QUANTUM SIMULATION AND COMPUTING · QUANTUM SIMULATION Simulate interesting physical situations Trotzky, Chen, Flesch, McCulloch, Schollwöck, Eisert, Bloch, Nature Physics 8, 325

QUANTUM SIMULATION AND COMPUTINGA NEW WAY OF COMPUTING BEYOND SUPERCOMPUTERS

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MOORE’S LAW

▸ Gordon Moore (Intel, 1965): Number of transistors in integrated circuits doubles approximately every two years

Zuse Z3 (1941)

▸ Gordon Moore (Intel, 1965): Number of transistors in integrated circuits doubles approximately every two years

MOORE’S LAW

ENIAC, EDVAC, ORDVAC, BRLESC-I (1945-62)

Zuse Z3 (1941)

▸ Gordon Moore (Intel, 1965): Number of transistors in integrated circuits doubles approximately every two years

Zuse Z3 (1941)

Tran

sist

or c

ount

MOORE’S LAW

ENIAC, EDVAC, ORDVAC, BRLESC-I (1945-62)

Zuse Z3 (1941)

▸ Gordon Moore (Intel, 1965): Number of transistors in integrated circuits doubles approximately every two years

Zuse Z3 (1941)

Tran

sist

or c

ount

▸ Minimum feature size down to that of single atoms

MOORE’S LAW

▸ Gordon Moore (Intel, 1965): Number of transistors in integrated circuits doubles approximately every two years

Tran

sist

or c

ount

▸ Minimum feature size down to that of single atoms

▸ Different physical laws matter

MOORE’S LAW

QUANTUM MECHANICS

QUANTUM MECHANICS

▸ Quantum mechanics is a physical theory

QUANTUM MECHANICS

▸ Quantum mechanics is a physical theory

▸ Theory of atoms, molecules, and light quanta

QUANTUM MECHANICS

▸ Developed 1925-1928

QUANTUM MECHANICS

▸ Developed 1925-1928

▸ Basis of semi-conductors, materials science, lasers

QUANTUM MECHANICS

▸ Developed 1925-1928

▸ Basis of semi-conductors, materials science, lasers

▸ Fine structure constant: 7,297.352.566.4(17) x 10-3

QUANTUM MECHANICS

▸ Developed 1925-1928

▸ Basis of semi-conductors, materials science, lasers

▸ Radically different from classical mechanics▸ Fine structure constant: 7,297.352.566.4(17) x 10-3

RANDOMNESS

RANDOMNESS IN QUANTUM MECHANICS

▸ Measurement outcomes are random

Präparation

0 1 1 0 1 0

RANDOMNESS IN QUANTUM MECHANICS

▸ Measurement outcomes are random

Präparation

0 1 1 0 1 0

▸ We are used to randomness…

▸ … but this has an explanation

RANDOMNESS IN QUANTUM MECHANICS

▸ Measurement outcomes are random

Präparation

RANDOMNESS IN QUANTUM MECHANICS

▸ The randomness of quantum mechanics is absolute

Präparation

RANDOMNESS IN QUANTUM MECHANICS

▸ The randomness of quantum mechanics is absolute

Präparation

▸ Bell inequality violated under assumption of local hidden variables

P (a, b|A,B) =

�d�p(�)⇥A(a,�)⇥B(b,�)

UNCERTAINTY

UNCERTAINTY PRINCIPLE

UNCERTAINTY PRINCIPLE

▸ No measurement without disturbance

SUPERPOSITION

SUPERPOSITION PRINCIPLE

0

1

|1i

|0i

SUPERPOSITION PRINCIPLE

|0i+ |1i

|1i

|0i

SUPERPOSITION PRINCIPLE

|0i+ |1i

▸ Systems can be in “many states at once”

|1i

|0i

SUPERPOSITION PRINCIPLE

|0i+ |1i

▸ State space over complex vector space

▸ For spins

{⇢ : ⇢ � 0, tr(⇢) = 1} HH = C⌦n

2n

QUANTUM TECHNOLOGIES

QUANTUM TECHNOLOGIES

▸ Make use of quantum effects on the single quantum system level to think of new technologies in communication, sensing, computation, simulation

SECURE COMMUNICATION

▸ Classical key distribution

010101 010101

Need to share key

SECURE COMMUNICATION

▸ Classical key distribution

010101 010101

Need to share key

▸ Classical key distribution

010101 010101

Need to share key

SECURE COMMUNICATION

010101 010101

▸ Quantum key distribution for secure communication

SECURE COMMUNICATION

010101 010101No information gain without disturbance

▸ Quantum key distribution for secure communication

SECURE COMMUNICATION

010101 010101No information gain without disturbance

▸ Quantum key distribution for secure communication

SECURE COMMUNICATION

+ + ⇥ + ⇥ ⇥ ⇥ +" ! & " & % % !+ ⇥ ⇥ ⇥ + ⇥ + +" % & % ! % !!

0 1 1 0 1 0 0 1

0 1 0 1

Alice’s bitAlice’s basis

Bob’s basis

Bob’s result

Public partKey

State

! Basis +

⇥! Basis

"= |1i, != |0i

&= |0i+ |1i, %= |0i � |1i

010101 010101No information gain without disturbance

▸ Quantum key distribution for secure communication

▸ Security can be proven

SECURE COMMUNICATION

QUANTUM COMPUTERS

QUANTUM COMPUTING

▸ Computational devices with single quantum systems

QUANTUM COMPUTING

▸ Computational devices with single quantum systems

‣ E.g., 01010011 (bits) replaced by (qubits) ↵|0, 1, 0, 1, 0, 0, 1, 1i + �|1, 1, 0, 0, 1, 1, 1, 0i + �|0, 0, 1, 0, 0, 1, 1, 1i + . . .

QUANTUM COMPUTING

▸ Could solve some problems supercomputers cannot

BQP

Classical probabilistic algorithms

Poly time quantum algorithms

BPP

QUANTUM ALGORITHMS

▸ E.g., factoring of large products of prime numbers

‣ A factor of a large number can be found if the period of

can be identified

N p

f(x) = a

x

modN

‣ Periods can be found using the quantum Fourier transformn�1X

i=0

xi|iin�1X

i=0

yi|ii yk =1pn

n�1X

j=0

xje2⇡ijk/n7! with

Shor, SIAM J Comp 26, 148 (1997)

QUANTUM ALGORITHMS

▸ E.g., factoring of large products of prime numbers

‣ A factor of a large number can be found if the period of

can be identified

N p

f(x) = a

x

modN

‣ Periods can be found using the quantum Fourier transformn�1X

i=0

xi|iin�1X

i=0

yi|ii yk =1pn

n�1X

j=0

xje2⇡ijk/n7! with

‣ Solves NP problem in poly time: Runtime

Shor, SIAM J Comp 26, 148 (1997)

‣ Best known classical algorithm

‣ Generalised to hidden subgroup problem

▸ E.g., factoring of large products of prime numbers

QUANTUM ALGORITHMS

▸ E.g., factoring of large products of prime numbers

▸ Solving linear systemsHarrow, Hassidim, Lloyd, Phys Rev Lett 15, 150502 (2009)

Shor, SIAM J Comp 26, 148 (1997)

▸ E.g., factoring of large products of prime numbers

▸ Solving linear systems

▸ Spectral analysis

▸ Semi-definite programming

QUANTUM ALGORITHMS

Harrow, Hassidim, Lloyd, Phys Rev Lett 15, 150502 (2009)

Steffens, Rebenstrost, Marvian, Eisert, Lloyd, New J Phys 19, 033005 (2017)

Brandão, Kalev, Li, Lin, Svore, Wu, arXiv:1710.02581

Shor, SIAM J Comp 26, 148 (1997)

▸ Can tolerate small errors in all steps (at high cost)

FAULT TOLERANT QUANTUM COMPUTING

E.g., Litinski, Kesselring, Eisert, von Oppen, arXiv:1704.01589

▸ The race for building quantum computers

FAULT TOLERANT QUANTUM COMPUTING

▸ Not there, but with 50 superconducting qubits taking shape

(IBM)

(Google)

(Rigetti)(D-wave)

QUANTUM SIMULATORS

QUANTUM SIMULATION

▸ Quantum simulators: Not all strongly correlated quantum systems/materials can be classically simulated

QUANTUM SIMULATION

▸ Quantum simulators: Not all strongly correlated quantum systems/materials can be classically simulated

▸ Idea: Simulate quantum systems with quantum systems

Richard Feynman

QUANTUM SIMULATION

Cold atoms in optical lattices

▸ Quantum simulators: Not all strongly correlated quantum systems/materials can be classically simulated

▸ Idea: Simulate quantum systems with quantum systems

QUANTUM SIMULATION

▸ Simulate interesting physical situations

Trotzky, Chen, Flesch, McCulloch, Schollwöck, Eisert, Bloch, Nature Physics 8, 325 (2012)

Gring, Kuhnert, Langen, Kitagawa,Rauer, Schreitl, Mazets, Smith, Demler, Schmiedmayer, ,Science 337, 1318 (2012)

Kaufman, Tai, Lukin, Rispoli, Schittko, Preiss, Greiner, Science 353, 794 (2016)

Choi, Hild, Zeiher, Schauß,Rubio-Abadal, Yefsah, Khemani, Huse, Gross, Science 352, 1547 (2016)

Equilibration Pre-thermalization Thermalization

Many-body localization

QUANTUM SIMULATION

▸ Some properties can be obtained beyond supercomputers

QUANTUM SIMULATION

▸ Some properties can be obtained beyond supercomputers

‣ Imbalance as function of time for under Bose-Hubbard Hamiltonian

Trotzky, Chen, Flesch, McCulloch, Schollwoeck, Eisert, Bloch, Nature Phys 8, 325 (2012)

nodd

| (0)i = |0, 1, . . . , 0, 1i

QUANTUM SIMULATION

▸ Some properties can be obtained beyond supercomputers

‣ Imbalance as function of time for under Bose-Hubbard Hamiltonian

Trotzky, Chen, Flesch, McCulloch, Schollwoeck, Eisert, Bloch, Nature Phys 8, 325 (2012)

nodd

Best available classical matrix-product state simulation, bond dimension 5000

| (0)i = |0, 1, . . . , 0, 1i

QUANTUM SIMULATION

▸ Some properties can be obtained beyond supercomputers

‣ Imbalance as function of time for under Bose-Hubbard Hamiltonian

Trotzky, Chen, Flesch, McCulloch, Schollwoeck, Eisert, Bloch, Nature Phys 8, 325 (2012)

nodd

Best available classical matrix-product state simulation, bond dimension 5000

‣ The approximation of dynamics with matrix-product states requires exponential resources in time

| (0)i = |0, 1, . . . , 0, 1i

QUANTUM SIMULATION

▸ Some properties can be obtained beyond supercomputers

Random Periodic Translationally invariant

Bermejo-Vega, Hangleiter, Schwarz, Raussendorf, Eisert, Phys Rev X 8, 021010 (2018)

▸ Simple Ising nearest-neighbor architectures

QUANTUM SIMULATION

▸ Some properties can be obtained beyond supercomputers

Random Periodic Translationally invariant

Relate to logical circuits

Bermejo-Vega, Hangleiter, Schwarz, Raussendorf, Eisert, Phys Rev X 8, 021010 (2018)

▸ Simple Ising nearest-neighbor architectures

BPP

BQP

U U 0Additive error✏

A

x Stockmeyer

Multiplicative

error

sU 0(x)

1/poly(n)

Use complexity theory tools

QUANTUM SIMULATION

▸ Some properties can be obtained beyond supercomputers

Random Periodic Translationally invariant

Relate to logical circuits

Bermejo-Vega, Hangleiter, Schwarz, Raussendorf, Eisert, Phys Rev X 8, 021010 (2018)

▸ Simple Ising nearest-neighbor architectures

BPP

BQP

U U 0Additive error✏

A

x Stockmeyer

Multiplicative

error

sU 0(x)

1/poly(n)

Use complexity theory tools

‣ Present technology (basic) quantum simulators already outperform supercomputers on some tasks (and can be verified)

GETTING GOING…

FLAGSHIP PROGRAM FOR QUANTUM TECHNOLOGIES

▸ 1G€ Euros-Flagship for quantum technologies

OUTLOOK

▸ “Quantum computing is exciting even if you restrict yourself to saying things that are true.”

OUTLOOK

▸ “Quantum computing is exciting even if you restrict yourself to saying things that are true.”

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