Qccc Seminar

7/31/2019 Qccc Seminar

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Classical simulations and

quantum Fourier transforms

QCCC Seminar, 19th Dec 2011Juan Bermejo Vega

MPQ Theory Division

Quantum VS Classical ?


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Introduction Classical Simulations Quantum Fourier transforms Hidden Subgroup Problem Simulation of QFTs Credits

Main Questions and Motivation

What do we know about Q-algorithms?

Faster (exponentially) for somecomputational problems.

E.g. Factorisation, computing discretelogarithms, solving Pellsequation, Q-simulation.

More examples: Q Algorithm Zoo

24 algorithms show exponentialquantum speed-ups!

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C S Q S S Q C


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Fundamental Q-uestions

Practical

How to design good quantum algorithms?

1 Important physical effects?

Entanglement? Interence? Simmetries?2 Structure?

Killer-apps (beyond simulation)?

Fundamental

What are the limits of Q Computation?

BPP=BQP?

?

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I d i Cl i l Si l i Q F i f Hidd S b P bl Si l i f QFT C di


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Practical





Fundamental


BPP=BQP?

?

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I t d ti Cl i l Si l ti Q t F i t f Hidd S b P bl Si l ti f QFT C dit


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Practical





Fundamental


BPP=BQP?

?

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Practical





Fundamental


BPP=BQP?

?

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Practical





Fundamental


BPP=BQP?

?

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Practical





Fundamental


BPP=BQP?

?

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An approach

Classical simulation of quantum computers

|0 H . . .

|0 R2

F2n1

......

...|0 Rni

...|0 Rn2

|0 Rn1

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Q g p Q


Outline

1 Classical simulation of quantum computers

2 Quantum Fourier transforms

3 The Hidden Subgroup Problem

4 Simulation of non-abelian QFTs

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Outline





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Main ideas

Scheme of a quantum computation

1 Initialise state.

2 Apply quantum circuit (unitary evolution).

3 Measure, obtain outcome x with probability px

Px.

|0 H . . .

|0 R2

F2n1

......

...

|0 Rni

...

|0 Rn2

|0 Rn1

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Main ideas

Simulation = Sampling

If we can efficiently sample from the probability distribution Pxof the final outcomes with a classical algorithm then we canefficiently simulate the quantum computation.

|0 H . . .

|0 R2

F2n1

......

...|0 Rni

...

|0 Rn2

|0 Rn1

Measure

|x

with

px Px

From now simulation = efficient classical simulation

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Clasical simulation results

An (incomplete) list of illustrative results

Entanglement is necessary

|p = |1 . . . |k[Jozsa-Linden,03] Given a Q circuit

where in each step of the computationthe state is p-blocked with constant p,

such process can be efficiently simulated

classically.

Interference is necessary

|UZU|[Van den Nest,09] Given a s-sparse

unitaryU

(withs

constant) acting on aproduct state |, then a Z-measurement

can be simulated classically efficiently.

But not sufficient 9 / 32 Introduction Classical Simulations Quantum Fourier transforms Hidden Subgroup Problem Simulation of QFTs Credits


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Clasical simulation results

Entanglement Interference

Today: quantum Fourier transforms

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Outline





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The QFT in Quantum Algorithms

Standard view of Q Computation

Choose a quiet environment

|Create HUGE interference

U |But how do we design the unitary U?

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Quantum Fourier transform FN

FN :=1

N

1 1 1 1 . . . 1

1 := e2iN 2 3 . . . N1

1 2 4 6 . . . 2(N1)

1

3

6

9

. . .

3(N1)

......

.... . . . . .

...1 N1 2(N1) . . . . . . (N1)(N1)

Not sparse

Role in computation

1 Case N = 2. It is the Hadamard gate!2 Case N = 2n. Used in phase estimation

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Quantum phase estimation

Example of usage:

Given U| = e2i | find Returns an approximation of the phase up to n-digits

n.

1 QFT at the end

2 Measures a hiddenproperty.

|0 H

FN

|0 H

.

.

.

.

.

.

.

.

.|0 H

| /m U U2 U2n1

Applications

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Quantum phase estimation

Example of usage:

Given U| = e2i | find Returns an approximation of the phase up to n-digits n.

1 QFT at the end

2 Measures a hiddenproperty.

|0 H

FN

|0 H

.

.

.

.

.

.

.

.

.|0 H

| /m U U2 U2n1

Applications

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Q Q g

Applications

General purpose:Measure the energy of an eigenstate of H(Q-Metropolis) Create and sample Gibbs states

G :=1

Z exp(H)Prepare some injective PEPS in a quantum computer.

Computation:

Estimating xTMx where x fulfils Ax = b (exp speed-up).Shors algorithm (exp speed-up).Generalisation: the hidden subgroup problem

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g

Applications


G :=1


Computation:


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Applications


G :=1


Computation:


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Definition


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Choose a group

Multiplication rule

1 gh = g

2 gg1 = 1

Definition of subgroup H:A small group inside G

with the same operation

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Definition


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The Hidden Subgroup Problem

Promise problem.

f(g1) = f(g2)

g1

g2H

Goal: find H

Classically hard:

# queries scales as

|G

|Easier for a Q computer:

exponentially less

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Definition


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Promise problem.

f(g1) = f(g2)

g1

g2H

Goal: find H

Classically hard:

# queries scales as

|G


exponentially less

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Definition


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Promise problem.

f(g1) = f(g2)

g1

g2H

Goal: find H

Classically hard:

# queries scales as

|G


exponentially less

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Definition


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Promise problem.

f(g1) = f(g2)

g1

g2H

Goal: find H

Classically hard:

# queries scales as

|G


exponentially less

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Definition


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Relation to computational problems

Abelian groups

Efficient Q algorithm

Subsumes other algorithms

1 Shors: factoring, discrete log2 Exponential speed-ups!

Non-abelian groups

Role in computation

Lattice problems, Graphs

No general Q algorithm

Q advantage!

Abelian

8 + 7 = 7 + 8

N = p11 . . . pdd

Non-abelian

A

B

= B

A

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Definition


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Abelian groups

Efficient Q algorithm


1 Shors: factoring, discrete log2 Exponential speed-ups!

Non-abelian groups

Role in computation

Lattice problems, Graphs


Q advantage!

Abelian

8 + 7 = 7 + 8

N = p11 . . . pdd

Non-abelian

A

B

= B

A

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Definition

R l i i l bl


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Abelian groupsEfficient Q algorithm


1 Shors: factoring, discrete log

2 Exponential speed-ups!

Non-abelian groups

Role in computationLattice problems, Graphs


Q advantage!

Abelian

8 + 7 = 7 + 8

N = p11 . . . pdd

Non-abelian

A B= B A

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Quantum algorithms for the HSP

Th t h


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The quantum approach

1 Create a superpositionover a random coset gH

State picture

|gH = 1H

H

|gh

Density matrix

H = |gH gH|

2 Apply Fourier transform.

3 Measure.

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Th t h


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State picture

|gH = 1H

H

|gh

Density matrix

H = |gH gH|


3 Measure.

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Th t h


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State picture

|gH = 1H

H

|gh

Density matrix

H = |gH gH|


3 Measure.

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State picture

|gH = 1H

H

|gh

Density matrix

H = |gH gH|


3 Measure.

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Informal analysis: exploit simmetries


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H = |gH gH|


H = 1|G|H

R(h)


R(g) |h := hg1 R(gh) = R(g)R(h)

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H = |gH gH|


H = 1|G|H

R(h)


R(g) |h := hg1 R(gh) = R(g)R(h)

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Idea: diagonalise the symmetry


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H = 1|G|

H R(h)

4 A group representation can be block-diagonalised

5 The Fourier transform performs that change

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H = 1|G|

H R(h)

4 A group representation can be block-diagonalised

5 The Fourier transform performs that change

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That was not everything


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1 Algorithm showed does not return Hhidden...2 ... but some related information!

3 Abelian. Classical algorithms to post-process

4 Non-abelian.

For some easy instances. General problem hardHope for lattice problems!

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4 Non-abelian.


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4 Non-abelian.


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y g



4 Non-abelian.


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y g



4 Non-abelian.


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y g



4 Non-abelian.


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Transition

Connection with quantum algorithms


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q g

Quantum Fourier transforms help to solve the HSP

Natural question: can we simulate them classically?

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Abelian results

Abelian Fourier transforms FN


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|0 H . . .

|0 R2

F2n1

.

.

.

.

.

.

.

.

.|0 Rni

.

..

|0 Rn2

|0 Rn1

F2n =12n

abab

Rk =

1 00 e2i/2

k

[Aharonov, Landau, Makowsky, 06], [Yoran, Short, 06], [Browne, 06]

Constant number of QFTs

Inputs: product states (and some MPS, graph states)

Techniques: tensor contractions, adaptive operations.

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Abelian results

Abelian Fourier transforms FN


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|0 H . . .

|0 R2

F2n1

.

.

.

.

.

.

.

.

.|0 Rni

.

..

|0 Rn2

|0 Rn1

F2n =12n

abab

Rk =

1 00 e2i/2

k

[Aharonov, Landau, Makowsky, 06], [Yoran, Short, 06], [Browne, 06]

Constant number of QFTs

Inputs: product states (and some MPS, graph states)

Techniques: tensor contractions, adaptive operations.

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Non-Abelian Quantum Fourier Transforms

Our contribution


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We implemented and simulated quantum Fourier transformsfor a family of non-abelian groups.

We only simulate one shot.

Inputs: coset-states (aka HSP states).

Techniques: symmetries and group theory.

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Our contribution


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27 / 32 Introduction Classical Simulations Quantum Fourier transforms Hidden Subgroup Problem Simulation of QFTs Credits


Our contribution


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Our contribution


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Examples of groups


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Dihedral group

Lattice problems

Pauli matrices

x |m = |m + 1

z |

m

= m

|m

= exp (2i/p)

Q error correction

Model(m, a) (n, b) = (m, a) + m(n, b)



Examples of groups


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Dihedral group

Lattice problems

Pauli matrices

x |m = |m + 1

z|m

= m

|m

= exp (2i/p)

Q error correction

Model(m, a) (n, b) = (m, a) + m(n, b)



Examples of groups


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Dihedral group

Lattice problems

Pauli matrices

x |m = |m + 1

z|m

= m

|m

= exp (2i/p)

Q error correction

Model(m, a) (n, b) = (m, a) + m(n, b)



Why these groups?


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Non-commutative.

Related to computational problems.Quantum algorithms for some instances of the HSP.

Those algorithms use quantum Fourier transforms.



Why these groups?


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Non-commutative.





Why these groups?


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Non-commutative.





Why these groups?


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Non-commutative.



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Main results


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1 New circuits for quantum Fourier transforms

|mFp UR

|aFA

2 Input well-known coset-state |gH3 We can simulate classically and efficiently

hidden-subgroup measurements4 Techniques: adaptive measurements, group-theory.



Main results


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|mFp UR

|aFA





Main results


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|mFp UR

|aFA





Bonus


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Non-abelian QFTs are not uniquely defined.

We can also simulate changes of basis of the blocks

1 Dihedral groups: always2 Pauli matrices: sometimes.

Block basis|m Fp Ui V

|a FA



Bonus


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|a FA



Bonus


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|a FA



Bonus


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|a FA



Conclusions

Effi i t l i l i l ti f t F i t f


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Efficient classical simulations of quantum Fourier transforms

are possible1 Previously known: abelian.2 New: non-abelian.3 Remark: we can not simulate Shors algorithm.

Miau!

Thanks!32 / 32



Conclusions

Efficient classical sim lations of q ant m Fo rier transforms


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Miau!

Thanks!32 / 32



Conclusions



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Miau!

Thanks!32 / 32



Conclusions



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Miau!

Thanks!32 / 32



Conclusions



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Miau!

Thanks!32 / 32



Conclusions



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Miau!

Thanks!/

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