quantum resource (Entanglement, coherence, Magic)
  • Magic of RMPS

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  • About myself
  • Liyuan Chen (陈力源)

Wikipedia: Arthur Jaffe

BS, Mechanical Engineering

PhD Candidate,

Applied Math

Advisor:

Arthur Jaffe

Research Interests: All fields in Physics, but especially QI+HEP or QI+CMT

Recently: random unitaries + BH, operator algebras + AdS/CFT, Magic + Axiomatic QFT…

Basketball, Workout, Dota, War3, board game….

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Outline

  • Introduction
  • Random matrices
  • Randomness in recent physics (QI/HEP/CMP)
  • Quantum resource (Entanglement, coherence, Magic)
  • Magic of RMPS

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  • Random matrices
  • Gaussian Orthogonal Ensemble (GOE)
  • Wigner semicircle

Wikipedia: Eugene Wigner

Eugene Wigner

Nuclei of Heavy Atoms(1955)

Symmetric Matrix + Elements i.i.d. from Gaussian

Ariel Amir:10.1017/9781108855259

Randomness 🡪 Not so random?

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Outline

  • Introduction
  • Random matrices
  • Randomness in recent physics (QI/HEP/CMP)
  • Quantum resource (Entanglement, coherence, Magic)
  • Magic of RMPS

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  • Randomness in recent physics

Randomness

SYK model

Spin glass

Anderson localization

…

Random unitary Circuit

QNN

Classical Shadow

…

Black hole

Quantum Chaos

Wormhole

…

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  • SYK model + Wormhole
  • Sachdev-Ye-Kitaev (SYK) model

 

 

 

  • Traversable wormhole dynamics on a quantum processor

D. Jafferis et.al., Nature, 2022

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  • Black hole information loss paradox
  • Hawking radiation

star

blackhole

collapse

(J,Q,M)

radiate

thermal

 

unitary

 

(J,Q,M)

unitary

Mixed state

?

  • Possible solutions
  1. Still unitary (with some subtleties)
  2. No unitarity (QM is wrong)
  3. Radiation stops at some small M (and enters another universe)
  • Burning a coal

C(0)

C(t)

+

R(t)

 

+

 

R(T)

 

D. Page, 9306083, M. Hotta, A. Sugita, 1505.05870

Page curve

ETH

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  • Black hole information loss paradox (cont.)
  • Hayden-Preskill thought experiment

 

  • Mixed field Ising model

 

  • SYK model

 

  • Random unitary circuit
  • Random Tensor Network

Information scrambling

Holographic duality

P. Hayden, J. Preskill, JHEP(2007)

A. Nahum et.al., PRX(2018)

P. Hayden et.al., JHEP(2016)

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  • In quantum info. and quantum comp.
  • Barren Plateau (Quantum Machine Learning)
  • Classical shadow Tomography

Random unitary circuit

 

 

 

Google, NC (2018)

Wikipedia: Mesa

 

 

 

  • Quantum Neural Net +RMPS
  • Complexity
  • Entanglement
  • Unitary t-design
  • Op spreading

……

Z. Liu , PRL (2022)

Z. Liu , PRL (2022)

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  • Why Randomness is not so random?
  • Random state

 

 

Wikipedia: Bloch Sphere

 

 

 

(By symmetry)

 

 

 

 

 

 

 

 

 

(By symmetry)

Averaging over random states

🡪 permutations

Random 🡪 simpler

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  • Random state design (cont.)
  • Random state design

 

Wikipedia: Bloch Sphere

 

 

 

 

State 2-desgin

 

  • Random unitary t-design

 

t-th moment

q – local dimension

Clifford unitaries – Unitary 3-design

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Outline

  • Introduction
  • Random matrices
  • Randomness in recent physics (QI/HEP/CMP)
  • Quantum resource (Entanglement, coherence, Magic)
  • Magic of RMPS

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  • Quantum Resource
  • Resource

S. Bravyi, A. Kitaev, PRA (2005)

Gold

  • Magic state distillation
  • Quantum resource (for quantum comp.)
  1. Entanglement
  2. Magic
  3. Sensitivity
  4. Coherence
  5. Circuit complexity
  6. …

 

Mixed states

 

Universal quantum computation

Google, Sycamore

Quantum Advantage

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  • Quantum Advantage
  • Gottesman-Knill Theorem
  • Single-Qubit Stabilizer States

 

Stabilizer states (mixed)

Convex combination of them

  • Two-Qubit Clifford group

 

 

Hadamard gate

 

S gate

 

 

 

 

 

In a subspace

 

T gate

 

Universal quantum computation

 

Magic - Resource

D. Gottesman, 9807006

Wikipedia: Clifford gates

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Outline

  • Introduction
  • Random matrices
  • Randomness in recent physics (QI/HEP/CMP)
  • Quantum resource (Entanglement, coherence, Magic)
  • Magic of RMPS

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  • Random Matrix Product States

 

  • A typical quantum state

 

 

  • Graphical representation

I. Affleck et.al., PRL(1987)

  • Random Matrix Product State (RMPS)

 

L. Chen et.al., 2211.10350

  1. Applications in QNN
  2. Simplest Random TN
  3. Shallow random unitary circuit
  4. Approximate random states
  5. GS of disordered parent H
  6. Typical in trivial phase of matter

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  • Magic of Random Matrix Product States

 

  • Main result

L. Chen et.al., 2211.10350

 

 

 

 

 

 

 

 

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  • Summary and Outlook
  • Summary
  • Outlook

L. Chen et.al., 2211.10350

A simple RMPS can provide a great amount of quantum resource (e.g. QNN).

​

Cooling method 🡪 obtain it from trivial phase of matter.

  • MPS is classically simulable, but it’s magic is large – Beyond GK theorem paradigm?
  • Could magic be applied to other RTN? – QI properties of BH
  • Magic -- other exotic quantum states?
  • Magic of QFT? (Magic can bound circuit complexity, and the latter is studies for QFT)
  • …

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Thanks!

​

Liyuan Chen (陈力源)

PhD candidate in Applied Mathematics

School of Engineering and Applied Science, Harvard University

BS, Zhejiang University

​

Email: liyuanchen@fas.harvard.edu / chenlyfriedrich@gmail.com

​

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