Quantum algorithm for P3: polar decomposition, Petz ...stanford.edu/~yquek/Slides/Petz_slides.pdf · Quantum algorithm for P3: polar decomposition, Petz recovery channels and pretty

Quantum algorithm for P3: polar decomposition, Petzrecovery channels and pretty good measurements

Yihui Quek

[email protected]

PI QI seminar

Joint work with various subsets of: Andras Gilyen, Seth Lloyd, Iman Marvian, Mark M.Wilde, Patrick Rebentrost

October 7, 2020

Yihui Quek (Stanford) P3: Polar, Petz, PGM October 7, 2020 1 / 31

Summary of both results

Using Quantum Singular Value Transformation, we provide twoquantum algorithms for two theoretical tools:

1 From classical linear algebra: polar decomposition, matrix analog ofz = re iθ.

2 From quantum information: Petz recovery map, which approximately‘reverses’ a quantum noise channel.

Application: implementation of Pretty-Good Measurements,another ubiquitous proof tool now brought to life!


Summary of both results

Using Quantum Singular Value Transformation, we provide twoquantum algorithms for two theoretical tools:

1 From classical linear algebra: polar decomposition, matrix analog ofz = re iθ.

2 From quantum information: Petz recovery map, which approximately‘reverses’ a quantum noise channel.

Application: implementation of Pretty-Good Measurements,another ubiquitous proof tool now brought to life!


Quantum Singular ValueTransformation

Quantum Singular Value Transformation

Block-encodings

Unitary U is a block-encoding of A if

U =

[A/α ·· ·

]⇐⇒ A = α(〈0|⊗s ⊗ I )U(|0〉⊗s ⊗ I ). (1)

U (acts on a qubits +s ancillae) can be used to realize a probabilisticimplementation of A/α.

On a-qubit input |ψ〉,Apply U to |0〉⊗s ⊗ |ψ〉Measure ancillae; if outcome was |0〉⊗s , the first a qubits contain astate ∼ A|ψ〉.



Block-encodings

Unitary U is a block-encoding of A if

U =

[A/α ·· ·

]⇐⇒ A = α(〈0|⊗s ⊗ I )U(|0〉⊗s ⊗ I ). (1)

U (acts on a qubits +s ancillae) can be used to realize a probabilisticimplementation of A/α.

On a-qubit input |ψ〉,Apply U to |0〉⊗s ⊗ |ψ〉Measure ancillae; if outcome was |0〉⊗s , the first a qubits contain astate ∼ A|ψ〉.



Quantum singular value transformation (I)

QSVT: A method to transform singular values of block-encodings[Gilyen-Su-Low-Wiebe’18, Low-Chuang ’16]

This circuit composes U =

[ρ ·· ·

]into

[f (ρ) ·· ·

]for your choice of polynomial f .

f (ρ) means ‘apply f to the singular values of ρ’.

Usually a polynomial approximation of ideal function f .



Quantum singular value transformation (II)

Uρ =

[ρ ·· ·

]QSVT−→

[f (ρ) ·· ·

]Gate complexity measured in number of uses of Uρ.

Depends on approximation’s domain ([θ, 1]) and error (δ).

e.g. Can approximate x1/2 with error1

2

∥∥∥f (x)− x1/2∥∥∥

[λmin,1]≤ δ

Let κ := 1λmin(ρ) ∼ “condition number”, overall gate complexity

O(κ log

1

δ

)uses of Uρ.


Algorithm 1: Polar decomposition

The first P: Polar decomposition

Polar decomposition

Definition (Polar decomposition)

The polar decomposition of A ∈ CM×N is the factorization

A = UB = BU

where U is a unitary/isometry and B =√A†A, B =

√AA† Hermitian.

Task: enact U := Upolar(A) on an input quantum state.

|ψ〉 → NUpolar(A)|ψ〉



Geometrical intuition for the polar decomposition



Geometrical intuition for the polar decomposition



Application of polar decomposition: Procrustes problem

Given: r (input, output) quantum state pairs: (|φi 〉, |ψi 〉)ri=1. Which Ubest transforms each input to output?

Figure 1: Taken from http://atlasgeographica.com/the-bed-of-procrustes/

Solution: U∗ is the polar decomposition of FG †, where F has |φi 〉as columns and G has |ψi 〉 as columns.


http://atlasgeographica.com/the-bed-of-procrustes/


Our algorithm

[Lloyd’20] provided an algorithm for enacting Upolar(A) based ondensity matrix exponentiation and quantum phase estimation.

QSVT simplifies this dramatically: One realizes that

SVD(A) =r∑

i=1

σii |pi 〉〈qi | ⇒ Upolar(A) =r∑

i=1

|pi 〉〈qi |.

i.e. can simply use QSVT to set all non-zero singular values to 1.

Significant speedups

vs [Lloyd ’20]: exponentially faster in ε, with polynomial speedups in r , κ.


Algorithm 2: Petz recovery map

The second P: Petz recovery map

arxiv:2006.16924


The second P: Petz recovery map Background and intuition for the Petz map

Classical ‘reversal’ channel from Bayes’ theorem

Given input pX (x) and channel pY |X (y |x), what is pX |Y (x |y)?



Petz recovery

Classically, Bayes theorem yields ‘reverse channel’:

pX |Y (x |y) =pX (x)pY |X (y |x)

pY (y). (2)

Quantumly: Petz recovery map!Given a forward channel, N and an ‘implicit’ input state σA:

Pσ,NB→A(·) := σ1/2A N †

(N (σA)−1/2(·)N (σA)−1/2

)σ

1/2A (3)



Why should you care about the Petz map?

1 Universal recovery operation in error correction [Barnum-Knill’02,Ng-Mandayam’09]

2 Important proof tool in QI: [Beigi-Datta-Leditzky’16] as a decoderin quantum communication, achieves coherent information rate.

3

A wild Petz map has appeared in quan-tum gravity! [Cotler-Hayden-Penington-

Salton-Swingle-Walter ’18]

φ(1)ab

φ(2)bc

φ(3)ab

φ(4)bc

R∗AB

R∗AB

R∗BC

R∗BC

A

B

C

D

4 Is a type of quantum “Bayesian inference” [Leifer-Spekkens’13](see: ?-product).

5 Has pretty-good measurements as a special case (later)






3



φ(1)ab

φ(2)bc

φ(3)ab

φ(4)bc

R∗AB

R∗AB

R∗BC

R∗BC

A

B

C

D








3



φ(1)ab

φ(2)bc

φ(3)ab

φ(4)bc

R∗AB

R∗AB

R∗BC

R∗BC

A

B

C

D








3



φ(1)ab

φ(2)bc

φ(3)ab

φ(4)bc

R∗AB

R∗AB

R∗BC

R∗BC

A

B

C

D








3



φ(1)ab

φ(2)bc

φ(3)ab

φ(4)bc

R∗AB

R∗AB

R∗BC

R∗BC

A

B

C

D




The second P: Petz recovery map QIT crash course

Quantum channels I

Quantum channel (informal): a physically valid map bringing onequantum state to another.

Important use case: model for quantum noise, e.g. amplitudedamping channels

Theorem (Choi-Kraus theorem)

Any physically valid channel NA→B(·) can be decomposed as

NA→B(XA) =d−1∑l=0

VlXAV†l

where Vl are linear (‘Kraus’) operators and∑d−1

l=0 V †l Vl = IA.(and vice versa!)



Quantum channels II

Definition (Channel adjoint)

Given NA→B , the channel adjoint N †B→A satisfies

〈Y ,N (X )〉 = 〈N †(Y ),X 〉 ∀X ∈ HA,Y ∈ HB

Every channel can be replicated by an isometry acting on a larger input.

Definition (Isometric extension)

Given a channel NA→B , an isometric extensionU : HA ⊗HE → HB ⊗HE ′ of N satisfies

TrE ′(U(ρ⊗ |0〉〈0|E )U†) = NA→B(ρ) (4)



Assumptions

Assume we have the following quantum circuits to start with:

1 Block-encodings of two statesOne can efficiently block-encode density matrices (proof omitted).

The implicit state σA (Uσ =

[σA ·· ·

])

The state N (σA) (UN (σ) =

[N (σA) ·· ·

]).

2 UNE ′A→EB , a unitary extension of the forward channel NSetting: we have characterized the noise and can simulate it usingquantum gates.



Assumptions




[σA ·· ·

])


[N (σA) ·· ·

]).




Assumptions




[σA ·· ·

])


[N (σA) ·· ·

]).




A peek at the Petz recovery map

Given: quantum state σA (implicit ‘input’ to channel ∼ pX ), quantumchannel NA→B , Petz map is:

Pσ,NB→A(ωB) := σ1/2A N †

(N (σA)−1/2ωBN (σA)−1/2

)σ

1/2A ,

Composition of 3 CP maps (overall trace-preserving):

(·)→ [N (σA)]−1/2 (·) [N (σA)]−1/2

(·)→ N †(·),(·)→ σ

1/2A (·)σ1/2

A .



A peek at the Petz recovery map

Given: quantum state σA (implicit ‘input’ to channel ∼ pX ), quantumchannel NA→B , Petz map is:

Pσ,NB→A(ωB) := σ1/2A N †

(N (σA)−1/2ωBN (σA)−1/2

)σ

1/2A ,

Composition of 3 CP maps (overall trace-preserving):

(·)→ [N (σA)]−1/2 (·) [N (σA)]−1/2

(·)→ N †(·),(·)→ σ

1/2A (·)σ1/2

A .


The second P: Petz recovery map Algorithm

Re-writing the channel adjoint

Second step of map: (·)→ N †(·)Can write adjoint N † in terms of unitary extension UN :

N †(ωB) = 〈0|E ′UN † (IE ⊗ ωB)UN |0〉E ′

Problem: IE is not a quantum state.Solution: act on maximally-entangled state 1

dE

∑dE−1i=0 |i〉E |i〉E , whose

density matrix is ∼ identity.



Re-writing the channel adjoint

Second step of map: (·)→ N †(·)Can write adjoint N † in terms of unitary extension UN :

N †(ωB) = 〈0|E ′UN † (IE ⊗ ωB)UN |0〉E ′

Problem: IE is not a quantum state.Solution: act on maximally-entangled state 1

dE

∑dE−1i=0 |i〉E |i〉E , whose

density matrix is ∼ identity.



Overview of algorithm

Implement an isometric extension of the Petz map:

(I) [N (σA)]−1/2 (·) [N (σA)]−1/2

(II) N †(·)

(III) σ1/2A (·)σ1/2

A .

Finally: tracing over environment E implements the map.





I)

(I) [N (σA)]−1/2 (·) [N (σA)]−1/2

(II) N †(·)

(III) σ1/2A (·)σ1/2

A .






II) I)

(I) [N (σA)]−1/2 (·) [N (σA)]−1/2

(II) N †(·)

(III) σ1/2A (·)σ1/2

A .






II) I)III)

(I) [N (σA)]−1/2 (·) [N (σA)]−1/2

(II) N †(·)

(III) σ1/2A (·)σ1/2

A .






II) I)III)

(I) [N (σA)]−1/2 (·) [N (σA)]−1/2

(II) N †(·)

(III) σ1/2A (·)σ1/2

A .




Theorem

For a forward channel N and an implicit input state σA, we can realize anapproximation P of the associated Petz recovery channel P, such that:∥∥PσA,N − PσA,N∥∥� ≤ ε, (5)

with

O(√

dEκN (σ)

)uses of UNAE ′→BE (∼optimal ) (6)

O(poly(dE , κN (σ), κ(σ))

)uses of UσA and UN (σA) (7)

dE is the dimension of the system E , which is at least the Kraus rank ofthe channel N (·).



Maps I and III

Maps I and III are a matter of transforming block-encodings:

I)III)

Map I: UN (σ) =

[N (σA) ·· ·

]QSVT−→

[N (σA)−1/2 ·

· ·

]with O

(κN (σ)

)uses of UN (σ).

Map III: Uσ =

[σA ·· ·

]QSVT−→

[σ

1/2A ·· ·

]with O(κσ) uses of Uσ.



Map II: Channel adjoint N †(·)

1 Tensor in the maximally entangled state ΓEE/dE

2 Perform UN†.

3 Measure the system E ′, accepting if the all-zeros outcome occurs.

4 Ignore the system E (i.e. trace it out).

Easy peasy, no QSVT needed.

Not contiguous with steps 1 and 2! Do these steps after Map III

which applies σ12A.





2 Perform UN†.










2 Perform UN†.








More on measurement

Implementing in sequence the unitaries created through SVT obtains theoverall unitary

W =

[14

√1

dEκN (σ)V ·

· ·

](8)

where V= ideal Petz map, ‖V − V ‖ < O(ε).

This is a probabilistic implementation: Measuring E ′ system,probability psuccess = O( 1

dEκ) of getting |0〉.

Make this deterministic: use Oblivious Amplitude Amplification toboost probability by repeating W O

(1/√psuccess

)times .


Application of our algorithms:Pretty-Good Measurements

The last P: Pretty-Good Measurements

Pretty-Good Measurements

Given an ensemble of mixed states {σx}x∈X , and a quantum state ρ, weare promised that ρ is in state σx with probability p(x). What POVMmaximizes Pr(correctly identify ρ)?

No optimal strategy when |X | ≥ 3, but ‘pretty-good measurement’does pretty-well on this. [Belavkin’75, Hausladen-Wootters’94]

Special case of the Petz map with σXB =∑

x pX (x)|x〉〈x |X ⊗ σxB , andN the partial trace over X .

Our Petz map algorithm can implement this, with O(√|X |poly(κ)

)uses of unitary preparing σXB .






























Faster PGM

Special case: ensemble of r pure states over a uniform distribution{p(j) = 1/r , |φj〉}r−1

j=0 .

Not that uncommon: this setting was considered by [Holevo’79].

Let κ = 1/σmin(∑

j |j〉〈φj |).

Petz map algorithm: O(r2κ3

)uses of Uφ.

Polar decomposition algorithm can handle this, with polynomialspeedup: O(rκ) uses of Uφ.



Why should you care about Pretty-Good Measurements?

1 Used to approach the Holevo information rate[Hausladen-Jozsa-Schumacher-Westmoreland-Wootters’96]

2 Important proof technique: PGM ∼ optimal measurement todistinguish states

i Is the optimal measurement in q. algorithm for dihedral hiddensubgroup problem [Bacon-Childs-vanDam’06]

ii Bounds on sample complexity for Quantum Probably ApproximatelyCorrect (PAC) learning [Arunachalam-de Wolf’16]

iii Optimal for port-based teleportation [Leditzky’20]



Why should you care about Pretty-Good Measurements?

1 Used to approach the Holevo information rate[Hausladen-Jozsa-Schumacher-Westmoreland-Wootters’96]

2 Important proof technique: PGM ∼ optimal measurement todistinguish states

i Is the optimal measurement in q. algorithm for dihedral hiddensubgroup problem [Bacon-Childs-vanDam’06]

ii Bounds on sample complexity for Quantum Probably ApproximatelyCorrect (PAC) learning [Arunachalam-de Wolf’16]

iii Optimal for port-based teleportation [Leditzky’20]



Takeaways!

The quantum singular value transform allows us to be systematic, rigorousand fast.

Our algorithm for Polar decomposition(arxiv:20??.?????)

provides new tools for quantum linear algebra;and

speeds up PGM, for pure states and uniformdistribution.

Our algorithm for Petz recovery maps andgeneral-purpose PGM (arxiv:2006.16924)

brings these theoretical tools closer toimplementation; and

is almost optimal in gate complexity.



Takeaways!










Takeaways!









Documents

Quantum algorithm for P3: polar decomposition, Petz ...stanford.edu/~yquek/Slides/Petz_slides.pdf · Quantum algorithm for P3: polar decomposition, Petz recovery channels and pretty